“Frontmatter” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Structural Engineering Contents
1
Basic Theory of Plates and Elastic Stability
2
Structural Analysis
3
Structural Steel Design1
4
Structural Concrete Design2
5
Earthquake Engineering
Charles Scawthorn
6
Composite Construction
Edoardo Cosenza and Riccardo Zandonini
7
Cold-Formed Steel Structures
8
Aluminum Structures
9
Timber Structures
10
Bridge Structures
11
Shell Structures
12
Multistory Frame Structures
13
Space Frame Structures
14
Cooling Tower Structures
Phillip L. Gould and Wilfried B. Krätzig
15
Transmission Structures
Shu-jin Fang, Subir Roy, and Jacob Kramer
Eiki Yamaguchi
J.Y. Richard Liew,
N.E. Shanmugam, and
C.H. Yu
E. M. Lui Amy Grider and Julio A. Ramirez and Young Mook Yun
Wei-Wen Yu
Maurice L. Sharp
Kenneth J. Fridley Shouji Toma, Lian Duan, and Wai-Fah Chen Clarence D. Miller
15B Tunnel Structures
J. Y. Richard Liew and T. Balendra and W. F. Chen
Tien T. Lan
Birger Schmidt, Christian Ingerslev, Brian Brenner, and J.-N. Wang
16 Performance-Based Seismic Design Criteria For Bridges LianDuanandMarkReno 1999 by CRC Press LLC
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17
Effective Length Factors of Compression Members
18
Stub Girder Floor Systems
19
Plate and Box Girders
20
Steel Bridge Construction
21
Basic Principles of Shock Loading
22
Welded Connections
23
Composite Connections
24
Fatigue and Fracture
25
Underground Pipe
26
Structural Reliability3
27
Passive Energy Dissipation and Active Control
28
An Innnovative Design For Steel Frame Using Advanced Analysis4 W. F. Chen
29
Welded Tubular Connections—CHS Trusses
Reidar Bjorhovde
Mohamed Elgaaly Jackson Durkee
1999 by CRC Press LLC
O.W. Blodgett and D.K. Miller
O.W. Blodgett and D. K. Miller Roberto Leon
Robert J. Dexter and John W. Fisher J. M. Doyle and S.J. Fang D. V. Rosowsky
30 Earthquake Damage to Structures
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Lian Duan and W.F. Chen
T.T. Soong and G.F. Dargush
Peter W. Marshall
Mark Yashinsky
Seung-Eock Kim and
Yamaguchi, E. “Basic Theory of Plates and Elastic Stability” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Basic Theory of Plates and Elastic Stability 1.1 1.2 1.3
Eiki Yamaguchi Department of Civil Engineering, Kyushu Institute of Technology, Kitakyusha, Japan
1.1
Introduction Plates
Basic Assumptions • Governing Equations • Boundary Conditions • Circular Plate • Examples of Bending Problems
Stability
Basic Concepts • Structural Instability Walled Members • Plates
•
Columns
•
Thin-
1.4 Defining Terms References Further Reading
Introduction
This chapter is concerned with basic assumptions and equations of plates and basic concepts of elastic stability. Herein, we shall illustrate the concepts and the applications of these equations by means of relatively simple examples; more complex applications will be taken up in the following chapters.
1.2 1.2.1
Plates Basic Assumptions
We consider a continuum shown in Figure 1.1. A feature of the body is that one dimension is much smaller than the other two dimensions: t 0 : unstable Equation 1.41 implies that as P increases, the state of the system changes from stable equilibrium to unstable equilibrium. The critical load is kL, at which multiple equilibrium positions, i.e., θ = 0 and θ 6 = 0, are possible. Thus, the critical load serves also as a bifurcation point of the equilibrium path. The load at such a bifurcation is called the buckling load. 1999 by CRC Press LLC
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FIGURE 1.10: Rigid bar AB with a spring. For the present system, the buckling load of kL is stability limit as well as neutral equilibrium. In general, the buckling load corresponds to a state of neutral equilibrium, but not necessarily to stability limit. Nevertheless, the buckling load is often associated with the characteristic change of structural behavior, and therefore can be regarded as the limit state of serviceability. Linear Buckling Analysis
We can compute a buckling load by considering an equilibrium condition for a slightly deformed state. For the system of Figure 1.10, the moment equilibrium yields P L sin θ − (kL sin θ )(L cos θ ) = 0 Since θ is infinitesimal, we obtain
Lθ (P − kL) = 0
(1.42) (1.43)
It is obvious that this equation is satisfied for any value of P if θ is zero: θ = 0 is called the trivial solution. We are seeking the buckling load, at which the equilibrium condition is satisfied for θ 6 = 0. The trivial solution is apparently of no importance and from Equation 1.43 we can obtain the following buckling load PC : (1.44) PC = kL A rigorous buckling analysis is quite involved, where we need to solve nonlinear equations even when elastic problems are dealt with. Consequently, the linear buckling analysis is frequently employed. The analysis can be justified, if deformation is negligible and structural behavior is linear before the buckling load is reached. The way we have obtained Equation 1.44 in the above is a typical application of the linear buckling analysis. In mathematical terms, Equation 1.43 is called a characteristic equation and Equation 1.44 an eigenvalue. The linear buckling analysis is in fact regarded as an eigenvalue problem.
1.3.2 Structural Instability Three classes of instability phenomenon are observed in structures: bifurcation, snap-through, and softening. We have discussed a simple example of bifurcation in the previous section. Figure 1.11a depicts a schematic load-displacement relationship associated with the bifurcation: Point A is where the 1999 by CRC Press LLC
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bifurcation takes place. In reality, due to imperfection such as the initial crookedness of a member and the eccentricity of loading, we can rarely observe the bifurcation. Instead, an actual structural behavior would be more like the one indicated in Figure 1.11a. However, the bifurcation load is still an important measure regarding structural stability and most instabilities of a column and a plate are indeed of this class. In many cases we can evaluate the bifurcation point by the linear buckling analysis. In some structures, we observe that displacement increases abruptly at a certain load level. This can take place at Point A in Figure 1.11b; displacement increases from UA to UB at PA , as illustrated by a broken arrow. The phenomenon is called snap-through. The equilibrium path of Figure 1.11b is typical of shell-like structures, including a shallow arch, and is traceable only by the finite displacement analysis. The other instability phenomenon is the softening: as Figure 1.11c illustrates, there exists a peak load-carrying capacity, beyond which the structural strength deteriorates. We often observe this phenomenon when yielding takes place. To compute the associated equilibrium path, we need to resort to nonlinear structural analysis. Since nonlinear analysis is complicated and costly, the information on stability limit and ultimate strength is deduced in practice from the bifurcation load, utilizing the linear buckling analysis. We shall therefore discuss the buckling loads (bifurcation points) of some structures in what follows.
1.3.3
Columns
Simply Supported Column
As a first example, we evaluate the buckling load of a simply supported column shown in Figure 1.12a. To this end, the moment equilibrium in a slightly deformed configuration is considered. Following the notation in Figure 1.12b, we can readily obtain w00 + k 2 w = 0 where k2 =
P EI
(1.45)
(1.46)
EI is the bending rigidity of the column. The general solution of Equation 1.45 is w = A1 sin kx + A2 cos kx
(1.47)
The arbitrary constants A1 and A2 are to be determined by the following boundary conditions: w w
= 0 at x = 0 = 0 at x = L
(1.48a) (1.48b)
Equation 1.48a gives A2 = 0 and from Equation 1.48b we reach A1 sin kL = 0
(1.49)
A1 = 0 is a solution of the characteristic equation above, but this is the trivial solution corresponding to a perfectly straight column and is of no interest. Then we obtain the following buckling loads: PC = 1999 by CRC Press LLC
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n2 π 2 EI L2
(1.50)
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FIGURE 1.11: Unstable structural behaviors.
FIGURE 1.12: Simply-supported column.
Although n is any integer, our interest is in the lowest buckling load with n = 1 since it is the critical load from the practical point of view. The buckling load, thus, obtained is PC =
π 2 EI L2
(1.51)
which is often referred to as the Euler load. From A2 = 0 and Equation 1.51, Equation 1.47 indicates the following shape of the deformation: w = A1 sin
πx L
(1.52)
This equation shows the buckled shape only, since A1 represents the undetermined amplitude of the deflection and can have any value. The deflection curve is illustrated in Figure 1.12c. The behavior of the simply supported column is summarized as follows: up to the Euler load the column remains straight; at the Euler load the state of the column becomes the neutral equilibrium and it can remain straight or it starts to bend in the mode expressed by Equation 1.52. 1999 by CRC Press LLC
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Cantilever Column
For the cantilever column of Figure 1.13a, by considering the equilibrium condition of the free body shown in Figure 1.13b, we can derive the following governing equation: w00 + k 2 w = k 2 δ
(1.53)
where δ is the deflection at the free tip. The boundary conditions are w w0 w
= 0 at x = 0 = 0 at x = 0 = δ at x = L
(1.54)
FIGURE 1.13: Cantilever column.
From these equations we can obtain the characteristic equation as δ cos kL = 0 1999 by CRC Press LLC
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(1.55)
which yields the following buckling load and deflection shape: PC
=
w
=
π 2 EI 2 4L
δ 1 − cos
(1.56)
πx
(1.57)
2L
The buckling mode is illustrated in Figure 1.13c. It is noted that the boundary conditions make much difference in the buckling load: the present buckling load is just a quarter of that for the simply supported column. Higher-Order Differential Equation
We have thus far analyzed the two columns. In each problem, a second-order differential equation was derived and solved. This governing equation is problem-dependent and valid only for a particular problem. A more consistent approach is possible by making use of the governing equation for a beam-column with no laterally distributed load: EI wI V + P w 00 = q
(1.58)
Note that in this equation P is positive when compressive. This equation is applicable to any set of boundary conditions. The general solution of Equation 1.58 is given by w = A1 sin kx + A2 cos kx + A3 x + A4
(1.59)
where A1 ∼ A4 are arbitrary constants and determined from boundary conditions. We shall again solve the two column problems, using Equation 1.58. 1. Simply supported column (Figure 1.12a) Because of no deflection and no external moment at each end of the column, the boundary conditions are described as w w
= 0, = 0,
w00 = 0 at x = 0 w00 = 0 at x = L
(1.60)
From the conditions at x = 0, we can determine A2 = A4 = 0
(1.61)
Using this result and the conditions at x = L, we obtain
sin kL −k 2 sin kL
L 0
A1 A3
=
0 0
(1.62)
For the nontrivial solution to exist, the determinant of the coefficient matrix in Equation 1.62 must vanish, leading to the following characteristic equation: k 2 L sin kL = 0
(1.63)
from which we arrive at the same critical load as in Equation 1.51. By obtaining the corresponding eigenvector of Equation 1.62, we can get the buckled shape of Equation 1.52. 1999 by CRC Press LLC
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2. Cantilever column (Figure 1.13a) In this column, we observe no deflection and no slope at the fixed end; no external moment and no external shear force at the free end. Therefore, the boundary conditions are w = 0, w 00 = 0,
w000
w0 = 0 + k 2 w0 = 0
at x = 0 at x = L
(1.64)
Note that since we are dealing with a slightly deformed column in the linear buckling analysis, the axial force has a transverse component, which is why P comes in the boundary condition at x = L. The latter condition at x = L eliminates A3 . With this and the second condition at x = 0, we can claim A1 = 0. The remaining two conditions then lead to 1 1 A2 0 (1.65) = A4 0 k 2 cos kL 0 The smallest eigenvalue and the corresponding eigenvector of Equation 1.65 coincide with the buckling load and the buckling mode that we have obtained previously in Section 1.3.3. Effective Length
We have obtained the buckling loads for the simply supported and the cantilever columns. By either the second- or the fourth-order differential equation approach, we can compute buckling loads for a fixed-hinged column (Figure 1.14a) and a fixed-fixed column (Figure 1.14b) without much difficulty: PC
=
PC
=
π 2 EI (0.7L)2 π 2 EI (0.5L)2
for a fixed - hinged column for a fixed - hinged column
(1.66)
For all the four columns considered thus far, and in fact for the columns with any other sets of boundary conditions, we can express the buckling load in the form of PC =
π 2 EI (KL)2
(1.67)
where KL is called the effective length and represents presumably the length of the equivalent Euler column (the equivalent simply supported column). For design purposes, Equation 1.67 is often transformed into σC =
π 2E (KL/r)2
(1.68)
where r is the radius of gyration defined in terms of cross-sectional area A and the moment of inertia I by r I (1.69) r= A For an ideal elastic column, we can draw the curve of the critical stress σC vs. the slenderness ratio KL/r, as shown in Figure 1.15a. 1999 by CRC Press LLC
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FIGURE 1.14: (a) Fixed-hinged column; (b) fixed-fixed column. For a column of perfectly plastic material, stress never exceeds the yield stress σY . For this class of column, we often employ a normalized form of Equation 1.68 as 1 σC = 2 σY λ where λ=
1 KL π r
r
(1.70)
σY E
(1.71)
This equation is plotted in Figure 1.15b. For this column, with λ < 1.0, it collapses plastically; elastic buckling takes place for λ > 1.0. Imperfect Columns
In the derivation of the buckling loads, we have dealt with the idealized columns; the member is perfectly straight and the loading is concentric at every cross-section. These idealizations help simplify the problem, but perfect members do not exist in the real world: minor crookedness of shape and small eccentricities of loading are always present. To this end, we shall investigate the behavior of an initially bent column in this section. We consider a simply supported column shown in Figure 1.16. The column is initially bent and the initial crookedness wi is assumed to be in the form of wi = a sin
πx L
(1.72)
where a is a small value, representing the magnitude of the initial deflection at the midpoint. If we describe the additional deformation due to bending as w and consider the moment equilibrium in 1999 by CRC Press LLC
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FIGURE 1.15: (a) Relationship between critical stress and slenderness ratio; (b) normalized relationship.
FIGURE 1.16: Initially bent column. this configuration, we obtain
πx (1.73) L where k 2 is defined in Equation 1.46. The general solution of this differential equation is given by w00 + k 2 w = −k 2 a sin
w = A sin 1999 by CRC Press LLC
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πx πx P /PE πx a sin + B cos + L L 1 − P /PE L
(1.74)
where PE is the Euler load, i.e., π 2 EI /L2 . From the boundary conditions of Equation 1.48, we can determine the arbitrary constants A and B, yielding the following load-displacement relationship: w=
πx P /PE a sin 1 − P /PE L
(1.75)
By adding this expression to the initial deflection, we can obtain the total displacement wt as wt = wi + w =
a πx sin 1 − P /PE L
(1.76)
Figure 1.17 illustrates the variation of the deflection at the midpoint of this column wm .
FIGURE 1.17: Load-displacement curve of the bent column.
Unlike the ideally perfect column, which remains straight up to the Euler load, we observe in this figure that the crooked column begins to bend at the onset of the loading. The deflection increases slowly at first, and as the applied load approaches the Euler load, the increase of the deflection is getting more and more rapid. Thus, although the behavior of the initially bent column is different from that of bifurcation, the buckling load still serves as an important measure of stability. We have discussed the behavior of a column with geometrical imperfection in this section. However, the trend observed herein would be the same for general imperfect columns such as an eccentrically loaded column.
1.3.4
Thin-Walled Members
In the previous section, we assumed that a compressed column would buckle by bending. This type of buckling may be referred to as flexural buckling. However, a column may buckle by twisting or by a combination of twisting and bending. Such a mode of failure occurs when the torsional rigidity of the cross-section is low. Thin-walled open cross-sections have a low torsional rigidity in general and hence are susceptible of this type of buckling. In fact, a column of thin-walled open cross-section usually buckles by a combination of twisting and bending: this mode of buckling is often called the torsional-flexural buckling. 1999 by CRC Press LLC
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A bar subjected to bending in the plane of a major axis may buckle in yet another mode: at the critical load a compression side of the cross-section tends to bend sideways while the remainder is stable, resulting in the rotation and lateral movement of the entire cross-section. This type of buckling is referred to as lateral buckling. We need to use caution in particular, if a beam has no lateral supports and the flexural rigidity in the plane of bending is larger than the lateral flexural rigidity. In the present section, we shall briefly discuss the two buckling modes mentioned above, both of which are of practical importance in the design of thin-walled members, particularly of open cross-section. Torsional-Flexural Buckling
We consider a simply supported column subjected to compressive load P applied at the centroid of each end, as shown in Figure 1.18. Note that the x axis passes through the centroid of every crosssection. Taking into account that the cross-section undergoes translation and rotation as illustrated in Figure 1.19, we can derive the equilibrium conditions for the column deformed slightly by the torsional-flexural buckling EIy ν I V + P ν 00 + P zs φ 00 = 0 EIz w I V + P w00 − P ys φ 00 = 0 EIw φ I V + P rs2 φ 00 − GJ φ 00 + P zs ν 00 − P ys w 00 = 0 where ν, w φ EIw GJ ys , zs and
= = = = =
(1.77)
displacements in the y, z-directions, respectively rotation warping rigidity torsional rigidity coordinates of the shear center Z EIy
=
EIz
=
rs2
=
ZA A
y 2 dA z2 dA
(1.78)
Is A
where = polar moment of inertia about the shear center Is A = cross-sectional area We can obtain the buckling load by solving the eigenvalue problem governed by Equation 1.77 and the boundary conditions of ν = ν 00 = w = w00 = φ = φ 00 = 0 at x = 0, L
(1.79)
For doubly symmetric cross-section, the shear center coincides with the centroid. Therefore, ys , zs , and rs vanish and the three equations in Equation 1.77 become independent of each other, if the cross-section of the column is doubly symmetric. In this case, we can compute three critical loads as follows: 1999 by CRC Press LLC
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FIGURE 1.18: Simply-supported thin-walled column.
FIGURE 1.19: Translation and rotation of the cross-section.
PyC
=
PzC
=
PφC
=
π 2 EIy L2 2 π EIz L2 1 π 2 EIw GJ + rs2 L2
(1.80a) (1.80b) (1.80c)
The first two are associated with flexural buckling and the last one with torsional buckling. For all cases, the buckling mode is in the shape of sin πLx . The smallest of the three would be the critical load of practical importance: for a relatively short column with low GJ and EIw , the torsional buckling may take place. When the cross-section of a column is symmetric with respect only to the y axis, we rewrite Equation 1.77 as EIy ν I V + P ν 00 = 0
(1.81a)
EIz w I V + P w00 − P ys φ 00 = 0 EIw φ I V + P rs2 − GJ φ 00 − P ys w 00 = 0
(1.81b) (1.81c)
The first equation indicates that the flexural buckling in the x − y plane occurs independently and the corresponding critical load is given by PyC of Equation 1.80a. The flexural buckling in the x − z plane and the torsional buckling are coupled. By assuming that the buckling modes are described by πx w = A1 sin πx L and φ = A2 sin L , Equations 1.81b,c yields
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P − PzC −P ys
rs2
−P ys P − PφC
A1 A2
=
0 0
(1.82)
This eigenvalue problem leads to f (P ) = rs2 P − PφC (P − PzC ) − (P ys )2 = 0
(1.83)
The solution of this quadratic equation is the critical load associated with torsional-flexural buckling. Since f (0) = rs2 PφC PzC > 0, f (PφC = −(P ys )2 < 0, and f (PzC ) = −(P ys )2 < 0, it is obvious that the critical load is lower than PzC and PφC . If this load is smaller than PyC , then the torsionalflexural buckling will take place. If there is no axis of symmetry in the cross-section, all the three equations in Equation 1.77 are coupled. The torsional-flexural buckling occurs in this case, since the critical load for this buckling mode is lower than any of the three loads in Equation 1.80. Lateral Buckling
The behavior of a simply supported beam in pure bending (Figure 1.20) is investigated. The equilibrium condition for a slightly translated and rotated configuration gives governing equations for the bifurcation. For a cross-section symmetric with respect to the y axis, we arrive at the following equations: EIy ν I V + Mφ 00 = 0
(1.84a)
IV
EIz w = 0 EIw φ I V − (GJ + Mβ) φ 00 + Mν 00 = 0 where β=
1 Iz
(1.84b) (1.84c)
Z n A
o y 2 + (z − zs )2 zdA
(1.85)
FIGURE 1.20: Simply supported beam in pure bending.
Equation 1.84b is a beam equation and has nothing to do with buckling. From the remaining two equations and the associated boundary conditions of Equation 1.79, we can evaluate the critical load for the lateral buckling. By assuming the bucking mode is in the shape of sin πLx for both ν and φ, we obtain the characteristic equation M 2 − βPyC M − rs2 PyC PφC = 0
(1.86)
The smallest root of this quadratic equation is the critical load (moment) for the lateral buckling. For doubly symmetric sections where β is zero, the critical moment MC is given by q MC = rs2 PyC PφC =
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s
π 2 EIy L2
GJ +
π 2 EIw L2
(1.87)
1.3.5
Plates
Governing Equation
The buckling load of a plate is also obtained by the linear buckling analysis, i.e., by considering the equilibrium of a slightly deformed configuration. The plate counterpart of Equation 1.58, thus, derived is ∂ 2w ∂ 2w ∂ 2w + Ny 2 = 0 (1.88) D∇ 4 w + N x 2 + 2N xy ∂x∂y ∂x ∂y The definitions of N x , N y , and N xy are the same as those of Nx , Ny , and Nxy given in Equations 1.8a through 1.8c, respectively, except the sign; N x , N y , and N xy are positive when compressive. The boundary conditions are basically the same as discussed in Section 1.2.3 except the mechanical condition in the vertical direction: to include the effect of in-plane forces, we need to modify Equation 1.18 as ∂w ∂w + Nns = Sn (1.89) Sn + Nn ∂n ∂s where Z Nn
=
Nns
=
Zz z
σn dz τns dz
(1.90)
Simply Supported Plate
As an example, we shall discuss the buckling load of a simply supported plate under uniform compression shown in Figure 1.21. The governing equation for this plate is D∇ 4 w + N x
∂ 2w =0 ∂x 2
(1.91)
and the boundary conditions are w
=
0,
w
=
0,
∂ 2w = 0 along x = 0, a ∂x 2 ∂ 2w = 0 along y = 0, b ∂y 2
FIGURE 1.21: Simply supported plate subjected to uniform compression. 1999 by CRC Press LLC
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(1.92)
We assume that the solution is of the form w=
∞ ∞ X X
Amn sin
m=1 n=1
mπ x nπ x sin a b
(1.93)
where m and n are integers. Since this solution satisfies all the boundary conditions, we have only to ensure that it satisfies the governing equation. Substituting Equation 1.93 into 1.91, we obtain " # 2 2 n2 N x m2 π 2 4 m + 2 − (1.94) =0 Amn π D a2 a2 b Since the trivial solution is of no interest, at least one of the coefficients amn must not be zero, the consideration of which leads to 2 π 2D b n2 a Nx = 2 (1.95) m + a mb b As the lowest N x is crucial and N x increases with n, we conclude n = 1: the buckling of this plate occurs in a single half-wave in the y direction and kπ 2 D b2
(1.96)
1 N xC π 2E =k 2 t 12(1 − ν ) (b/t)2
(1.97)
1a 2 b k= m + a mb
(1.98)
N xC = or σxC = where
Note that Equation 1.97 is comparable to Equation 1.68, and k is called the buckling stress coefficient. The optimum value of m that gives the lowest N xC depends on the aspect ratio a/b, as can be realized in Figure 1.22. For example, the optimum m is 1 for a square plate while it is 2 for a plate of a/b = 2. For a plate with a large aspect ratio, k = 4.0 serves as a good approximation. Since the aspect ratio of a component of a steel structural member such as a web plate is large in general, we can often assume k is simply equal to 4.0.
1.4
Defining Terms
The following is a list of terms as defined in the Guide to Stability Design Criteria for Metal Structures, 4th ed., Galambos, T.V., Structural Stability Research Council, John Wiley & Sons, New York, 1988. Bifurcation: A term relating to the load-deflection behavior of a perfectly straight and perfectly centered compression element at critical load. Bifurcation can occur in the inelastic range only if the pattern of post-yield properties and/or residual stresses is symmetrically disposed so that no bending moment is developed at subcritical loads. At the critical load a member can be in equilibrium in either a straight or slightly deflected configuration, and a bifurcation results at a branch point in the plot of axial load vs. lateral deflection from which two alternative load-deflection plots are mathematically valid. Braced frame: A frame in which the resistance to both lateral load and frame instability is provided by the combined action of floor diaphragms and structural core, shear walls, and/or a diagonal K brace, or other auxiliary system of bracing. 1999 by CRC Press LLC
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FIGURE 1.22: Variation of the buckling stress coefficient k with the aspect ratio a/b.
Effective length: The equivalent or effective length (KL) which, in the Euler formula for a hinged-end column, results in the same elastic critical load as for the framed member or other compression element under consideration at its theoretical critical load. The use of the effective length concept in the inelastic range implies that the ratio between elastic and inelastic critical loads for an equivalent hinged-end column is the same as the ratio between elastic and inelastic critical loads in the beam, frame, plate, or other structural element for which buckling equivalence has been assumed. Instability: A condition reached during buckling under increasing load in a compression member, element, or frame at which the capacity for resistance to additional load is exhausted and continued deformation results in a decrease in load-resisting capacity. Stability: The capacity of a compression member or element to remain in position and support load, even if forced slightly out of line or position by an added lateral force. In the elastic range, removal of the added lateral force would result in a return to the prior loaded position, unless the disturbance causes yielding to commence. Unbraced frame: A frame in which the resistance to lateral loads is provided primarily by the bending of the frame members and their connections.
References [1] Chajes, A. 1974. Principles of Structural Stability Theory, Prentice-Hall, Englewood Cliffs, NJ. [2] Chen, W.F. and Atsuta, T. 1976. Theory of Beam-Columns, vol. 1: In-Plane Behavior and Design, and vol. 2: Space Behavior and Design, McGraw-Hill, NY. [3] Thompson, J.M.T. and Hunt, G.W. 1973. A General Theory of Elastic Stability, John Wiley & Sons, London, U.K. [4] Timoshenko, S.P. and Woinowsky-Krieger, S. 1959. Theory of Plates and Shells, 2nd ed., McGraw-Hill, NY. [5] Timoshenko, S.P. and Gere, J.M. 1961. Theory of Elastic Stability, 2nd ed., McGraw-Hill, NY. 1999 by CRC Press LLC
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Further Reading [1] Chen, W.F. and Lui, E.M. 1987. Structural Stability Theory and Implementation, Elsevier, New York. [2] Chen, W.F. and Lui, E.M. 1991. Stability Design of Steel Frames, CRC Press, Boca Raton, FL. [3] Galambos, T.V. 1988. Guide to Stability Design Criteria for Metal Structures, 4th ed., Structural Stability Research Council, John Wiley & Sons, New York.
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Richard Liew, J.Y.; Shanmugam, N.W. and Yu, C.H. “Structural Analysis” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Structural Analysis
J.Y. Richard Liew, N.E. Shanmugam, and C.H. Yu Department of Civil Engineering The National University of Singapore, Singapore
2.1
2.1 Fundamental Principles 2.2 Flexural Members 2.3 Trusses 2.4 Frames 2.5 Plates 2.6 Shell 2.7 Influence Lines 2.8 Energy Methods in Structural Analysis 2.9 Matrix Methods 2.10 The Finite Element Method 2.11 Inelastic Analysis 2.12 Frame Stability 2.13 Structural Dynamic 2.14 Defining Terms References Further Reading
Fundamental Principles
Structural analysis is the determination of forces and deformations of the structure due to applied loads. Structural design involves the arrangement and proportioning of structures and their components in such a way that the assembled structure is capable of supporting the designed loads within the allowable limit states. An analytical model is an idealization of the actual structure. The structural model should relate the actual behavior to material properties, structural details, and loading and boundary conditions as accurately as is practicable. All structures that occur in practice are three-dimensional. For building structures that have regular layout and are rectangular in shape, it is possible to idealize them into two-dimensional frames arranged in orthogonal directions. Joints in a structure are those points where two or more members are connected. A truss is a structural system consisting of members that are designed to resist only axial forces. Axially loaded members are assumed to be pin-connected at their ends. A structural system in which joints are capable of transferring end moments is called a frame. Members in this system are assumed to be capable of resisting bending moment axial force and shear force. A structure is said to be two dimensional or planar if all the members lie in the same plane. Beams are those members that are subjected to bending or flexure. They are usually thought of as being in horizontal positions and loaded with vertical loads. Ties are members that are subjected to axial tension only, while struts (columns or posts) are members subjected to axial compression only. 1999 by CRC Press LLC
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2.1.1
Boundary Conditions
A hinge represents a pin connection to a structural assembly and it does not allow translational movements (Figure 2.1a). It is assumed to be frictionless and to allow rotation of a member with
FIGURE 2.1: Various boundary conditions.
respect to the others. A roller represents a kind of support that permits the attached structural part to rotate freely with respect to the foundation and to translate freely in the direction parallel to the foundation surface (Figure 2.1b) No translational movement in any other direction is allowed. A fixed support (Figure 2.1c) does not allow rotation or translation in any direction. A rotational spring represents a support that provides some rotational restraint but does not provide any translational restraint (Figure 2.1d). A translational spring can provide partial restraints along the direction of deformation (Figure 2.1e).
2.1.2
Loads and Reactions
Loads may be broadly classified as permanent loads that are of constant magnitude and remain in one position and variable loads that may change in position and magnitude. Permanent loads are also referred to as dead loads which may include the self weight of the structure and other loads such as walls, floors, roof, plumbing, and fixtures that are permanently attached to the structure. Variable loads are commonly referred to as live or imposed loads which may include those caused by construction operations, wind, rain, earthquakes, snow, blasts, and temperature changes in addition to those that are movable, such as furniture and warehouse materials. Ponding load is due to water or snow on a flat roof which accumulates faster than it runs off. Wind loads act as pressures on windward surfaces and pressures or suctions on leeward surfaces. Impact loads are caused by suddenly applied loads or by the vibration of moving or movable loads. They are usually taken as a fraction of the live loads. Earthquake loads are those forces caused by the acceleration of the ground surface during an earthquake. A structure that is initially at rest and remains at rest when acted upon by applied loads is said to be in a state of equilibrium. The resultant of the external loads on the body and the supporting forces or reactions is zero. If a structure or part thereof is to be in equilibrium under the action of a system 1999 by CRC Press LLC
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of loads, it must satisfy the six static equilibrium equations, such as P P P Fx = 0, Fy = 0, Fz = 0 P P P My = 0, Mz = 0 Mx = 0,
(2.1)
The summation in these equations is for all the components of the forces (F ) and of the moments (M) about each of the three axes x, y, and z. If a structure is subjected to forces that lie in one plane, say x-y, the above equations are reduced to: X X X Fy = 0, Mz = 0 (2.2) Fx = 0, Consider, for example, a beam shown in Figure 2.2a under the action of the loads shown. The
FIGURE 2.2: Beam in equilibrium. reaction at support B must act perpendicular to the surface on which the rollers are constrained to roll upon. The support reactions and the applied loads, which are resolved in vertical and horizontal directions, are shown in Figure 2.2b. √ From geometry, it can be calculated that By = 3Bx . Equation 2.2 can be used to determine the magnitude of the support reactions. Taking moment about B gives 10Ay − 346.4x5 = 0 from which Equating the sum of vertical forces,
P
Ay = 173.2 kN. Fy to zero gives
173.2 + By − 346.4 = 0 and, hence, we get By = 173.2 kN. Therefore,
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√ Bx = By / 3 = 100 kN.
Equilibrium in the horizontal direction,
P
Fx = 0 gives,
Ax − 200 − 100 = 0 and, hence, Ax = 300 kN. There are three unknown reaction components at a fixed end, two at a hinge, and one at a roller. If, for a particular structure, the total number of unknown reaction components equals the number of equations available, the unknowns may be calculated from the equilibrium equations, and the structure is then said to be statically determinate externally. Should the number of unknowns be greater than the number of equations available, the structure is statically indeterminate externally; if less, it is unstable externally. The ability of a structure to support adequately the loads applied to it is dependent not only on the number of reaction components but also on the arrangement of those components. It is possible for a structure to have as many or more reaction components than there are equations available and yet be unstable. This condition is referred to as geometric instability.
2.1.3
Principle of Superposition
The principle states that if the structural behavior is linearly elastic, the forces acting on a structure may be separated or divided into any convenient fashion and the structure analyzed for the separate cases. Then the final results can be obtained by adding up the individual results. This is applicable to the computation of structural responses such as moment, shear, deflection, etc. However, there are two situations where the principle of superposition cannot be applied. The first case is associated with instances where the geometry of the structure is appreciably altered under load. The second case is in situations where the structure is composed of a material in which the stress is not linearly related to the strain.
2.1.4
Idealized Models
Any complex structure can be considered to be built up of simpler components called members or elements. Engineering judgement must be used to define an idealized structure such that it represents the actual structural behavior as accurately as is practically possible. Structures can be broadly classified into three categories: 1. Skeletal structures consist of line elements such as a bar, beam, or column for which the length is much larger than the breadth and depth. A variety of skeletal structures can be obtained by connecting line elements together using hinged, rigid, or semi-rigid joints. Depending on whether the axes of these members lie in one plane or in different planes, these structures are termed as plane structures or spatial structures. The line elements in these structures under load may be subjected to one type of force such as axial force or a combination of forces such as shear, moment, torsion, and axial force. In the first case the structures are referred to as the truss-type and in the latter as frame-type. 2. Plated structures consist of elements that have length and breadth of the same order but are much larger than the thickness. These elements may be plane or curved in plane, in which case they are called plates or shells, respectively. These elements are generally used in combination with beams and bars. Reinforced concrete slabs supported on beams, box-girders, plate-girders, cylindrical shells, or water tanks are typical examples of plate and shell structures. 3. Three-dimensional solid structures have all three dimensions, namely, length, breadth, and depth, of the same order. Thick-walled hollow spheres, massive raft foundation, and dams are typical examples of solid structures. 1999 by CRC Press LLC
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Recent advancement in finite element methods of structural analysis and the advent of more powerful computers have enabled the economic analysis of skeletal, plated, and solid structures.
2.2
Flexural Members
One of the most common structural elements is a beam; it bends when subjected to loads acting transversely to its centroidal axis or sometimes by loads acting both transversely and parallel to this axis. The discussions given in the following subsections are limited to straight beams in which the centroidal axis is a straight line with shear center coinciding with the centroid of the cross-section. It is also assumed that all the loads and reactions lie in a simple plane that also contains the centroidal axis of the flexural member and the principal axis of every cross-section. If these conditions are satisfied, the beam will simply bend in the plane of loading without twisting.
2.2.1
Axial Force, Shear Force, and Bending Moment
Axial force at any transverse cross-section of a straight beam is the algebraic sum of the components acting parallel to the axis of the beam of all loads and reactions applied to the portion of the beam on either side of that cross-section. Shear force at any transverse cross-section of a straight beam is the algebraic sum of the components acting transverse to the axis of the beam of all the loads and reactions applied to the portion of the beam on either side of the cross-section. Bending moment at any transverse cross-section of a straight beam is the algebraic sum of the moments, taken about an axis passing through the centroid of the cross-section. The axis about which the moments are taken is, of course, normal to the plane of loading.
2.2.2
Relation Between Load, Shear, and Bending Moment
When a beam is subjected to transverse loads, there exist certain relationships between load, shear, and bending moment. Let us consider, for example, the beam shown in Figure 2.3 subjected to some arbitrary loading, p.
FIGURE 2.3: A beam under arbitrary loading.
Let S and M be the shear and bending moment, respectively, for any point ‘m’ at a distance x, which is measured from A, being positive when measured to the right. Corresponding values of shear and bending moment at point ‘n’ at a differential distance dx to the right of m are S + dS and M + dM, respectively. It can be shown, neglecting the second order quantities, that p= 1999 by CRC Press LLC
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dS dx
(2.3)
and
dM (2.4) dx Equation 2.3 shows that the rate of change of shear at any point is equal to the intensity of load applied to the beam at that point. Therefore, the difference in shear at two cross-sections C and D is Z xD pdx (2.5) SD − SC = S=
xC
We can write in the same way for moment as MD − MC =
2.2.3
Z
xD
xC
Sdx
(2.6)
Shear and Bending Moment Diagrams
In order to plot the shear force and bending moment diagrams it is necessary to adopt a sign convention for these responses. A shear force is considered to be positive if it produces a clockwise moment about a point in the free body on which it acts. A negative shear force produces a counterclockwise moment about the point. The bending moment is taken as positive if it causes compression in the upper fibers of the beam and tension in the lower fiber. In other words, sagging moment is positive and hogging moment is negative. The construction of these diagrams is explained with an example given in Figure 2.4.
FIGURE 2.4: Bending moment and shear force diagrams.
The section at E of the beam is in equilibrium under the action of applied loads and internal forces acting at E as shown in Figure 2.5. There must be an internal vertical force and internal bending moment to maintain equilibrium at Section E. The vertical force or the moment can be obtained as the algebraic sum of all forces or the algebraic sum of the moment of all forces that lie on either side of Section E. 1999 by CRC Press LLC
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FIGURE 2.5: Internal forces. The shear on a cross-section an infinitesimal distance to the right of point A is +55 k and, therefore, the shear diagram rises abruptly from 0 to +55 at this point. In the portion AC, since there is no additional load, the shear remains +55 on any cross-section throughout this interval, and the diagram is a horizontal as shown in Figure 2.4. An infinitesimal distance to the left of C the shear is +55, but an infinitesimal distance to the right of this point the 30 k load has caused the shear to be reduced to +25. Therefore, at point C there is an abrupt change in the shear force from +55 to +25. In the same manner, the shear force diagram for the portion CD of the beam remains a rectangle. In the portion DE, the shear on any cross-section a distance x from point D is S = 55 − 30 − 4x = 25 − 4x which indicates that the shear diagram in this portion is a straight line decreasing from an ordinate of +25 at D to +1 at E. The remainder of the shear force diagram can easily be verified in the same way. It should be noted that, in effect, a concentrated load is assumed to be applied at a point and, hence, at such a point the ordinate to the shear diagram changes abruptly by an amount equal to the load. In the portion AC, the bending moment at a cross-section a distance x from point A is M = 55x. Therefore, the bending moment diagram starts at 0 at A and increases along a straight line to an ordinate of +165 k-ft at point C. In the portion CD, the bending moment at any point a distance x from C is M = 55(x + 3) − 30x. Hence, the bending moment diagram in this portion is a straight line increasing from 165 at C to 265 at D. In the portion DE, the bending moment at any point a distance x from D is M = 55(x + 7) − 30(X + 4) − 4x 2 /2. Hence, the bending moment diagram in this portion is a curve with an ordinate of 265 at D and 343 at E. In an analogous manner, the remainder of the bending moment diagram can be easily constructed. Bending moment and shear force diagrams for beams with simple boundary conditions and subject to some simple loading are given in Figure 2.6.
2.2.4
Fix-Ended Beams
When the ends of a beam are held so firmly that they are not free to rotate under the action of applied loads, the beam is known as a built-in or fix-ended beam and it is statically indeterminate. The bending moment diagram for such a beam can be considered to consist of two parts, namely the free bending moment diagram obtained by treating the beam as if the ends are simply supported and the fixing moment diagram resulting from the restraints imposed at the ends of the beam. The solution of a fixed beam is greatly simplified by considering Mohr’s principles which state that: 1. the area of the fixing bending moment diagram is equal to that of the free bending moment diagram 2. the centers of gravity of the two diagrams lie in the same vertical line, i.e., are equidistant from a given end of the beam The construction of bending moment diagram for a fixed beam is explained with an example shown in Figure 2.7. P Q U T is the free bending moment diagram, Ms , and P Q R S is the fixing 1999 by CRC Press LLC
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FIGURE 2.6: Shear force and bending moment diagrams for beams with simple boundary conditions subjected to selected loading cases.
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FIGURE 2.6: (Continued) Shear force and bending moment diagrams for beams with simple boundary conditions subjected to selected loading cases.
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FIGURE 2.6: (Continued) Shear force and bending moment diagrams for beams with simple boundary conditions subjected to selected loading cases.
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FIGURE 2.7: Fixed-ended beam. moment diagram, Mi . The net bending moment diagram, M, is shown shaded. If As is the area of the free bending moment diagram and Ai the area of the fixing moment diagram, then from the first Mohr’s principle we have As = Ai and 1 W ab × ×L 2 L
=
MA + MB
=
1 (MA + MB ) × L 2 W ab L
(2.7)
From the second principle, equating the moment about A of As and Ai , we have, MA + 2MB =
W ab 2 2 + 3ab + b 2a L3
(2.8)
Solving Equations 2.7 and 2.8 for MA and MB , we get MA
=
MB
=
W ab2 L2 W a2b L2
Shear force can be determined once the bending moment is known. The shear force at the ends of the beam, i.e., at A and B are SA
=
SB
=
Wb MA − MB + L L MB − MA Wa + L L
Bending moment and shear force diagrams for some typical loading cases are shown in Figure 2.8.
2.2.5 Continuous Beams Continuous beams, like fix-ended beams, are statically indeterminate. Bending moments in these beams are functions of the geometry, moments of inertia and modulus of elasticity of individual members besides the load and span. They may be determined by Clapeyron’s Theorem of three moments, moment distribution method, or slope deflection method. 1999 by CRC Press LLC
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FIGURE 2.8: Shear force and bending moment diagrams for built-up beams subjected to typical loading cases.
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FIGURE 2.8: (Continued) Shear force and bending moment diagrams for built-up beams subjected to typical loading cases.
An example of a two-span continuous beam is solved by Clapeyron’s Theorem of three moments. The theorem is applied to two adjacent spans at a time and the resulting equations in terms of unknown support moments are solved. The theorem states that A1 x1 A2 x2 + (2.9) MA L1 + 2MB (L1 + L2 ) + MC L2 = 6 L1 L2 in which MA , MB , and MC are the hogging moment at the supports A, B, and C, respectively, of two adjacent spans of length L1 and L2 (Figure 2.9); A1 and A2 are the area of bending moment diagrams produced by the vertical loads on the simple spans AB and BC, respectively; x1 is the centroid of A1 from A, and x2 is the distance of the centroid of A2 from C. If the beam section is constant within a
FIGURE 2.9: Continuous beams. 1999 by CRC Press LLC
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span but remains different for each of the spans, Equation 2.9 can be written as L2 A1 x1 A2 x2 L1 L1 L2 + 2MB + =6 + + MC MA I1 I1 I2 I2 L1 I1 L2 I2
(2.10)
in which I1 and I2 are the moments of inertia of beam section in span L1 and L2 , respectively.
EXAMPLE 2.1:
The example in Figure 2.10 shows the application of this theorem. For spans AC and BC
FIGURE 2.10: Example—continuous beam.
MA × 10 + 2MC (10 + 10) + MB × 10 # " 2 1 3 × 250 × 10 × 5 2 × 500 × 10 × 5 + =6 10 10 Since the support at A is simply supported, MA = 0. Therefore, 4MC + MB = 1250
(2.11)
Considering an imaginary span BD on the right side of B, and applying the theorem for spans CB and BD ×2 MC × 10 + 2MB (10) + MD × 10 = 6 × (2/3)×10×5 10 MC + 2MB = 500 (because MC = MD ) Solving Equations 2.11 and 2.12 we get MB MC 1999 by CRC Press LLC
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= =
107.2 kNm 285.7 kNm
(2.12)
Shear force at A is SA = Shear force at C is
MA − MC + 100 = −28.6 + 100 = 71.4 kN L
MC − MB MC − MA + 100 + + 100 = L L = (28.6 + 100) + (17.9 + 100) = 246.5 kN
SC
Shear force at B is
SB =
MB − MC + 100 = −17.9 + 100 = 82.1 kN L
The bending moment and shear force diagrams are shown in Figure 2.10.
2.2.6
Beam Deflection
There are several methods for determining beam deflections: (1) moment-area method, (2) conjugatebeam method, (3) virtual work, and (4) Castigliano’s second theorem, among others. The elastic curve of a member is the shape the neutral axis takes when the member deflects under load. The inverse of the radius of curvature at any point of this curve is obtained as M 1 = R EI
(2.13)
in which M is the bending moment at the point and EI is the flexural rigidity of the beam section. 2 Since the deflection is small, R1 is approximately taken as ddxy2 , and Equation 2.13 may be rewritten as: d 2y (2.14) M = EI 2 dx In Equation 2.14, y is the deflection of the beam at distance x measured from the origin of coordinate. The change in slope in a distance dx can be expressed as Mdx/EI and hence the slope in a beam is obtained as Z B M (2.15) dx θB − θA = A EI Equation 2.15 may be stated as the change in slope between the tangents to the elastic curve at two points is equal to the area of the M/EI diagram between the two points. Once the change in slope between tangents to the elastic curve is determined, the deflection can be obtained by integrating further the slope equation. In a distance dx the neutral axis changes in direction by an amount dθ. The deflection of one point on the beam with respect to the tangent at another point due to this angle change is equal to dδ = xdθ , where x is the distance from the point at which deflection is desired to the particular differential distance. To determine the total deflection from the tangent at one point A to the tangent at another point B on the beam, it is necessary to obtain a summation of the products of each dθ angle (from A to B) times the distance to the point where deflection is desired, or Z B Mx dx (2.16) δB − δA = EI A The deflection of a tangent to the elastic curve of a beam with respect to a tangent at another point is equal to the moment of M/EI diagram between the two points, taken about the point at which deflection is desired. 1999 by CRC Press LLC
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Moment Area Method
Moment area method is most conveniently used for determining slopes and deflections for beams in which the direction of the tangent to the elastic curve at one or more points is known, such as cantilever beams, where the tangent at the fixed end does not change in slope. The method is applied easily to beams loaded with concentrated loads because the moment diagrams consist of straight lines. These diagrams can be broken down into single triangles and rectangles. Beams supporting uniform loads or uniformly varying loads may be handled by integration. Properties of M diagrams designers usually come across are given in Figure 2.11. some of the shapes of EI
FIGURE 2.11: Typical M/EI diagram.
It should be understood that the slopes and deflections that are obtained using the moment area theorems are with respect to tangents to the elastic curve at the points being considered. The theorems do not directly give the slope or deflection at a point in the beam as compared to the horizontal axis (except in one or two special cases); they give the change in slope of the elastic curve from one point to another or the deflection of the tangent at one point with respect to the tangent at another point. There are some special cases in which beams are subjected to several concentrated loads or the combined action of concentrated and uniformly distributed loads. In such cases it is advisable to separate the concentrated loads and uniformly distributed loads and the moment area method can be applied separately to each of these loads. The final responses are obtained by the principle of superposition. For example, consider a simply supported beam subjected to uniformly distributed load q as shown in Figure 2.12. The tangent to the elastic curve at each end of the beam is inclined. The deflection δ1 of the tangent at the left end from the tangent at the right end is found as ql 4 /24EI . The distance from the original chord between the supports and the tangent at right end, δ2 , can be computed as ql 4 /48EI . The deflection of a tangent at the center from a tangent at right end, δ3 , is determined in ql 4 5 ql 4 . The difference between δ2 and δ3 gives the centerline deflection as 384 this step as 128EI EI .
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FIGURE 2.12: Deflection-simply supported beam under UDL.
2.2.7
Curved Flexural Members
The flexural formula is based on the assumption that the beam to which bending moment is applied is initially straight. Many members, however, are curved before a bending moment is applied to them. Such members are called curved beams. It is important to determine the effect of initial curvature of a beam on the stresses and deflections caused by loads applied to the beam in the plane of initial curvature. In the following discussion, all the conditions applicable to straight-beam formula are assumed valid except that the beam is initially curved. Let the curved beam DOE shown in Figure 2.13 be subjected to the loads Q. The surface in which the fibers do not change in length is called the neutral surface. The total deformations of the fibers between two normal sections such as AB and A1 B1 are assumed to vary proportionally with the distances of the fibers from the neutral surface. The top fibers are compressed while those at the bottom are stretched, i.e., the plane section before bending remains plane after bending. In Figure 2.13 the two lines AB and A1 B1 are two normal sections of the beam before the loads are applied. The change in the length of any fiber between these two normal sections after bending is represented by the distance along the fiber between the lines A1 B1 and A0 B 0 ; the neutral surface is represented by NN1 , and the stretch of fiber P P1 is P 1P10 , etc. For convenience it will be assumed that the line AB is a line of symmetry and does not change direction. The total deformations of the fibers in the curved beam are proportional to the distances of the fibers from the neutral surface. However, the strains of the fibers are not proportional to these distances because the fibers are not of equal length. Within the elastic limit the stress on any fiber in the beam is proportional to the strain of the fiber, and hence the elastic stresses in the fibers of a curved beam are not proportional to the distances of the fibers from the neutral surface. The resisting moment in a curved beam, therefore, is not given by the expression σ I /c. Hence, the neutral axis in a curved beam does not pass through the centroid of the section. The distribution of stress over the section and the relative position of the neutral axis are shown in Figure 2.13b; if the beam were straight, the stress would be zero at the centroidal axis and would vary proportionally with the distance from the 1999 by CRC Press LLC
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FIGURE 2.13: Bending of curved beams. centroidal axis as indicated by the dot-dash line in the figure. The stress on a normal section such as AB is called the circumferential stress. Sign Conventions
The bending moment M is positive when it decreases the radius of curvature, and negative when it increases the radius of curvature; y is positive when measured toward the convex side of the beam, and negative when measured toward the concave side, that is, toward the center of curvature. With these sign conventions, σ is positive when it is a tensile stress. Circumferential Stresses
Figure 2.14 shows a free body diagram of the portion of the body on one side of the section; the equations of equilibrium are applied to the forces acting on this portion. The equations obtained are Z X σ da = 0 (2.17) Fz = 0 or Z X Mz = 0 or M = yσ da (2.18) Figure 2.15 represents the part ABB1 A1 of Figure 2.13a enlarged; the angle between the two sections AB and A1 B1 is dθ . The bending moment causes the plane A1 B1 to rotate through an angle 1dθ, thereby changing the angle this plane makes with the plane BAC from dθ to (dθ + 1dθ ); the center of curvature is changed from C to C 0 , and the distance of the centroidal axis from the center of curvature is changed from R to ρ. It should be noted that y, R, and ρ at any section are measured from the centroidal axis and not from the neutral axis. It can be shown that the bending stress σ is given by the relation 1 y M (2.19) 1+ σ = aR ZR+y in which Z=−
1 a
Z
y da R+y
σ is the tensile or compressive (circumferential) stress at a point at the distance y from the centroidal axis of a transverse section at which the bending moment is M; R is the distance from the centroidal 1999 by CRC Press LLC
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FIGURE 2.14: Free-body diagram of curved beam segment.
FIGURE 2.15: Curvature in a curved beam. axis of the section to the center of curvature of the central axis of the unstressed beam; a is the area of the cross-section; Z is a property of the cross-section, the values of which can be obtained from the expressions for various areas given in Table 2.1. Detailed information can be obtained from [51].
EXAMPLE 2.2:
The bent bar shown in Figure 2.16 is subjected to a load P = 1780 N. Calculate the circumferential stress at A and B assuming that the elastic strength of the material is not exceeded. We know from Equation 2.19 M 1 y P + 1+ σ = a aR ZR+y
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TABLE 2.1
Analytical Expressions for Z
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TABLE 2.1
Analytical Expressions for Z (continued)
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TABLE 2.1
Analytical Expressions for Z (continued)
From Seely, F.B. and Smith, J.O., Advanced Mechanics of Materials, John Wiley & Sons, New York, 1952. With permission.
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FIGURE 2.16: Bent bar. in which a = area of rectangular section = 40 × 12 = 480 mm2 R = 40 mm yA = −20 yB = +20 P = 1780 N M = −1780 × 120 = −213600 N mm From Table 2.1, for rectangular section R+c R loge h R−c = 40 mm = 20 mm
Z
=
h c Hence,
−1 +
40 + 20 40 loge = 0.0986 Z = −1 + 40 40 − 20
Therefore,
2.3
σA
=
1780 480
+
−213600 480×40
σB
=
1780 480
+
−213600 480×40
−20 1 0.0986 40−20
1+
20 1 0.0986 40+20
1+
= 105.4 N mm2 (tensile) = −45 N mm2 (compressive)
Trusses
A structure that is composed of a number of bars pin connected at their ends to form a stable framework is called a truss. If all the bars lie in a plane, the structure is a planar truss. It is generally assumed that loads and reactions are applied to the truss only at the joints. The centroidal axis of each member is straight, coincides with the line connecting the joint centers at each end of the member, and lies in a plane that also contains the lines of action of all the loads and reactions. Many truss structures are three dimensional in nature and a complete analysis would require consideration of the full spatial interconnection of the members. However, in many cases, such as bridge structures and simple roof systems, the three-dimensional framework can be subdivided into planar components for analysis as planar trusses without seriously compromising the accuracy of the results. Figure 2.17 shows some typical idealized planar truss structures. 1999 by CRC Press LLC
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FIGURE 2.17: Typical planar trusses.
There exists a relation between the number of members, m, number of joints, j , and reaction components, r. The expression is m = 2j − r
(2.20)
which must be satisfied if it is to be statically determinate internally. The least number of reaction components required for external stability is r. If m exceeds (2j − r), then the excess members are called redundant members and the truss is said to be statically indeterminate. Truss analysis gives the bar forces in a truss; for a statically determinate truss, these bar forces can be found by employing the laws of statics to assure internal equilibrium of the structure. The process requires repeated use of free-body diagrams from which individual bar forces are determined. The method of joints is a technique of truss analysis in which the bar forces are determined by the sequential isolation of joints—the unknown bar forces at one joint are solved and become known bar forces at subsequent joints. The other method is known as method of sections in which equilibrium of a part of the truss is considered.
2.3.1
Method of Joints
An imaginary section may be completely passed around a joint in a truss. has become a P The jointP free body in equilibrium under the forces applied to it. The equations H = 0 and V = 0 may be applied to the joint to determine the unknown forces in members meeting there. It is evident that no more than two unknowns can be determined at a joint with these two equations. 1999 by CRC Press LLC
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EXAMPLE 2.3:
A truss shown in Figure 2.18 is symmetrically loaded, and it is sufficient to solve half the truss by considering the joints 1 through 5. At Joint 1, there are two unknown forces. Summation of the
FIGURE 2.18: Example—methods of joints, planar truss.
vertical components of all forces at Joint 1 gives 135 − F12 sin 45 = 0 which in turn gives the force in the member 1-2, F12 = 190.0 kN (compressive). Similarly, summation of the horizontal components gives F13 − F12 cos 45◦ = 0 Substituting for F12 gives the force in the member 1-3 as F13 = 135 kN (tensile). 1999 by CRC Press LLC
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Now, Joint 2 is cut completely and it is found that there are two unknown forces F25 and F23 . Summation of the vertical components gives F12 cos 45◦ − F23 = 0. Therefore, F23 = 135 kN (tensile). Summation of the horizontal components gives F12 sin 45◦ − F25 = 0 and hence F25 = 135 kN (compressive). After solving for Joints 1 and 2, one proceeds to take a section around Joint 3 at which there are now two unknown forces, namely, F34 and F35 . Summation of the vertical components at Joint 3 gives F23 − F35 sin 45◦ − 90 = 0 Substituting for F23 , one obtains F35 = 63.6 kN (compressive). Summing the horizontal components and substituting for F13 one gets −135 − 45 + F34 = 0 Therefore, F34 = 180 kN (tensile). The next joint involving two unknowns is Joint 4. When we consider a section around it, the summation of the vertical components at Joint 4 gives F45 = 90 kN (tensile). Now, the forces in all the members on the left half of the truss are known and by symmetry the forces in the remaining members can be determined. The forces in all the members of a truss can also be determined by making use of the method of section.
2.3.2
Method of Sections
If only a few member forces of a truss are needed, the quickest way to find these forces is by the Method of Sections. In this method, an imaginary cutting line called a section is drawn through a stable and determinate truss. Thus, a section subdivides the truss into two separate parts. Since the entire truss is in equilibrium, any part of it must also be in equilibrium. Either P of the two parts P of the P truss can be considered and the three equations of equilibrium Fx = 0, Fy = 0, and M = 0 can be applied to solve for member forces. The example considered in Section 2.3.1 (Figure 2.19) is once again considered. To calculate the force in the member 3-5, F35 , a section AA should be run to cut the member 3-5 as shown in the figure. It is only required to consider the equilibrium of one of the two parts of the truss. In this case, the portion of the truss on the left of the section is considered. The left portion of the truss as shown in Figure 2.19 is in equilibrium under the action of the forces, namely, the external and internal forces. Considering the equilibrium of forces in the vertical direction, one can obtain 135 − 90 + F35 sin 45◦ = 0 1999 by CRC Press LLC
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FIGURE 2.19: Example—method of sections, planar truss. Therefore, F35 is obtained as
√ F35 = −45 2 kN
The negative sign indicates that the member force is compressive. This result is the same as the one obtained by the Method of Joints. The other memberP forces cut by the section can be obtained by considering the other equilibrium equations, namely, M = 0. More sections can be taken in the same way so as to solve for other member forces in the truss. The most important advantage of this method is that one can obtain the required member force without solving for the other member forces.
2.3.3
Compound Trusses
A compound truss is formed by interconnecting two or more simple trusses. Examples of compound trusses are shown in Figure 2.20. A typical compound roof truss is shown in Figure 2.20a in which
FIGURE 2.20: Compound truss.
two simple trusses are interconnected by means of a single member and a common joint. The compound truss shown in Figure 2.20b is commonly used in bridge construction and in this case, 1999 by CRC Press LLC
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three members are used to interconnect two simple trusses at a common joint. There are three simple trusses interconnected at their common joints as shown in Figure 2.20c. The Method of Sections may be used to determine the member forces in the interconnecting members of compound trusses similar to those shown in Figure 2.20a and b. However, in the case of cantilevered truss, the middle simple truss is isolated as a free body diagram to find its reactions. These reactions are reversed and applied to the interconnecting joints of the other two simple trusses. After the interconnecting forces between the simple trusses are found, the simple trusses are analyzed by the Method of Joints or the Method of Sections.
2.3.4
Stability and Determinacy
A stable and statically determinate plane truss should have at least three members, three joints, and three reaction components. To form a stable and determinate plane truss of ‘n’ joints, the three members of the original triangle plus two additional members for each of the remaining (n−3) joints are required. Thus, the minimum total number of members, m, required to form an internally stable plane truss is m = 2n−3. If a stable, simple, plane truss of n joints and (2n−3) members is supported by three independent reaction components, the structure is stable and determinate when subjected to a general loading. If the stable, simple, plane truss has more than three reaction components, the structure is externally indeterminate. That means not all of the reaction components can be determined from the three available equations of statics. If the stable, simple, plane truss has more than (2n − 3) members, the structure is internally indeterminate and hence all of the member forces cannot be determined from the 2n available equations of statics in the Method of Joints. The analyst must examine the arrangement of the truss members and the reaction components to determine if the simple plane truss is stable. Simple plane trusses having (2n − 3) members are not necessarily stable.
2.4
Frames
Frames are statically indeterminate in general; special methods are required for their analysis. Slope deflection and moment distribution methods are two such methods commonly employed. Slope deflection is a method that takes into account the flexural displacements such as rotations and deflections and involves solutions of simultaneous equations. Moment distribution on the other hand involves successive cycles of computation, each cycle drawing closer to the “exact” answers. The method is more labor intensive but yields accuracy equivalent to that obtained from the “exact” methods. This method, however, remains the most important hand-calculation method for the analysis of frames.
2.4.1
Slope Deflection Method
This method is a special case of the stiffness method of analysis, and it is convenient for hand analysis of small structures. Moments at the ends of frame members are expressed in terms of the rotations and deflections of the joints. Members are assumed to be of constant section between each pair of supports. It is further assumed that the joints in a structure may rotate or deflect, but the angles between the members meeting at a joint remain unchanged. The member force-displacement equations that are needed for the slope deflection method are written for a member AB in a frame. This member, which has its undeformed position along the x axis is deformed into the configuration shown in Figure 2.21. The positive axes, along with the positive member-end force components and displacement components, are shown in the figure. 1999 by CRC Press LLC
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FIGURE 2.21: Deformed configuration of a beam.
The equations for end moments are written as MAB
=
MBA
=
2EI (2θA + θB − 3ψAB ) + MF AB l 2EI (2θB + θA − 3ψAB ) + MF BA l
(2.21)
in which MFAB and MFBA are fixed-end moments at supports A and B, respectively, due to the applied load. ψAB is the rotation as a result of the relative displacement between the member ends A and B given as 1AB yA + yB (2.22) = ψAB = l l where 1AB is the relative deflection of the beam ends. yA and yB are the vertical displacements at ends A and B. Fixed-end moments for some loading cases may be obtained from Figure 2.8. The slope deflection equations in Equation 2.21 show that the moment at the end of a member is dependent on member properties EI , dimension l, and displacement quantities. The fixed-end moments reflect the transverse loading on the member.
2.4.2
Application of Slope Deflection Method to Frames
The slope deflection equations may be applied to statically indeterminate frames with or without sidesway. A frame may be subjected to sidesway if the loads, member properties, and dimensions of the frame are not symmetrical about the centerline. Application of slope deflection method can be illustrated with the following example.
EXAMPLE 2.4:
Consider the frame shown in Figure 2.22. subjected to sidesway 1 to the right of the frame. Equation 2.21 can be applied to each of the members of the frame as follows: Member AB: MAB 1999 by CRC Press LLC
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=
2EI 20
31 2θA + θB − + MF AB 20
FIGURE 2.22: Example—slope deflection method.
θA = 0,
MBA
=
MFAB
=
2EI 31 2θB + θA − + MF BA 20 20 MF BA = 0
Hence, MAB
=
MBA
=
2EI (θB − 3ψ) 20 2EI (2θB − 3ψ) 20
in which ψ=
(2.23) (2.24)
1 20
Member BC: MBC
=
MCB
=
MFBC
=
MF CB
=
2EI (2θB + θC − 3 × 0) + MF BC 30 2EI (2θC + θB − 3 × 0) + MF CB 30 40 × 10 × 202 − = −178 ft-kips 302 40 × 102 × 20 − = 89 ft-kips 302
Hence, MBC
=
MCB
=
Member CD:
1999 by CRC Press LLC
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MCD
=
MDC
=
MF CD
=
2EI (2θB + θC ) − 178 30 2EI (2θC + θB ) + 89 30
2EI 31 2θC + θD − + MF CD 30 30 2EI 31 2θD + θC − + MF DC 30 30 MF DC = 0
(2.25) (2.26)
Hence, MDC
=
MDC
=
2EI θC − 3 × 30 2EI θC − 3 × 30
2 ψ = 3 2 ψ = 3
2EI (2θC − 2ψ) 30 2EI (θC − 2ψ) 30
(2.27) (2.28)
Considering moment equilibrium at Joint B X MB = MBA + MBC = 0 Substituting for MBA and MBC , one obtains EI (10θB + 2θC − 9ψ) = 178 30 or 10θB + 2θC − 9ψ =
267 K
(2.29)
where K = EI 20 . Considering moment equilibrium at Joint C X MC = MCB + MCD = 0 Substituting for MCB and MCD we get 2EI (4θC + θB − 2ψ) = −89 30 or θB + 4θC − 2ψ = −
66.75 K
(2.30)
Summation of base shear equals to zero, we have X H = HA + HD = 0 or
MCD + MDC MAB + MBA + =0 1AB 1CD
Substituting for MAB , MBA , MCD , and MDC and simplifying 2θB + 12θC − 70ψ = 0
(2.31)
Solution of Equations 2.29 to 2.31 results in θB
=
θC
=
and ψ= 1999 by CRC Press LLC
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42.45 K 20.9 K 12.8 K
(2.32)
Substituting for θB , θC , and ψ from Equations 2.32 into Equations 2.23 to 2.28 we get, MAB MBA MBC MCB MCD MDC
2.4.3
= = = = = =
10.10 ft-kips 93 ft-kips −93 ft-kips 90 ft-kips −90 ft-kips −62 ft-kips
Moment Distribution Method
The moment distribution method involves successive cycles of computation, each cycle drawing closer to the “exact” answers. The calculations may be stopped after two or three cycles, giving a very good approximate analysis, or they may be carried on to whatever degree of accuracy is desired. Moment distribution remains the most important hand-calculation method for the analysis of continuous beams and frames and it may be solely used for the analysis of small structures. Unlike the slope deflection method, this method does require the solution to simultaneous equations. The terms constantly used in moment distribution are fixed-end moments, unbalanced moment, distributed moments, and carry-over moments. When all of the joints of a structure are clamped to prevent any joint rotation, the external loads produce certain moments at the ends of the members to which they are applied. These moments are referred to as fixed-end moments. Initially the joints in a structure are considered to be clamped. When the joint is released, it rotates if the sum of the fixed-end moments at the joint is not zero. The difference between zero and the actual sum of the end moments is the unbalanced moment. The unbalanced moment causes the joint to rotate. The rotation twists the ends of the members at the joint and changes their moments. In other words, rotation of the joint is resisted by the members and resisting moments are built up in the members as they are twisted. Rotation continues until equilibrium is reached—when the resisting moments equal the unbalanced moment—at which time the sum of the moments at the joint is equal to zero. The moments developed in the members resisting rotation are the distributed moments. The distributed moments in the ends of the member cause moments in the other ends, which are assumed fixed, and these are the carry-over moments. Sign Convention
The moments at the end of a member are assumed to be positive when they tend to rotate the member clockwise about the joint. This implies that the resisting moment of the joint would be counter-clockwise. Accordingly, under gravity loading condition the fixed-end moment at the left end is assumed as counter-clockwise (−ve) and at the right end as clockwise (+ve). Fixed-End Moments
Fixed-end moments for several cases of loading may be found in Figure 2.8. Application of moment distribution may be explained with reference to a continuous beam example as shown in Figure 2.23. Fixed-end moments are computed for each of the three spans. At Joint B the unbalanced moment is obtained and the clamp is removed. The joint rotates, thus distributing the unbalanced moment to the B-ends of spans BA and BC in proportion to their distribution factors. The values of these distributed moments are carried over at one-half rate to the other ends of the members. When equilibrium is reached, Joint B is clamped in its new rotated position and Joint C is released afterwards. Joint C rotates under its unbalanced moment until it reaches equilibrium, the rotation causing distributed moments in the C-ends of members CB and CD and their resulting carry-over 1999 by CRC Press LLC
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FIGURE 2.23: Example—continuous beam by moment distribution. moments. Joint C is now clamped and Joint B is released. This procedure is repeated again and again for Joints B and C, the amount of unbalanced moment quickly diminishing, until the release of a joint causes negligible rotation. This process is called moment distribution. The stiffness factors and distribution factors are computed as follows: DFBA
=
DFBC
=
DFCB
=
DFCD
=
KBA P K KBC P K KCB P K KCD P K
I /20 I /20 + I /30 I /30 = I /20 + I /30 I /30 = I /30 + I /25 I /25 = I /30 + I /25 =
= 0.6 = 0.4 = 0.45 = 0.55
Fixed-end moments MFAB MFBA
= −50 ft-kips; = 50 ft-kips;
MF BC MF CB
= −150 ft-kips; = 150 ft-kips;
MF CD MF DC
= −104 ft-kips = 104 ft-kips
When a clockwise couple is applied at the near end of a beam, a clockwise couple of half the magnitude is set up at the far end of the beam. The ratio of the moments at the far and near ends is defined as carry-over factor, and it is 21 in the case of a straight prismatic member. The carry-over factor was developed for carrying over to fixed ends, but it is applicable to simply supported ends, which must have final moments of zero. It can be shown that the beam simply supported at the far end is only three-fourths as stiff as the one that is fixed. If the stiffness factors for end spans that are simply supported are modified by three-fourths, the simple end is initially balanced to zero and no carry-overs are made to the end afterward. This simplifies the moment distribution process significantly. 1999 by CRC Press LLC
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FIGURE 2.24: Example—non-sway frame by moment distribution.
Moment Distribution for Frames
Moment distribution for frames without sidesway is similar to that for continuous beams. The example shown in Figure 2.24 illustrates the applications of moment distribution for a frame without sidesway. EI /20 = 0.25 DFBA = EI EI 2EI 20 + 20 + 20 Similarly DFBE MFBC MFBE
= 0.50; = −100 ft-kips; = 50 ft-kips;
DFBC MF CB MF EB
= 0.25 = 100 ft-kips = −50 ft-kips.
Structural frames are usually subjected to sway in one direction or the other due to asymmetry of the structure and eccentricity of loading. The sway deflections affect the moments resulting in unbalanced moment. These moments could be obtained for the deflections computed and added to the originally distributed fixed-end moments. The sway moments are distributed to columns. Should a frame have columns all of the same length and the same stiffness, the sidesway moments will be the same for each column. However, should the columns have differing lengths and/or stiffness, this will not be the case. The sidesway moments should vary from column to column in proportion to their I / l 2 values. 1999 by CRC Press LLC
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The frame in Figure 2.25 shows a frame subjected to sway. The process of obtaining the final moments is illustrated for this frame. The frame sways to the right and the sidesway moment can be assumed in the ratio 400 300 : (or) 1 : 0.7 202 202 Final moments are obtained by adding distributed fixed-end moments and tributed assumed sidesway moments.
2.4.4
13.06 2.99
times the dis-
Method of Consistent Deformations
The method of consistent deformations makes use of the principle of deformation compatibility to analyze indeterminate structures. This method employs equations that relate the forces acting on the structure to the deformations of the structure. These relations are formed so that the deformations are expressed in terms of the forces and the forces become the unknowns in the analysis. Let us consider the beam shown in Figure 2.26a. The first step, in this method, is to determine the degree of indeterminacy or the number of redundants that the structure possesses. As shown in the figure, the beam has three unknown reactions, RA , RC , and MA . Since there are only two equations of equilibrium available for calculating the reactions, the beam is said to be indeterminate to the first degree. Restraints that can be removed without impairing the load-supporting capacity of the structure are referred to as redundants. Once the number of redundants is known, the next step is to decide which reaction is to be removed in order to form a determinate structure. Any one of the reactions may be chosen to be the redundant provided that a stable structure remains after the removal of that reaction. For example, let us take the reaction RC as the redundant. The determinate structure obtained by removing this restraint is the cantilever beam shown in Figure 2.26b. We denote the deflection at end C of this beam, due to P , by 1CP . The first subscript indicates that the deflection is measured at C and the second subscript that the deflection is due to the applied load P . Using the moment area method, it can be shown that 1CP = 5P L3 /48EI . The redundant RC is then applied to the determinate cantilever beam, as shown in Figure 2.26c. This gives rise to a deflection 1CR at point C the magnitude of which can be shown to be RC L3 /3EI . In the actual indeterminate structure, which is subjected to the combined effects of the load P and the redundant RC , the deflection at C is zero. Hence the algebraic sum of the deflection 1CP in Figure 2.26b and the deflection 1CR in Figure 2.26c must vanish. Assuming downward deflections to be positive, we write (2.33) 1CP − 1CR = 0 or
RC L3 5P L3 − =0 48EI 3EI
from which
5 P 16 Equation 2.33, which is used to solve for the redundant, is referred to as an equation of consistent of deformation. reactions by applying Once the redundant RC has been evaluated, one can determine the remaining P the equations of equilibrium to the structure in Figure 2.26a. Thus, Fy = 0 leads to RC =
RA = P − 1999 by CRC Press LLC
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11 5 P = P 16 16
FIGURE 2.25: Example—sway frame by moment distribution.
1999 by CRC Press LLC
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FIGURE 2.25: (Continued) Example—sway frame by moment distribution.
1999 by CRC Press LLC
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FIGURE 2.26: Beam with one redundant reaction. and
P
MA = 0 gives
PL 5 3 − PL = PL 2 16 16 A free body of the beam, showing all the forces acting on it, is shown in Figure 2.26d. The steps involved in the method of consistent deformations are: MA =
1. The number of redundants in the structure is determined. 2. Enough redundants are removed to form a determinate structure. 3. The displacements that the applied loads cause in the determinate structure at the points where the redundants have been removed are then calculated. 4. The displacements at these points in the determinate structure due to the redundants are obtained. 5. At each point where a redundant has been removed, the sum of the displacements calculated in Steps 3 and 4 must be equal to the displacement that exists at that point in the actual indeterminate structure. The redundants are evaluated using these relationships. 6. Once the redundants are known, the remaining reactions are determined using the equations of equilibrium. Structures with Several Redundants
The method of consistent deformations can be applied to structures with two or more redundants. For example, the beam in Figure 2.27a is indeterminate to the second degree and has two redundant reactions. If we let the reactions at B and C be the redundants, then the determinate structure obtained by removing these supports is the cantilever beam shown in Figure 2.27b. To this determinate structure we apply separately the given load (Figure 2.27c) and the redundants RB and RC one at a time (Figures 2.27d and e). Since the deflections at B and C in the original beam are zero, the algebraic sum of the deflections in Figures 2.27c, d, and e at these same points must also vanish. Thus, 1BP − 1BB − 1BC 1CP − 1CB − 1CC 1999 by CRC Press LLC
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= =
0 0
(2.34)
FIGURE 2.27: Beam with two redundant reactions.
1999 by CRC Press LLC
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It is useful in the case of complex structures to write the equations of consistent deformations in the form 1BP − δBB RB − δBC RC 1CP − δCB RB − δCC RC
= =
0 0
(2.35)
in which δBC , for example, denotes the deflection at B due to a unit load at C in the direction of RC . Solution of Equation 2.35 gives the redundant reactions RB and RC .
EXAMPLE 2.5:
Determine the reactions for the beam shown in Figure 2.28 and draw its shear force and bending moment diagrams. It can be seen from the figure that there are three reactions, namely, MA , RA , and RC one more than that required for a stable structure. The reaction RC can be removed to make the structure determinate. We know that the deflection at support C of the beam is zero. One can determine the deflection δCP at C due to the applied load on the cantilever in Figure 2.28b. The deflection δCR at C due to the redundant reaction on the cantilever (Figure 2.28c) can be determined in the same way. The compatibility equation gives δCP − δCR = 0 By moment area method, δCP
=
δCR
=
1 20 2 20 ×2×1+ × ×2× ×2 EI 2 EI 3 1 60 2 40 ×2×3+ × ×2× ×2+2 + EI 2 EI 3 1520 = 3EI 1 4RC 2 64RC × ×4× ×4= 2 EI 3 3EI
Substituting for δCP and δCR in the compatibility equation one obtains 1520 64RC − =0 3EI 3EI from which RC = 23.75 kN ↑ By using statical equilibrium equations we get RA = 6.25 kN ↑ and MA = 5 kNm. The shear force and bending moment diagrams are shown in Figure 2.28d. 1999 by CRC Press LLC
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FIGURE 2.28: Example 2.5.
1999 by CRC Press LLC
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2.5 2.5.1
Plates Bending of Thin Plates
When the thickness of an object is small compared to the other dimensions, it is called a thin plate. The plane parallel to the faces of the plate and bisecting the thickness of the plate, in the undeformed state, is called the middle plane of the plate. When the deflection of the middle plane is small compared with the thickness, h, it can be assumed that 1. There is no deformation in the middle plane. 2. The normal of the middle plane before bending is deformed into the normals of the middle plane after bending. 3. The normal stresses in the direction transverse to the plate can be neglected. Based on these assumptions, all stress components can be expressed by deflection w0 of the plate. is a function of the two coordinates (x, y) in the plane of the plate. This function has to satisfy a linear partial differential equation, which, together with the boundary conditions, completely defines w0 . Figure 2.29a shows a plate element cut from a plate whose middle plane coincides with the xy plane. The middle plane of the plate subjected to a lateral load of intensity ‘q’ is shown in Figure 2.29b. It can be shown, by considering the equilibrium of the plate element, that the stress resultants are given as 2 ∂ w ∂ 2w +ν 2 Mx = −D ∂x 2 ∂y 2 ∂ w ∂ 2w + ν My = −D ∂y 2 ∂x 2
w0
where Mx and My Mxy and Myx Qx and Qy Vx and Vy R D E
= = = = = = =
1999 by CRC Press LLC
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Mxy
=
Vx
=
Vy
=
Qx
=
Qy
=
R
=
∂ 2w ∂x∂y 3 3 ∂ w ∂ w + (2 − ν) ∂x 3 ∂x∂y 2 ∂ 3w ∂ 3w + (2 − ν) ∂y 3 ∂y∂x 2 2 ∂ ∂ w ∂ 2w −D + ∂x ∂x 2 ∂y 2 2 ∂ ∂ w ∂ 2w −D + ∂y ∂x 2 ∂y 2 −Myx = D(1 − ν)
2D(1 − ν)
∂ 2w ∂x∂y
bending moments per unit length in the x and y directions, respectively twisting moments per unit length shearing forces per unit length in the x and y directions, respectively supplementary shear forces in the x and y directions, respectively corner force Eh3 , flexural rigidity of the plate per unit length 12(1−ν 2 ) modulus of elasticity
(2.36) (2.37) (2.38) (2.39) (2.40) (2.41)
FIGURE 2.29: (a) Plate element; (b) stress resultants.
ν
= Poisson’s Ratio The governing equation for the plate is obtained as ∂ 4w ∂ 4w q ∂ 4w + 2 + = 4 2 2 4 D ∂x ∂x ∂y ∂y
(2.42)
Any plate problem should satisfy the governing Equation 2.42 and boundary conditions of the plate.
2.5.2
Boundary Conditions
There are three basic boundary conditions for plate problems. These are the clamped edge, the simply supported edge, and the free edge. 1999 by CRC Press LLC
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Clamped Edge
For this boundary condition, the edge is restrained such that the deflection and slope are zero along the edge. If we consider the edge x = a to be clamped, we have (w)x=a = 0
∂w ∂x
x=a
=0
(2.43)
Simply Supported Edge
If the edge x = a of the plate is simply supported, the deflection w along this edge must be zero. At the same time this edge can rotate freely with respect to the edge line. This means that (w)x=a = 0;
∂ 2w ∂x 2
x=a
=0
(2.44)
Free Edge
If the edge x = a of the plate is entirely free, there are no bending and twisting moments or vertical shearing forces. This can be written in terms of w, the deflection as
2.5.3
∂ 2w ∂ 2w + ν ∂x 2 ∂y 2
x=a
∂ 3w ∂ 3w + (2 − ν) 3 ∂x ∂x∂y 2
=0
x=a
=0
(2.45)
Bending of Simply Supported Rectangular Plates
A number of the plate bending problems may be solved directly by solving the differential Equation 2.42. The solution, however, depends on the loading and boundary condition. Consider a simply supported plate subjected to a sinusoidal loading as shown in Figure 2.30. The differential
FIGURE 2.30: Rectangular plate under sinusoidal loading.
Equation 2.42 in this case becomes πx πy ∂ 4w ∂ 4w qo ∂ 4w sin sin + 2 + = 4 2 2 4 D a b ∂x ∂x ∂y ∂y 1999 by CRC Press LLC
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(2.46)
The boundary conditions for the simply supported edges are w
= 0,
w
= 0,
∂ 2w = 0 for x ∂x 2 ∂ 2w = 0 for y ∂y 2
= 0 and x = a = 0 and y = b
(2.47)
The deflection function becomes
πy πx sin (2.48) a b which satisfies all the boundary conditions in Equation 2.47. w0 must be chosen to satisfy Equation 2.46. Substitution of Equation 2.48 into Equation 2.46 gives w = w0 sin
π4
1 1 + 2 a2 b
2 w0 =
qo D
The deflection surface for the plate can, therefore, be found as w=
π 4D
qo 1 a2
+
1 b2
πx πy sin a b
2 sin
(2.49)
Using Equations 2.49 and 2.36, we find expression for moments as πy qo ν 1 πx sin + sin Mx = 2 2 2 a b a b π 2 a12 + b12 qo 1 πy ν πx + 2 sin sin My = 2 2 a b a b π 2 a12 + b12 Mxy
=
πy πx qo (1 − ν) cos cos 2 a b π 2 a12 + b12 ab
(2.50)
Maximum deflection and maximum bending moments that occur at the center of the plate can be written by substituting x = a/2 and y = b/2 in Equation 2.50 as wmax (Mx )max (My )max
=
π 4D
=
π2
=
π2
qo 1 a2
+
qo 1 a2
+
1 b2
qo 1 a2
+
1 b2
1 b2
2
2 2
(2.51)
ν 1 + 2 a2 b 1 ν + 2 2 a b
If the plate is square, then a = b and Equation 2.51 becomes
1999 by CRC Press LLC
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wmax
=
qo a 4 4π 4 D 0
(Mx )max
=
(My )max =
(1 + ν) qo a 2 4π 2
(2.52)
If the simply supported rectangular plate is subjected to any kind of loading given by q = q(x, y)
(2.53)
the function q(x, y) should be represented in the form of a double trigonometric series as q(x, y) =
∞ ∞ X X
qmn sin
m=1 n=1
mπ x nπy sin a b
(2.54)
in which qmn is given by qmn =
Z
4 ab
a
Z
0
b
q(x, y) sin
0
nπy mπ x sin dxdy a b
(2.55)
From Equations 2.46, 2.53, 2.54, and 2.55 we can obtain the expression for deflection as w=
∞ ∞ mπ x qmn nπy 1 XX sin 2 2 sin 2 a b π 4D m=1 n=1 m2 + n2 a b
(2.56)
If the applied load is uniformly distributed of intensity qo , we have q(x, y) = qo and from Equation 2.55 we obtain qmn =
4qo ab
Z
a
Z
b
sin 0
0
nπy 16qo mπ x sin dxdy = 2 a b π mn
(2.57)
in which ‘m’ and ‘n’ are odd integers. qmn = 0 if ‘m’ or ‘n’ or both of them are even numbers. We can, therefore, write the expression for deflection of a simply supported plate subjected to uniformly distributed load as ∞ ∞ nπy 16qo X X sin mπa x sin b (2.58) w= 6 2 2 2 π D m n m=1 n=1 mn + b2 a2 where m = 1, 3, 5, . . . and n = 1, 3, 5, . . . The maximum deflection occurs at the center and it can be written by substituting x = y = b2 in Equation 2.58 as wmax
∞ ∞ 16qo X X = 6 π D m=1 n=1
a 2
and
m+n
(−1) 2 −1 2 2 n2 mn m + a2 b2
(2.59)
Equation 2.59 is a rapid converging series and a satisfactory approximation can be obtained by taking only the first term of the series; for example, in the case of a square plate, wmax =
qo a 4 4qo a 4 = 0.00416 D π 6D
Assuming ν = 0.3, we get for the maximum deflection wmax = 0.0454 1999 by CRC Press LLC
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qo a 4 Eh3
FIGURE 2.31: Typical loading on plates and loading functions.
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FIGURE 2.31: (Continued) Typical loading on plates and loading functions.
FIGURE 2.32: Rectangular plate.
The expressions for bending and twisting moments can be obtained by substituting Equation 2.58 into Equation 2.36. Figure 2.31 shows some loading cases and the corresponding loading functions. The above solution for uniformly loaded cases is known as Navier solution. If two opposite sides (say x = 0 and x = a) of a rectangular plate are simply supported, the solution taking the deflection function as ∞ X mπ x Ym sin (2.60) w= a m=1
1999 by CRC Press LLC
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can be adopted. This solution was proposed by Levy [53]. Equation 2.60 satisfies the boundary 2 conditions w = 0 and ∂∂xw2 = 0 on the two simply supported edges. Ym should be determined such that it satisfies the boundary conditions along the edges y = ± b2 of the plate shown in Figure 2.32 and also the equation of the deflection surface ∂ 4w ∂ 4w qo ∂ 4w +2 2 2 + = 4 D ∂x ∂x ∂y ∂y 4
(2.61)
qo being the intensity of uniformly distributed load. The solution for Equation 2.61 can be taken in the form w = w1 + w2
(2.62)
for a uniformly loaded simply supported plate. w1 can be taken in the form qo 4 x − 2ax 3 + a 3 x w1 = 24D
(2.63)
representing the deflection of a uniformly loaded strip parallel to the x axis. It satisfies Equation 2.61 and also the boundary conditions along x = 0 and x = a. The expression w2 has to satisfy the equation ∂ 4 w2 ∂ 4 w2 ∂ 4 w2 + 2 + =0 ∂x 4 ∂x 2 ∂y 2 ∂y 4
(2.64)
and must be chosen such that Equation 2.62 satisfies all boundary conditions of the plate. Taking w2 in the form of series given in Equation 2.60 it can be shown that the deflection surface takes the form wψ =
∞ q a4 X mπy qo 4 o x − 2ax 3 + a 3 x + Am cosh 24D 24D a m=1
mπy mπy mπy sinh + Cm sinh + Bm a a a mπy mπy mπ x +Dm cosh sin a a a
(2.65)
Observing that the deflection surface of the plate is symmetrical with respect to the x axis, we keep in Equation 2.65 only an even function of y; therefore, Cm = Dm = 0. The deflection surface takes the form ∞ q a4 X mπy qo 4 o 3 3 x − 2ax + a x + Am cosh w= 24D 24D a m=1 mπy mπ x mπy sinh sin +Bm a a a
(2.66)
Developing the expression in Equation 2.63 into a trigonometric series, the deflection surface in Equation 2.66 is written as ∞ mπy mπx mπy 4 mπy qo a 4 X + Bm sin sin + Am cosh (2.67) w= 5 5 D a a a a π m m=1
Substituting Equation 2.67 in the boundary conditions w = 0, 1999 by CRC Press LLC
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∂ 2w =0 ∂y 2
(2.68)
one obtains the constants of integration Am and Bm and the expression for deflection may be written as ∞ 2αm y 1 αm tanh αm + 2 4qo a 4 X cosh 1− w= 5 5 2 cosh αm b π D m m=1,3,5... 2αm y mπ x 2y αm sinh sin (2.69) + 2 cosh αm b b a b in which αm = mπ 2a . Maximum deflection occurs at the middle of the plate, x = a2 , y = 0 and is given by
4qo a 4 w= 5 π D
∞ X m=1,3,5...
m−1
(−1) 2 m5
αm tanh αm + 2 1− 2 cosh αm
(2.70)
Solution of plates with arbitrary boundary conditions are complicated. It is possible to make some simplifying assumptions for plates with the same boundary conditions along two parallel edges in order to obtain the desired solution. Alternately, the energy method can be applied more efficiently to solve plates with complex boundary conditions. However, it should be noted that the accuracy of results depends upon the deflection function chosen. These functions must be so chosen that they satisfy at least the kinematics boundary conditions. Figure 2.33 gives formulas for deflection and bending moments of rectangular plates with typical boundary and loading conditions.
2.5.4
Bending of Circular Plates
In the case of symmetrically loaded circular plate, the loading is distributed symmetrically about the axis perpendicular to the plate through its center. In such cases, the deflection surface to which the middle plane of the plate is bent will also be symmetrical. The solution of circular plates can be conveniently carried out by using polar coordinates. Stress resultants in a circular plate element are shown in Figure 2.34. The governing differential equation is expressed in polar coordinates as d 1 d dw q 1 d r r = (2.71) r dr dr r dr dr D in which q is the intensity of loading. In the case of uniformly loaded circular plates, Equation 2.71 can be integrated successively and the deflection at any point at a distance r from the center can be expressed as w=
r C1 r 2 qo r 4 + + C2 log + C3 64D 4 a
(2.72)
in which qo is the intensity of loading and a is the radius of the plate. C1 , C2 , and C3 are constants of integration to be determined using the boundary conditions. For a plate with clamped edges under uniformly distributed load qo , the deflection surface reduces to 2 qo 2 a − r2 (2.73) w= 64D The maximum deflection occurs at the center where r = 0, and is given by w= 1999 by CRC Press LLC
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qo a 4 64D
(2.74)
FIGURE 2.33: Typical loading and boundary conditions for rectangular plates. Bending moments in the radial and tangential directions are respectively given by i qo h 2 a (1 + ν) − r 2 (3 + ν) Mr = 16 h i qo 2 a (1 + ν) − r 2 (1 + 3ν) Mt = 16
(2.75)
The method of superposition can be applied in calculating the deflections for circular plates with 1999 by CRC Press LLC
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FIGURE 2.34: (a) Circular plate; (b) stress resultants. simply supported edges. The expressions for deflection and bending moment are given as follows: qo (a 2 − r 2 ) 5 + ν 2 a − r2 w = 64D 1+ν wmax
=
Mr
=
Mt
=
5 + ν qo a 4 64(1 + ν) D qo (3 + ν)(a 2 − r 2 ) 16 h i qo 2 a (3 + ν) − r 2 (1 + 3ν) 16
(2.76)
(2.77)
This solution can be used to deal with plates with circular holes at the center and subjected to concentric moment and shearing forces. Plates subjected to concentric loading and concentrated loading also can be solved by this method. More rigorous solutions are available to deal with irregular loading on circular plates. Once again energy method can be employed advantageously to solve circular plate problems. Figure 2.35 gives deflection and bending moment expressions for typical cases of loading and boundary conditions on circular plates. 1999 by CRC Press LLC
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FIGURE 2.35: Typical loading and boundary conditions for circular plates.
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FIGURE 2.35: (Continued) Typical loading and boundary conditions for circular plates.
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2.5.5 Strain Energy of Simple Plates The strain energy expression for a simple rectangular plate is given by ( 2 Z Z ∂ 2w ∂ 2w D + Uψ = 2 ∂x 2 ∂y 2 area " 2 2 #) ∂ 2w ∂ 2w ∂ w dxdyψ −2(1 − ν) − ∂x∂y ∂x 2 ∂y 2
(2.78)
Suitable deflection function w(x, y) satisfying the boundary conditions of the given plate may be chosen. The strain energy, U , and the work done by the given load, q(x, y), Z Z (2.79) W =− q(x, y)w(x, y)dxdyψ area can be calculated. The total potential energy is, therefore, given as V = U + W . Minimizing the total potential energy the plate problem can be solved. " 2 2 # ∂ w ∂ 2w ∂ 2w − ∂x∂y ∂x 2 ∂y 2 The term is known as the Gaussian curvature. If the function w(x, y) = f (x) · φ(y) (product of a function of x only and a function of y only) and w = 0 at the boundary are assumed, then the integral of the Gaussian curvature over the entire plate equals zero. Under these conditions U=
D 2
Z Z
∂ 2w ∂ 2w + ∂x 2 ∂y 2
area
2 dxdy
If polar coordinates instead of rectangular coordinates are used and axial symmetry of loading and deformation is assumed, the equation for strain energy, U , takes the form ( ) 2 Z Z 2(1 − ν) ∂w ∂ 2 w ∂ 2 w 1 ∂w D + − (2.80) rdrdθψ U= 2 r ∂r r ∂r ∂r 2 ∂r 2 area and the work done, W , is written as
Z Z W =−
area
qwrdrdθψ
(2.81)
Detailed treatment of the Plate Theory can be found in [56].
2.5.6
Plates of Various Shapes and Boundary Conditions
Simply Supported Isosceles Triangular Plate Subjected to a Concentrated Load
Plates of shapes other than circle and rectangle are used in some situations. A rigorous solution of the deflection for a plate with a more complicated shape is likely to be very difficult. Consider, for example, the bending of an isosceles triangular plate with simply supported edges under concentrated load P acting at an arbitrary point (Figure 2.36). A solution can be obtained for this plate by considering a mirror image of the plate as shown in the figure. The deflection of OBC of the square 1999 by CRC Press LLC
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FIGURE 2.36: Isosceles triangular plate. plate is identical with that of a simply supported triangular plate OBC. The deflection owing to the force P can be written as w1 =
∞ ∞ 4P a 2 X X sin(mπ x1 /a) sin(nπy1 /a) mπ x nπy sin sin a a π 4D (m2 + n2 )2
(2.82)
m=1 n=1
Upon substitution of −P for P , (a − y1 ) for x1 , and (a − x1) for y1 in Equation 2.82 we obtain the deflection due to the force −P at Ai : ∞ ∞ 4P a 2 X X mπ x sin(mπ x1 /a) sin(nπy1 /a) nπy (−1)m+n sin sin w2 = − 4 a a π D (m2 + n2 )2
(2.83)
m=1 n=1
The deflection surface of the triangular plate is then w = w1 + w2
(2.84)
Equilateral Triangular Plates
The deflection surface of a simply supported plate loaded by uniform moment Mo along its boundary and the surface of a uniformly loaded membrane, uniformly stretched over the same triangular boundary, are identical. The deflection surface for such a case can be obtained as 4 Mo x 3 − 3xy 2 − a(x 2 + y 2 ) + a 3 (2.85) w= 4aD 27 If the simply supported plate is subjected to uniform load po the deflection surface takes the form 4 3 4 2 po 3 2 2 2 2 2 x − 3xy − a(x + y ) + a a −x −y (2.86) w= 64aD 27 9 For the equilateral triangular plate (Figure 2.37) subjected to uniform load and supported at the corners approximate solutions based on the assumption that the total bending moment along each 1999 by CRC Press LLC
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FIGURE 2.37: Equilateral triangular plate with coordinate axes. side of the triangle vanishes were obtained by Vijakkhana et al. [58] who derived an equation for deflection surface as 8 qa 4 (7 + ν)(2 − ν) − (7 + ν)(1 − ν) w = 2 144(1 − ν )D 27 2 3 y2 xy 2 x x + − 3 − (5 − ν)(1 + ν) a2 a2 a3 a3 4 x x2y2 y4 9 + 2 + (2.87) + (1 − ν 2 ) 4 a4 a4 a4 The errors introduced by the approximate boundary condition, i.e., the total bending moment along each side of the triangle vanishes, are not significant because its influence on the maximum deflection and stress resultants is small for practical design purposes. The value of the twisting moment on the edge at the corner given by this solution is found to be exact. The details of the mathematical treatment may be found in [58]. Rectangular Plate Supported at Corners
Approximate solutions for rectangular plates supported at the corners and subjected to uniformly distributed load were obtained by Lee and Ballesteros [36]. The approximate deflection surface is given as b4 b2 qa 4 2 (10 + ν − ν ) 1 + − 2(7ν − 1) 2 w = 2 4 48(1 − ν )D a a 2 b x + 2 (1 + 5ν) 2 − (6 + ν − ν 2 ) a a 2 2 b y + 2 (1 + 5ν) − (6 + ν − ν 2 ) 2 a a2 4 4 x +y x2y2 +(2 + ν − ν 2 ) − 6(1 + ν) (2.88) a4 a4 The details of the mathematical treatment may be found in [36]. 1999 by CRC Press LLC
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2.5.7
Orthotropic Plates
Plates of anisotropic materials have important applications owing to their exceptionally high bending stiffness. A nonisotropic or anisotropic material displays direction-dependent properties. Simplest among them are those in which the material properties differ in two mutually perpendicular directions. A material so described is orthotropic, e.g., wood. A number of manufactured materials are approximated as orthotropic. Examples include corrugated and rolled metal sheets, fillers in sandwich plate construction, plywood, fiber reinforced composites, reinforced concrete, and gridwork. The latter consists of two systems of equally spaced parallel ribs (beams), mutually perpendicular, and attached rigidly at the points of intersection. The governing equation for orthotropic plates similar to that of isotropic plates (Equation 2.42) takes the form δ4w δ4w δ4w (2.89) Dx 4 + 2H 2 2 + Dy 4 = q δx δx δy δy In which h3 Ey h3 Exy h3 Ex h3 G , Dy = , H = Dxy + 2Gxy , Dxy = , Gxy = 12 12 12 12 The expressions for Dx , Dy , Dxy , and Gxy represent the flexural rigidities and the torsional rigidity of an orthotropic plate, respectively. Ex , Ey , and G are the orthotropic plate moduli. Practical considerations often lead to assumptions, with regard to material properties, resulting in approximate expressions for elastic constants. The accuracy of these approximations is generally the most significant factor in the orthotropic plate problem. Approximate rigidities for some cases that are commonly encountered in practice are given in Figure 2.38. General solution procedures applicable to the case of isotropic plates are equally applicable to the orthotropic plates as well. Deflections and stress-resultants can thus be obtained for orthotropic plates of different shapes with different support and loading conditions. These problems have been researched extensively and solutions concerning plates of various shapes under different boundary and loading conditions may be found in the references, namely [37, 52, 53, 56, 57]. Dx =
2.5.8
Buckling of Thin Plates
Rectangular Plates
Buckling of a plate involves bending in two planes and is therefore fairly complicated. From a mathematical point of view, the main difference between columns and plates is that quantities such as deflections and bending moments, which are functions of a single independent variable, in columns become functions of two independent variables in plates. Consequently, the behavior of plates is described by partial differential equations, whereas ordinary differential equations suffice for describing the behavior of columns. A significant difference between columns and plates is also apparent if one compares their buckling characteristics. For a column, buckling terminates the ability of the member to resist axial load, and the critical load is thus the failure load of the member. However, the same is not true for plates. These structural elements can, subsequently to reaching the critical load, continue to resist increasing axial force, and they do not fail until a load considerably in excess of the critical load is reached. The critical load of a plate is, therefore, not its failure load. Instead, one must determine the load-carrying capacity of a plate by considering its postbuckling behavior. To determine the critical in-plane loading of a plate by the concept of neutral equilibrium, a governing equation in terms of biaxial compressive forces Nx and Ny and constant shear force Nxy as shown in Figure 2.39 can be derived as 4 δ4w δ4w δ2 w δ2 w δ2 w δ w =0 + 2 + + N + 2N (2.90) + N D x y xy δxδy δx 4 δx 2 δy 2 δy 4 δx 2 δy 2 1999 by CRC Press LLC
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FIGURE 2.38: Various orthotropic plates.
The critical load for uniaxial compression can be determined from the differential equation 4 δ4w δ4w δ2 w δ w + 2 + (2.91) + Nx 2 = 0 D 4 2 2 4 δx δx δy δy δx which is obtained by setting Ny = Nxy = 0 in Equation 2.90. For example, in the case of a simply supported plate Equation 2.91 can be solved to give 2 π 2 a 2 D m2 n2 + 2 Nx = m2 a2 b 1999 by CRC Press LLC
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(2.92)
FIGURE 2.39: Plate subjected to in-plane forces. The critical value of Nx , i.e., the smallest value, can be obtained by taking n equal to 1. The physical meaning of this is that a plate buckles in such a way that there can be several half-waves in the direction of compression but only one half-wave in the perpendicular direction. Thus, the expression for the critical value of the compressive force becomes 2 π 2D 1 a2 (2.93) m+ (Nx )cr = 2 m b2 a The first factor in this expression represents the Euler load for a strip of unit width and of length a. The second factor indicates in what proportion the stability of the continuous plate is greater than the stability of an isolated strip. The magnitude of this factor depends on the magnitude of the ratio a/b and also on the number m, which gives the number of half-waves into which the plate buckles. If ‘a’ is smaller than ‘b’, the second term in the parenthesis of Equation 2.93 is always smaller than the first and the minimum value of the expression is obtained by taking m = 1, i.e., by assuming that the plate buckles in one half-wave. The critical value of Nx can be expressed as Ncr =
kπ 2 D b2
(2.94)
The factor k depends on the aspect ratio a/b of the plate and m, the number of half-waves into which the plate buckles in the x direction. The variation of k with a/b for different values of m can be plotted, as shown in Figure 2.40. The critical value of Nx is the smallest value that is obtained for m = 1 and the corresponding value of k is equal to 4.0. This formula is analogous to Euler’s formula for buckling of a column. In the more general case in which normal forces Nx and Ny and the shearing forces Nxy are acting on the boundary of the plate, the same general method can be used. The critical stress for the case of a uniaxially compressed simply supported plate can be written as 2 h π 2E (2.95) σcr = 4 12(1 − ν 2 ) b The critical stress values for different loading and support conditions can be expressed in the form 2 h π 2E (2.96) fcr = k 12(1 − ν 2 ) b in which fcr is the critical value of different loading cases. Values of k for plates with several different boundary and loading conditions are given in Figure 2.41. 1999 by CRC Press LLC
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FIGURE 2.40: Buckling stress coefficients for unaxially compressed plate.
Circular Plates
The critical value of the compressive forces Nr uniformly distributed around the edge of a circular plate of radius ro , clamped along the edge (Figure 2.42) can be determined by using the governing equation Qr 2 dφ d 2φ −φ =− (2.97) r2 2 + r dr D dr in which φ is the angle between the axis of revolution of the plate surface and any normal to the plate, r is the distance of any point measured from the center of the plate, and Q is the shearing force per unit of length. When there are no lateral forces acting on the plate, the solution of Equation 2.97 involves a Bessel function of the first order of the first and second kind and the resulting critical value of Nr is obtained as 14.68D (2.98) (Nr )cr = r02 The critical value of Nr for the plate when the edge is simply supported can be obtained in the same way as 4.20D (2.99) (Nr )cr = r02
2.6 2.6.1
Shell Stress Resultants in Shell Element
A thin shell is defined as a shell with a thickness that is relatively small compared to its other dimensions. Also, deformations should not be large compared to the thickness. The primary difference between a shell structure and a plate structure is that the former has a curvature in the unstressed state, whereas the latter is assumed to be initially flat. The presence of initial curvature is of little consequence as far as flexural behavior is concerned. The membrane behavior, however, is affected significantly by the curvature. Membrane action in a surface is caused by in-plane forces. These forces may be primary forces caused by applied edge loads or edge deformations, or they may be secondary forces resulting from flexural deformations. 1999 by CRC Press LLC
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FIGURE 2.41: Values of K for plate with different boundary and loading conditions.
In the case of the flat plates, secondary in-plane forces do not give rise to appreciable membrane action unless the bending deformations are large. Membrane action due to secondary forces is, therefore, neglected in small deflection theory. If the surface, as in the case of shell structures, has an initial curvature, membrane action caused by secondary in-plane forces will be significant regardless of the magnitude of the bending deformations. A plate is likened to a two-dimensional beam and resists transverse loads by two dimensional bending and shear. A membrane is likened to a two-dimensional equivalent of the cable and resists loads through tensile stresses. Imagine a membrane with large deflections (Figure 2.43a), reverse the load and the membrane and we have the structural shell (Figure 2.43b) provided that the shell is stable for the type of load shown. The membrane resists the load through tensile stresses but the ideal thin shell must be capable of developing both tension and compression. Consider an infinitely small shell element formed by two pairs of adjacent planes which are normal to the middle surface of the shell and which contain its principal curvatures as shown in Figure 2.44a. The thickness of the shell is denoted as h. Coordinate axes x and y are taken tangent at ‘O’ to the lines of principal curvature and the axis z normal to the middle surface. rx and ry are the principal 1999 by CRC Press LLC
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FIGURE 2.42: Circular plate under compressive loading.
FIGURE 2.43: Membrane with large deflections. radii of curvature lying in the xz and yz planes, respectively. The resultant forces per unit length of the normal sections are given as Z =
Nx
h/2
−h/2 h/2
σx
z 1− ry
dz,
z τxy 1 − dz, ry −h/2 Z h/2 z = τxz 1 − dz, ry −h/2 Z
=
Nxy Qx
z = σy 1 − dz rx −h/2 Z h/2 z = τyx 1 − dz rx −h/2 Z h/2 z = τyz 1 − dz rx −h/2 Z
Ny Nyx Qy
h/2
(2.100)
The bending and twisting moments per unit length of the normal sections are given by z σx z 1 − dz, My ry −h/2 Z h/2 z =− τxy z 1 − dz, Myx ry −h/2 Z
Mx Mxy
=
h/2
z σy z 1 − dz rx −h/2 Z h/2 z = τyx z 1 − dz rx −h/2 Z
=
h/2
(2.101)
It is assumed, in bending of the shell, that linear elements as AD and BC (Figure 2.44), which are normal to the middle surface of the shell, remain straight and become normal to the deformed middle surface of the shell. If the conditions of a shell are such that bending can be neglected, the 1999 by CRC Press LLC
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FIGURE 2.44: A shell element. problem of stress analysis is greatly simplified because the resultant moments (Equation 2.101) vanish along with shearing forces Qx and Qy in Equation 2.100. Thus, the only unknowns are Nx , Ny , and Nxy = Nyx and these are called membrane forces.
2.6.2
Membrane Theory of Shells of Revolution
Shells having the form of surfaces of revolution find extensive application in various kinds of containers, tanks, and domes. Consider an element of a shell cut by two adjacent meridians and two parallel circles as shown in Figure 2.45. There will be no shearing forces on the sides of the element because of the symmetry of loading. By considering the equilibrium in the direction of the tangent to the meridian and z, two equations of equilibrium are written, respectively, as d (Nφ r0 ) − Nθ r1 cos φ + Y r1 r0 = 0 dφ Nφ r0 + Nθ r1 sin φ + Zr1 r0 = 0
(2.102)
The force Nθ and Nφ can be calculated from Equation 2.102 if the radii r0 and r1 and the components Y and Z of the intensity of the external load are given.
2.6.3
Spherical Dome
The spherical shell shown in Figure 2.46 is assumed to be subjected to its own weight; the intensity of the self weight is assumed as a constant value qo per unit area. Considering an element of the shell at an angle φ, the self weight of the portion of the shell above this element is obtained as Z φ a 2 qo sin φdφ r = 2π 0
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FIGURE 2.45: An element from shells of revolution—symmetrical loading.
FIGURE 2.46: Spherical dome. =
2π a 2 qo (1 − cos φ)
Considering the equilibrium of the portion of the shell above the parallel circle defined by the angle φ, we can write (2.103) 2π r0 Nφ sin φ + R = 0 Therefore, Nφ = −
aq(1 − cos φ) sin φ 2
=−
aq 1 + cos φ
We can write from Equation 2.102
Nφ Nθ + = −Z r1 r2 Substituting for Nφ and R into Equation 2.104 1 − cos φ Nθ = −aq 1 + cos φ 1999 by CRC Press LLC
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(2.104)
It is seen that the forces Nφ are always negative. Thus, there is a compression along the meridians that increases as the angle φ increases. The forces Nθ are also negative for small angles φ. The stresses as calculated above will represent the actual stresses in the shell with great accuracy if the supports are of such a type that the reactions are tangent to meridians as shown in the figure.
2.6.4
Conical Shells
If a force P is applied in the direction of the axis of the cone as shown in Figure 2.47, the stress distribution is symmetrical and we obtain Nφ = −
P 2π r0 cos α
By Equation 2.104, one obtains Nθ = 0.
FIGURE 2.47: Conical shell. In the case of a conical surface in which the lateral forces are symmetrically distributed, the membrane stresses can be obtained by using Equations 2.103 and 2.104. The curvature of the meridian in the case of a cone is zero and hence r1 = ∞; Equations 2.103 and 2.104 can, therefore, be written as R Nφ = − 2π r0 sin φ and Nθ = −r2 Z = −
Zr0 sin φ
If the load distribution is given, Nφ and Nθ can be calculated independently. For example, a conical tank filled with a liquid of specific weight γ is considered as shown in Figure 2.48. The pressure at any parallel circle mn is p = −Z = γ (d − y) For the tank, φ = α + Therefore,
π 2
and r0 = y tan α. Nθ =
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γ (d − y)y tan α cos α
FIGURE 2.48: Inverted conical tank. Nθ is maximum when y =
d 2
and hence (Nθ )max =
γ d 2 tan α 4 cos α
The term R in the expression for Nφ is equal to the weight of the liquid in the conical part mno and the cylindrical part must be as shown in Figure 2.47. Therefore, 1 3 2 2 2 R = − πy tan α + πy (d − y) tan α γ 3 2 2 = −πγ y d − y tan2 α 3 Hence, Nφ = Nφ is maximum when y =
3 4d
γ y d − 23 y tan α 2 cos α
and (Nφ )max =
3 d 2 γ tan α 16 cos α
The horizontal component of Nφ is taken by the reinforcing ring provided along the upper edge of the tank. The vertical components constitute the reactions supporting the tank.
2.6.5
Shells of Revolution Subjected to Unsymmetrical Loading
Consider an element cut from a shell by two adjacent meridians and two parallel circles (Figure 2.49). In the general case, shear forces Nϕθ = Nθ ϕ in addition to normal forces Nϕ and Nθ will act on the sides of the element. Projecting the forces on the element in the y direction we obtain the equation ∂Nθ ϕ ∂ (Nϕ r0 ) + r1 − Nθ r1 cos ϕ + Y r1 r0 = 0 ∂ϕ ∂θ
(2.105)
Similarly the forces in the x direction can be summed up to give ∂Nθ ∂ (r0 Nϕθ ) + r1 + Nθ ϕ r1 cos ϕ + Xr0 r1 = 0 ∂ϕ ∂θ 1999 by CRC Press LLC
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(2.106)
FIGURE 2.49: An element from shells of revolution—unsymmetrical loading. Since the projection of shearing forces on the z axis vanishes, the third equation is the same as Equation 2.104. The problem of determining membrane stresses under unsymmetrical loading reduces to the solution of Equations 2.104, 2.105, and 2.106 for given values of the components X, Y , and Z of the intensity of the external load.
2.6.6
Membrane Theory of Cylindrical Shells
It is assumed that the generator of the shell is horizontal and parallel to the x axis. An element is cut from the shell by two adjacent generators and two cross-sections perpendicular to the x axis, and its position is defined by the coordinate x and the angle ϕ. The forces acting on the sides of the element are shown in Figure 2.50b. The components of the distributed load over the surface of the element are denoted as X, Y, and Z. Considering the equilibrium of the element and summing up the forces in the x direction, we obtain ∂Nϕx ∂Nx rdϕdx + dϕdx + Xrdϕdx = 0 ∂x ∂ϕ The corresponding equations of equilibrium in the y and z directions are given, respectively, as ∂Nϕ ∂Nxϕ rdϕdx + dϕdx + Y rdϕdx = 0 ∂x ∂ϕ Nϕ dϕdx + Zrdϕdx = 0 The three equations of equilibrium can be simplified and represented in the following form: 1 ∂Nxϕ ∂Nx + = −X ∂x r ∂ϕ ∂Nxϕ 1 ∂Nϕ + = −Y ∂x r ∂ϕ Nϕ = −Zr
(2.107)
In each particular case we readily find the value of Nϕ . Substituting this value in the second of the equations, we then obtain Nxϕ by integration. Using the value of Nxϕ thus obtained we find Nx by integrating the first equation. 1999 by CRC Press LLC
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FIGURE 2.50: Membrane forces on a cylindrical shell element.
2.6.7
Symmetrically Loaded Circular Cylindrical Shells
In practical applications problems in which a circular shell is subjected to the action of forces distributed symmetrically with respect to the axis of the cylinder are common. To establish the equations required for the solution of these problems, we consider an element, as shown in Figures 2.50a and 2.51, and consider the equations of equilibrium. From symmetry, the membrane shearing forces
FIGURE 2.51: Stress resultants in a cylindrical shell element. Nxϕ = Nϕx vanish in this case; forces Nϕ are constant along the circumference. From symmetry, only the forces Qz do not vanish. Considering the moments acting on the element in Figure 2.51, 1999 by CRC Press LLC
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from symmetry it can be concluded that the twisting moments Mxϕ = Mϕx vanish and the bending moments Mϕ are constant along the circumference. Under such conditions of symmetry three of the six equations of equilibrium of the element are identically satisfied. We have to consider only the equations obtained by projecting the forces on the x and z axes and by taking the moment of the forces about the y axis. For example, consider a case in which external forces consist only of a pressure normal to the surface. The three equations of equilibrium are dN adxdϕ = 0 dx
dQx adxdϕ + Nϕ dxdϕ + Zadxdϕ = 0 dx dMx adxdϕ − Qx adxdϕ = 0 dx
(2.108)
The first one indicates that the forces Nx are constant, and they are taken equal to zero in the further discussion. If they are different from zero, the deformation and stress corresponding to such constant forces can be easily calculated and superposed on stresses and deformations produced by lateral load. The remaining two equations are written in the simplified form: 1 dQx + Nϕ = −Z dx a dMx − Qx = 0 dx
(2.109)
These two equations contain three unknown quantities: Nϕ , Qx , and Mx . We need, therefore, to consider the displacements of points in the middle surface of the shell. The component v of the displacement in the circumferential direction vanishes because of symmetry. Only the components u and w in the x and z directions, respectively, are to be considered. The expressions for the strain components then become εx =
du dx
εϕ = −
w a
(2.110)
By Hooke’s law, we obtain Nx Nϕ
w Eh Eh du − ν =0 (ε + νε ) = x ϕ a 1 − ν2 1 − ν 2 dx Eh Eh du w = (ε + νε ) = + ν =0 − ϕ x a dx 1 − ν2 1 − ν2 =
(2.111)
From the first of these equation it follows that w du =ν dx a and the second equation gives
Ehw (2.112) a Considering the bending moments, we conclude from symmetry that there is no change in curvature in the circumferential direction. The curvature in the x direction is equal to −d 2 w/dx 2 . Using the same equations as for plates, we then obtain Nϕ = −
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Mϕ
=
νMx
Mx
=
−D
d 2w dx 2
(2.113)
where D=
Eh3 12(1 − ν 2 )
is the flexural rigidity per unit length of the shell. Eliminating Qx from Equation 2.109, we obtain 1 d 2 Mx + Nϕ = −Z 2 a dx from which, by using Equations 2.112 and 2.113, we obtain d 2w Eh d2 D + 2w=Z dx 2 dx 2 a
(2.114)
All problems of symmetrical deformation of circular cylindrical shells thus reduce to the integration of Equation 2.114. The simplest application of this equation is obtained when the thickness of the shell is constant. Under such conditions, Equation 2.114 becomes D
d 4 w Eh + 2w=Z dx 4 a
Using the notation 3(1 − ν 2 ) Eh = 4a 2 D a 2 h2 Equation 2.115 can be represented in the simplified form β4 =
Z d 4w + 4β 4 w = 4 D dx
(2.115)
(2.116)
The general solution of this equation is w
=
eβx (C1 cos βx + C2 sin βx) + e−βx (C3 cos βx + C4 sin βx) + f (x)
(2.117)
Detailed treatment of shell theory can be obtained from Timoshenko and Woinowsky-Krieger [56].
2.6.8
Buckling of Shells
If a circular cylindrical shell is uniformly compressed in the axial direction, buckling symmetrical with respect to the axis of the cylinder (Figure 2.52) may occur at a certain value of the compressive load. The critical value of the compressive force Ncr per unit length of the edge of the shell can be obtained by solving the differential equation D
d 2w w d 4w + N + Eh 2 = 0 4 2 dx dx a
(2.118)
in which a is the radius of the cylinder and h is the wall thickness. Alternatively, the critical force per unit length may also be obtained by using the energy method. For a cylinder of length L simply supported at both ends one obtains 2 2 m π EhL2 + (2.119) Ncr = D L2 Da 2 m2 π 2 1999 by CRC Press LLC
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FIGURE 2.52: Buckling of a cylindrical shell. For each value of m there is a unique buckling mode shape and a unique buckling load. The lowest value is of greatest interest and is thus found by setting the derivative of Ncr with respect to L equal to zero for m = 1. With Poisson’s Ratio, = 0.3, the buckling load is obtained as Ncr = 0.605
Eh2 a
(2.120)
It is possible for a cylindrical shell be subjected to uniform external pressure or to the combined action of axial and uniform lateral pressure. In such cases the mathematical treatment is more involved and it requires special considerations. More detailed treatment of such cases may be found in Timoshenko and Gere [55].
2.7
Influence Lines
Structures such as bridges, industrial buildings with travelling cranes, and frames supporting conveyor belts are subjected to moving loads. Each member of these structures must be designed for the most severe conditions that can possibly be developed in that member. Live loads should be placed at the position where they will produce these severe conditions. The critical positions for placing live loads will not be the same for every member. On some occasions it is possible to determine by inspection where to place the loads to give the most critical forces, but on many other occasions it is necessary to resort to certain criteria to find the locations. The most useful of these methods is influence lines. An influence line for a particular response such as reaction, shear force, bending moment, and axial force is defined as a diagram the ordinate to which at any point equals the value of that response attributable to a unit load acting at that point on the structure. Influence lines provide a systematic procedure for determining how the force in a given part of a structure varies as the applied load moves about on the structure. Influence lines of responses of statically determinate structures consist only of straight lines whereas they are curves for statically indeterminate structures. They are primarily used to determine where to place live loads to cause maximum force and to compute the magnitude of those forces. The knowledge of influence lines helps to study the structural response under different moving load conditions.
2.7.1
Influence Lines for Shear in Simple Beams
Figure 2.53 shows influence lines for shear at two sections of a simply supported beam. It is assumed that positive shear occurs when the sum of the transverse forces to the left of a section is in the upward direction or when the sum of the forces to the right of the section is downward. A unit force is placed at various locations and the shear force at sections 1-1 and 2-2 are obtained for each position of the 1999 by CRC Press LLC
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FIGURE 2.53: Influence line for shear force.
unit load. These values give the ordinate of influence line with which the influence line diagrams for shear force at sections 1-1 and 2-2 can be constructed. Note that the slope of the influence line for shear on the left of the section is equal to the slope of the influence line on the right of the section. This information is useful in drawing shear force influence line in other cases.
2.7.2 Influence Lines for Bending Moment in Simple Beams Influence lines for bending moment at the same sections, 1-1 and 2-2 of the simple beam considered in Figure 2.53, are plotted as shown in Figure 2.54. For a section, when the sum of the moments of
FIGURE 2.54: Influence line for bending moment. 1999 by CRC Press LLC
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all the forces to the left is clockwise or when the sum to the right is counter-clockwise, the moment is taken as positive. The values of bending moment at sections 1-1 and 2-2 are obtained for various positions of unit load and plotted as shown in the figure. It should be understood that a shear or bending moment diagram shows the variation of shear or moment across an entire structure for loads fixed in one position. On the other hand, an influence line for shear or moment shows the variation of that response at one particular section in the structure caused by the movement of a unit load from one end of the structure to the other. Influence lines can be used to obtain the value of a particular response for which it is drawn when the beam is subjected to any particular type of loading. If, for example, a uniform load of intensity qo per unit length is acting over the entire length of the simple beam shown in Figure 2.53, then the shear force at section 1-1 is given by the product of the load intensity, qo , and the net area under the influence line diagram. The net area is equal to 0.3l and the shear force at section 1-1 is, therefore, equal to 0.3qo l. In the same way, the bending moment at the section can be found as the area of the corresponding influence line diagram times the intensity of loading, qo . The bending moment at the section is, therefore, (0.08l 2 × qo =)0.08qo l 2 .
2.7.3
Influence Lines for Trusses
Influence lines for support reactions and member forces may be constructed in the same manner as those for various beam functions. They are useful to determine the maximum load that can be applied to the truss. The unit load moves across the truss, and the ordinates for the responses under consideration may be computed for the load at each panel point. Member force, in most cases, need not be calculated for every panel point because certain portions of influence lines can readily be seen to consist of straight lines for several panels. One method used for calculating the forces in a chord member of a truss is by the Method of Sections discussed earlier. The truss shown in Figure 2.55 is considered for illustrating the construction of influence lines for trusses. The member forces in U1 U2 , L1 L2 , and U1 L2 are determined by passing a section 1-1 and considering the equilibrium of the free body diagram of one of the truss segments. Unit load is placed at L1 first and the force in U1 U2 is obtained by taking moment about L2 of all the forces acting on the right-hand segment of the truss and dividing the resulting moment by the lever arm (the perpendicular distance of the force in U1 U2 from L2 ). The value thus obtained gives the ordinate of the influence diagram at L1 in the truss. The ordinate at L2 obtained similarly represents the force in U1 U2 for unit load placed at L2 . The influence line can be completed with two other points, one at each of the supports. The force in the member L1 L2 due to unit load placed at L1 and L2 can be obtained in the same manner and the corresponding influence line diagram can be completed. By considering the horizontal component of force in the diagonal of the panel, the influence line for force in U1 L2 can be constructed. Figure 2.55 shows the respective influence diagram for member forces in U1 U2 , L1 L2 , and U1 L2 . Influence line ordinates for the force in a chord member of a “curved-chord” truss may be determined by passing a vertical section through the panel and taking moments at the intersection of the diagonal and the other chord.
2.7.4
Qualitative Influence Lines
One of the most effective methods of obtaining influence lines is by the use of M¨uller-Breslau’s principle, which states that “the ordinates of the influence line for any response in a structure are equal to those of the deflection curve obtained by releasing the restraint corresponding to this response and introducing a corresponding unit displacement in the remaining structure”. In this way, the shape of the influence lines for both statically determinate and indeterminate structures can be easily obtained especially for beams. 1999 by CRC Press LLC
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FIGURE 2.55: Influence line for truss. To draw the influence lines of 1. Support reaction: Remove the support and introduce a unit displacement in the direction of the corresponding reaction to the remaining structure as shown in Figure 2.56 for a symmetrical overhang beam.
FIGURE 2.56: Influence line for support reaction.
2. Shear: Make a cut at the section and introduce a unit relative translation (in the direction of positive shear) without relative rotation of the two ends at the section as shown in Figure 2.57. 3. Bending moment: Introduce a hinge at the section (releasing the bending moment) and apply bending (in the direction corresponding to positive moment) to produce a unit relative rotation of the two beam ends at the hinged section as shown in Figure 2.58. 1999 by CRC Press LLC
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FIGURE 2.57: Influence line for midspan shear force.
FIGURE 2.58: Influence line for midspan bending moment.
2.7.5
Influence Lines for Continuous Beams
Using M¨uller-Breslau’s principle, the shape of the influence line of any response of a continuous beam can be sketched easily. One of the methods for beam deflection can then be used for determining the ordinates of the influence line at critical points. Figures 2.59 to 2.61 show the influence lines of bending moment at various points of two, three, and four span continuous beams.
FIGURE 2.59: Influence lines for bending moments—two span beam.
2.8
Energy Methods in Structural Analysis
Energy methods are a powerful tool in obtaining numerical solutions of statically indeterminate problems. The basic quantity required is the strain energy, or work stored due to deformations, of the structure. 1999 by CRC Press LLC
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FIGURE 2.60: Influence lines for bending moments—three span beam.
FIGURE 2.61: Influence lines for bending moments—four span beam.
2.8.1
Strain Energy Due to Uniaxial Stress
In an axially loaded bar with constant cross-section, the applied load causes normal stress σy as shown in Figure 2.62. The tensile stress σy increases from zero to a value σy as the load is gradually applied. The original, unstrained position of any section such as C − C will be displaced by an amount v. A section D − D located a differential length below C − C will have been displaced by an amount v + ∂v ∂y dy. As σy varies with the applied load, from zero to σy , the work done by the forces external to the element can be shown to be dV =
1 1 2 σy Ady = σy εy Ady 2E 2
in which A is the area of cross-section of the bar and εy is the strain in the direction of σy .
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(2.121)
FIGURE 2.62: Axially loaded bar.
2.8.2
Strain Energy in Bending
It can be shown that the strain energy of a differential volume dxdydz stressed in tension or compression in the x direction only by a normal stress σx will be dV =
1 1 2 σ dxdydz = σx εx dxdydz 2E x 2
(2.122)
1 M y When σx is the bending stress given by σx = My I (see Figure 2.63), then dV = 2E I 2 dxdydz, where I is the moment of inertia of the cross-sectional area about the neutral axis. 2 2
FIGURE 2.63: Beam under arbitrary bending load.
The total strain energy of bending of a beam is obtained as Z Z Z V = volume
where
1 M2 2 y dzdydx 2E I 2
Z Z y 2 dzdy
I= area
1999 by CRC Press LLC
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Therefore,
Z V = length
2.8.3
M2 dx 2EI
(2.123)
Strain Energy in Shear
Figure 2.64 shows an element of volume dxdydz subjected to shear stress τxy and τyx .
FIGURE 2.64: Shear loading.
For static equilibrium, it can readily be shown that τxy = τyx The shear strain, γ is defined as AB/AC. For small deformations, it follows that γxy =
AB AC
Hence, the angle of deformation γxy is a measure of the shear strain. The strain energy for this differential volume is obtained as dV =
1 1 τxy dzdx γxy dy = τxy γxy dxdydz 2 2
(2.124)
Hooke’s Law for shear stress and strain is γxy =
τxy G
(2.125)
where G is the shear modulus of elasticity of the material. The expression for strain energy in shear reduces to 1 2 τ dxdydz (2.126) dV = 2G xy 1999 by CRC Press LLC
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2.8.4
The Energy Relations in Structural Analysis
The energy relations or laws such as (1) Law of Conservation of Energy, (2) Theorem of Virtual Work, (3) Theorem of Minimum Potential Energy, and (4) Theorem of Complementary Energy are of fundamental importance in structural engineering and are used in various ways in structural analysis. The Law of Conservation of Energy
There are many ways of stating this law. For the purpose of structural analysis it will be sufficient to state it in the following way: If a structure and the external loads acting on it are isolated so that these neither receive nor give out energy, then the total energy of this system remains constant. A typical application of the Law of Conservation of Energy can be made by referring to Figure 2.65 which shows a cantilever beam of constant cross-sections subjected to a concentrated load at its end. If only bending strain energy is considered, External work Pδ 2
= =
Internal work Z L 2 M dx 2EI 0
Substituting M = −P x and integrating along the length gives δ=
P L3 3EI
(2.127)
FIGURE 2.65: Cantilever beam.
The Theorem of Virtual Work
The Theorem of Virtual Work can be derived by considering the beam shown in Figure 2.66. The full curved line represents the equilibrium position of the beam under the given loads. Assume the beam to be given an additional small deformation consistent with the boundary conditions. This is called a virtual deformation and corresponds to increments of deflection 1y1 , 1y2 , ..., 1yn at loads P1 , P2 , ..., Pn as shown by the broken line. The change in potential energy of the loads is given by 1(P .E.) =
n X
Pi 1yi
(2.128)
i=1
By the Law of Conservation of Energy this must be equal to the internal strain energy stored in the beam. Hence, we may state the Theorem of Virtual Work in the following form: 1999 by CRC Press LLC
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FIGURE 2.66: Equilibrium of a simply supported beam under loading.
If a body in equilibrium under the action of a system of external loads is given any small (virtual) deformation, then the work done by the external loads during this deformation is equal to the increase in internal strain energy stored in the body. The Theorem of Minimum Potential Energy
Let us consider the beam shown in Figure 2.67. The beam is in equilibrium under the action
FIGURE 2.67: Simply supported beam under point loading. of loads, P1 , P2 , P3 , ..., Pi , ..., Pn . The curve ACB defines the equilibrium positions of the loads and reactions. Now apply by some means an additional small displacement to the curve so that it is defined by AC 0 B. Let yi be the original equilibrium displacement of the curve beneath a particular load Pi . The additional small displacement is called δyi . The potential energy of the system while it is in the equilibrium configuration is found by comparing the potential energy of the beam and loads in equilibrium and in the undeflected position. If the change in potential energy of the loads is W and the strain energy of the beam is V , the total energy of the system is U =W +V
(2.129)
δU = δ(W + V ) = 0
(2.130)
If we neglect the second-order terms, then
The above is expressed as the Principle or Theorem of Minimum Potential Energy which can be stated as Of all displacements satisfying given boundary conditions, those that satisfy the equilibrium conditions make the potential energy a minimum. 1999 by CRC Press LLC
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Castigliano’s Theorem
An example of application of energy methods to the field of structural engineering is Castigliano’s Theorem. The theorem applies only to structures stressed within the elastic limit. Also, all deformations must be linear homogeneous functions of the loads. Castigliano’s Theorem can be derived using the expression for total potential energy as follows: For a beam in equilibrium loaded as in Figure 2.66, the total energy is U = −[P1 y1 + P2 y2 + ...Pj yj + ...Pn yn ] + V
(2.131)
For an elastic system, the strain energy, V , turns out to be one half the change in the potential energy of the loads. i=n 1X Pi yi (2.132) V = 2 i=1
Castigliano’s Theorem results from studying the variation in the strain energy, V , produced by a differential change in one of the loads, say Pj . If the load Pj is changed by a differential amount δPj and if the deflections y are linear functions of the loads, then i=n 1 X ∂yi 1 ∂V = Pi + yj = yj (2.133) ∂Pj 2 ∂Pj 2 i=1
Castigliano’s Theorem is stated as follows: The partial derivatives of the total strain energy of any structure with respect to any one of the applied forces is equal to the displacement of the point of application of the force in the direction of the force. To find the deflection of a point in a beam that is not the point of application of a concentrated load, one should apply a load P = 0 at that point and carry the term P into the strain energy equation. Finally, introduce the true value of P = 0 into the expression for the answer.
EXAMPLE 2.6:
For example, it is required to determine the bending deflection at the free end of a cantilever loaded as shown in Figure 2.68. Solution Z L M2 V = dx 0 2EI Z L M ∂M ∂V = dx 1 = ∂W1 EI ∂W1 0 L M = W1 x 0 Cc 23(Kl/r)2 1999 by CRC Press LLC
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FIGURE 3.6: Definition of width-thickness ratio of selected cross-sections.
1999 by CRC Press LLC
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TABLE 3.4 Limiting Width-Thickness Ratios for Compression Elements Under Pure Compression Width-thickness ratio
Component element Flanges of I-shaped sections; plates projecting from compression elements; outstanding legs of pairs of angles in continuous contact; flanges of channels. Flanges of square and rectangular box and hollow structural sections of uniform thickness; flange cover plates and diaphragm plates between lines of fasteners or welds. Unsupported width of cover plates perforated with a succession of access holes. Legs of single angle struts; legs of double angle struts with separators; unstiffened elements (i.e., elements supported along one edge). Flanges projecting from built-up members. Stems of tees. All other uniformly compressed elements (i.e., elements supported along two edges). Circular hollow sections.
ak c Fy
= =
b/t
Limiting value, λr p 95/ fy
b/t
p 238/ fy
b/t
p 317/ fy
b/t
p 76/ fy
b/t d/t b/t h/tw D/t D = outside diameter t = wall thickness
q 109/ (Fy /kca ) p 127/pFy 253/ Fy
3,300/Fy
√ 4/ (h/tw ), and 0.35 ≤ kc ≤ 0.763 for I-shaped sections, kc = 0.763 for other sections. specified minimum yield stress, in ksi.
where Kl/r is the slenderness ratio, K is the effective length factor of the compression member (see Section 3.4.3), l is the unbraced memberqlength, r is the radius of gyration of the cross-section,
E is the modulus of elasticity, and Cc = (2π 2 E/Fy ) is the slenderness ratio that demarcates between inelastic member buckling from elastic member buckling. Kl/r should be evaluated for both buckling axes and the larger value used in Equation 3.16 to compute Fa . The first of Equation 3.16 is the allowable stress for inelastic buckling, and the second of Equation 3.16 is the allowable stress for elastic buckling. In ASD, no distinction is made between flexural, torsional, and flexural-torsional buckling.
3.4.2
Load and Resistance Factor Design
Compression members are to be designed so that the design compressive strength φc Pn will exceed the required compressive strength Pu . φc Pn is to be calculated as follows for the different types of overall buckling modes. Flexural Buckling (with width-thickness ratio < λr ): h i 2 0.85 Ag (0.658λc )Fy , if λc ≤ 1.5 (3.17) φ c Pn = i h 0.85 Ag 0.877 Fy , > 1.5 if λ c 2 λ c
where λc = Ag = Fy = E = K = l = r =
p (KL/rπ) (Fy /E) is the slenderness parameter gross cross-sectional area specified minimum yield stress modulus of elasticity effective length factor unbraced member length radius of gyration of the cross-section
1999 by CRC Press LLC
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The first of Equation 3.17 is the design strength for inelastic buckling and the second of Equation 3.17 is the design strength for elastic buckling. The slenderness parameter λc = 1.5 is therefore the value that demarcates between inelastic and elastic behavior. Torsional Buckling (with width-thickness ratio < λr ): φc Pn is to be calculated from Equation 3.17, but with λc replaced by λe given by λe = where
q (Fy /Fe )
(3.18)
π 2 ECw 1 + GJ Fe = 2 Ix + Iy (Kz L)
(3.19)
in which = warping constant Cw G = shear modulus = 11,200 ksi (77,200 MPa) Ix , Iy = moment of inertia about the major and minor principal axes, respectively J = torsional constant = effective length factor for torsional buckling Kz The warping constant Cw and the torsional constant J are tabulated for various steel shapes in the AISC-LRFD Manual [22]. Equations for calculating approximate values for these constants for some commonly used steel shapes are shown in Table 3.5. TABLE 3.5
Approximate Equations for Cw and J
Structural shape
Warping constant, Cw
I
h02 Ic It /(Ic + It )
C
(b0 − 3Eo )h02 b02 tf /6 + Eo2 Ix where Eo = b02 tf /(2b0 tf + h0 tw /3)
b0 h0 h00 l1 , l2 t1 , t2 bf tf tw Ic It Ix
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T
3 )/36 (bf3 tf3 /4 + h003 tw (≈ 0 for small t )
L
(l13 t13 + l23 t23 )/36 (≈ 0 for small t )
= = = = = = = = = = =
Torsional constant, J P Ci (bi ti3 /3)
where bi = width of component element i ti = thickness of component element i Ci = correction factor for component element i (see values below) bi /ti 1.00 1.20 1.50 1.75 2.00 2.50 3.00 4.00 5.00 6.00 8.00 10.00 ∞
Ci 0.423 0.500 0.588 0.642 0.687 0.747 0.789 0.843 0.873 0.894 0.921 0.936 1.000
distance measured from toe of flange to center line of web distance between centerline lines of flanges distance from centerline of flange to tip of stem length of the legs of the angle thickness of the legs of the angle flange width average thickness of flange thickness of web moment of inertia of compression flange taken about the axis of the web moment of inertia of tension flange taken about the axis of the web moment of inertia of the cross-section taken about the major principal axis
Flexural-Torsional Buckling (with width-thickness ratio ≤ λr ): Same as for torsional buckling except Fe is now given by For singly symmetric sections: s " # Fes + Fez 4Fes Fez H 1− 1− Fe = 2H (Fes + Fez )2
(3.20)
where Fes = Fex if the x-axis is the axis of symmetry of the cross-section, or Fey if the y-axis is the axis of symmetry of the cross-section Fex = π 2 E/(Kl/r)2x Fey = π 2 E/(Kl/r)2x H = 1 − (xo2 + yo2 )/ro2 in which Kx , Ky = effective length factors for buckling about the x and y axes, respectively l = unbraced member length = radii of gyration about the x and y axes, respectively rx , ry xo , yo = the shear center coordinates with respect to the centroid Figure 3.7 = xo2 + yo2 + rx2 + ry2 ro2 Numerical values for ro and H are given for hot-rolled W, channel, tee, and single- and double-angle sections in the AISC-LRFD Manual [22]. For unsymmetric sections: Fe is to be solved from the cubic equation (Fe − Fex )(Fe − Fey )(Fe − Fez ) − Fe2 (Fe
xo − Fey ) ro
2 − Fe2 (Fe
yo − Fex ) ro
2 =0
(3.21)
The terms in the above equations are defined the same as in Equation 3.20. Local Buckling (with width-thickness ratio ≥ λr ): Local buckling in a component element of the cross-section is accounted for in design by introducing a reduction factor Q in Equation 3.17 as follows: h i √ 2 0.85 Ag Q 0.658Qλ Fy , if λ Q ≤ 1.5 (3.22) φ c Pn = i h √ 0.85 Ag 0.877 Fy , Q > 1.5 if λ λ2 where λ = λc for flexural buckling, and λ = λe for flexural-torsional buckling. The Q factor is given by Q = Qs Qa
(3.23)
where Qs is the reduction factor for unstiffened compression elements of the cross-section (see Table 3.6); and Qa is the reduction factor for stiffened compression elements of the cross-section (see Table 3.7)
3.4.3
Built-Up Compression Members
Built-up members are members made by bolting and/or welding together two or more standard structural shapes. For a built-up member to be fully effective (i.e., if all component structural shapes are to act as one unit rather than as individual units), the following conditions must be satisfied: 1999 by CRC Press LLC
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FIGURE 3.7: Location of shear center for selected cross-sections. 1. The ends of the built-up member must be prevented from slippage during buckling. 2. Adequate fasteners must be provided along the length of the member. 3. The fasteners must be able to provide sufficient gripping force on all the component shapes being connected. Condition 1 is satisfied if all component shapes in contact at the ends of the member are connected by a weld having a length not less than the maximum width of the member or by fully tightened bolts spaced longitudinally not more than four diameters apart for a distance equal to 1-1/2 times the maximum width of the member. Condition 2 is satisfied if continuous welds are used throughout the length of the built-up compression member. Condition 3 is satisfied if either welds or fully tightened bolts are used as the fasteners. While condition 1 is mandatory, conditions 2 and 3 can be violated in design. If condition 2 or 3 is violated, the built-up member is not fully effective and slight slippage among component shapes 1999 by CRC Press LLC
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TABLE 3.6
Formulas for Qs
Range of b/t
Qs
p 76.0/ Fy < b/t < 155/ Fy
p 1.340 − 0.00447(b/t) fy
Structural element p
Single angles
p b/t ≥ 155/ fy p p 95.0/ Fy < b/t < 176/ fy
p 1.415 − 0.00437(b/t) fy
p b/t ≥ 176/ Fy
20, 000/[Fy (b/t)2 ]
q p 109/ (Fy /kca ) < b/t < 200/ (Fy /kc )
p 1.415 − 0.00381(b/t) (Fy /kc )
p b/t ≥ 200/ (Fy /kc )
26, 200kc/[Fy (b/t)2 ]
p p 127/ Fy < b/t < 176/ Fy
p 1.908 − 0.00715(b/t) Fy
p b/t ≥ 176/ fy
20, 000/[Fy (b/t)2 ]
Flanges, angles, and plates projecting from columns or other compression members
Flanges, angles, and plates projecting from built-up columns or other compression members
Stems of tees
15, 500/[Fy (b/t)2 ]
a see footnote a in Table 3.4
Fy b t
TABLE 3.7
= = =
specified minimum yield stress, in ksi width of the component element thickness of the component element
Formula for Qa Qs = effective area actual area
The effective area is equal to the summation of the effective areas of the stiffened elements of the crosssection. The effective area of a stiffened element is equal to the product of its thickness t and its effective width be given by: a √ For flanges of square and rectangular sections of uniform thickness: when b/t ≥ 238 f
√ be = 326t f
h
√ 1 − 64.9 (b/t) f
i
≤b
a √ For other uniformly compressed elements: when b/t ≥ 253 f
h √ 1− be = 326t f
57.2 √ (b/t) f
i
≤b
where b = actual width of the stiffened element f = computed elastic compressive stress in the stiffened elements, in ksi ab e
=
b otherwise.
may occur. To account for the decrease in capacity due to slippage, a modified slenderness ratio is used for the computation of the design compressive strength when buckling of the built-up member is about an axis coincide or parallel to at least one plane of contact for the component shapes. The modified slenderness ratio (KL/r)m is given as follows: If condition 2 is violated:
1999 by CRC Press LLC
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KL r
s
m
=
KL r
2
0.82α 2 + (1 + α 2 ) o
a rib
2 (3.24)
If conditions 2 and 3 are violated:
KL r
s
m
=
KL r
2 a + r i o
2
(3.25)
In the above equations, (KL/r)o = (KL/r)x if the buckling axis is the x-axis and at least one plane of contact between component shapes is parallel to that axis; (KL/r)o = (KL/r)y if the buckling axis is the y axis and at least one plane of contact is parallel to that axis. a is the longitudinal spacing of the fasteners, ri is the minimum radius of gyration of any component element of the built-up cross-section, rib is the radius of gyration of an individual component relative to its centroidal axis parallel to the axis of buckling of the member, h is the distance between centroids of component elements measured perpendicularly to the buckling axis of the built-up member. No modification to (KL/r) is necessary if the buckling axis is perpendicular to the planes of contact of the component shapes. Modifications to both (KL/r)x and (KL/r)y are required if the built-up member is so constructed that planes of contact exist in both the x and y directions of the cross-section. Once the modified slenderness ratio is computed, it is to be used in the appropriate equation to calculate Fa in allowable stress design, or φc Pn in load and resistance factor design. An additional requirement for the design of built-up members is that the effective slenderness ratio, Ka/ri , of each component shape, where K is the effective length factor of the component shape between adjacent fasteners, does not exceed 3/4 of the governing slenderness ratio of the builtup member. This provision is provided to prevent component shape buckling between adjacent fasteners from occurring prior to overall buckling of the built-up member.
EXAMPLE 3.2:
Using LRFD, determine the size of a pair of cover plates to be bolted, using snug-tight bolts, to the flanges of a W24x229 section as shown in Figure 3.8 so that its design strength, φc Pn , will be increased by 15%. Also, determine the spacing of the bolts in the longitudinal direction of the built-up column.
FIGURE 3.8: Design of cover plates for a compression member. 1999 by CRC Press LLC
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The effective lengths of the section about the major (KL)x and minor (KL)y axes are both equal to 20 ft. A36 steel is to be used. Determine design strength for the W24x229 section: Since (KL)x = (KL)y and rx > ry , (KL/r)y will be greater than (KL/r)x and the design strength will be controlled by flexural buckling about the minor axis. Using section properties, ry = 3.11 in. and A = 67.2 in.2 , obtained from the AISC-LRFD Manual [22], the slenderness parameter λc about the minor axis can be calculated as follows: 1 (λc )y = π
KL r
r y
Fy 1 = E 3.142
20 × 12 3.11
s
36 = 0.865 29, 000
Substituting λc = 0.865 into Equation 3.17, the design strength of the section is h i 2 φc Pn = 0.85 67.2 0.6580.865 36 = 1503 kips Alternatively, the above value of φc Pn can be obtained directly from the column tables contained in the AISC-LRFD Manual. Determine design strength for the built-up section: The built-up section is expected to possess a design strength which is 15% in excess of the design strength of the W24x229 section, so (φc Pn )req 0 d = (1.15)(1503) = 1728 kips Determine size of the cover plates: After cover plates are added, the resulting section is still doubly symmetric. Therefore, the overall failure mode is still flexural buckling. For flexural buckling about the minor axis (y-y), no modification to (KL/r) is required because the buckling axis is perpendicular to the plane of contact of the component shapes and no relative movement between the adjoining parts is expected. However, for flexural buckling about the major (x-x) axis, modification to (KL/r) is required because the buckling axis is parallel to the plane of contact of the adjoining structural shapes and slippage between the component pieces will occur. We shall design the cover plates assuming flexural buckling about the minor axis will control and check for flexural buckling about the major axis later. A W24x229 section has a flange width of 13.11 in.; so, as a trial, use cover plates with widths of 13 in. as shown in Figure 3.8a. Denoting t as the thickness of the plates, we have s (ry )built-up = and (λc )y,built-up
1 = π
(Iy )W-shape + (Iy )plates = AW-shape + Aplates
KL r
y,built-up
r
r
651 + 183.1t 67.2 + 26t
r Fy 67.2 + 26t = 2.69 E 651 + 183.1t
Assuming (λ)y,built−up is less than 1.5, one can substitute the above expression for λc in Equation 3.17. With φc Pn equals 1728, we can solve for t. The result is t = 1/2 in. Backsubstituting t = 1/2 into the above expression, we obtain (λ)c,built−up = 0.884 which is indeed 0.6Fy Afg = 0.6(36)(7.005)(0.505) = 76.4 kips
Cover Plates
0.5Fu Af n = 0.5(58)(7 − 2 × 1/2)(1/2) = 87 kips
> 0.6Fy Af g = 0.6(36)(7)(1/2) = 75.6 kips so the use of the gross cross-sectional area to compute section properties is justified. In the event that the condition is violated, cross-sectional properties should be evaluated using an effective tension flange area Af e given by 5 Fu Af n Af e = 6 Fy Use 1/2” diameter A325N bolts spaced 4.5” apart longitudinally in two lines 4” apart to connect the cover plates to the beam flanges. 1999 by CRC Press LLC
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3.5.2
Load and Resistance Factor Design
Flexural Strength Criterion
Flexural members must be designed to satisfy the flexural strength criterion of φb Mn ≥ Mu
(3.36)
where φb Mn is the design flexural strength and Mu is the required strength. The design flexural strength is determined as follows: Compact Section Members Bent About Their Major Axes For Lb ≤ Lp , (Plastic hinge formation) φb Mn = 0.90Mp
(3.37)
For Lp < Lb ≤ Lr , (Inelastic lateral torsional buckling) Lb − Lp ≤ 0.90Mp φb Mn = 0.90Cb Mp − (Mp − Mr ) Lr − Lp
(3.38)
For Lb > Lr , (Elastic lateral torsional buckling) For I-shaped members and channels: π φb Mn = 0.90Cb Lb
s
EIy GJ +
πE Lb
2
Iy Cw ≤ 0.90Mp
(3.39)
For solid rectangular bars and symmetric box sections: √ 57, 000 J A ≤ 0.90Mp φb Mn = 0.90Cb Lb /ry The variables used in the above equations are defined in the following. Lb
= lateral unsupported length of the member
Lp , Lr = limiting lateral unsupported lengths given in the following table 1999 by CRC Press LLC
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(3.40)
Lp
Structural shape I-shaped chanels
p 300ry / Fyf
sections,
Solid rectangular bars, symmetric box sections
Lr ry X1 /FL
(s 1+
r
1 + X2 FL2
)
where
where
ry = radius of gyration about minor axis, in. Fyf = flange yield stress, ksi
ry = radius of√gyration about minor axis, in. X1 = (π/Sx ) (EGJ A/2) X2 = (4Cw /Iy )(Sx /GJ )2 FL = smaller of (Fyf − Fr ) or Fyw Fyf = flange yield stress, ksi Fyw = web yield stress, ksi Fr = 10 ksi for rolled shapes, 16.5 ksi for welded shapes Sx = elastic section modulus about the major axis, in.3 (use Sxc , the elastic section modulus about the major axis with respect to the compression flange if the compression flange is larger than the tension flange) Iy = moment of inertia about the minor axis, in.4 J = torsional constant, in.4 Cw = warping constant, in.6 E = modulus of elasticity, ksi G = shear modulus, ksi
√ 3, 750ry (J A) /Mp
√ 57, 000ry (J A) /Mr
where
where
ry = radius of gyration about minor axis, in. J = torsional constant, in.4 A = cross-sectional area, in.2 Mp = plastic moment capacity = Fy Zx Fy = yield stress, ksi Zx = plastic section modulus about the major axis, in.3
ry = radius of gyration about minor axis, in. J = torsional constant, in.4 A = cross-sectional area, in.2 Mr = Fy Sx for solid rectangular bar, Fyf Seff for box sections Fy = yield stress, ksi Fyf = flange yield stress, ksi Sx = plastic section modulus about the major axis, in.3
Note: Lp given in this table are valid only if the bending coefficient Cb is equal to unity. If Cb > 1, the value of Lp can be increased. However, using the Lp expressions given above for Cb > 1 will give a conservative value for the flexural design strength.
and Mp = Fy Zx Mr = FL Sx for I-shaped sections and channels, Fy Sx for solid rectangular bars, Fyf Seff for box sections FL = smaller of (Fyf − Fr ) or Fyw Fyf = flange yield stress, ksi Fyw = web yield stress Fr = 10 ksi for rolled sections, 16.5 ksi for welded sections Fy = specified minimum yield stress Sx = elastic section modulus about the major axis Seff = effective section modular, calculated using effective width be , in Table 3.7 Zx = plastic section modulus about the major axis Iy = moment of inertia about the minor axis J = torsional constant Cw = warping constant E = modulus of elasticity G = shear modulus Cb = 12.5Mmax /(2.5Mmax + 3MA + 4MB + 3MC ) 1999 by CRC Press LLC
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Mmax , MA , MB , MC
= maximum moment, quarter-point moment, midpoint moment, and three-quarter point moment along the unbraced length of the member, respectively.
Cb is a factor that accounts for the effect of moment gradient on the lateral torsional buckling strength of the beam. Lateral torsional buckling strength increases for a steep moment gradient. The worst loading case as far as lateral torsional buckling is concerned is when the beam is subjected to a uniform moment resulting in single curvature bending. For this case Cb =1. Therefore, the use of Cb =1 is conservative for the design of beams. Compact Section Members Bent About Their Minor Axes Regardless of Lb , the limit state will be a plastic hinge formation φb Mn = 0.90Mpy = 0.90Fy Zy
(3.41)
Noncompact Section Members Bent About Their Major Axes For Lb ≤ L0p , (Flange or web local buckling) φb Mn =
φb Mn0
where L0p
λ − λp = 0.90 Mp − (Mp − Mr ) λr − λp
Mp − Mn0 = Lp + (Lr − Lp ) Mp − Mr
(3.42)
(3.43)
Lp , Lr , Mp , Mr are defined as before for compact section members, and For flange local buckling: λ = bf /2t p f for I-shaped members, bf /tf for channels λp = 65/ Fy p λr = 141/ (Fy − 10) For web local buckling: λ = hc /twp λp = 640/ Fy p λr = 970/ Fy in which bf = flange width tf = flange thickness hc = twice the distance from the neutral axis to the inside face of the compression flange less the fillet or corner radius tw = web thickness For L0p < Lb ≤ Lr , (Inelastic lateral torsional buckling), φb Mn is given by Equation 3.38 except that the limit 0.90Mp is to be replaced by the limit 0.90Mn0 . For Lb > Lr , (Elastic lateral torsional buckling), φb Mn is the same as for compact section members as given in Equation 3.39 or Equation 3.40. Noncompact Section Members Bent About Their Minor Axes Regardless of the value of Lb , the limit state will be either flange or web local buckling, and φb Mn is given by Equation 3.42. 1999 by CRC Press LLC
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Slender Element Sections Refer to the section on Plate Girder. Tees and Double Angle Bent About Their Major Axes The design flexural strength for tees and double-angle beams with flange and web slenderness ratios less than the corresponding limiting slenderness ratios λr shown in Table 3.8 is given by " p # p π EIy GJ 2 (B + 1 + B ) ≤ 0.90(CMy ) φb Mn = 0.90 Lb where
B = ±2.3
d Lb
r
Iy J
(3.44)
(3.45)
C = 1.5 for stems in tension, and 1.0 for stems in compression. Use the plus sign for B if the entire length of the stem along the unbraced length of the member is in tension. Otherwise, use the minus sign. The other variables in Equation 3.44 are defined as before in Equation 3.39. Shear Strength Criterion
For a satisfactory design, the design shear strength of the webs must exceed the factored shear acting on the cross-section, i.e., (3.46) φv Vn ≥ Vu Depending on the slenderness ratios of the webs, three limit states can be identified: shear yielding, inelastic shear buckling, and elastic shear buckling. The design shear strength that corresponds to each of these limit states is given as follows: p For h/tw ≤ 418/ Fyw , (Shear yielding of web)
p For 418/ Fyw
φv Vn = 0.90[0.60Fyw Aw ] p < h/tw ≤ 523/ Fyw , (Inelastic shear buckling of web) # p 418/ Fyw φv Vn = 0.90 0.60Fyw Aw h/tw
(3.47)
"
(3.48)
p For 523/ Fyw < h/tw ≤ 260, (Elastic shear buckling of web) φv Vn = 0.90
132,000Aw (h/tw )2
The variables used in the above equations are defined in the following: h tw Fyw Aw d
= = = = =
clear distance between flanges less the fillet or corner radius, in. web thickness, in. yield stress of web, ksi dtw , in.2 overall depth of section, in.
1999 by CRC Press LLC
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(3.49)
Criteria for Concentrated Loads
When concentrated loads are applied normal to the flanges in planes parallel to the webs of flexural members, the flange(s) and web(s) must be checked to ensure that they have sufficient strengths φRn to withstand the concentrated forces Ru , i.e., φRn ≥ Ru
(3.50)
The design strength for a variety of limit states are given below: Local Flange Bending The design strength for local flange bending is given by φRn ≥ 0.90[6.25tf2 Fyf ]
(3.51)
where = flange thickness of the loaded flange, in. tf Fyf = flange yield stress, ksi Local Web Yielding The design strength for yielding of a beam web at the toe of the fillet under tensile or compressive loads acting on one or both flanges are: If the load acts at a distance from the beam end which exceeds the depth of the member φRn = 1.00[(5k + N )Fyw tw ]
(3.52)
If the load acts at a distance from the beam end which does not exceed the depth of the member φRn = 1.00[(2.5k + N )Fyw tw ] where k = N = Fyw = tw =
(3.53)
distance from outer face of flange to web toe of fillet length of bearing on the beam flange web yield stress web thickness
Web Crippling The design strength for crippling of a beam web under compressive loads acting on one or both flanges are: If the load acts at a distance from the beam end which exceeds half the depth of the beam ( φRn = 0.75
"
N 1+3 d
135tw2
tw tf
1.5 # s
Fyw tf tw
) (3.54)
If the load acts at a distance from the beam end which does not exceed half the depth of the beam and if N/d ≤ 0.2 ( φRn = 0.75 1999 by CRC Press LLC
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" 68tw2
N 1+3 d
tw tf
1.5 # s
Fyw tf tw
) (3.55)
If the load acts at a distance from the beam end which does not exceed half the depth of the beam and if N/d>0.2 ( " ) 1.5 # s Fyw tf tw 4N 2 − 0.2 (3.56) φRn = 0.75 68tw 1 + d tf tw where d = overall depth of the section, in. = flange thickness, in. tf The other variables are the same as those defined in Equations 3.52 and 3.53. Sidesway Web Buckling Sidesway web buckling may occur in the web of a member if a compressive concentrated load is applied to a flange which is not restrained against relative movement by stiffeners or lateral bracings. The sidesway web buckling design strength for the member is: If the loaded flange is restrained against rotation about the longitudinal member axis and (hc /tw )(l/bf ) ≤ 2.3 ( " #) Cr tw3 tf h/tw 3 (3.57) 1 + 0.4 φRn = 0.85 l/bf h2 If the loaded flange is not restrained against rotation about the longitudinal member axis and (hc /tw )(l/bf ) ≤ 1.7 " ( #) Cr tw3 tf h/tw 3 0.4 (3.58) φRn = 0.85 l/bf h2 where = flange thickness, in. tf tw = web thickness, in. h = clear distance between flanges less the fillet or corner radius for rolled shapes; distance between adjacent lines of fasteners or clear distance between flanges when welds are used for built-up shapes, in. bf = flange width, in. l = largest laterally unbraced length along either flange at the point of load, in. Cr = 960,000 if Mu /My 234
r
Cv r
kv Fyw
kv Fyw
√
187 kv /Fyw h/tw 44,000kv (h/tw )2 Fyw
Flexure-Shear Interaction
Plate girders designed for tension field action must satisfy the flexure-shear interaction criterion in regions where 0.60φVn ≤ Vu ≤ φVn and 0.75φMn ≤ Mu ≤ φMn Vu Mu + 0.625 ≤ 1.375 φMn φVn
(3.89)
where φ = 0.90. Bearing Stiffeners
Bearing stiffeners must be provided for a plate girder at unframed girder ends and at points of concentrated loads where the web yielding or the web crippling criterion is violated (see section on Concentrated Load Criteria). Bearing stiffeners shall be provided in pairs and extended from the upper flange to the lower flange of the girder. Denoting bst as the width of one stiffener and tst as its thickness, bearing stiffeners shall be portioned to satisfy the following limit states: For the limit state of local buckling
95 bst ≤p tst Fy
(3.90)
For the limit state of compression The design compressive strength, φc Pn , must exceed the required compressive force acting on the stiffeners. φc Pn is to be determined based on an effective length factor K of 0.75 and an effective area, Aeff , equal to the area of the bearing stiffeners plus a portion of the web. For end bearing, this effective area is equal to 2(bst tst ) + 12tw2 ; and for interior bearing, this effective area is equal to 2(bst tst ) + 25tw2 . tw is the p web thickness. The slenderness parameter, λc , is to be calculated using a radius of gyration, r = (Ist /Aeff ), where Ist = tst (2bst + tw )3 /12. For the limit state of bearing The bearing strength, φRn , must exceed the required compression force acting on the stiffeners. φRn is given by (3.91) φRn ≥ 0.75[1.8Fy Apb ] where Fy is the yield stress and Apb is the bearing area. 1999 by CRC Press LLC
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Intermediate Stiffeners
Intermediate stiffeners shall be provided if (1) the shear strength capacity is calculated based on tension field action, (2) the shearpcriterion is violated (i.e., when the Vu exceeds φv Vn ), or (3) the web slenderness h/tw exceeds 418/ Fyw . Intermediate stiffeners can be provided in pairs or on one side of the web only in the form of plates or angles. They should be welded to the compression flange and the web but they may be stopped short of the tension flange. The following requirements apply to the design of intermediate stiffeners: Local Buckling The width-thickness ratio of the stiffener must be proportioned so that Equation 3.90 is satisfied to prevent failure by local buckling. Stiffener Area The cross-section area of the stiffener must satisfy the following criterion: Fyw Vu 2 − 18tw ≥ 0 0.15Dhtw (1 − Cv ) Ast ≥ Fy φv Vn
(3.92)
where Fy = yield stress of stiffeners D = 1.0 for stiffeners in pairs, 1.8 for single angle stiffeners, and 2.4 for single plate stiffeners The other terms in Equation 3.92 are defined as before in Equation 3.87 and Equation 3.88. Stiffener Moment of Inertia The moment of inertia for stiffener pairs taken about an axis in the web center or for single stiffeners taken in the face of contact with the web plate must satisfy the following criterion: Ist ≥
atw3
2.5 − 2 ≥ 0.5atw3 (a/ h)2
(3.93)
Stiffener Length The length of the stiffeners, lst , should fall within the range h − 6tw < lst < h − 4tw
(3.94)
where h is the clear distance between the flanges less the widths of the flange-to-web welds and tw is the web thickness. If intermittent welds are used to connect the stiffeners to the girder web, the clear distance between welds shall not exceed 16tw , or 10 in. If bolts are used, their spacing shall not exceed 12 in. Stiffener Spacing The spacing of the stiffeners, a, shall be determined from the shear criterion φv Vn ≥ Vu . This spacing shall not exceed the smaller of 3h and [260/(h/tw )]2 h.
EXAMPLE 3.7:
Using LRFD, design the cross-section of an I-shaped plate girder shown in Figure 3.12a to support a factored moment Mu of 4600 kip-ft (6240 kN-m), dead weight of the girder is included. The girder is a 60-ft (18.3-m) long simply supported girder. It is laterally supported at every 20-ft (6.1-m) interval. Use A36 steel. 1999 by CRC Press LLC
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FIGURE 3.12: Design of a plate girder cross-section.
Proportion of the girder web Ordinarily, the overall depth-to-span ratio d/L of a building girder is in the prange 1/12 to 1/10. p So, let us try h =70 in. Also, knowing h/tw of a plate girder is in the range 970/ Fyf and 2,000/ Fyf , let us try tw = 5/16 in. Proportion of the girder flanges For a preliminary design, the required area of the flange can be determined using the flange area method 4600 kip-ft x12 in./ft Mu = = 21.7 in.2 Af ≈ Fy h (36 ksi )(70 in.) So, let bf = 20 in. and tf = 1-1/8 in. giving Af = 22.5 in.2 1999 by CRC Press LLC
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Determine the design flexural strength φb Mn of the girder: Calculate Ix : X [Ii + Ai yi2 ] Ix = = [8932 + (21.88)(0)2 ] + 2[2.37 + (22.5)(35.56)2 ] = 65840 in.4 Calculate Sxt , Sxc : Sxt = Sxc =
Ix Ix 65840 = 1823 in.3 = = ct cc 35 + 1.125
Calculate rT : Refer to Figure 3.12b, s s IT (1.125)(20)3 /12 + (11.667)(5/16)3 /12 = 5.36 in. = rT = 1 Af + 6 Aw 22.5 + 16 (21.88) Calculate Fcr : For Flange Local Buckling (FLB),
# " bf 65 20 65 = 8.89 < p = = √ = 10.8 so, Fcr = Fyf = 36 ksi 2tf 2(1.125) Fyf 36
For Lateral Torsional Buckling (LTB), # " 300 20 × 12 300 Lb = 44.8 < p = = √ = 50 so, Fcr = Fyf = 36 ksi rT 5.36 Fyf 36 Calculate RP G : RP G
√ √ 0.972[70/(5/16) − 970/ 36] ar (hc /tw − 970/ Fcr ) =1− = 0.96 =1− (1,200 + 300ar ) [1,200 + 300(0.972)]
Calculate φb Mn : φb Mn
=
0.90 Sxt Re Fy t = (0.90)(1823)(1)(36) = 59,065 kip-in. 0.90 Sxc RP G Re Fcr = (0.90)(1823)(0.96)(1)(36) = 56,700 kip-in. 56,700 kip-in.
=
4725 kip-ft.
=
smaller of
Since [φb Mn = 4725 kip-ft ] > [Mu = 4600 kip-ft ], the cross-section is acceptable. Use web plate 5/16”x70” and two flange plates 1-1/8”x20” for the girder cross-section.
EXAMPLE 3.8:
Design bearing stiffeners for the plate girder of the preceding example for a factored end reaction of 260 kips. Since the girder end is unframed, bearing stiffeners are required at the supports. The size of the stiffeners must be selected to ensure that the limit states of local buckling, compression, and bearing are not violated. 1999 by CRC Press LLC
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Limit state of local buckling Refer to p Figure 3.13, try bst = 8 in. To avoid problems with local buckling, bst /2tst must not exceed 95/ Fy = 15.8. Therefore, try tst = 1/2 in. So, bst /2tst = 8 which is less than 15.8.
FIGURE 3.13: Design of bearing stiffeners.
Limit state of compression Aeff
=
2(bst tst ) + 12tw2 = 2(8)(0.5) + 12(5/16)2 = 9.17 in.2
Ist
=
rst
=
tst (2bst + tw )3 /12 = 0.5[2(8) + 5/16]3 /12 = 181 in.4 q p (Ist /Aeff ) = (181/9.17) = 4.44 in.
Kh/rst
=
λc
=
0.75(70)/4.44 = 11.8 q p (Kh/πrst ) (Fy /E) = (11.8/3.142) (36/29,000) = 0.132
and from Equation 3.17 φc Pn = 0.85(0.658λc )Fy Ast = 0.85(0.658)0.132 (36)(9.17) = 279 kips 2
2
Since φc Pn > 260 kips, the design is satisfactory for compression. Limit state of bearing Assuming there is a 1/4-in. weld cutout at the corners of the bearing stiffeners at the junction of the stiffeners and the girder flanges, the bearing area for the stiffener pairs is Apb = (8 − 0.25)(0.5)(2) = 7.75 in.2 . Substitute this into Equation 3.91, we have φRn = 0.75(1.8)(36)(7.75) = 377 kips, which exceeds the factored reaction of 260 kips. So, bearing is not a problem. Use two 1/2”x 8” plates for bearing stiffeners.
3.11
Connections
Connections are structural elements used for joining different members of a framework. Connections can be classified according to: 1999 by CRC Press LLC
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• the type of connecting medium used: bolted connections, welded connections, boltedwelded connections, riveted connections • the type of internal forces the connections are expected to transmit: shear (semi-rigid, simple) connections, moment (rigid) connections • the type of structural elements that made up the connections: single plate angle connections, double web angle connections, top and seated angle connections, seated beam connections, etc. • the type of members the connections are joining: beam-to-beam connections (beam splices), column-to-column connections (column splices), beam-to-column connections, hanger connections, etc. To properly design a connection, a designer must have a thorough understanding of the behavior of the joint under loads. Different modes of failure can occur depending on the geometry of the connection and the relative strengths and stiffnesses of the various components of the connection. To ensure that the connection can carry the applied loads, a designer must check for all perceivable modes of failure pertinent to each component of the connection and the connection as a whole.
3.11.1
Bolted Connections
Bolted connections are connections whose components are fastened together primarily by bolts. The four basic types of bolts commonly used for steel construction are discussed in the section on Structural Fasteners. Depending on the direction and line of action of the loads relative to the orientation and location of the bolts, the bolts may be loaded in tension, shear, or a combination of tension and shear. For bolts subjected to shear forces, the design shear strength of the bolts also depends on whether or not the threads of the bolts are excluded from the shear planes. A letter X or N is placed at the end of the ASTM designation of the bolts to indicate whether the threads are excluded or not excluded from the shear planes, respectively. Thus, A325X denotes A325 bolts whose threads are excluded from the shear planes and A490N denotes A490 bolts whose threads are not excluded from the shear planes. Because of the reduced shear areas for bolts whose threads are not excluded from the shear planes, these bolts have lower design shear strengths than their counterparts whose threads are excluded from the shear planes. Bolts can be used in both bearing-type connections and slip-critical connections. Bearing-type connections rely on bearing between the bolt shanks and the connecting parts to transmit forces. Some slippage between the connected parts is expected to occur for this type of connection. Slipcritical connections rely on the frictional force developing between the connecting parts to transmit forces. No slippage between connecting elements is expected for this type of connection. Slipcritical connections are used for structures designed for vibratory or dynamic loads such as bridges, industrial buildings, and buildings in regions of high seismicity. Bolts used in slip-critical connections are denoted by the letter F after their ASTM designation, e.g., A325F, A490F. Bolt Holes
Holes made in the connected parts for bolts may be standard size, oversized, short slotted, or long slotted. Table 3.10 gives the maximum hole dimension for ordinary construction usage. Standard holes can be used for both bearing-type and slip-critical connections. Oversized holes shall be used only for slip-critical connections. Short- and long-slotted holes can be used for both bearing-type and slip-critical connections provided that when such holes are used for bearing, the direction of slot is transverse to the direction of loading. 1999 by CRC Press LLC
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TABLE 3.10
Nominal Hole Dimensions
Bolt
Hole dimensions
diameter, d (in.)
Standard (dia.)
Oversize (dia.)
Short-slot (width × length)
Long-slot (width × length)
1/2 5/8 3/4 7/8 1 ≥ 1-1/8
9/16 11/16 13/16 15/16 1-1/16 d+1/16
5/8 13/16 15/16 1-1/16 1-1/4 d+5/16
9/16×11/16 11/16×7/8 13/16×1 15/16×1-1/8 1-1/16×1-5/16 (d+1/16)×(d+3/8)
9/16×1-1/4 11/16×1-9/16 13/16×1-7/8 15/16×2-3/16 1-1/16×2-1/2 (d+1/16)×(2.5d)
Note: 1 in. = 25.4 mm.
Bolts Loaded in Tension
If a tensile force is applied to the connection such that the direction of the load is parallel to the longitudinal axes of the bolts, the bolts will be subjected to tension. The following condition must be satisfied for bolts under tensile stresses. Allowable Stress Design: ft ≤ Ft
(3.95)
where ft = computed tensile stress in the bolt, ksi Ft = allowable tensile stress in bolt (see Table 3.11) Load and Resistance Factor Design: φt Ft ≥ ft
(3.96)
where φt = 0.75 ft = tensile stress produced by factored loads, ksi Ft = nominal tensile strength given in Table 3.11 TABLE 3.11
Ft of Bolts, ksi ASD
Bolt type
Ft , ksi (static loading)
A307 A325
20 44.0
Ft , ksi (fatigue loading)
Not allowed If N ≤ 20,000: Ft = same as for static loading
LRFD Ft , ksi (static loading)
45.0 90.0
If 20,000 < N ≤ 500,000: Ft = 40 (A325) = 49 (A490)
Ft , ksi
(fatigue loading) Not allowed If N ≤ 20,000: Ft = same as for static loading If 20,000 < N ≤ 500,000: Ft = 0.30Fu (at service loads)
If N > 500,000: A490
54.0
Ft = 31(A325) = 38 (A490)
where N = number of cycles Fu = minimum specified tensile strength, ksi Note: 1 ksi = 6.895 MPa.
1999 by CRC Press LLC
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113
If N > 500,000: Ft = 0.25Fu (at service loads) where N = number of cycles Fu = minimum specified tensile strength, ksi
Bolts Loaded in Shear
When the direction of load is perpendicular to the longitudinal axes of the bolts, the bolts will be subjected to shear. The condition that needs to be satisfied for bolts under shear stresses is as follows. Allowable Stress Design: fv ≤ Fv
(3.97)
where fv = computed shear stress in the bolt, ksi Fv = allowable shear stress in bolt (see Table 3.12) Load and Resistance Factor Design: φv Fv ≥ fv
(3.98)
where φv = 0.75 (for bearing-type connections), 1.00 (for slip-critical connections when standard, oversized, short-slotted, or long-slotted holes with load perpendicular to the slots are used), 0.85 (for slip-critical connections when long-slotted holes with load in the direction of the slots are used) fv = shear stress produced by factored loads (for bearing-type connections), or by service loads (for slip-critical connections), ksi Fv = nominal shear strength given in Table 3.12 TABLE 3.12
Fv of Bolts, ksi
Bolt type A307 A325N A325X A325Fb
A490N A490X A490Fb
Fv , ksi
ASD
LRFD
10.0a (regardless of whether or not threads
24.0a (regardless of whether or not threads
are excluded from shear planes) 21.0a 30.0a 17.0 (for standard size holes) 15.0 (for oversized and short-slotted holes) 12.0 (for long-slotted holes when direction of load is transverse to the slots) 10.0 (for long-slotted holes when direction of load is parallel to the slots) 28.0a 40.0a 21.0 (for standard size holes) 18.0 (for oversized and short-slotted holes) 15.0 (for long-slotted holes when direction of load is transverse to the slots) 13.0 (for long-slotted holes when direction of load is parallel to the slots)
are excluded from shear planes) 48.0a 60.0a 17.0 (for standard size holes) 15.0 (for oversized and short-slotted holes) 12.0 (for long-slotted holes)
60.0a 75.0a 21.0 (for standard size holes) 18.0 (for oversized and short-slotted holes) 15.0 (for long-slotted holes)
a tabulated values shall be reduced by 20% if the bolts are used to splice tension members having a fastener pattern whose length,
measured parallel to the line of action of the force, exceeds 50 in.
b tabulated values are applicable only to class A surface, i.e., clean mill surface and blast cleaned surface with class A coatings (with
slip coefficient = 0.33). For design strengths with other coatings, see RCSC “Load and Resistance Factor Design Specification to Structural Joints Using ASTM A325 or A490 Bolts” [28] Note: 1 ksi = 6.895 MPa.
Bolts Loaded in Combined Tension and Shear
If a tensile force is applied to a connection such that its line of action is at an angle with the longitudinal axes of the bolts, the bolts will be subjected to combined tension and shear. The conditions that need to be satisfied are given as follows. Allowable Stress Design: fv ≤ Fv and ft ≤ Ft 1999 by CRC Press LLC
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(3.99)
where fv , Fv ft Ft
= as defined in Equation 3.97 = computed tensile stress in the bolt, ksi = allowable tensile stress given in Table 3.13
Load and Resistance Factor Design: φv Fv ≥ fv and φt Ft ≥ ft
(3.100)
where φv , Fv , fv = as defined in Equation 3.98 = 1.0 φt = tensile stress due to factored loads (for bearing-type connection), or due to service ft loads (for slip-critical connections), ksi = nominal tension stress limit for combined tension and shear given in Table 3.13 Ft TABLE 3.13
Ft for Bolts Under Combined Tension and Shear, ksi Bearing-type connections ASD
Threads not excluded from the shear plane
Bolt type
LRFD Threads excluded from the shear plane
26-1.8fv ≤ 20 q q (442 − 4.39fv2 ) (442 − 2.15fv2 ) q q (542 − 3.75fv2 ) (542 − 1.82fv2 )
A307 A325 A490
Threads not excluded from the shear plane
Threads excluded from the shear plane
59-1.9fv ≤ 45 117 − 1.9fv ≤ 90
117 − 1.5fv ≤ 90
147 − 1.9fv ≤ 113
147 − 1.5fv ≤ 113
Slip-critical connections For ASD: Ft = Fv =
values given above [1 − (ft Ab /Tb )]× (values of Fv given in Table 3.12)
where ft Tb Fu Ab
computed tensile stress in the bolt, ksi pretension load = 0.70Fu Ab , kips minimum specified tensile strength, ksi nominal cross-sectional area of bolt, in.2
= = = =
For LRFD: Ft = Fv =
values given above [1 − (T /Tb )]× (values of Fv given in Table 3.12)
where T Tb Fu Ab
service tensile force, kips pretension load = 0.70Fu Ab , kips minimum specified tensile strength, ksi nominal cross-sectional area of bolt, in.2
= = = =
Note: 1 ksi = 6.895 MPa.
Bearing Strength at Fastener Holes
Connections designed on the basis of bearing rely on the bearing force developed between the fasteners and the holes to transmit forces and moments. The limit state for bearing must therefore be checked to ensure that bearing failure will not occur. Bearing strength is independent of the type of fastener. This is because the bearing stress is more critical on the parts being connected than on the fastener itself. The AISC specification provisions for bearing strength are based on preventing 1999 by CRC Press LLC
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excessive hole deformation. As a result, bearing capacity is expressed as a function of the type of holes (standard, oversized, slotted), bearing area (bolt diameter times the thickness of the connected parts), bolt spacing, edge distance (Le ), strength of the connected parts (Fu ) and the number of fasteners in the direction of the bearing force. Table 3.14 summarizes the expressions used in ASD and LRFD for calculating the bearing strength and the conditions under which each expression is valid. TABLE 3.14
Bearing Capacity
Conditions 1. For standard or short-slotted holes with Le ≥ 1.5d, s ≥ 3d and number of fasteners in the direction of bearing ≥ 2 2. For long-slotted holes with direction of slot transverse to the direction of bearing and Le ≥ 1.5d, s ≥ 3d and the number of fasteners in the direction of bearing ≥ 2 3. If neither condition 1 nor 2 above is satisfied
ASD
LRFD
Allowable bearing stress, Fp , ksi
Design bearing strength, φRn , ksi
1.2Fu
0.75[2.4dtFu ]
1.0Fu
0.75[2.0dtFu ]
Le Fu /2d ≤ 1.2Fu
For the bolt hole nearest the edge: 0.75[Le tFu ] ≤ 0.75[2.4dtFu ]a For the remaining bolt holes: 0.75[(s − d/2)tFu ] ≤ 0.75[2.4dtFu ]a
1.5Fu
For the bolt hole nearest the edge: 0.75[Le tFu ] ≤ 0.75[3.0dtFu ] For the remaining bolt holes: 0.75[(s − d/2)tFu ] ≤ 0.75[3.0dtFu ]
4. If hole deformation is not a design consideration and adequate spacing and edge distance is provided (see sections on Minimum Fastener Spacing and Minimum Edge Distance)
a For long-slotted bolt holes with direction of slot transverse to the direction of bearing, this limit is
0.75[2.0dtFu ] = edge distance (i.e., distance measured from the edge of the connected part to the center of Le a standard hole or the center of a short- and long-slotted hole perpendicular to the line of force. For oversized holes and short- and long-slotted holes parallel to the line of force, Le shall be increased by the edge distance increment C2 given in Table 3.16) s = fastener spacing (i.e., center to center distance between adjacent fasteners measured in the direction of bearing. For oversized holes and short- and long-slotted holes parallel to the line of force, s shall be increased by the spacing increment C1 given in Table 3.15) d = nominal bolt diameter, in. t = thickness of the connected part, in. Fu = specified minimum tensile strength of the connected part, ksi
TABLE 3.15
Values of Spacing Increment, C1 , in. Slotted Holes
Nominal
Parallel to line of force
diameter of fastener (in.)
Standard holes
Oversized holes
Transverse to line of force
Shortslots
Long-slotsa
≤ 7/8 1 ≥ 1-1/8
0 0 0
1/8 3/16 1/4
0 0 0
3/16 1/4 5/16
3d /2-1/16 23/16 3d /2-1/16
a When length of slot is less than the value shown in Table 3.10, C may be reduced by the 1
difference between the value shown and the actual slot length. Note: 1 in. = 25.4 mm.
1999 by CRC Press LLC
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Minimum Fastener Spacing
To ensure safety, efficiency, and to maintain clearances between bolt nuts as well as to provide room for wrench sockets, the fastener spacing, s, should not be less than 3d where d is the nominal fastener diameter. TABLE 3.16
Values of Edge Distance Increment, C2 , in.
Nominal diameter
Slotted holes
of fastener
Slot transverse to edge
Slot parallel to
(in.)
Oversized holes
Short-slot
Long-slota
edge
≤ 7/8 1 ≤ 1-1/8
1/16 1/8 1/8
1/8 1/8 3/16
3d/4 3d/4 3d/4
0
a If the length of the slot is less than the maximum shown in Table 3.10, the value shown may
be reduced by one-half the difference between the maximum and the actual slot lengths. Note: 1 in. = 25.4 mm.
Minimum Edge Distance
To prevent excessive deformation and shear rupture at the edge of the connected part, a minimum edge distance Le must be provided in accordance with the values given in Table 3.17 for standard holes. For oversized and slotted holes, the values shown must be incremented by C2 given in Table 3.16. TABLE 3.17
Minimum Edge Distance for Standard Holes, in.
Nominal fastener diameter (in.)
At sheared edges
At rolled edges of plates, shapes, and bars or gas cut edges
1/2 5/8 3/4 7/8 1 1-1/8 1-1/4 over 1-1/4
7/8 1-1/8 1-1/4 1-1/2 1-3/4 2 2-1/4 1-3/4 x diameter
3/4 7/8 1 1-1/8 1-1/4 1-1/2 1-5/8 1-1/4 x diameter
Note: 1 in. = 25.4 mm.
Maximum Fastener Spacing
A limit is placed on the maximum value for the spacing between adjacent fasteners to prevent the possibility of gaps forming or buckling from occurring in between fasteners when the load to be transmitted by the connection is compressive. The maximum fastener spacing measured in the direction of the force is given as follows. For painted members or unpainted members not subject to corrosion: smaller of 24t where t is the thickness of the thinner plate and 12 in. For unpainted members of weathering steel subject to atmospheric corrosion: smaller of 14t where t is the thickness of the thinner plate and 7 in. 1999 by CRC Press LLC
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Maximum Edge Distance
A limit is placed on the maximum value for edge distance to prevent prying action from occurring. The maximum edge distance shall not exceed the smaller of 12t where t is the thickness of the connected part and 6 in.
EXAMPLE 3.9:
Check the adequacy of the connection shown in Figure 3.4a. The bolts are 1-in. diameter A325N bolts in standard holes. Check bolt capacity All bolts are subjected to double shear. Therefore, the design shear strength of the bolts will be twice that shown in Table 3.12. Assuming each bolt carries an equal share of the factored applied load, we have from Equation 3.98 208 2 = 44.1 ksi [φv Fv = 0.75(2 × 48) = 72 ksi] > fv = (6) π41 The shear capacity of the bolt is therefore adequate. Check bearing capacity of the connected parts With reference to Table 3.14, it can be seen that condition 1 applies for the present problem. Therefore, we have 3 208 (58) = 39.2 kips] > Ru = = 34.7 kips [φRn = 0.75(2.4)(1) 8 6 and so bearing is not a problem. Note that bearing on the gusset plate is more critical than bearing on the webs of the channels because the thickness of the gusset plate is less than the combined thickness of the double channels. Check bolt spacing The minimum bolt spacing is 3d = 3(1) = 3 in. The maximum bolt spacing is the smaller of 14t = 14(.303) = 4.24 in. or 7 in. The actual spacing is 3 in. which falls within the range of 3 to 4.24 in., so bolt spacing is adequate. Check edge distance From Table 3.17, it can be determined that the minimum edge distance is 1.25 in. The maximum edge distance allowed is the smaller of 12t = 12(0.303) = 3.64 in. or 6 in. The actual edge distance is 3 in. which falls within the range of 1.25 to 3.64 in., so edge distance is adequate. The connection is adequate. Bolted Hanger Type Connections
A typical hanger connection is shown in Figure 3.14. In the design of such connections, the designer must take into account the effect of prying action. Prying action results when flexural deformation occurs in the tee flange or angle leg of the connection (Figure 3.15). Prying action tends to increase the tensile force, called prying force, in the bolts. To minimize the effect of prying, the fasteners should be placed as close to the tee stem or outstanding angle leg as the wrench clearance will permit (see Tables on Entering and Tightening Clearances in Volume II-Connections of the AISC-LRFD Manual [22]). In addition, the flange and angle thickness should be proportioned so that the full tensile capacities of the bolts can be developed. 1999 by CRC Press LLC
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FIGURE 3.14: Hanger connections.
Two failure modes can be identified for hanger type connections: formation of plastic hinges in the tee flange or angle leg at cross-sections 1 and 2, and tensile failure of the bolts when the tensile force including prying action Bc (= T + Q) exceeds the tensile capacity of the bolt B. Since the determination of the actual prying force is rather complex, the design equation for the required thickness for the tee flange or angle leg is semi-empirical in nature. It is given by the following. If ASD is used:
s treq 0 d =
8T b0 pFy (1 + δα 0 )
(3.101)
where T = tensile force per bolt due to service load exclusive of initial tightening and prying force, kips The other variables are as defined in Equation 3.102 except that B in the equation for α 0 is defined as the allowable tensile force per bolt. A design is considered satisfactory if the thickness of the tee flange or angle leg tf exceeds treq 0 d and B > T . If LRFD is used:
s treq 0 d =
4Tu b0 φb pFy (1 + δα 0 )
(3.102)
where φb = 0.90 Tu = factored tensile force per bolt exclusive of initial tightening and prying force, kips p = length of flange tributary to each bolt measured along the longitudinal axis of the tee or double angle section, in. δ = ratio of net area at bolt line to gross area at angle leg or stem face = (p − d 0 )/p d 0 = diameter of bolt hole = bolt diameter +1/800 , in. α 0 = [(B/Tu − 1)(a 0 /b0 )]/{δ[1 − (B/Tu − 1)(a 0 /b0 )]}, but not larger than 1 (if α 0 is less than zero, use α 0 = 1) B = design tensile strength of one bolt = φFt Ab , kips (φFt is given in Table 3.11 and Ab is the nominal diameter of the bolt) a 0 = a + d/2 b0 = b − d/2 1999 by CRC Press LLC
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FIGURE 3.15: Prying action in hanger connections. a b
= distance from bolt centerline to edge of tee flange or angle leg but not more than 1.25b, in. = distance from bolt centerline to face of tee stem or outstanding leg, in.
A design is considered satisfactory if the thickness of the tee flange or angle leg tf exceeds treg 0 d and B > Tu . Note that if tf is much larger than treg 0 d , the design will be too conservative. In this case α 0 should be recomputed using the equation # " 1 4Tu b0 0 −1 (3.103) α = δ φb ptf2 Fy As before, the value of α 0 should be limited to the range 0 ≤ α 0 ≤ 1. This new value of α 0 is to be used in Equation 3.102 to recalculate treg 0 d . Bolted Bracket Type Connections
Figure 3.16 shows three commonly used bracket type connections. The bracing connection shown in Figure 3.16a should be designed so that the line of action the force passes through is the centroid of the bolt group. It is apparent that the bolts connecting the bracket to the column flange are subjected to combined tension and shear. As a result, the capacity of the connection is limited 1999 by CRC Press LLC
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FIGURE 3.16: Bolted bracket-type connections.
1999 by CRC Press LLC
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to the combined tensile-shear capacities of the bolts in accordance with Equation 3.99 in ASD and Equation 3.100 in LRFD. For simplicity, fv and ft are to be computed assuming that both the tensile and shear components of the force are distributed evenly to all bolts. In addition to checking for the bolt capacities, the bearing capacities of the column flange and the bracket should also be checked. If the axial component of the force is significant, the effect of prying should also be considered. In the design of the eccentrically loaded connections shown in Figure 3.16b, it is assumed that the neutral axis of the connection lies at the center of gravity of the bolt group. As a result, the bolts above the neutral axis will be subjected to combined tension and shear and so Equation 3.99 or Equation 3.100 needs to be checked. The bolts below the neutral axis are subjected to shear only and so Equation 3.97 or Equation 3.98 applies. In calculating fv , one can assume that all bolts in the bolt group carry an equal share of the shear force. In calculating ft , one can assume that the tensile force varies linearly from a value of zero at the neutral axis to a maximum value at the bolt farthest away from the neutral axis. Using this assumption, ft can be calculated from the equation P ey/I wherePy is the distance from the neutral axis to the location of the bolt above the neutral axis and I = Ab y 2 is the moment of inertia of the bolt areas with Ab equal to the cross-sectional area of each bolt. The capacity of the connection is determined by the capacities of the bolts and the bearing capacity of the connected parts. For the eccentrically loaded bracket connection shown in Figure 3.16c, the bolts are subjected to shear. The shear force in each bolt can be obtained by adding vectorally the shear caused by the applied load P and the moment P χo . The design of this type of connection is facilitated by the use of tables contained in the AISC Manuals for Allowable Stress Design and Load and Resistance Factor Design [21, 22]. In addition to checking for bolt shear capacity, one needs to check the bearing and shear rupture capacities of the bracket plate to ensure that failure will not occur in the plate. Bolted Shear Connections
Shear connections are connections designed to resist shear force only. These connections are not expected to provide appreciable moment restraint to the connection members. Examples of these connections are shown in Figure 3.17. The framed beam connection shown in Figure 3.17a consists of two web angles which are often shop-bolted to the beam web and then field-bolted to the column flange. The seated beam connection shown in Figure 3.17b consists of two flange angles often shop-bolted to the beam flange and field-bolted to the column flange. To enhance the strength and stiffness of the seated beam connection, a stiffened seated beam connection shown in Figure 3.17c is sometimes used to resist large shear force. Shear connections must be designed to sustain appreciable deformation and yielding of the connections is expected. The need for ductility often limits the thickness of the angles that can be used. Most of these connections are designed with angle thickness not exceeding 5/8 in. The design of the connections shown in Figure 3.17 is facilitated by the use of design tables contained in the AISC-ASD and AISC-LRFD Manuals. These tables give design loads for the connections with specific dimensions based on the limit states of bolt shear, bearing strength of the connection, bolt bearing with different edge distances, and block shear (for coped beams). Bolted Moment-Resisting Connections
Moment-resisting connections are connections designed to resist both moment and shear. These connections are often referred to as rigid or fully restrained connections as they provide full continuity between the connected members and are designed to carry the full factored moments. Figure 3.18 shows some examples of moment-resisting connections. Additional examples can be found in the AISC-ASD and AISC-LRFD Manuals and Chapter 4 of the AISC Manual on Connections [20]. 1999 by CRC Press LLC
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FIGURE 3.17: Bolted shear connections. (a) Bolted frame beam connection. (b) Bolted seated beam connection. (c) Bolted stiffened seated beam connection.
1999 by CRC Press LLC
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FIGURE 3.18: Bolted moment connections.
1999 by CRC Press LLC
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Design of Moment-Resisting Connections
An assumption used quite often in the design of moment connections is that the moment is carried solely by the flanges of the beam. The moment is converted to a couple Ff given by Ff = M/(d − tf ) acting on the beam flanges as shown in Figure 3.19.
FIGURE 3.19: Flange forces in moment connections.
The design of the connection for moment is considered satisfactory if the capacities of the bolts and connecting plates or structural elements are adequate to carry the flange force Ff . Depending on the geometry of the bolted connection, this may involve checking: (a) the shear and/or tensile capacities of the bolts, (b) the yield and/or fracture strength of the moment plate, (c) the bearing strength of the connected parts, and (d) bolt spacing and edge distance as discussed in the foregoing sections. As for shear, it is common practice to assume that all the shear resistance is provided by the shear plates or angles. The design of the shear plates or angles is governed by the limit states of bolt shear, bearing of the connected parts, and shear rupture. If the moment to be resisted is large, the flange force may cause bending of the column flange, or local yielding, crippling, or buckling of the column web. To prevent failure due to bending of the column flange or local yielding of the column web (for a tensile Ff ) as well as local yielding, crippling or buckling of the column web (for a compressive Ff ), column stiffeners should be provided if any one of the conditions discussed in the section on Criteria on Concentrated Loads is violated. Following is a set of guidelines for the design of column web stiffeners [21, 22]: 1. If local web yielding controls, the area of the stiffeners (provided in pairs) shall be determined based on any excess force beyond that which can be resisted by the web alone. The stiffeners need not extend more than one-half the depth of the column web if the concentrated beam flange force Ff is applied at only one column flange. 1999 by CRC Press LLC
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2. If web crippling or compression buckling of the web controls, the stiffeners shall be designed as axially loaded compression members (see section on Compression Members). The stiffeners shall extend the entire depth of the column web. 3. The welds that connect the stiffeners to the column shall be designed to develop the full strength of the stiffeners. In addition, the following recommendations are given: 1. The width of the stiffener plus one-half of the column web thickness should not be less than one-half the width of the beam flange nor the moment connection plate which applies the force. 2. The stiffener thickness should not be less than one-half the thickness of the beam flange. 3. If only one flange of the column is connected by a moment connection, the length of the stiffener plate does not have to exceed one-half the column depth. 4. If both flanges of the column are connected by moment connections, the stiffener plate should extend through the depth of the column web and welds should be used to connect the stiffener plate to the column web with sufficient strength to carry the unbalanced moment on opposite sides of the column. 5. If column stiffeners are required on both the tension and compression sides of the beam, the size of the stiffeners on the tension side of the beam should be equal to that on the compression size for ease of construction. In lieu of stiffener plates, a stronger column section could be used to preclude failure in the column flange and web. For a more thorough discussion of bolted connections, the readers are referred to the book by Kulak et al. [16]. Examples on the design of a variety of bolted connections can be found in the AISC-LRFD Manual [22] and the AISC Manual on Connections [20]
3.11.2
Welded Connections
Welded connections are connections whose components are joined together primarily by welds. The four most commonly used welding processes are discussed in the section on Structural Fasteners. Welds can be classified according to: • types of welds: groove, fillet, plug, and slot welds. • positions of the welds: horizontal, vertical, overhead, and flat welds. • types of joints: butt, lap, corner, edge, and tee. Although fillet welds are generally weaker than groove welds, they are used more often because they allow for larger tolerances during erection than groove welds. Plug and slot welds are expensive to make and they do not provide much reliability in transmitting tensile forces perpendicular to the faying surfaces. Furthermore, quality control of such welds is difficult because inspection of the welds is rather arduous. As a result, plug and slot welds are normally used just for stitching different parts of the members together. Welding Symbols
A shorthand notation giving important information on the location, size, length, etc. for the various types of welds was developed by the American Welding Society [6] to facilitate the detailing of welds. This system of notation is reproduced in Figure 3.20. 1999 by CRC Press LLC
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FIGURE 3.20: Basic weld symbols.
1999 by CRC Press LLC
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Strength of Welds
In ASD, the strength of welds is expressed in terms of allowable stress. In LRFD, the design strength of welds is taken as the smaller of the design strength of the base material φFBM and the design strength of the weld electrode φFW . These allowable stresses and design strengths are summarized in Table 3.18 [18, 21]. When a design uses ASD, the computed stress in the weld shall not exceed its allowable value. When a design uses LRFD, the design strength of welds should exceed the required strength obtained by dividing the load to be transmitted by the effective area of the welds. TABLE 3.18
Strength of Welds
Types of weld and stressa
ASD allowable stress
Material
LRFD φFBM or φFW
Required weld strength levelb,c
Full penetration groove weld Tension normal to effective area Compression normal to effective area Tension of compression parallel to axis of weld Shear on effective area
Base
Same as base metal
0.90Fy
Base
Same as base metal
0.90Fy
Base
Same as base metal
0.90Fy
Base weld electrode
0.30× nominal tensile strength of weld metal
0.90[0.60Fy ] 0.80[0.60FEXX ]
“Matching” weld must be used Weld metal with a strength level equal to or less than “matching” must be used
Partial penetration groove welds Compression normal to effective area Tension or compression parallel to axis of weldd Shear parallel to axis of weld
Base
Same as base metal
0.90Fy
Base weld electrode
0.75[0.60FEXX ]
Tension normal to effective area
Base weld electrode
0.30× nominal tensile strength of weld metal 0.30× nominal tensile strength of weld metal ≤ 0.18× yield stress of base metal
Weld metal with a strength level equal to or less than “matching” weld metal may be used
0.90Fy 0.80[0.60FEXX ]
Fillet welds Stress on effective area
Tension or compression parallel to axis of weldd
Base weld electrode
0.30× nominal tensile strength of weld metal
0.75[0.60FEXX ] 0.90Fy
Base
Same as base metal
0.90Fy
Weld metal with a strength level equal to or less than “matching” weld metal may be used
Plug or slot welds Shear parallel to faying surfaces (on effective area)
Base weld electrode
0.30×nominal tensile strength of weld metal
0.75[0.60FEXX ]
Weld metal with a strength level equal to or less than “matching” weld metal may be used
a see below for effective area b see AWS D1.1 for “matching”weld material c weld metal one strength level stronger than “matching” weld metal will be permitted
d fillet welds partial-penetration groove welds joining component elements of built-up members such as flange-to-web con-
nections may be designed without regard to the tensile or compressive stress in these elements parallel to the axis of the welds
1999 by CRC Press LLC
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Effective Area of Welds
The effective area of groove welds is equal to the product of the width of the part joined and the effective throat thickness. The effective throat thickness of a full-penetration groove weld is taken as the thickness of the thinner part joined. The effective throat thickness of a partial-penetration groove weld is taken as the depth of the chamfer for J, U, bevel, or V (with bevel ≥ 60◦ ) joints and it is taken as the depth of the chamfer minus 1/8 in. for bevel or V joints if the bevel is between 45◦ and 60◦ . For flare bevel groove welds the effective throat thickness is taken as 5R/16 and for flare V-groove the effective throat thickness is taken as R/2 (or 3R/8 for GMAW process when R ≥ 1 in.). R is the radius of the bar or bend. The effective area of fillet welds is equal to the product of length of the fillets including returns and the effective throat thickness. The effective throat thickness of a fillet weld is the shortest distance from the root of the joint to the face of the diagrammatic weld as shown in Figure 3.21. Thus, for
FIGURE 3.21: Effective throat of fillet welds. an equal leg fillet weld, the effective throat is given by 0.707 times the leg dimension. For fillet weld made by the submerged arc welding process (SAW), the effective throat thickness is taken as the leg size (for 3/8-in. and smaller fillet welds) or as the theoretical throat plus 0.11-in. (for fillet weld over 3/8-in.). A larger value for the effective throat thickness is permitted for welds made by the SAW process to account for the inherently superior quality of such welds. The effective area of plug and slot welds is taken as the nominal cross-sectional area of the hole or slot in the plane of the faying surface. 1999 by CRC Press LLC
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Size and Length Limitations of Welds
To ensure effectiveness, certain size and length limitations are imposed for welds. For partialpenetration groove welds, minimum values for the effective throat thickness are given in Table 3.19. TABLE 3.19 Minimum Effective Throat Thickness of Partial-Penetration Groove Welds Thickness of the thicker part joined, t (in.)
Minimum effective throat thickness (in.)
t ≤ 1/4 1/4 < t ≤ 1/2 1/2 < t ≤ 3/4 3/4 < t ≤ 1-1/2 1-1/2 < t ≤ 2-1/4 2-1/4 < t ≤ 6 >6
1/8 3/16 1/4 5/16 3/8 1/2 5/8
Note: 1 in. = 25.4 mm.
For fillet welds, the following size and length limitations apply: Minimum Size of Leg—The minimum leg size is given in Table 3.20. TABLE 3.20
Minimum Leg Size of Fillet Welds
Thickness of thicker part joined, t (in.)
Minimum leg size (in.)
≤ 1/4 1/4 < t ≤ 1/2 1/2 < t ≤ 3/4 > 3/4
1/8 3/16 1/4 5/16
Note: 1 in. = 25.4 mm.
Maximum Size of Leg—Along the edge of a connected part less than 1/4 thick, the maximum leg size is equal to the thickness of the connected part. For thicker parts, the maximum leg size is t minus 1/16 in. where t is the thickness of the part. Minimum effective length of weld—The minimum effective length of a fillet weld is four times its nominal size. If a shorter length is used, the leg size of the weld shall be taken as 1/4 its effective length for purpose of stress computation. The length of fillet welds used for flat bar tension members shall not be less than the width of the bar if the welds are provided in the longitudinal direction only. The transverse distance between longitudinal welds should not exceed 8 in. unless the effect of shear lag is accounted for by the use of an effective net area. Maximum effective length of weld—The maximum effective length of a fillet weld loaded by forces parallel to the weld shall not exceed 70 times the size of the fillet weld leg. End returns—End returns must be continued around the corner and must have a length of at least two times the size of the weld leg. Welded Connections for Tension Members
Figure 3.22 shows a tension angle member connected to a gusset plate by fillet welds. The applied tensile force P is assumed to act along the center of gravity of the angle. To avoid eccentricity, the lengths of the two fillet welds must be proportioned so that their resultant will also act along the 1999 by CRC Press LLC
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FIGURE 3.22: An eccentrically loaded welded tension connection.
center of gravity of the angle. For example, if LRFD is used, the following equilibrium equations can be written: Summing force along the axis of the angle (φFM )teff L1 + (φFm )teff L2 = Pu
(3.104)
Summing moment about the center of gravity of the angle (φFM )teff L1 d1 = (φFM )teff L2 d2
(3.105)
where Pu is the factored axial force, φFM is the design strength of the welds as given in Table 3.18, teff is the effective throat thickness, L1 , L2 are the lengths of the welds, and d1 , d2 are the transverse distances from the center of gravity of the angle to the welds. The two equations can be used to solve for L1 and L2 . If end returns are used, the added strength of the end returns should also be included in the calculations. Welded Bracket Type Connections
A typical welded bracket connection is shown in Figure 3.23. Because the load is eccentric with respect to the center of gravity of the weld group, the connection is subjected to both moment and shear. The welds must be designed to resist the combined effect of direct shear for the applied load and any additional shear from the induced moment. The design of the welded bracket connection is facilitated by the use of design tables in the AISC-ASD and AISC-LRFD Manuals. In both ASD and LRFD, the load capacity for the connection is given by P = CC1 Dl where P = l = D = C1 = C =
allowable load (in ASD), or factored load, Pu (in LRFD), kips length of the vertical weld, in. number of sixteenths of an inch in fillet weld size coefficients for electrode used (see table below) coefficients tabulated in the AISC-ASD and AISC-LRFD Manuals. In the tables, values of C for a variety of weld geometries and dimensions are given
1999 by CRC Press LLC
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(3.106)
FIGURE 3.23: An eccentrically loaded welded bracket connection.
Electrode ASD LRFD
Fv (ksi) C1 FEXX (ksi) C1
E60
E70
E80
E90
E100
E110
18 0.857 60 0.857
21 1.0 70 1.0
24 1.14 80 1.03
27 1.29 90 1.16
30 1.43 100 1.21
33 1.57 110 1.34
Welded Connections with Welds Subjected to Combined Shear and Flexure
Figure 3.24 shows a welded framed connection and a welded seated connection. The welds for these connections are subjected to combined shear and flexure. For purpose of design, it is common practice to assume that the shear force per unit length, RS , acting on the welds is a constant and is given by P (3.107) RS = 2l where P is the allowable load (in ASD), or factored load, Pu (in LRFD), and l is the length of the vertical weld. In addition to shear, the welds are subjected to flexure as a result of load eccentricity. There is no general agreement on how the flexure stress should be distributed on the welds. One approach is to assume that the stress distribution is linear with half the weld subjected to tensile flexure stress and half is subjected to compressive flexure stress. Based on this stress distribution and ignoring the returns, the flexure tension force per unit length of weld, RF , acting at the top of the weld can be written as Pe (l/2) 3Pe Mc = 3 = 2 (3.108) RF = I 2l /12 l where e is the load eccentricity. The resultant force per unit length acting on the weld, R, is then q R = RS2 + RF2 1999 by CRC Press LLC
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(3.109)
FIGURE 3.24: Welds subjected to combined shear and flexure.
1999 by CRC Press LLC
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For a satisfactory design, the value R/teff where teff is the effective throat thickness of the weld should not exceed the allowable values or design strengths given in Table 3.18. Welded Shear Connections
Figure 3.25 shows three commonly used welded shear connections: a framed beam connection, a seated beam connection, and a stiffened seated beam connection. These connections can be designed by using the information presented in the earlier sections on welds subjected to eccentric shear and welds subjected to combined tension and flexure. For example, the welds that connect the angles to the beam web in the framed beam connection can be considered as eccentrically loaded welds and so Equation 3.106 can be used for their design. The welds that connect the angles to the column flange can be considered as welds subjected to combined tension and flexure and so Equation 3.109 can be used for their design. Like bolted shear connections, welded shear connections are expected to exhibit appreciable ductility and so the use of angles with thickness in excess of 5/8 in. should be avoided. To prevent shear rupture failure, the shear rupture strength of the critically loaded connected parts should be checked. To facilitate the design of these connections, the AISC-ASD and AISC-LRFD Manuals provide design tables by which the weld capacities and shear rupture strengths for different connection dimensions can be checked readily. Welded Moment-Resisting Connections
Welded moment-resisting connections (Figure 3.26), like bolted moment-resisting connections, must be designed to carry both moment and shear. To simplify the design procedure, it is customary to assume that the moment, to be represented by a couple Ff as shown in Figure 3.19, is to be carried by the beam flanges and that the shear is to be carried by the beam web. The connected parts (e.g., the moment plates, welds, etc.) are then designed to resist the forces Ff and shear. Depending on the geometry of the welded connection, this may include checking: (a) the yield and/or fracture strength of the moment plate, (b) the shear and/or tensile capacity of the welds, and (c) the shear rupture strength of the shear plate. If the column to which the connection is attached is weak, the designer should consider the use of column stiffeners to prevent failure of the column flange and web due to bending, yielding, crippling, or buckling (see section on Design of Moment-Resisting Connections). Examples on the design of a variety of welded shear and moment-resisting connections can be found in the AISC Manual on Connections [20] and the AISC-LRFD Manual [22].
3.11.3
Shop Welded-Field Bolted Connections
A large percentage of connections used for construction are shop welded and field bolted types. These connections are usually more cost effective than fully welded connections and their strength and ductility characteristics often rival those of fully welded connections. Figure 3.27 shows some of these connections. The design of shop welded–field bolted connections is also covered in the AISC Manual on Connections and the AISC-LRFD Manual. In general, the following should be checked: (a) Shear/tensile capacities of the bolts and/or welds, (b) bearing strength of the connected parts, (c) yield and/or fracture strength of the moment plate, and (d) shear rupture strength of the shear plate. Also, as for any other types of moment connections, column stiffeners shall be provided if any one of the following criteria is violated: column flange bending, local web yielding, crippling, and compression buckling of the column web. 1999 by CRC Press LLC
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FIGURE 3.25: Welded shear connections. (a) Framed beam connection, (b) seated beam connection, (c) stiffened beam connection.
1999 by CRC Press LLC
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FIGURE 3.26: Welded moment connections.
3.11.4
Beam and Column Splices
Beam and column splices (Figure 3.28) are used to connect beam or column sections of different sizes. They are also used to connect beams or columns of the same size if the design calls for an extraordinarily long span. Splices should be designed for both moment and shear unless it is the intention of the designer to utilize the splices as internal hinges. If splices are used for internal hinges, provisions must be made to ensure that the connections possess adequate ductility to allow for large hinge rotation. Splice plates are designed according to their intended functions. Moment splices should be designed to resist the flange force Ff = M/(d − tf ) (Figure 3.19) at the splice location. In particular, the following limit states need to be checked: yielding of gross area of the plate, fracture of net area of the plate (for bolted splices), bearing strengths of connected parts (for bolted splices), shear capacity of bolts (for bolted splices), and weld capacity (for welded splices). Shear splices should be designed to resist the shear forces acting at the locations of the splices. The limit states that need to be checked include: shear rupture of the splice plates, shear capacity of bolts under an eccentric load (for bolted splices), bearing capacity of the connected parts (for bolted splices), shear capacity of bolts (for bolted splices), and weld capacity under an eccentric load (for welded splices). Design examples of beam and column splices can be found in the AISC Manual of Connections [20] and the AISC-LRFD Manuals [22].
1999 by CRC Press LLC
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FIGURE 3.27: Shop-welded field-bolted connections.
1999 by CRC Press LLC
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FIGURE 3.28: Bolted and welded beam and column splices.
3.12
Column Base Plates and Beam Bearing Plates (LRFD Approach)
3.12.1
Column Base Plates
Column base plates are steel plates placed at the bottom of columns whose function is to transmit column loads to the concrete pedestal. The design of column base plates involves two major steps: (1) determining the size N × B of the plate, and (2) determining the thickness tp of the plate. Generally, the size of the plate is determined based on the limit state of bearing on concrete and the thickness of the plate is determined based on the limit state of plastic bending of critical sections in the plate. Depending on the types of forces (axial force, bending moment, shear force) the plate will be subjected to, the design procedures differ slightly. In all cases, a layer of grout should be placed between the base plate and its support for the purpose of leveling and anchor bolts should be provided to stabilize the column during erection or to prevent uplift for cases involving large bending moment. 1999 by CRC Press LLC
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Axially Loaded Base Plates
Base plates supporting concentrically loaded columns in frames in which the column bases are assumed pinned are designed with the assumption that the column factored load Pu is distributed uniformly to the area of concrete under the base plate. The size of the base plate is determined from the limit state of bearing on concrete. The design bearing strength of concrete is given by the equation s # " A2 0 (3.110) φc Pp = 0.60 0.85fc A1 A1 where fc0 = compressive strength of concrete A1 = area of base plate A2 = area of concrete pedestal that is geometrically similar to and concentric with the loaded area, A1 ≤ A2 ≤ 4A1 From Equation 3.110, it can be seen that the bearing capacity increases when the concrete area is greater than the plate area. This accounts for the beneficial effect of confinement. The upper limit of the bearing strength is obtained when A2 = 4A1 . Presumably, the concrete area in excess of 4A1 is not effective in resisting the load transferred through the base plate. Setting the column factored load, Pu , equal to the bearing capacity of the concrete pedestal, φc Pp , and solving for A1 from Equation 3.110, we have A1 =
1 A2
Pu 0.6(0.85fc0 )
2 (3.111)
The length, N, and width, B, of the plate should be established so that N × B > A1 . For an efficient design, the length can be determined from the equation p (3.112) N ≈ A1 + 0.50(0.95d − 0.80bf ) where 0.95d and 0.80bf define the so-called effective load bearing area shown cross-hatched in Figure 3.29a. Once N is obtained, B can be solved from the equation B=
A1 N
(3.113)
Both N and B should be rounded up to the nearest full inches. The required plate thickness, treg 0 d , is to be determined from the limit state of yield line formation along the most severely stressed sections. A yield line develops when the cross-section moment capacity is equal to its plastic moment capacity. Depending on the size of the column relative to the plate and the magnitude of the factored axial load, yield lines can form in various patterns on the plate. Figure 3.29 shows three models of plate failure in axially loaded plates. If the plate is large compared to the column, yield lines are assumed to form around the perimeter of the effective load bearing area (the cross-hatched area) as shown in Figure 3.29a. If the plate is small and the column factored load is light, yield lines are assumed to form around the inner perimeter of the I-shaped area as shown in Figure 3.29b. If the plate is small and the column factored load is heavy, yield lines are assumed to form around the inner edge of the column flanges and both sides of the column web as shown in Figure 3.29c. The following equation can be used to calculate the required plate thickness s 2Pu (3.114) treq 0 d = l 0.90Fy BN 1999 by CRC Press LLC
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FIGURE 3.29: Failure models for centrally loaded column base plates.
where l is the larger of m, n, and λn0 given by
m
=
n = n0 1999 by CRC Press LLC
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=
(N − 0.95d) 2 (B − 0.80bf ) 2 p dbf 4
and
√ 2 X ≤1 λ= √ 1+ 1−X
in which
X=
4dbf (d + bf )2
Pu φc Pp
Base Plates for Tubular and Pipe Columns
The design concept for base plates discussed above for I-shaped sections can be applied to the design of base plates for rectangular tubes and circular pipes. The critical section used to determine the plate thickness should be based on 0.95 times the outside column dimension for rectangular tubes and 0.80 times the outside dimension for circular pipes [11]. Base Plates with Moments
For columns in frames designed to carry moments at the base, base plates must be designed to support both axial forces and bending moments. If the moment is small compared to the axial force, the base plate can be designed without consideration of the tensile force which may develop in the anchor bolts. However, if the moment is large, this effect should be considered. To quantify the relative magnitude of this moment, an eccentricity e = Mu /Pu is used. The general procedures for the design of base plates for different values of e will be given in the following [11]. Small eccentricity, e ≤ N/6 If e is small, the bearing stress is assumed to distribute linearly over the entire area of the base plate (Figure 3.30). The maximum bearing stress is given by fmax =
Pu Mu c + BN I
(3.115)
where c = N/2 and I = BN 3 /12.
FIGURE 3.30: Eccentrically loaded column base plate (small load eccentricity).
The size of the plate is to be determined by a trial and error process. The size of the base plate should be such that the bearing stress calculated using Equation 3.115 does not exceed φc Pp /A1 , 1999 by CRC Press LLC
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given by
s
" 0.60
0.85fc0
A2 A1
# ≤ 0.60[1.7fc0 ]
The thickness of the plate is to be determined from s 4Mplu tp = 0.90Fy
(3.116)
(3.117)
where Mplu is the moment per unit width of critical section in the plate. Mplu is to be determined by assuming that the portion of the plate projecting beyond the critical section acts as an inverted cantilever loaded by the bearing pressure. The moment calculated at the critical section divided by the length of the critical section (i.e., B) gives Mplu . Moderate eccentricity, N/6 < e ≤ N/2 For plates subjected to moderate moments, only portions of the plate will be subjected to bearing stress (Figure 3.31). Ignoring the tensile force in the anchor bolt in the region of the plate where no
FIGURE 3.31: Eccentrically loaded column base plate (moderate load eccentricity). bearing occurs and denoting A as the length of the plate in bearing, the maximum bearing stress can be calculated from force equilibrium consideration as fmax =
2Pu AB
(3.118)
where A = 3(N/2 − e) is determined from moment equilibrium. The plate should be portioned such that fmax does not exceed the value calculated using Equation 3.116. tp is to be determined from Equation 3.117. Large eccentricity, e > N/2 For plates subjected to large bending moments so that e > N/2, one needs to take into consideration the tensile force developing in the anchor bolts (Figure 3.32). Denoting T as the resultant force in the anchor bolts, force equilibrium requires that T + Pu = 1999 by CRC Press LLC
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fmax AB 2
(3.119)
FIGURE 3.32: Eccentrically loaded column base plate (large load eccentricity). and moment equilibrium requires that fmax AB N A +M = N0 − Pu N 0 − 2 2 3
(3.120)
The above equations can be used to solve for A and T . The size of the plate is to be determined using a trial-and-error process. The size should be chosen such that fmax does not exceed the value calculated using Equation 3.116, A should be smaller than N 0 and T should not exceed the tensile capacity of the bolts. Once the size of the plate is determined, the plate thickness tp is to be calculated using Equation 3.117. Note that there are two critical sections on the plate, one on the compression side of the plate and the other on the tension side of the plate. Two values of Mplu are to be calculated and the larger value should be used to calculate tp . Base Plates with Shear
Under normal circumstances, the factored column base shear is adequately resisted by the frictional force developed between the plate and its support. Additional shear capacity is also provided by the anchor bolts. For cases in which exceptionally high shear force is expected, such as in a bracing connection or in which uplift occurs which reduces the frictional resistance, the use of shear lugs may be necessary. Shear lugs can be designed based on the limit states of bearing on concrete and bending of the lugs. The size of the lug should be proportioned such that the bearing stress on concrete does not exceed 0.60(0.85fc0 ). The thickness of the lug can be determined from Equation 3.117. Mplu is the moment per unit width at the critical section of the lug. The critical section is taken to be at the junction of the lug and the plate (Figure 3.33).
3.12.2
Anchor Bolts
Anchor bolts are provided to stabilize the column during erection and to prevent uplift for cases involving large moments. Anchor bolts can be cast-in-place bolts or drilled-in bolts. The latter are placed after the concrete is set and are not too often used. Their design is governed by the manufacturer’s specifications. Cast-in-place bolts are hooked bars, bolts, or threaded rods with nuts (Figure 3.34) placed before the concrete is set. Of the three types of cast-in-place anchors shown in the figure, the hooked bars are recommended for use only in axially loaded base plates. They are not normally relied upon to carry significant tensile force. Bolts and threaded rods with nuts can be used 1999 by CRC Press LLC
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FIGURE 3.33: Column base plate subjected to shear.
FIGURE 3.34: Base plate anchors.
for both axially loaded base plates or base plates with moments. Threaded rods with nuts are used when the length and size required for the specific design exceed those of standard size bolts. Failure of bolts or threaded rods with nuts occur when their tensile capacities are reached. Failure is also considered to occur when a cone of concrete is pulled out from the pedestal. This cone pull-out type of failure is depicted schematically in Figure 3.35. The failure cone is assumed to radiate out from the bolt head or nut at p an angle of 45◦ with tensile failure occurring along the surface of the cone at an average stress of 4 fc0 where fc0 is the compressive strength of concrete in psi. The load that will cause this cone pull-out failure is given by the product of this average stress and the projected area the cone Ap [23, 24]. The design of anchor bolts is thus governed by the limit states of tensile fracture of the anchors and cone pull-out. 1999 by CRC Press LLC
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FIGURE 3.35: Cone pullout failure.
Limit State of Tensile Fracture The area of the anchor should be such that Ag ≥
Tu φt 0.75Fu
(3.121)
where Ag is the required gross area of the anchor, Fu is the minimum specified tensile strength, and φt is the resistance factor for tensile fracture which is equal to 0.75. Limit State of Cone Pull-Out From Figure 3.35, it is clear that the size of the cone is a function of the length of the anchor. Provided that there is sufficient edge distance and spacing between adjacent anchors, the amount of tensile force required to cause cone pull-out failure increases with the embedded length of the anchor. This concept can be used to determine the required embedded length of the anchor. Assuming that the failure cone does not intersect with another failure cone nor the edge of the pedestal, the required embedded length can be calculated from the equation r L≥
s Ap = π
p (Tu /φt 4 fc0 ) π
(3.122)
where Ap is the projected area of the failure cone, Tu is the required bolt force in pounds, fc0 is the compressive strength of concrete in psi and φt is the resistance factor assumed to be equal to 0.75. If failure cones from adjacent anchors overlap one another or intersect with the pedestal edge, the projected area Ap must be adjusted according (see, for example [23, 24]). The length calculated using the above equation should not be less than the recommended values given by [29]. These values are reproduced in the following table. Also shown in the table are the recommended minimum edge distances for the anchors. 1999 by CRC Press LLC
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Bolt type (material)
Minimum embedded length
Minimum edge distance
A307 (A36)
12d
5d > 4 in.
A325 (A449)
17d
7d > 4 in.
d = nominal diameter of the anchor
3.12.3
Beam Bearing Plates
Beam bearing plates are provided between main girders and concrete pedestals to distribute the girder reactions to the concrete supports (Figure 3.36). Beam bearing plates may also be provided between cross beams and girders if the cross beams are designed to sit on the girders.
FIGURE 3.36: Beam bearing plate.
Beam bearing plates are designed based on the limit states of web yielding, web crippling, bearing on concrete, and plastic bending of the plate. The dimension of the plate along the beam axis, i.e., N, is determined from the web yielding or web crippling criterion (see section on Concentrated Load Criteria), whichever is more critical. The dimension B of the plate is determined from Equation 3.113 with A1 calculated using Equation 3.111. Pu in Equation 3.111 is to be replaced by Ru , the factored reaction at the girder support. 1999 by CRC Press LLC
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Once the size B × N is determined, the plate thickness tp can be calculated using the equation s tp =
2Ru n2 0.90Fy BN
(3.123)
where Ru is the factored girder reaction, Fy is the yield stress of the plate and n = (B − 2k)/2 in which k is the distance from the web toe of the fillet to the outer surface of the flange. The above equation was developed based on the assumption that the critical sections for plastic bending in the plate occur at a distance k from the centerline of the web.
3.13
Composite Members (LRFD Approach)
Composite members are structural members made from two or more materials. The majority of composite sections used for building constructions are made from steel and concrete. Steel provides strength and concrete provides rigidity. The combination of the two materials often results in efficient load-carrying members. Composite members may be concrete-encased or concrete-filled. For concrete-encased members (Figure 3.37a), concrete is casted around steel shapes. In addition to enhancing strength and providing rigidity to the steel shapes, the concrete acts as a fire-proofing material to the steel shapes. It also serves as a corrosion barrier shielding the steel from corroding under adverse environmental conditions. For concrete-filled members (Figure 3.37b), structural steel tubes are filled with concrete. In both concrete-encased and concrete-filled sections, the rigidity of the concrete often eliminates the problem of local buckling experienced by some slender elements of the steel sections. Some disadvantages associated with composite sections are that concrete creeps and shrinks. Furthermore, uncertainties with regard to the mechanical bond developed between the steel shape and the concrete often complicate the design of beam-column joints.
3.13.1
Composite Columns
According to the LRFD Specification [18], a compression member is regarded as a composite column if (1) the cross-sectional area of the steel shape is at least 4% of the total composite area. If this condition is not satisfied, the member should be designed as a reinforced concrete column. (2) Longitudinal reinforcements and lateral ties are provided for concrete-encased members. The cross-sectional area of the reinforcing bars shall be 0.007 in.2 per inch of bar spacing. To avoid spalling, lateral ties shall be placed at a spacing not greater than 2/3 the least dimension of the composite cross-section. For fire and corrosion resistance, a minimum clear cover of 1.5 in. shall be provided. (3) The compressive strength of concrete fc0 used for the composite section falls within the range 3 to 8 ksi for normal weight concrete and not less than 4 ksi for light weight concrete. These limits are set because they represent the range of test data available for the development of the design equations. (4) The specified minimum yield stress for the steel shapes and reinforcing bars used in calculating the strength of the composite column does not exceed 55 ksi. This limit is set because this stress corresponds to a strain below which the concrete remains unspalled andp stable. (5) The minimum wall thickness of the steel shapes for concrete filled members is equal to b (Fy /3E) for rectangular sections of width b and p D (Fy /8E) for circular sections of outside diameter D. Design Compressive Strength
The design compressive strength, φc Pn , shall exceed the factored compressive force, Pu . The design compressive strength is given as follows: 1999 by CRC Press LLC
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FIGURE 3.37: Composite columns. For λc ≤ 1.5
h i 0.85 0.658λ2c As Fmy , i h φc Pn = F A , 0.85 0.877 s my 2 λ c
where λc =
KL rm π
Fmy = Fy + c1 Fyr
q Ar As
Fmy Em
Em = E + c3 Ec Ac Ar As E Ec Fy Fyr
= = = = = = =
if λc > 1.5
Ac As
Ac As
area of concrete, in.2 area of longitudinal reinforcing bars, in.2 area of steel shape, in.2 modulus of elasticity of steel, ksi modulus of elasticity of concrete, ksi specified minimum yield stress of steel shape, ksi specified minimum yield stress of longitudinal reinforcing bars, ksi
1999 by CRC Press LLC
c
+ c2 fc0
if λc ≤ 1.5
(3.124)
(3.125) (3.126) (3.127)
fc0 = specified compressive strength of concrete, ksi c1 , c2 , c3 = coefficients given in table below Type of composite section Concrete encased shapes Concrete-filled pipes and tubings
c1
c2
c3
0.7
0.6
0.2
1.0
0.85
0.4
In addition to satisfying the condition φc Pn ≥ Pu , the bearing condition for concrete must also be satisfied. Denoting φc Pnc (= φc Pn,composite section −φc Pn,steel shape alone ) as the portion of compressive strength resisted by the concrete and AB as the loaded area (the condition), then if the supporting concrete area is larger than the loaded area, the bearing condition that needs to be satisfied is φc Pnc ≤ 0.60[1.7fc0 AB ]
3.13.2
(3.128)
Composite Beams
For steel beams fully encased in concrete, no additional anchorage for shear transfer is required if (1) at least 1.5 in. concrete cover is provided on top of the beam and at least 2 in. cover is provided over the sides and at the bottom of the beam, and (2) spalling of concrete is prevented by adequate mesh or other reinforcing steel. The design flexural strength φb Mn can be computed using either an elastic or plastic analysis. If an elastic analysis is used, φb shall be taken as 0.90. A linear strain distribution is assumed for the cross-section with zero strain at the neutral axis and maximum strains at the extreme fibers. The stresses are then computed by multiplying the strains by E (for steel) or Ec (for concrete). Maximum stress in steel shall be limited to Fy , and maximum stress in concrete shall be limited to 0.85fc0 . Tensile strength of concrete shall be neglected. Mn is to be calculated by integrating the resulting stress block about the neutral axis. If a plastic analysis is used, φc shall be taken as 0.90, and Mn shall be assumed to be equal to Mp , the plastic moment capacity of the steel section alone.
3.13.3 Composite Beam-Columns Composite beam-columns shall be designed to satisfy the interaction equation of Equation 3.68 or Equation 3.69, whichever is applicable, with φc Pn calculated based on Equations 3.124 to 3.127, Pe calculated using the equation Pe = As Fmy /λ2c , and φb Mn calculated using the following equation [14]: Aw Fy 1 h2 − (3.129) Aw Fy φb Mn = 0.90 ZFy + (h2 − 2cr )Ar Fyr + 3 2 1.7fc0 h1 where Z = plastic section modulus of the steel section, in.3 cr = average of the distance measured from the compression face to the longitudinal reinforcement in that face and the distance measured from the tension face to the longitudinal reinforcement in that face, in. h1 = width of the composite section perpendicular to the plane of bending, in. h2 = width of the composite section parallel to the plane of bending, in. Ar = cross-sectional area of longitudinal reinforcing bars, in.2 Aw = web area of the encased steel shape (= 0 for concrete-filled tubes) 1999 by CRC Press LLC
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If 0 < (Pu /φc Pn ) ≤ 0.3, a linear interpolation of φb Mn calculated using the above equation assuming Pu /φc Pn = 0.3 and that for beams with Pu /φc Pn = 0 (see section on Composite Beams) should be used.
3.13.4
Composite Floor Slabs
Composite floor slabs (Figure 3.38) can be designed as shored or unshored. In shored construction,
FIGURE 3.38: Composite floor slabs. 1999 by CRC Press LLC
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temporary shores are used during construction to support the dead and accidental live loads until the concrete cures. The supporting beams are designed on the basis of their ability to develop composite action to support all factored loads after the concrete cures. In unshored construction, temporary shores are not used. As a result, the steel beams alone must be designed to support the dead and accidental live loads before the concrete has attained 75% of its specified strength. After the concrete is cured, the composite section should have adequate strength to support all factored loads. Composite action for the composite floor slabs shown in Figure 3.38 is developed as a result of the presence of shear connectors. If sufficient shear connectors are provided so that the maximum flexural strength of the composite section can be developed, the section is referred to as fully composite. Otherwise, the section is referred to as partially composite. The flexural strength of a partially composite section is governed by the shear strength of the shear connectors. The horizontal shear force Vh , which should be designed for at the interface of the steel beam and the concrete slab, is given by: In regions of positive moment Vh = min(0.85fc0 Ac , As Fy , In regions of negative moment Vh = min(Ar Fyr , where fc0 Ac tc beff
X
X
Qn )
Qn )
(3.130)
(3.131)
= = = = = =
compressive strength of concrete, ksi effective area of the concrete slab = tc beff , in.2 thickness of the concrete slab, in. effective width of the concrete slab, in. min(L/4, s), for an interior beam min(L/8+ distance from beam centerline to edge of slab, s/2+ distance from beam centerline to edge of slab), for an exterior beam L = beam span measured from center-to-center of supports, in. s = spacing between centerline of adjacent beams, in. = cross-sectional area of the steel beam, in.2 As Fy = yield stress of the steel beam, ksi = area of reinforcing steel within the effective area of the concrete slab, in.2 Ar Fyr = yield stress of the reinforcing steel, ksi 6Qn = sum of nominal shear strengths of the shear connectors, kips The nominal shear strength of a shear connector (used without a formed steel deck) is given by: For a stud shear connector
p Qn = 0.5Asc fc0 Ec ≤ Asc Fu
(3.132)
p Qn = 0.3(tf + 0.5tw )Lc fc0 Ec
(3.133)
For a channel shear connector
where Asc = cross-sectional area of the shear stud, in.2 fc0 = compressive strength of concrete, ksi Ec = modulus of elasticity of concrete, ksi 1999 by CRC Press LLC
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Fu = minimum specified tensile strength of the shear stud, ksi tf = flange thickness of the channel, in. tw = web thickness of the channel, in. Lc = length of the channel, in. If a formed steel deck is used, Qn must be reduced by a reduction factor. The reduction factor depends on whether the deck ribs are perpendicular or parallel to the steel beam. Expressions for the reduction factor are given in the AISC-LRFD Specification [18]. For full composite action, the number of connectors required between the maximum moment point and the zero moment point of the beam is given by N=
Vh Qn
(3.134)
For partial composite action, the number of connectors required is governed by the condition φb Mn ≥ Mu , where φb Mn is governed by the shear strength of the connectors. The placement and spacing of the shear connectors should comply with the following guidelines: 1. The shear connectors shall be uniformly spaced with the region of maximum moment and zero moment. However, the number of shear connectors placed between a concentrated load point and the nearest zero moment point must be sufficient to resist the factored moment Mu . 2. Except for connectors installed in the ribs of formed steel decks, shear connectors shall have at least 1 in. of lateral concrete cover. 3. Unless located over the web, diameter of shear studs must not exceed 2.5 times the thickness of the beam flange. 4. The longitudinal spacing of the studs should fall in the range 6 times the stud diameter to 8 times the slab thickness if a solid slab is used or 4 times the stud diameter to 8 times the slab thickness if a formed steel deck is used. The design flexural strength φb Mn of the composite beam with shear connectors is determined as follows: In regions of positive moments p For hc /tw ≤ 640/ Fyf , φb = 0.85, Mn = moment capacity determined using a plastic stress distribution assuming concrete crushes at a stress of 0.85fc0 and steel yields at a stress of Fy . If a portion of the concrete slab is in tension, the strength contribution of that portion of concrete is ignored. The determination of Mn using this method is very similar to the technique used for computing the moment capacity of a reinforced concrete beam according to the ultimate strength method. p For hc /tw > 640/ Fyf , φb = 0.90, Mn = moment capacity determined using superposition of elastic stress, considering the effect of shoring. The determination of Mn using this method is quite similar to the technique used for computing the moment capacity of a reinforced concrete beam according to the working stress method. In regions of negative moments φb Mn is to be determined for the steel section alone in accordance with the requirements discussed in the section on Flexural Members. To facilitate design, numerical values of φb Mn for composite beams with shear studs in solid slabs are given in tabulated form by the AISC-LRFD Manual. Values of φb Mn for composite beams with formed steel decks are given in a publication by the Steel Deck Institute [19]. 1999 by CRC Press LLC
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3.14
Plastic Design
Plastic analysis and design is permitted only for steels with yield stress not exceeding 65 ksi. The reason for this is that steels with high yield stress lack the ductility required for inelastic rotation at hinge locations. Without adequate inelastic rotation, moment redistribution (which is an important characteristic for plastic design) cannot take place. In plastic design, the predominant limit state is the formation of plastic hinges. Failure occurs when sufficient plastic hinges have formed for a collapse mechanism to develop. To ensure that plastic hinges can form and can undergo large inelastic rotation, the following conditions must be satisfied: 1. Sections must be compact. That is, the width-thickness ratios of flanges in compression and webs must not exceed λp in Table 3.8. 2. For columns, the slenderness parameter λc (see section on Compression Members) shall not exceed 1.5K where K is the effective length factor, and Pu from gravity and horizontal loads shall not exceed 0.75Ag Fy . 3. For beams, the lateral unbraced length Lb shall not exceed Lpd where For doubly and singly symmetric I-shaped members loaded in the plane of the web Lpd =
3,600 + 2,200(M1 /M2 ) ry Fy
(3.135)
and for solid rectangular bars and symmetric box beams Lpd =
3,000ry 5,000 + 3,000(M1 /M2 ) ry ≥ Fy Fy
(3.136)
In the above equations, M1 is the smaller end moment within the unbraced length of the beam. M2 = Mp is the plastic moment (= Zx Fy ) of the cross-section. ry is the radius of gyration about the minor axis, in inches, and Fy is the specified minimum yield stress, in ksi. Lpd is not defined for beams bent about their minor axes nor for beams with circular and square cross-sections because these beams do not experience lateral torsional bucking when loaded.
3.14.1
Plastic Design of Columns and Beams
Provided that the above limitations are satisfied, the design of columns shall meet the condition 1.7Fa A ≥ Pu where Fa is the allowable compressive stress given in Equation 3.16, A is the gross cross-sectional area, and Pu is the factored axial load. The design of beams shall satisfy the conditions Mp ≥ Mu and 0.55Fy tw d ≥ Vu where Mu and Vu are the factored moment and shear, respectively. Mp is the plastic moment capacity Fy is the minimum specified yield stress, tw is the beam web thickness, and d is the beam depth. For beams subjected to concentrated loads, all failure modes associated with concentrated loads (see section on Concentrated Load Criteria) should also be prevented. Except at the location where the last hinge forms, a beam bending about its major axis must be braced to resist lateral and torsional displacements at plastic hinge locations. The distance between adjacent braced points should not exceed lcr given by 1375 + 25 ry , if − 0.5 < MMp < 1.0 F y (3.137) lcr = 1375 ry , if − 1.0 < MMp ≤ −0.5 Fy
1999 by CRC Press LLC
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where ry M Mp M/Mp
3.14.2
= = = =
radius of gyration about the weak axis smaller of the two end moments of the unbraced segment plastic moment capacity is taken as positive if the unbraced segment bends in reverse curvature, and it is taken as negative if the unbraced segment bends in single curvature
Plastic Design of Beam-Columns
Beam-columns designed on the basis of plastic analysis shall satisfy the following interaction equations for stability (Equation 3.138) and for strength (Equation 3.139). Pu Pcr
+
Pu Py
Cm Mu 1− PPue Mm
+
Mu 1.18Mp
≤ 1.0
≤ 1.0
(3.138) (3.139)
where Pu = Pcr = Py = Pe = Cm = Mu = Mp = Mm = = = l = ry = Mpx = Fy =
factored axial load 1.7Fa A, Fa is defined in Equation 3.16 and A is the cross-sectional area yield load = AFy Euler buckling load = π 2 EI /(Kl)2 coefficient defined in the section on Compression Members factored moment plastic moment = ZFy maximum moment that can be resisted by the member in the absence of axial load Mpx if the member p is braced in the weak direction {1.07 − [(l/ry ) Fy ]/3160}Mpx ≤ Mpx if the member is unbraced in the weak direction unbraced length of the member radius of gyration about the minor axis plastic moment about the major axis = Zx Fy minimum specified yield stress
3.15
Defining Terms
ASD: Acronym for Allowable Stress Design. Beamxcolumns: Structural members whose primary function is to carry loads both along and transverse to their longitudinal axes. Biaxial bending: Simultaneous bending of a member about two orthogonal axes of the crosssection. Builtxup members: Structural members made of structural elements jointed together by bolts, welds, or rivets. Composite members: Structural members made of both steel and concrete. Compression members: Structural members whose primary function is to carry loads along their longitudinal axes Design strength: Resistance provided by the structural member obtained by multiplying the nominal strength of the member by a resistance factor. Drift: Lateral deflection of a building. Factored load: The product of the nominal load and a load factor. 1999 by CRC Press LLC
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Flexural members: Structural members whose primary function is to carry loads transverse to their longitudinal axes. Limit state: A condition in which a structural or structural component becomes unsafe (strength limit state) or unfit for its intended function (serviceability limit state). Load factor: A factor to account for the unavoidable deviations of the actual load from its nominal value and uncertainties in structural analysis in transforming the applied load into a load effect (axial force, shear, moment, etc.) LRFD: Acronym for Load and Resistance Factor Design. PD: Acronym for Plastic Design. Plastic hinge: A yielded zone of a structural member in which the internal moment is equal to the plastic moment of the cross-section. Resistance factor: A factor to account for the unavoidable deviations of the actual resistance of a member from its nominal value. Service load: Nominal load expected to be supported by the structure or structural component under normal usage. Sidesway inhibited frames: Frames in which lateral deflections are prevented by a system of bracing. Sidesway uninhibited frames: Frames in which lateral deflections are not prevented by a system of bracing. Shear lag: The phenomenon in which the stiffer (or more rigid) regions of a structure or structural component attract more stresses than the more flexible regions of the structure or structural component. Shear lag causes stresses to be unevenly distributed over the cross-section of the structure or structural component. Tension field action: Post-buckling shear strength developed in the web of a plate girder. Tension field action can develop only if sufficient transverse stiffeners are provided to allow the girder to carry the applied load using truss-type action after the web has buckled.
References [1] AASHTO. 1992. Standard Specification for Highway Bridges. 15th ed., American Association of State Highway and Transportation Officials, Washington D.C. [2] ASTM. 1988. Specification for Carbon Steel Bolts and Studs, 60000 psi Tensile Strength (A30788a). American Society for Testing and Materials, Philadelphia, PA. [3] ASTM. 1986. Specification for High Strength Bolts for Structural Steel Joints (A325-86). American Society for Testing and Materials, Philadelphia, PA. [4] ASTM. 1985. Specification for Heat-Treated Steel Structural Bolts, 150 ksi Minimum Tensile Strength (A490-85). American Society for Testing and Materials, Philadelphia, PA. [5] ASTM. 1986. Specification for Quenched and Tempered Steel Bolts and Studs (A449-86). American Society for Testing and Materials, Philadelphia, PA. [6] AWS. 1987. Welding Handbook. 8th ed., 1, Welding Technology, American Welding Society, Miami, FL. [7] AWS. 1996. Structural Welding Code-Steel. American Welding Society, Miami, FL. [8] Blodgett, O.W. Distortion... How to Minimize it with Sound Design Practices and Controlled Welding Procedures Plus Proven Methods for Straightening Distorted Members. Bulletin G261, The Lincoln Electric Company, Cleveland, OH. [9] Chen, W.F. and Lui, E.M. 1991. Stability Design of Steel Frames, CRC Press, Boca Raton, FL. 1999 by CRC Press LLC
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[10] CSA. 1994. Limit States Design of Steel Structures. CSA Standard CAN/CSA S16.1-94, Canadian Standards Association, Rexdale, Ontantio. [11] Dewolf, J.T. and Ricker, D.T. 1990. Column Base Plates. Steel Design Guide Series 1, American Institute of Steel Construction, Chicago, IL. [12] Disque, R.O. 1973. Inelastic K-factor in column design. AISC Eng. J., 10(2):33-35. [13] Galambos, T.V., Ed. 1988. Guide to Stability Design Criteria for Metal Structures. 4th ed., John Wiley & Sons, New York. [14] Galambos, T.V. and Chapuis, J. 1980. LRFD Criteria for Composite Columns and Beam Columns. Washington University, Department of Civil Engineering, St. Louis, MO. [15] Gaylord, E.H., Gaylord, C.N., and Stallmeyer, J.E. 1992. Design of Steel Structures, 3rd ed., McGraw-Hill, New York. [16] Kulak, G.L., Fisher, J.W., and Struik, J.H.A. 1987. Guide to Design Criteria for Bolted and Riveted Joints, 2nd ed., John Wiley & Sons, New York. [17] Lee, G.C., Morrel, M.L., and Ketter, R.L. 1972. Design of Tapered Members. WRC Bulletin No.
173. [18] Load and Resistance Factor Design Specification for Structural Steel Buildings. 1993. American Institute of Steel Construction, Chicago, IL. [19] LRFD Design Manual for Composite Beams and Girders with Steel Deck. 1989. Steel Deck Institute, Canton, OH. [20] Manual of Steel Construction-Volume II Connections. 1992. ASD 1st ed./LRFD 1st ed., American Institute of Steel Construction, Chicago, IL. [21] Manual of Steel Construction-Allowable Stress Design. 1989. 9th ed., American Institute of Steel Construction, Chicago, IL. [22] Manual of Steel Construction-Load and Resistance Factor Design. 1994. Vol. I and II, 2nd ed., American Institute of Steel Construction, Chicago, IL. [23] Marsh, M.L. and Burdette, E.G. 1985. Multiple bolt anchorages: Method for determining the effective projected area of overlapping stress cones. AISC Eng. J., 22(1):29-32. [24] Marsh, M.L. and Burdette, E.G. 1985. Anchorage of steel building components to concrete. AISC Eng. J., 22(1):33-39. [25] Munse, W.H. and Chesson E., Jr. 1963. Riveted and Bolted Joints: Net Section Design. ASCE J. Struct. Div., 89(1):107-126. [26] Rains, W.A. 1976. A new era in fire protective coatings for steel. Civil Eng., ASCE, September:8083. [27] RCSC. 1985. Allowable Stress Design Specification for Structural Joints Using ASTM A325 or A490 Bolts. American Institute of Steel Construction, Chicago, IL. [28] RCSC. 1988. Load and Resistance Factor Design Specification for Structural Joints Using ASTM A325 or A490 Bolts. American Institute of Steel Construction, Chicago, IL. [29] Shipp, J.G. and Haninge, E.R. 1983. Design of headed anchor bolts. AISC Eng. J., 20(2):58-69. [30] SSRC. 1993. Is Your Structure Suitably Braced? Structural Stability Research Council, Bethlehem, PA.
Further Reading The following publications provide additional sources of information for the design of steel structures:
General Information [1] Chen, W.F. and Lui, E.M. 1987. Structural Stability—Theory and Implementation, Elsevier, New York. 1999 by CRC Press LLC
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[2] Englekirk, R. 1994. Steel Structures—Controlling Behavior Through Design, John Wiley & Sons, New York. [3] Stability of Metal Structures—A World View. 1991. 2nd ed., Lynn S. Beedle (editor-in-chief), Structural Stability Research Council, Lehigh University, Bethlehem, PA. [4] Trahair, N.S. 1993. Flexural-Torsional Buckling of Structures, CRC Press, Boca Raton, FL.
Allowable Stress Design [5] Adeli, H. 1988. Interactive Microcomputer-Aided Structural Steel Design, Prentice-Hall, Englewood Cliffs, NJ. [6] Cooper S.E. and Chen A.C. 1985. Designing Steel Structures—Methods and Cases, PrenticeHall, Englewood Cliffs, NJ. [7] Crawley S.W. and Dillon, R.M. 1984. Steel Buildings Analysis and Design, 3rd ed., John Wiley & Sons, New York. [8] Fanella, D.A., Amon, R., Knobloch, B., and Mazumder, A. 1992. Steel Design for Engineers and Architects, 2nd ed., Van Nostrand Reinhold, New York. [9] Kuzmanovic, B.O. and Willems, N. 1983. Steel Design for Structural Engineers, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ. [10] McCormac, J.C. 1981. Structural Steel Design, 3rd ed., Harper & Row, New York. [11] Segui, W.T. 1989. Fundamentals of Structural Steel Design, PWS-KENT, Boston, MA. [12] Spiegel, L. and Limbrunner, G.F. 1986. Applied Structural Steel Design, Prentice-Hall, Englewood Cliffs, NJ.
Plastic Design [13] Horne, M.R. and Morris, L.J. 1981. Plastic Design of Low-Rise Frames, Constrado Monographs, Collins, London, England. [14] Plastic Design in Steel-A Guide and Commentary. 1971. 2nd ed., ASCE Manual No. 41, ASCEWRC, New York. [15] Chen, W.F. and Sohal, I.S. 1995. Plastic Design and Second-Order Analysis of Steel Frames, Springer-Verlag, New York.
Load and Resistance Factor Design [16] Geschwindner, L.F., Disque, R.O., and Bjorhovde, R. 1994. Load and Resistance Factor Design of Steel Structures, Prentice-Hall, Englewood Cliffs, NJ. [17] McCormac, J.C. 1995. Structural Steel Design—LRFD Method, 2nd ed., Harper & Row, New York. [18] Salmon C.G. and Johnson, J.E. 1990. Steel Structures—Design and Behavior, 3rd ed., Harper & Row, New York. [19] Segui, W.T. 1994. LRFD Steel Design, PWS, Boston, MA. [20] Smith, J.C. 1996. Structural Steel Design—LRFD Approach, 2nd ed., John Wiley & Sons, New York. [21] Chen, W.F. and Kim, S.E. 1997. LRFD Steel Design Using Advanced Analysis, CRC Press, Boca Raton, FL. [22] Chen, W.F., Goto, Y., and Liew, J.Y.R. 1996. Stability Design of Semi-Rigid Frames, John Wiley & Sons, New York.
1999 by CRC Press LLC
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Grider, A.; Ramirez, J.A. and Yun, Y.M. “Structural Concrete Design” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Structural Concrete Design
1
4.1 4.2 4.3 4.4
4.5
Properties of Concrete and Reinforcing Steel
Properties of Concrete • Lightweight Concrete • Heavyweight Concrete • High-Strength Concrete • Reinforcing Steel
Proportioning and Mixing Concrete
Proportioning Concrete Mix • Admixtures • Mixing
Flexural Design of Beams and One-Way Slabs
Reinforced Concrete Strength Design • Prestressed Concrete Strength Design
Columns under Bending and Axial Load
Short Columns under Minimum Eccentricity • Short Columns under Axial Load and Bending • Slenderness Effects • Columns under Axial Load and Biaxial Bending
Shear and Torsion Reinforced Concrete Beams and One-Way Slabs Strength Design • Prestressed Concrete Beams and One-Way Slabs Strength Design
4.6
4.7
4.8
Development of Reinforcement
Development of Bars in Tension • Development of Bars in Compression • Development of Hooks in Tension • Splices, Bundled Bars, and Web Reinforcement
Two-Way Systems
Definition • Design Procedures • Minimum Slab Thickness and Reinforcement • Direct Design Method • Equivalent Frame Method • Detailing
Frames
Analysis of Frames • Design for Seismic Loading
4.9 Brackets and Corbels 4.10 Footings
Amy Grider and Julio A. Ramirez School of Civil Engineering, Purdue University, West Lafayette, IN
Young Mook Yun Department of Civil Engineering, National University, Taegu, South Korea
Types of Footings • Design Considerations • Wall Footings • Single-Column Spread Footings • Combined Footings • TwoColumn Footings • Strip, Grid, and Mat Foundations • Footings on Piles
4.11 Walls
Panel, Curtain, and Bearing Walls • Basement Walls • Partition Walls • Shears Walls
4.12 Defining Terms References Further Reading
1 The material in this chapter was previously published by CRC Press in The Civil Engineering Handbook, W.F. Chen, Ed.,
1995. 1999 by CRC Press LLC
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At this point in the history of development of reinforced and prestressed concrete it is necessary to reexamine the fundamental approaches to design of these composite materials. Structural engineering is a worldwide industry. Designers from one nation or a continent are faced with designing a project in another nation or continent. The decades of efforts dedicated to harmonizing concrete design approaches worldwide have resulted in some successes but in large part have led to further differences and numerous different design procedures. It is this abundance of different design approaches, techniques, and code regulations that justifies and calls for the need for a unification of design approaches throughout the entire range of structural concrete, from plain to fully prestressed [5]. The effort must begin at all levels: university courses, textbooks, handbooks, and standards of practice. Students and practitioners must be encouraged to think of a single continuum of structural concrete. Based on this premise, this chapter on concrete design is organized to promote such unification. In addition, effort will be directed at dispelling the present unjustified preoccupation with complex analysis procedures and often highly empirical and incomplete sectional mechanics approaches that tend to both distract the designers from fundamental behavior and impart a false sense of accuracy to beginning designers. Instead, designers will be directed to give careful consideration to overall structure behavior, remarking the adequate flow of forces throughout the entire structure.
4.1
Properties of Concrete and Reinforcing Steel
The designer needs to be knowledgeable about the properties of concrete, reinforcing steel, and prestressing steel. This part of the chapter summarizes the material properties of particular importance to the designer.
4.1.1
Properties of Concrete
Workability is the ease with which the ingredients can be mixed and the resulting mix handled, transported, and placed with little loss in homogeneity. Unfortunately, workability cannot be measured directly. Engineers therefore try to measure the consistency of the concrete by performing a slump test. The slump test is useful in detecting variations in the uniformity of a mix. In the slump test, a mold shaped as the frustum of a cone, 12 in. (305 mm) high with an 8 in. (203 mm) diameter base and 4 in. (102 mm) diameter top, is filled with concrete (ASTM Specification C143). Immediately after filling, the mold is removed and the change in height of the specimen is measured. The change in height of the specimen is taken as the slump when the test is done according to the ASTM Specification. A well-proportioned workable mix settles slowly, retaining its original shape. A poor mix crumbles, segregates, and falls apart. The slump may be increased by adding water, increasing the percentage of fines (cement or aggregate), entraining air, or by using an admixture that reduces water requirements; however, these changes may adversely affect other properties of the concrete. In general, the slump specified should yield the desired consistency with the least amount of water and cement. Concrete should withstand the weathering, chemical action, and wear to which it will be subjected in service over a period of years; thus, durability is an important property of concrete. Concrete resistance to freezing and thawing damage can be improved by increasing the watertightness, entraining 2 to 6% air, using an air-entraining agent, or applying a protective coating to the surface. Chemical agents damage or disintegrate concrete; therefore, concrete should be protected with a resistant coating. Resistance to wear can be obtained by use of a high-strength, dense concrete made with hard aggregates. 1999 by CRC Press LLC
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Excess water leaves voids and cavities after evaporation, and water can penetrate or pass through the concrete if the voids are interconnected. Watertightness can be improved by entraining air or reducing water in the mix, or it can be prolonged through curing. Volume change of concrete should be considered, since expansion of the concrete may cause buckling and drying shrinkage may cause cracking. Expansion due to alkali-aggregate reaction can be avoided by using nonreactive aggregates. If reactive aggregates must be used, expansion may be reduced by adding pozzolanic material (e.g., fly ash) to the mix. Expansion caused by heat of hydration of the cement can be reduced by keeping cement content as low as possible; using Type IV cement; and chilling the aggregates, water, and concrete in the forms. Expansion from temperature increases can be reduced by using coarse aggregate with a lower coefficient of thermal expansion. Drying shrinkage can be reduced by using less water in the mix, using less cement, or allowing adequate moist curing. The addition of pozzolans, unless allowing a reduction in water, will increase drying shrinkage. Whether volume change causes damage usually depends on the restraint present; consideration should be given to eliminating restraints or resisting the stresses they may cause [8]. Strength of concrete is usually considered its most important property. The compressive strength at 28 d is often used as a measure of strength because the strength of concrete usually increases with time. The compressive strength of concrete is determined by testing specimens in the form of standard cylinders as specified in ASTM Specification C192 for research testing or C31 for field testing. The test procedure is given in ASTM C39. If drilled cores are used, ASTM C42 should be followed. The suitability of a mix is often desired before the results of the 28-d test are available. A formula proposed by W. A. Slater estimates the 28-d compressive strength of concrete from its 7-d strength: p (4.1) S28 = S7 + 30 S7 where S28 = 28-d compressive strength, psi S7 = 7-d compressive strength, psi Strength can be increased by decreasing water-cement ratio, using higher strength aggregate, using a pozzolan such as fly ash, grading the aggregates to produce a smaller percentage of voids in the concrete, moist curing the concrete after it has set, and vibrating the concrete in the forms. The short-time strength can be increased by using Type III portland cement, accelerating admixtures, and by increasing the curing temperature. The stress-strain curve for concrete is a curved line. Maximum stress is reached at a strain of 0.002 in./in., after which the curve descends. The modulus of elasticity, Ec , as given in ACI 318-89 (Revised 92), Building Code Requirements for Reinforced Concrete [1], is: Ec
=
Ec
=
p wc1.5 33 fc0 lb/ft3 and psi p wc1.5 0.043 fc0 kg/m3 and MPa
(4.2a) (4.2b)
where wc = unit weight of concrete fc0 = compressive strength at 28 d p Tensile strength of concrete is much lower than the compressive strength—about 7 fc0 for the p higher-strength concretes and 10 fc0 for the lower-strength concretes. Creep is the increase in strain with time under a constant load. Creep increases with increasing water-cement ratio and decreases with an increase in relative humidity. Creep is accounted for in design by using a reduced modulus of elasticity of the concrete. 1999 by CRC Press LLC
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4.1.2 Lightweight Concrete Structural lightweight concrete is usually made from aggregates conforming to ASTM C330 that are usually produced in a kiln, such as expanded clays and shales. Structural lightweight concrete has a density between 90 and 120 lb/ft3 (1440 to 1920 kg/m3 ). Production of lightweight concrete is more difficult than normal-weight concrete because the aggregates vary in absorption of water, specific gravity, moisture content, and amount of grading of undersize. Slump and unit weight tests should be performed often to ensure uniformity of the mix. During placing and finishing of the concrete, the aggregates may float to the surface. Workability can be improved by increasing the percentage of fines or by using an air-entraining admixture to incorporate 4 to 6% air. Dry aggregate should not be put into the mix because it will continue to absorb moisture and cause the concrete to harden before placement is completed. Continuous water curing is important with lightweight concrete. No-fines concrete is obtained by using pea gravel as the coarse aggregate and 20 to 30% entrained air instead of sand. It is used for low dead weight and insulation when strength is not important. This concrete weighs from 105 to 118 lb/ft3 (1680 to 1890 kg/m3 ) and has a compressive strength from 200 to 1000 psi (1 to 7 MPa). A porous concrete made by gap grading or single-size aggregate grading is used for low conductivity or where drainage is needed. Lightweight concrete can also be made with gas-forming of foaming agents which are used as admixtures. Foam concretes range in weight from 20 to 110 lb/ft3 (320 to 1760 kg/m3 ). The modulus of elasticity of lightweight concrete can be computed using the same formula as normal concrete. The shrinkage of lightweight concrete is similar to or slightly greater than for normal concrete.
4.1.3
Heavyweight Concrete
Heavyweight concretes are used primarily for shielding purposes against gamma and x-radiation in nuclear reactors and other structures. Barite, limonite and magnetite, steel punchings, and steel shot are typically used as aggregates. Heavyweight concretes weigh from 200 to 350 lb/ft3 (3200 to 5600 kg/m3 ) with strengths from 3200 to 6000 psi (22 to 41 MPa). Gradings and mix proportions are similar to those for normal weight concrete. Heavyweight concretes usually do not have good resistance to weathering or abrasion.
4.1.4
High-Strength Concrete
Concretes with strengths in excess of 6000 psi (41 MPa) are referred to as high-strength concretes. Strengths up to 18,000 psi (124 MPa) have been used in buildings. Admixtures such as superplasticizers, silica fume, and supplementary cementing materials such as fly ash improve the dispersion of cement in the mix and produce workable concretes with lower water-cement ratios, lower void ratios, and higher strength. Coarse aggregates should be strong fine-grained gravel with rough surfaces. For concrete strengths in excess of 6000 psi (41 MPa), the modulus of elasticity should be taken as p Ec = 40,000 fc0 + 1.0 × 106 where fc0 = compressive strength at 28 d, psi [4] The shrinkage of high-strength concrete is about the same as that for normal concrete. 1999 by CRC Press LLC
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(4.3)
4.1.5
Reinforcing Steel
Concrete can be reinforced with welded wire fabric, deformed reinforcing bars, and prestressing tendons. Welded wire fabric is used in thin slabs, thin shells, and other locations where space does not allow the placement of deformed bars. Welded wire fabric consists of cold drawn wire in orthogonal patterns—square or rectangular and resistance-welded at all intersections. The wire may be smooth (ASTM A185 and A82) or deformed (ASTM A497 and A496). The wire is specified by the symbol W for smooth wires or D for deformed wires followed by a number representing the cross-sectional area in hundredths of a square inch. On design drawings it is indicated by the symbol WWF followed by spacings of the wires in the two 90◦ directions. Properties for welded wire fabric are given in Table 4.1. TABLE 4.1
Wire and Welded Wire Fabric Steels Wire size AST designation
Minimum yield stress,a fy
Minimum tensile strength
designation
ksi
MPa
ksi
MPa
A82-79 (cold-drawn wire) (properties apply when material is to be used for fabric)
W1.2 and largerb Smaller than W1.2
65 56
450 385
75 70
520 480
A185-79 (welded wire fabric)
Same as A82; this is A82 material fabricated into sheet (so-called “mesh”) by the process of electric welding
A496-78 (deformed steel wire) (properties apply when material is to be used for fabric)
D1-D31c
A497-79
Same as A82 or A496; this specification applies for fabric made from A496, or from a combination of A496 and A82 wires
70
480
80
550
a The term “yield stress” refers to either yield point, the well-defined deviation from perfect elasticity, or yield strength,
the value obtained by a specified offset strain for material having no well-defined yield point.
b The W number represents the nominal cross-sectional area in square inches multiplied by 100, for smooth wires. c The D number represents the nominal cross-sectional area in square inches multiplied by 100, for deformed wires.
The deformations on a deformed reinforcing bar inhibit longitudinal movement of the bar relative to the concrete around it. Table 4.2 gives dimensions and weights of these bars. Reinforcing bar steel can be made of billet steel of grades 40 and 60 having minimum specific yield stresses of 40,000 and 60,000 psi, respectively (276 and 414 MPa) (ASTM A615) or low-alloy steel of grade 60, which is intended for applications where welding and/or bending is important (ASTM A706). Presently, grade 60 billet steel is the most predominantly used for construction. Prestressing tendons are commonly in the form of individual wires or groups of wires. Wires of different strengths and properties are available with the most prevalent being the 7-wire lowrelaxation strand conforming to ASTM A416. ASTM A416 also covers a stress-relieved strand, which is seldom used in construction nowadays. Properties of standard prestressing strands are given in Table 4.3. Prestressing tendons could also be bars; however, this is not very common. Prestressing bars meeting ASTM A722 have been used in connections between members. The modulus of elasticity for non-prestressed steel is 29,000,000 psi (200,000 MPa). For prestressing steel, it is lower and also variable, so it should be obtained from the manufacturer. For 7-wires strands conforming to ASTM A416, the modulus of elasticity is usually taken as 27,000,000 psi (186,000 MPa). 1999 by CRC Press LLC
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TABLE 4.2
Reinforcing Bar Dimensions and Weights Nominal dimensions
Bar number 3 4 5 6 7 8 9 10 11 14 18
Diameter (in.) 0.375 0.500 0.625 0.750 0.875 1.000 1.128 1.270 1.410 1.693 2.257
TABLE 4.3
(mm) 9.5 12.7 15.9 19.1 22.2 25.4 28.7 32.3 35.8 43.0 57.3
(cm2 )
0.11 0.20 0.31 0.44 0.60 0.79 1.00 1.27 1.56 2.25 4.00
0.71 1.29 2.00 2.84 3.87 5.10 6.45 8.19 10.06 14.52 25.81
(lb/ft) 0.376 0.668 1.043 1.502 2.044 2.670 3.400 4.303 5.313 7.65 13.60
(kg/m) 0.559 0.994 1.552 2.235 3.041 3.973 5.059 6.403 7.906 11.38 20.24
Standard Prestressing Strands, Wires, and Bars Nominal dimension
fpu ksi
Diameter in.
Area in.2
Weight plf
Seven-wire strand
250 270 250 270 250 270 250
1/4 3/8 3/8 1/2 1/2 0.6 0.6
0.036 0.085 0.080 0.153 0.144 0.215 0.216
0.12 0.29 0.27 0.53 0.49 0.74 0.74
Prestressing wire
250 240 235
0.196 0.250 0.276
0.0302 0.0491 0.0598
0.10 0.17 0.20
Deformed prestressing bars
157 150 150 150
5/8 1 1 1/4 1 3/8
0.28 0.85 1.25 1.58
0.98 3.01 4.39 5.56
Tendon type
4.2.1
. Weight
(in.2 )
Grade
4.2
. Area
Proportioning and Mixing Concrete Proportioning Concrete Mix
A concrete mix is specified by the weight of water, sand, coarse aggregate, and admixture to be used per 94-pound bag of cement. The type of cement (Table 4.4), modulus of the aggregates, and maximum size of the aggregates (Table 4.5) should also be given. A mix can be specified by the weight ratio of cement to sand to coarse aggregate with the minimum amount of cement per cubic yard of concrete. In proportioning a concrete mix, it is advisable to make and test trial batches because of the many variables involved. Several trial batches should be made with a constant water-cement ratio but varying ratios of aggregates to obtain the desired workability with the least cement. To obtain results similar to those in the field, the trial batches should be mixed by machine. When time or other conditions do not allow proportioning by the trial batch method, Table 4.6 may be used. Start with mix B corresponding to the appropriate maximum size of aggregate. Add just enough water for the desired workability. If the mix is undersanded, change to mix A; if oversanded, change to mix C. Weights are given for dry sand. For damp sand, increase the weight of sand 10 lb, and for very wet sand, 20 lb, per bag of cement. 1999 by CRC Press LLC
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TABLE 4.4
Types of Portland Cementa
Type I
Usage Ordinary construction where special properties are not required Ordinary construction when moderate sulfate resistance or moderate heat of hydration is desired When high early strength is desired When low heat of hydration is desired When high sulfate resistance is desired
II III IV V
a According to ASTM C150.
TABLE 4.5
Recommended Maximum Sizes of Aggregatea Maximum size, in., of aggregate for:
Minimum dimension of section, in. 5 or less 6–11 12–29 30 or more
Reinforced-concrete beams, columns, walls
Heavily reinforced slabs
Lightly reinforced or unreinforced slabs
··· 3/4 – 1 1/2 1 1/2 – 3 1 1/2 – 3
3/4 – 1 1/2 1 1/2 3 3
3/4 – 1 1/2 1 1/2 – 3 3–6 6
a Concrete Manual. U.S. Bureau of Reclamation.
TABLE 4.6
Typical Concrete Mixesa Aggregate, lb per bag of cement
Maximum size of
Bags of cement
Sand
aggregate, in.
Mix designation
per yd3 of concrete
Air-entrained concrete
Concrete without air
Gravel or crushed stone
1/2
A B C
7.0 6.9 6.8
235 225 225
245 235 235
170 190 205
3/4
A B C
6.6 6.4 6.3
225 225 215
235 235 225
225 245 265
1
A B C
6.4 6.2 6.1
225 215 205
235 225 215
245 275 290
1 1/2
A B C
6.0 5.8 5.7
225 215 205
235 225 215
290 320 345
2
A B C
5.7 5.6 5.4
225 215 205
235 225 215
330 360 380
a Concrete Manual. U.S. Bureau of Reclamation.
4.2.2
Admixtures
Admixtures may be used to modify the properties of concrete. Some types of admixtures are set accelerators, water reducers, air-entraining agents, and waterproofers. Admixtures are generally helpful in improving quality of the concrete. However, if admixtures are not properly used, they could have undesirable effects; it is therefore necessary to know the advantages and limitations of the proposed admixture. The ASTM Specifications cover many of the admixtures. Set accelerators are used (1) when it takes too long for concrete to set naturally; such as in cold weather, or (2) to accelerate the rate of strength development. Calcium chloride is widely used as a set accelerator. If not used in the right quantities, it could have harmful effects on the concrete and reinforcement. Water reducers lubricate the mix and permit easier placement of the concrete. Since the workability of a mix can be improved by a chemical agent, less water is needed. With less water but the same 1999 by CRC Press LLC
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cement content, the strength is increased. Since less water is needed, the cement content could also be decreased, which results in less shrinkage of the hardened concrete. Some water reducers also slow down the concrete set, which is useful in hot weather and integrating consecutive pours of the concrete. Air-entraining agents are probably the most widely used type of admixture. Minute bubbles of air are entrained in the concrete, which increases the resistance of the concrete to freeze-thaw cycles and the use of ice-removal salts. Waterproofing chemicals are often applied as surface treatments, but they can be added to the concrete mix. If applied properly and uniformly, they can prevent water from penetrating the concrete surface. Epoxies can also be used for waterproofing. They are more durable than silicone coatings, but they may be more costly. Epoxies can also be used for protection of wearing surfaces, patching cavities and cracks, and glue for connecting pieces of hardened concrete.
4.2.3
Mixing
Materials used in making concrete are stored in batch plants that have weighing and control equipment and bins for storing the cement and aggregates. Proportions are controlled by automatic or manually operated scales. The water is measured out either from measuring tanks or by using water meters. Machine mixing is used whenever possible to achieve uniform consistency. The revolving drumtype mixer and the countercurrent mixer, which has mixing blades rotating in the opposite direction of the drum, are commonly used. Mixing time, which is measured from the time all ingredients are in the drum, “should be at least 1.5 minutes for a 1-yd3 mixer, plus 0.5 min for each cubic yard of capacity over 1 yd3 ” [ACI 304-73, 1973]. It also is recommended to set a maximum on mixing time since overmixing may remove entrained air and increase fines, thus requiring more water for workability; three times the minimum mixing time can be used as a guide. Ready-mixed concrete is made in plants and delivered to job sites in mixers mounted on trucks. The concrete can be mixed en route or upon arrival at the site. Concrete can be kept plastic and workable for as long as 1.5 hours by slow revolving of the mixer. Mixing time can be better controlled if water is added and mixing started upon arrival at the job site, where the operation can be inspected.
4.3 4.3.1
Flexural Design of Beams and One-Way Slabs Reinforced Concrete Strength Design
The basic assumptions made in flexural design are: 1. Sections perpendicular to the axis of bending that are plane before bending remain plane after bending. 2. A perfect bond exists between the reinforcement and the concrete such that the strain in the reinforcement is equal to the strain in the concrete at the same level. 3. The strains in both the concrete and reinforcement are assumed to be directly proportional to the distance from the neutral axis (ACI 10.2.2) [1]. 4. Concrete is assumed to fail when the compressive strain reaches 0.003 (ACI 10.2.3). 5. The tensile strength of concrete is neglected (ACI 10.2.5). 6. The stresses in the concrete and reinforcement can be computed from the strains using stressstrain curves for concrete and steel, respectively. 1999 by CRC Press LLC
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7. The compressive stress-strain relationship for concrete may be assumed to be rectangular, trapezoidal, parabolic, or any other shape that results in prediction of strength in substantial agreement with the results of comprehensive tests (ACI 10.2.6). ACI 10.2.7 outlines the use of a rectangular compressive stress distribution which is known as the Whitney rectangular stress block. For other stress distributions see Reinforced Concrete Mechanics and Design by James G. MacGregor [8]. Analysis of Rectangular Beams with Tension Reinforcement Only Equations for Mn and φMn : Tension Steel Yielding Consider the beam shown in Figure 4.1. The compressive force, C, in the concrete is (4.4) C = 0.85fc0 ba
The tension force, T , in the steel is
T = As fy
(4.5)
For equilibrium, C = T , so the depth of the equivalent rectangular stress block, a, is a=
As fy 0.85fc0 b
(4.6)
Noting that the internal forces C and T form an equivalent force-couple system, the internal moment is (4.7) Mn = T (d − a/2) or Mn = C(d − a/2) φMn is then
φMn = φT (d − a/2)
or φMn = φC(d − a/2) where φ =0.90.
FIGURE 4.1: Stresses and forces in a rectangular beam.
1999 by CRC Press LLC
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(4.8)
Equation for Mn and φMn : Tension Steel Elastic
The internal forces and equilibrium are
given by: C
= T = As fs = ρbdEs εs
0.85fc0 ba 0.85fc0 ba From strain compatibility (see Figure 4.1),
εs = εcu
d −c c
(4.9)
(4.10)
Substituting εs into the equilibrium equation, noting that a = β1 c, and simplifying gives 0.85fc0 a 2 + (d)a − β1 d 2 = 0 ρEs εcu
(4.11)
which can be solved for a. Equations 4.7 and 4.8 can then be used to obtain Mn and φMn . Reinforcement Ratios The reinforcement ratio, ρ, is used to represent the relative amount of tension reinforcement in a beam and is given by ρ=
As bd
(4.12)
At the balanced strain condition the maximum strain, εcu , at the extreme concrete compression fiber reaches 0.003 just as the tension steel reaches the strain εy = fy /Es . The reinforcement ratio in the balanced strain condition, ρb , can be obtained by applying equilibrium and compatibility conditions. From the linear strain condition, Figure 4.1, εcu cb = = d εcu + εy
0.003 fy 0.003 + 29,000,000
87,000 87,000 + fy
=
(4.13)
The compressive and tensile forces are: Cb Tb
0.85fc0 bβ1 cb fy Asb = ρb bdfy
= =
(4.14)
Equating Cb to Tb and solving for ρb gives 0.85fc0 β1 cb fy d
ρb = which on substitution of Equation 4.13 gives
0.85fc0 β1 ρb = fy
87,000 87,000 + fy
(4.15)
(4.16)
ACI 10.3.3 limits the amount of reinforcement in order to prevent nonductile behavior: maxρ = 0.75ρb
(4.17)
ACI 10.5 requires a minimum amount of flexural reinforcement: ρmin =
1999 by CRC Press LLC
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200 fy
(4.18)
Analysis of Beams with Tension and Compression Reinforcement
For the analysis of doubly reinforced beams, the cross-section will be divided into two beams. Beam 1 consists of the compression reinforcement at the top and sufficient steel at the bottom so that T1 = Cs ; beam 2 consists of the concrete web and the remaining tensile reinforcement, as shown in Figure 4.2
FIGURE 4.2: Strains, stresses, and forces in beam with compression reinforcement.
Equation for Mn : Compression Steel Yields The area of tension steel in beam 1 is obtained by setting T1 = Cs , which gives As1 = A0s . The nominal moment capacity of beam 1 is then (4.19) Mn1 = A0s fy d − d 0
Beam 2 consists of the concrete and the remaining steel, As2 = As −As1 = As −A0s . The compression force in the concrete is (4.20) C = 0.85fc0 ba and the tension force in the steel for beam 2 is
T = As − A0s fy
(4.21)
The depth of the compression stress block is then
As − A0s fy a= 0.85fc0 b
Therefore, the nominal moment capacity for beam 2 is Mn2 = As − A0s fy (d − a/2)
(4.22)
(4.23)
The total moment capacity for a doubly reinforced beam with compression steel yielding is the summation of the moment capacity for beam 1 and beam 2; therefore, (4.24) Mn = A0s fy d − d 0 + As − A0s fy (d − a/2) 1999 by CRC Press LLC
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Equation for Mn : Compression Steel Does Not Yield the internal forces in the beam are
T Cc Cs where
Assuming that the tension steel yields,
= As fy = 0.85fc0 ba = A0s Es εs0
(4.25)
β1 d 0 (0.003) εs0 = 1 − a
(4.26)
β1 d 0 (0.003) = As fy 0.85fc0 ba + A0s Es 1 − a
(4.27)
From equilibrium, Cs + Cc = T or
This can be rewritten in quadratic form as 0.85fc0 b a 2 + 0.003A0s Es − As Fy a − 0.003A0s Es β1 d 0 = 0
(4.28)
where a can be calculated by means of the quadratic equation. Therefore, the nominal moment capacity in a doubly reinforced concrete beam where the compression steel does not yield is a + Cs d − d 0 (4.29) Mn = Cc d − 2 Reinforcement Ratios The reinforcement ratio at the balanced strain condition can be obtained in a similar manner as that for beams with tension steel only. For compression steel yielding, the balanced ratio is 0.85fc0 β1 87,000 0 (4.30) ρ−ρ b = fy 87,000 + fy
For compression steel not yielding, the balanced ratio is 0.85fc0 β1 87,000 ρ 0 fs0 = ρ− fy b fy 87,000 + fy
(4.31)
The maximum and minimum reinforcement ratios as given in ACI 10.3.3 and 10.5 are ρmax
=
ρmin
=
0.75ρb 200 fy
(4.32)
4.3.2 Prestressed Concrete Strength Design Elastic Flexural Analysis
In developing elastic equations for prestress, the effects of prestress force, dead load moment, and live load moment are calculated separately, and then the separate stresses are superimposed, giving f =− 1999 by CRC Press LLC
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F ey My F ± ± A I I
(4.33)
where (−) indicates compression and (+) indicates tension. It is necessary to check that the stresses in the extreme fibers remain within the ACI-specified limits under any combination of loadings that many occur. The stress limits for the concrete and prestressing tendons are specified in ACI 18.4 and 18.5 [1]. ACI 18.2.6 states that the loss of area due to open ducts shall be considered when computing section properties. It is noted in the commentary that section properties may be based on total area if the effect of the open duct area is considered negligible. In pretensioned members and in post-tensioned members after grouting, section properties can be based on gross sections, net sections, or effective sections using the transformed areas of bonded tendons and nonprestressed reinforcement. Flexural Strength
The strength of a prestressed beam can be calculated using the methods developed for ordinary reinforced concrete beams, with modifications to account for the differing nature of the stress-strain relationship of prestressing steel compared with ordinary reinforcing steel. A prestressed beam will fail when the steel reaches a stress fps , generally less than the tensile strength fpu . For rectangular cross-sections the nominal flexural strength is Mn = Aps fps d − where a=
a 2
Aps fps 0.85fc0 b
(4.34)
(4.35)
The steel stress fps can be found based on strain compatibility or by using approximate equations such as those given in ACI 18.7.2. The equations in ACI are applicable only if the effective prestress in the steel, fse , which equals Pe /Aps , is not less than 0.5 fpu . The ACI equations are as follows. (a) For members with bonded tendons: γp fpu d 0 (4.36) ω−ω ρ 0 + fps = fpu 1 − β1 fc dp If any compression reinforcement is taken into account when calculating fps with Equation 4.36, the following applies: fpu d 0 (4.37) ω−ω ≥ 0.17 ρp 0 + fc dp and
d 0 ≤ 0.15dp
(b) For members with unbonded tendons and with a span-to-depth ratio of 35 or less: fc0 fpy fps = fse + 10,000 + ≤ fse + 60,000 100ρp
(4.38)
(c) For members with unbonded tendons and with a span-to-depth ratio greater than 35: fc0 fpy ≤ (4.39) fps = fse + 10,000 + fse + 30,000 300ρp The flexural strength is then calculated from Equation 4.34. The design strength is equal to φMn , where φ = 0.90 for flexure. 1999 by CRC Press LLC
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Reinforcement Ratios
ACI requires that the total amount of prestressed and nonprestressed reinforcement be adequate to develop a factored load at least 1.2 times the cracking load calculated on the basis of a modulus of p rupture of 7.5 fc0 . To control cracking in members with unbonded tendons, some bonded reinforcement should be uniformly distributed over the tension zone near the extreme tension fiber. ACI specifies the minimum amount of bonded reinforcement as As = 0.004A
(4.40)
where A is the area of the cross-section between the flexural tension face and the center of gravity of the gross cross-section. ACI 19.9.4 gives the minimum length of the bonded reinforcement. To ensure adequate ductility, ACI 18.8.1 provides the following requirement: ω p d 0 ωp + ω−ω dp ≤ 0.36β1 (4.41) d 0 ωw − ωw ωpw + dp ACI allows each of the terms on the left side to be set equal to 0.85 a/dp in order to simplify the equation. When a reinforcement ratio greater than 0.36 β1 is used, ACI 18.8.2 states that the design moment strength shall not be greater than the moment strength based on the compression portion of the moment couple.
4.4
Columns under Bending and Axial Load
4.4.1
Short Columns under Minimum Eccentricity
When a symmetrical column is subjected to a concentric axial load, P , longitudinal strains develop uniformly across the section. Because the steel and concrete are bonded together, the strains in the concrete and steel are equal. For any given strain it is possible to compute the stresses in the concrete and steel using the stress-strain curves for the two materials. The forces in the concrete and steel are equal to the stresses multiplied by the corresponding areas. The total load on the column is the sum of the forces in the concrete and steel: (4.42) Po = 0.85fc0 Ag − Ast + fy Ast To account for the effect of incidental moments, ACI 10.3.5 specifies that the maximum design axial load on a column be, for spiral columns, (4.43) φPn(max) = 0.85φ .85fc0 Ag − Ast + fy Ast and for tied columns, φPn(max) = 0.80φ .85fc0 Ag − Ast + fy Ast
(4.44)
For high values of axial load, φ values of 0.7 and 0.75 are specified for tied and spiral columns, respectively (ACI 9.3.2.2b) [1]. Short columns are sufficiently stocky such that slenderness effects can be ignored. 1999 by CRC Press LLC
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4.4.2
Short Columns under Axial Load and Bending
Almost all compression members in concrete structures are subjected to moments in addition to axial loads. Although it is possible to derive equations to evaluate the strength of columns subjected to combined bending and axial loads, the equations are tedious to use. For this reason, interaction diagrams for columns are generally computed by assuming a series of strain distributions, each corresponding to a particular point on the interaction diagram, and computing the corresponding values of P and M. Once enough such points have been computed, the results are summarized in an interaction diagram. For examples on determining the interaction diagram, see Reinforced Concrete Mechanics and Design by James G. MacGregor [8] or Reinforced Concrete Design by Chu-Kia Wang and Charles G. Salmon [11]. Figure 4.3 illustrates a series of strain distributions and the resulting points on the interaction diagram. Point A represents pure axial compression. Point B corresponds to crushing at one face and zero tension at the other. If the tensile strength of concrete is ignored, this represents the onset of cracking on the bottom face of the section. All points lower than this in the interaction diagram represent cases in which the section is partially cracked. Point C, the farthest right point, corresponds to the balanced strain condition and represents the change from compression failures for higher loads and tension failures for lower loads. Point D represents a strain distribution where the reinforcement has been strained to several times the yield strain before the concrete reaches its crushing strain. The horizontal axis of the interaction diagram corresponds to pure bending where φ = 0.9. A transition is required from φ = 0.7 or 0.75 for high axial loads to φ = 0.9 for pure bending. The change in φ begins at a capacity φPa , which equals the smaller of the balanced load, φPb , or 0.1 fc0 Ag . Generally, φPb exceeds 0.1 fc0 Ag except for a few nonrectangular columns. ACI publication SP-17A(85), A Design Handbook for Columns, contains nondimensional interaction diagrams as well as other design aids for columns [2].
4.4.3
Slenderness Effects
ACI 10.11 describes an approximate slenderness-effect design procedure based on the moment magnifier concept. The moments are computed by ordinary frame analysis and multiplied by a moment magnifier that is a function of the factored axial load and the critical buckling load of the column. The following gives a summary of the moment magnifier design procedure for slender columns in frames. 1. Length of Column. The unsupported length, lu , is defined in ACI 10.11.1 as the clear distance between floor slabs, beams, or other members capable of giving lateral support to the column. 2. Effective length. The effective length factors, k, used in calculating δb shall be between 0.5 and 1.0 (ACI 10.11.2.1). The effective length factors used to compute δs shall be greater than 1 (ACI 10.11.2.2). The effective length factors can be estimated using ACI Fig. R10.11.2 or using ACI Equations (A)–(E) given in ACI R10.11.2. These two procedures require that the ratio, ψ, of the columns and beams be known: P (Ec Ic / lc ) ψ=P (Eb Ib / lb )
(4.45)
In computing ψ it is acceptable to take the EI of the column as the uncracked gross Ec Ig of the columns and the EI of the beam as 0.5 Ec Ig . 3. Definition of braced and unbraced frames. The ACI Commentary suggests that a frame is braced if either of the following are satisfied: 1999 by CRC Press LLC
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FIGURE 4.3: Strain distributions corresponding to points on interaction diagram. (a) If the stability index, Q, for a story is less than 0.04, where P Q=
Pu 1u ≤ 0.04 Hu hs
(4.46)
(b) If the sum of the lateral stiffness of the bracing elements in a story exceeds six times the lateral stiffness of all the columns in the story. 4. Radius of gyration. For a rectangular cross-section √ r equals 0.3 h, and for a circular cross-section r equals 0.25 h. For other sections, r equals I /A. 5. Consideration of slenderness effects. ACI 10.11.4.1 allows slenderness effects to be neglected for columns in braced frames when M1b klu < 34 − 12 r M2b
(4.47)
ACI 10.11.4.2 allows slenderness effects to be neglected for columns in unbraced frames when klu < 22 r
(4.48)
If klu /r exceeds 100, ACI 10.11.4.3 states that design shall be based on second-order analysis. 1999 by CRC Press LLC
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6. Minimum moments. For columns in a braced frame, M2b shall be not less than the value given in ACI 10.11.5.4. In an unbraced frame ACI 10.11.5.5 applies for M2s . 7. Moment magnifier equation. ACI 10.11.5.1 states that columns shall be designed for the factored axial load, Pu , and a magnified factored moment, Mc , defined by Mc = δb M2b + δs M2s
(4.49)
where M2b is the larger factored end moment acting on the column due to loads causing no appreciable sidesway (lateral deflections less than l/1500) and M2s is the larger factored end moment due to loads that result in an appreciable sidesway. The moments are computed from a conventional first-order elastic frame analysis. For the above equation, the following apply:
δb
=
δs
=
Cm ≥ 1.0 1 − Pu /φPc 1 P P ≥ 1.0 1 − Pu /φ Pc
(4.50)
For members braced against sidesway, ACI 10.11.5.1 gives δs = 1.0. Cm = 0.6 + 0.4
M1b ≥ 0.4 M2b
(4.51)
The ratio M1b /M2b is taken as positive if the member is bent in single curvature and negative if the member is bent in double curvature. Equation 4.51 applies only to columns in braced frames. In all other cases, ACI 10.11.5.3 states that Cm = 1.0.
Pc =
π 2 EI (klu )2
(4.52)
where EI =
Ec Ig /5 + Es Ise 1 + βd
(4.53)
Ec Ig /2.5 1 + βd
(4.54)
or, approximately
EI =
1999 by CRC Press LLC
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When computing δb , Axial load due to factored dead load Total factored axial load
(4.55)
Factored sustained lateral shear in the story Total factored lateral shear in the story
(4.56)
βd = When computing δs , βd =
If δb or δs is found to be negative, the column should be enlarged. If either δb or δs exceeds 2.0, consideration should be given to enlarging the column.
4.4.4
Columns under Axial Load and Biaxial Bending
The nominal ultimate strength of a section under biaxial bending and compression is a function of three variables, Pn , Mnx , and Mny , which may also be expressed as Pn acting at eccentricities ey = Mnx /Pn and ex = Mny /Pn with respect to the x and y axes. Three types of failure surfaces can be defined. In the first type, S1 , the three orthogonal axes are defined by Pn , ex , and ey ; in the second type, S2 , the variables defining the axes are 1/Pn , ex , and ey ; and in the third type, S3 the axes are Pn , Mnx , and Mny . In the presentation that follows, the Bresler reciprocal load method makes use of the reciprocal failure surface S2 , and the Bresler load contour method and the PCA load contour method both use the failure surface S3 . Bresler Reciprocal Load Method
Using a failure surface of type S2 , Bresler proposed the following equation as a means of approximating a point on the failure surface corresponding to prespecified eccentricities ex and ey : 1 1 1 1 = + − Pni Pnx Pny P0
(4.57)
where Pni = nominal axial load strength at given eccentricity along both axes Pnx = nominal axial load strength at given eccentricity along x axis Pny = nominal axial load strength at given eccentricity along y axis P0 = nominal axial load strength for pure compression (zero eccentricity) Test results indicate that Equation 4.57 may be inappropriate when small values of axial load are involved, such as when Pn /P0 is in the range of 0.06 or less. For such cases the member should be designed for flexure only. Bresler Load Contour Method The failure surface S3 can be thought of as a family of curves (load contours) each corresponding to a constant value of Pn . The general nondimensional equation for the load contour at constant Pn may be expressed in the following form: Mny α2 Mnx α1 + = 1.0 (4.58) Mox Moy
where Mnx = Pn ey ; Mny = Pn ex 1999 by CRC Press LLC
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Mox = Mnx capacity at axial load Pn when Mny (or ex ) is zero Moy = Mny capacity at axial load Pn when Mnx (or ey ) is zero The exponents α1 and α2 depend on the column dimensions, amount and arrangement of the reinforcement, and material strengths. Bresler suggests taking α1 = α2 = α. Calculated values of α vary from 1.15 to 1.55. For practical purposes, α can be taken as 1.5 for rectangular sections and between 1.5 and 2.0 for square sections. PCA (Parme-Gowens) Load Contour Method
This method has been developed as an extension of the Bresler load contour method in which the Bresler interaction Equation 4.58 is taken as the basic strength criterion. In this approach, a point on the load contour is defined in such a way that the biaxial moment strengths Mnx and Mny are in the same ratio as the uniaxial moment strengths Mox and Moy , Moy Mny = =β Mnx Mox
(4.59)
The actual value of β depends on the ratio of Pn to P0 as well as the material and cross-sectional properties, with the usual range of values between 0.55 and 0.70. Charts for determining β can be found in ACI Publication SP-17A(85), A Design Handbook for Columns [2]. Substituting Equation 4.59 into Equation 4.58, βMoy α βMox α + = 1 Mox Moy 2β α = 1 (4.60) β α = 1/2 log 0.5 α = log β thus,
Mnx Mox
log0.5/logβ
+
Mny Moy
log0.5/logβ
=1
(4.61)
For more information on columns subjected to biaxial bending, see Reinforced Concrete Design by Chu-Kia Wang and Charles G. Salmon [11].
4.5 4.5.1
Shear and Torsion Reinforced Concrete Beams and One-Way Slabs Strength Design
The cracks that form in a reinforced concrete beam can be due to flexure or a combination of flexure and shear. Flexural cracks start at the bottom of the beam, where the flexural stresses are the largest. Inclined cracks, also called shear cracks or diagonal tension cracks, are due to a combination of flexure and shear. Inclined cracks must exist before a shear failure can occur. Inclined cracks form in two different ways. In thin-walled I-beams in which the shear stresses in the web are high while the flexural stresses are low, a web-shear crack occurs. The inclined cracking shear can be calculated as the shear necessary to cause a principal tensile stress equal to the tensile strength of the concrete at the centroid of the beam. In most reinforced concrete beams, however, flexural cracks occur first and extend vertically in the beam. These alter the state of stress in the beam and cause a stress concentration near the tip of the crack. In time, the flexural cracks extend to become flexure-shear cracks. Empirical equations have 1999 by CRC Press LLC
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been developed to calculate the flexure-shear cracking load, since this cracking cannot be predicted by calculating the principal stresses. In the ACI Code, the basic design equation for the shear capacity of concrete beams is as follows: Vu ≤ φVn
(4.62)
where Vu is the shear force due to the factored loads, φ is the strength reduction factor equal to 0.85 for shear, and Vn is the nominal shear resistance, which is given by Vn = Vc + Vs
(4.63)
where Vc is the shear carried by the concrete and Vs is the shear carried by the shear reinforcement. The torsional capacity of a beam as given in ACI 11.6.5 is as follows: Tu ≤ φTn
(4.64)
where Tu is the torsional moment due to factored loads, φ is the strength reduction factor equal to 0.85 for torsion, and Tn is the nominal torsional moment strength given by Tn = Tc + Tc
(4.65)
where Tc is the torsional moment strength provided by the concrete and Ts is the torsional moment strength provided by the torsion reinforcement. Design of Beams and One-Way Slabs Without Shear Reinforcement: for Shear
The critical section for shear in reinforced concrete beams is taken at a distance d from the face of the support. Sections located at a distance less than d from the support are designed for the shear computed at d. Shear Strength Provided by Concrete Beams without web reinforcement will fail when inclined cracking occurs or shortly afterwards. For this reason the shear capacity is taken equal to the inclined cracking shear. ACI gives the following equations for calculating the shear strength provided by the concrete for beams without web reinforcement subject to shear and flexure: p (4.66) Vc = 2 fc0 bw d or, with a more detailed equation: p p Vu d 0 bw d ≤ 3.5 fc0 bw d Vc = 1.9 fc + 2500ρw Mu
(4.67)
The quantity Vu d/Mu is not to be taken greater than 1.0 in computing Vc where Mu is the factored moment occurring simultaneously with Vu at the section considered. Combined Shear, Moment, and Axial Load For members that are also subject to axial compression, ACI modifies Equation 4.66 as follows (ACI 11.3.1.2): p Nu fc0 bw d (4.68) Vc = 2 1 + 2000Ak where Nu is positive in compression. ACI 11.3.2.2 contains a more detailed calculation for the shear strength of members subject to axial compression. For members subject to axial tension, ACI 11.3.1.3 states that shear reinforcement shall be designed to carry total shear. As an alternative, ACI 11.3.2.3 gives the following for the shear strength of members subject to axial tension: p Nu fc0 bw d (4.69) Vc = 2 1 + 500Ag 1999 by CRC Press LLC
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p where Nu is negative in tension. In Equation 4.68 and 4.69 the terms fc0 , Nu /Ag , 2000, and 500 all have units of psi. Combined Shear, Moment, and Torsion For members subject to torsion, ACI 11.3.1.4 gives the equation for the shear strength of the concrete as the following: p 2 fc0 bw d (4.70) Vc = q 1 + (2.5Ct Tu /Vu )2 where
p X x2y Tu ≥ φ 0.5 fc0
Design of Beams and One-Way Slabs Without Shear Reinforcements: for Torsion
ACI 11.6.1 requires that torsional moments be considered in design if p X x2y Tu ≥ φ 0.5 fc0
(4.71)
Otherwise, torsion effects may be neglected. The critical section for torsion is taken at a distance d from the face of support, and sections located at a distance less than d are designed for the torsion at d. If a concentrated torque occurs within this distance, the critical section is taken at the face of the support. Torsional Strength Provided by Concrete Torsion seldom occurs by itself; bending moments and shearing forces are typically present also. In an uncracked member, shear forces as well as torques produce shear stresses. Flexural shear forces and torques interact in a way that reduces the strength of the member compared with what it would be if shear or torsion were acting alone. The interaction between shear and torsion is taken into account by the use of a circular interaction equation. For more information, refer to Reinforced Concrete Mechanics and Design by James G. MacGregor [8]. The torsional moment strength provided by the concrete is given in ACI 11.6.6.1 as p 0.8 fc0 x 2 y (4.72) Tc = q 1 + (0.4Vu /Ct Tu )2 Combined Torsion and Axial Load For members subject to significant axial tension, ACI 11.6.6.2 states that the torsion reinforcement must be designed to carry the total torsional moment, or as an alternative modify Tc as follows: p 0.8 fc0 x 2 y Nu (4.73) 1+ Tc = q 500Ag 1 + (0.4Vu /Ct Tu )2
where Nu is negative for tension. Design of Beams and One-Way Slabs without Shear Reinforcement: Minimum Reinforcement ACI 11.5.5.1 requires a minimum amount of web reinforcement to be provided for shear and torsion if the factored shear force Vu exceeds one half the shear strength provided by the concrete (Vu ≥ 0.5φVc ) except in the following:
(a) Slabs and footings (b) Concrete joist construction 1999 by CRC Press LLC
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(c) Beams with total depth not greater than 10 inches, 2 1/2 times the thickness of the flange, or 1/2 the width of the web, whichever is greatest The minimum area of shear reinforcement shall be at least p X 50bw s x2y for Tu < φ 0.5 fc0 Av(min) = fy
(4.74)
When torsion is to be considered in design, the sum of the closed stirrups for shear and torsion must satisfy the following: 50bw s (4.75) Av + 2At ≥ fy where Av is the area of two legs of a closed stirrup and At is the area of only one leg of a closed stirrup. Design of Stirrup Reinforcement for Shear and Torsion Shear Reinforcement
Shear reinforcement is to be provided when Vu ≥ φVc , such that Vs ≥
Vu − Vc φ
(4.76)
The design yield strength of the shear reinforcement is not to exceed 60,000 psi. When the shear reinforcement is perpendicular to the axis of the member, the shear resisted by the stirrups is Av fy d (4.77) Vs = s If the shear reinforcement is inclined at an angle α, the shear resisted by the stirrups is Vs =
Av fy (sin α + cos α) d s
(4.78)
Tu − Tc φ
(4.79)
p The maximum shear strength of the shear reinforcement is not to exceed 8 fc0 bw d as stated in ACI 11.5.6.8. Spacing Limitations for Shear Reinforcement ACI 11.5.4.1 sets the maximum spacing of vertical stirrups as the smaller of d/2 or 24 inches. The maximum spacing of inclined stirrups is such that a 45◦ line extending from midheight of the member to the tension reinforcement will intercept at least one stirrup.p If Vs exceeds 4 fc0 bw d, the maximum allowable spacings are reduced to one half of those just described. Torsion Reinforcement Torsion reinforcement is to be provided when Tu ≥ φTc , such that Ts ≥
The design yield strength of the torsional reinforcement is not to exceed 60,000 psi. The torsional moment strength of the reinforcement is computed by Ts =
At αt x1 y1 fy s
(4.80)
where αt = [0.66 + 0.33 (yt /xt )] ≥ 1.50 1999 by CRC Press LLC
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(4.81)
where At is the area of one leg of a closed stirrup resisting torsion within a distance s. The torsional moment strength is not to exceed 4 Tc as given in ACI 11.6.9.4. Longitudinal reinforcement is to be provided to resist axial tension that develops as a result of the torsional moment (ACI 11.6.9.3). The required area of longitudinal bars distributed around the perimeter of the closed stirrups that are provided as torsion reinforcement is to be Al
≥
Al
≥
(x1 + y1 ) 2At s " ! # 400xs Tu x1 + y1 = 2A t fy s Tu + Vu
(4.82)
3Ct
Spacing Limitations for Torsion Reinforcement ACI 11.6.8.1 gives the maximum spacing of closed stirrups as the smaller of (x1 + y1 )/4 or 12 inches. The longitudinal bars are to spaced around the circumference of the closed stirrups at not more than 12 inches apart. At least one longitudinal bar is to be placed in each corner of the closed stirrups (ACI 11.6.8.2). Design of Deep Beams
ACI 11.8 covers the shear design of deep beams. This section applies to members with ln /d 60,000 psi
(4.106) (4.107)
but not less than 12 inches. For fc0 less than 3000 psi, the lap length must be increased by one-third. When different size bars are lap spliced in compression, the splice length is to be the larger of: 1. Compression splice length of the smaller bar, or 2. Compression development length of larger bar. Compression lap splices are allowed for no. 14 and no. 18 bars to no. 11 or smaller bars (ACI 12.16.2). End-Bearing Splices End-bearing splices are allowed for compression only where the compressive stress is transmitted by bearing of square cut ends held in concentric contact by a suitable device. According to ACI 12.16.4.2 bar ends must terminate in flat surfaces within 1 1/2◦ of right angles to the axis of the bars and be fitted within 3◦ of full bearing after assembly. End-bearing splices are only allowed in members containing closed ties, closed stirrups, or spirals. Welded Splices or Mechanical Connections Bars stressed in tension or compression may be spliced by welding or by various mechanical connections. ACI 12.14.3, 12.15.3, 12.15.4, and 12.16.3 govern the use of such splices. For further information see Reinforced Concrete Design, by Chu-Kia Wang and Charles G. Salmon [11]. Bundled Bars
The requirements of ACI 12.4.1 specify that the development length for bundled bars be based on that for the individual bar in the bundle, increased by 20% for a three-bar bundle and 33% for a four-bar bundle. ACI 12.4.2 states that “a unit of bundled bars shall be treated as a single bar of a diameter derived from the equivalent total area” when determining the appropriate modification factors in ACI 12.2.3 and 12.2.4.3. Web Reinforcement
ACI 12.13.1 requires that the web reinforcement be as close to the compression and tension faces as cover and bar-spacing requirements permit. The ACI Code requirements for stirrup anchorage are illustrated in Figure 4.4. (a) ACI 12.13.3 requires that each bend away from the ends of a stirrup enclose a longitudinal bar, as seen in Figure 4.4(a). 1999 by CRC Press LLC
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FIGURE 4.4: Stirrup detailing requirements.
(b) For no. 5 or D31 wire stirrups and smaller with any yield strength and for no. 6, 7, and 8 bars with a yield strength of 40,000 psi or less, ACI 12.13.2.1 allows the use of a standard hook around longitudinal reinforcement, as shown in Figure 4.4(b). (c) For no. 6, 7, and 8 stirrups with fy greater than 40,000 psi, ACI 12.13.2.2 requires a standard hook around a longitudinal bar plus an embedment p between midheight of the member and the outside end of the hook of at least 0.014 db fy / fc0 . (d) Requirements for welded wire fabric forming U stirrups are given in ACI 12.13.2.3. 1999 by CRC Press LLC
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(e) Pairs of U stirrups that form a closed unit shall have a lap length of 1.3ld as shown in Figure 4.4(c). This type of stirrup has proven unsuitable in seismic areas. (f) Requirements for longitudinal bars bent to act as shear reinforcement are given in ACI 12.13.4.
4.7 4.7.1
Two-Way Systems Definition
When the ratio of the longer to the shorter spans of a floor panel drops below 2, the contribution of the longer span in carrying the floor load becomes substantial. Since the floor transmits loads in two directions, it is defined as a two-way system, and flexural reinforcement is designed for both directions. Two-way systems include flat plates, flat slabs, two-way slabs, and waffle slabs (see Figure 4.5). The choice between these different types of two-way systems is largely a matter of the architectural layout, magnitude of the design loads, and span lengths. A flat plate is simply a slab of uniform thickness supported directly on columns, generally suitable for relatively light loads. For larger loads and spans, a flat slab becomes more suitable with the column capitals and drop panels providing higher shear and flexural strength. A slab supported on beams on all sides of each floor panel is generally referred to as a two-way slab. A waffle slab is equivalent to a two-way joist system or may be visualized as a solid slab with recesses in order to decrease the weight of the slab.
FIGURE 4.5: Two-way systems.
4.7.2
Design Procedures
The ACI code [1] states that a two-way slab system “may be designed by any procedure satisfying conditions of equilibrium and geometric compatibility if shown that the design strength at every section is at least equal to the required strength . . . and that all serviceability conditions, including 1999 by CRC Press LLC
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specified limits on deflections, are met” (p.204). There are a number of possible approaches to the analysis and design of two-way systems based on elastic theory, limit analysis, finite element analysis, or combination of elastic theory and limit analysis. The designer is permitted by the ACI Code to adopt any of these approaches provided that all safety and serviceability criteria are satisfied. In general, only for cases of a complex two-way system or unusual loading would a finite element analysis be chosen as the design approach. Otherwise, more practical design approaches are preferred. The ACI Code details two procedures—the direct design method and the equivalent frame method—for the design of floor systems with or without beams. These procedures were derived from analytical studies based on elastic theory in conjunction with aspects of limit analysis and results of experimental tests. The primary difference between the direct design method and equivalent frame method is in the way moments are computed for two-way systems. The yield-line theory is a limit analysis method devised for slab design. Compared to elastic theory, the yield-line theory gives a more realistic representation of the behavior of slabs at the ultimate limit state, and its application is particularly advantageous for irregular column spacing. While the yield-line method is an upper-bound limit design procedure, strip method is considered to give a lower-bound design solution. The strip method offers a wide latitude of design choices and it is easy to use; these are often cited as the appealing features of the method. Some of the earlier design methods based on moment coefficients from elastic analysis are still favored by many designers. These methods are easy to apply and give valuable insight into slab behavior; their use is especially justified for many irregular slab cases where the preconditions of the direct design method are not met or when column interaction is not significant. Table 4.7 lists the moment coefficients taken from method 2 of the 1963 ACI Code. TABLE 4.7
Elastic Moment Coefficients for Two-Way Slabs Short span
Long span, all
Span ratio, short/long 1.0
0.9
0.8
0.7
0.6
0.5 and less
span ratios
Case 1—Interior panels Negative moment at: Continuous edge Discontinuous edge Positive moment at midspan
0.033 — 0.025
0.040 — 0.030
0.048 — 0.036
0.055 — 0.041
0.063 — 0.047
0.083 — 0.062
0.033 — 0.025
Case 2—One edge discontinuous Negative moment at: Continuous edge Discontinuous edge Positive moment at midspan
0.041 0.021 0.031
0.048 0.024 0.036
0.055 0.027 0.041
0.062 0.031 0.047
0.069 0.035 0.052
0.085 0.042 0.064
0.041 0.021 0.031
Case 3—Two edges discontinuous Negative moment at: Continuous edge Discontinuous edge Positive moment at midspan:
0.049 0.025 0.037
0.057 0.028 0.043
0.064 0.032 0.048
0.071 0.036 0.054
0.078 0.039 0.059
0.090 0.045 0.068
0.049 0.025 0.037
Case 4—Three edges discontinuous Negative moment at: Continuous edge Discontinuous edge Positive moment at midspan:
0.058 0.029 0.044
0.066 0.033 0.050
0.074 0.037 0.056
0.082 0.041 0.062
0.090 0.045 0.068
0.098 0.049 0.074
0.058 0.029 0.044
Case 5—Four edges discontinuous Negative moment at: Continuous edge Discontinuous edge Positive moment at midspan
— 0.033 0.050
— 0.038 0.057
— 0.043 0.064
— 0.047 0.072
— 0.053 0.080
— 0.055 0.083
— 0.033 0.050
Moments
As in the 1989 code, two-way slabs are divided into column strips and middle strips as indicated by Figure 4.6, where l1 and l2 are the center-to-center span lengths of the floor panel. A column strip is 1999 by CRC Press LLC
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FIGURE 4.6: Definitions of equivalent frame, column strip, and middle strip. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 318-89 (Revised 92) and ACI 318R-89 (Revised 92), Detroit, MI. With permission.)
a design strip with a width on each side of a column centerline equal to 0.25l2 or 0.25l1 , whichever is less. A middle strip is a design strip bounded by two column strips. Taking the moment coefficients from Table 4.7, bending moments per unit width M for the middle strips are computed from the formula M = (Coef.)wls2
(4.108)
where w is the total uniform load per unit area and ls is the shorter span length of l1 and l2 . The average moments per unit width in the column strip is taken as two-thirds of the corresponding moments in the middle strip.
4.7.3
Minimum Slab Thickness and Reinforcement
ACI Code Section 9.5.3 contains requirements to determine minimum slab thickness of a two-way system for deflection control. For slabs without beams, the thickness limits are summarized by Table 4.8, but thickness must not be less than 5 in. for slabs without drop panels or 4 in. for slabs with drop panels. In Table 4.8 ln is the length of clear span in the long direction and α is the ratio of flexural stiffness of beam section to flexural stiffness of a width of slab bounded laterally by centerline of adjacent panel on each side of beam. For slabs with beams, it is necessary to compute the minimum thickness h from fy ln 0.8 + 200, 000 h= 1 36 + 5β αm − 0.12 1 + β 1999 by CRC Press LLC
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(4.109)
but not less than
h=
fy 0.8 + 200, 000 36 + 9β
ln
and need not be more than
h=
fy 0.8 + 200, 000 36
ln
(4.110)
(4.111)
where β is the ratio of clear spans in long-to-short direction and αm is the average value of α for all beams on edges of a panel. In no case should the slab thickness be less than 5 in. for αm < 2.0 or less than 3 1/2 in. for αm ≥ 2.0. Minimum reinforcement in two-way slabs is governed by shrinkage and temperature controls to minimize cracking. The minimum reinforcement area stipulated by the ACI Code shall not be less than 0.0018 times the gross concrete area when grade 60 steel is used (0.0020 when grade 40 or grade 50 is used). The spacing of reinforcement in two-way slabs shall exceed neither two times the slab thickness nor 18 in. TABLE 4.8 Minimum Thickness of Two-Way Slabs without Beams Yield stress
Exterior panels
fy , psia
Without edge beams
40,000
ln /33
ln /36
ln /36
60,000
ln /30
ln /33
ln /33
With edge beamsb
Interior panels
Without drop panels
With drop panels 40,000
ln /36
ln /40
ln /40
ln /33 ln /36 ln /36 60,000 a For values of reinforcement yield stress between 40,000
and 60,000 psi minimum thickness shall be obtained by linear interpolation.
b Slabs with beams between columns along exterior
edges. The value of α for the edge beam shall not be less than 0.8. From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 31889 (Revised 92) and ACI 318R-89 (Revised 92), Detroit, MI. With permission.
4.7.4
Direct Design Method
The direct design method consists of a set of rules for the design of two-ways slabs with or without beams. Since the method was developed assuming simple designs and construction, its application is restricted by the code to two-way systems with a minimum of three continuous spans, successive span lengths that do not differ by more than one-third, columns with offset not more than 10% of the span, and all loads are due to gravity only and uniformly distributed with live load not exceeding three times dead load. The direct design method involves three fundamental steps: (1) determine the total factored static moment; (2) distribute the static moment to negative and positive sections; 1999 by CRC Press LLC
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and (3) distribute moments to column and middle strips and to beams, if any. The total factored static moment Mo for a span bounded laterally by the centerlines of adjacent panels (see Figure 4.6) is given by wu l2 ln2 (4.112) Mo = 8 In an interior span, 0.65 Mo is assigned to each negative section and 0.35 Mo is assigned to the positive section. In an end span, Mo is distributed according to Table 4.9. If the ratio of dead load to live load is less than 2, the effect of pattern loading is accounted for by increasing the positive moment following provisions in ACI Section 13.6.10. Negative and positive moments are then proportioned to the column strip following the percentages in Table 4.10, where βt is the ratio of the torsional stiffness of edge beam section to flexural stiffness of a width of slab equal to span length of beam. The remaining moment not resisted by the column strip is proportionately assigned to the corresponding half middle strip. If beams are present, they are proportioned to resist 85% of column strip moments. When (αl2 / l1 ) is less than 1.0, the proportion of column strip moments resisted by beams is obtained by linear interpolation between 85% and zero. The shear in beams is determined from loads acting on tributary areas projected from the panel corners at 45 degrees. TABLE 4.9
Direct Design Method—Distribution of Moment in End Span (1)
(2)
(3) (4) Slab without beams between interior supports
Slab with
Interior negativefactored moment Positive-factored moment Exterior negativefactored moment
(5)
Exterior edge unrestrained
beams between all supports
Without edge beam
With edge beam
Exterior edge fully restrained
0.75
0.70
0.70
0.70
0.65
0.63
0.57
0.52
0.50
0.35
0
0.16
0.26
0.30
0.65
From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 318-89 (Revised 92) and ACI 318R-89 (Revised 92), Detroit, MI. With permission.
TABLE 4.10 Proportion of Moment to Column Strip in Percent Interior negative-factored moment `2 /`1 (α1 `2 /`1 ) = 0 (α1 `2 /`1 ) ≥ 1.0
0.5 75 90
1.0 75 75
2.0 75 45
100 75 100 75
100 75 100 45
Positive-factored moment (α1 `2 /`1 ) = 0 (α1 `2 /`1 ) ≥ 1.0
Bt Bt Bt Bt
=0 ≥ 2.5 =0 = 2.5
100 75 100 90
Exterior negative-factored moment (α1 `2 /`1 ) = 0 (α1 `2 /`1 ) ≥ 1.0
60 90
60 75
60 45
From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 318-89 (Revised 92) and ACI 318R-89 (Revised 92), Detroit, MI. With permission.
1999 by CRC Press LLC
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4.7.5
Equivalent Frame Method
For two-way systems not meeting the geometric or loading preconditions of the direct design method, design moments may be computed by the equivalent frame method. This is a more general method and involves the representation of the three-dimensional slab system by dividing it into a series of two-dimensional “equivalent” frames (Figure 4.6). The complete analysis of a two-way system consists of analyzing the series of equivalent interior and exterior frames that span longitudinally and transversely through the system. Each equivalent frame, which is centered on a column line and bounded by the center lines of the adjacent panels, comprises a horizontal slab-beam strip and equivalent columns extending above and below the slab beam (Figure 4.7). This structure is analyzed
FIGURE 4.7: Equivalent column (columns plus torsional members).
as a frame for loads acting in the plane of the frame, and the moments obtained at critical sections across the slab-beam strip are distributed to the column strip, middle strip, and beam in the same manner as the direct design method (see Table 4.10). In its original development, the equivalent frame method assumed that analysis would be done by moment distribution. Presently, frame analysis is more easily accomplished in design practice with computers using general purpose programs based on the direct stiffness method. Consequently, the equivalent frame method is now often used as a method for modeling a two-way system for computer analysis. For the different types of two-way systems, the moment of inertias for modeling the slab-beam element of the equivalent frame are indicated in Figure 4.8. Moments of inertia of slab beams are based on the gross area of concrete; the variation in moment of inertia along the axis is taken into account, which in practice would mean that a node would be located on the computer model where a change of moment of inertia occurs. To account for the increased stiffness between the center of the column and the face of column, beam, or capital, the moment of inertia is divided by the quantity (1 − c2 / l2 )2 , where c2 and l2 are measured transverse to the direction of the span. For 1999 by CRC Press LLC
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FIGURE 4.8: Slab-beam stiffness by equivalent frame method. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 318-89 (Revised 92) and ACI 318R-89 (Revised 92), Detroit, MI. With permission.)
column modeling, the moment of inertia at any cross-section outside of joints or column capitals may be based on the gross area of concrete, and the moment of inertia from the top to bottom of the slab-beam joint is assumed infinite. Torsion members (Figure 4.7) are elements in the equivalent frame that provide moment transfer between the horizontal slab beam and vertical columns. The cross-section of torsional members are assumed to consist of the portion of slab and beam having a width according to the conditions depicted 1999 by CRC Press LLC
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FIGURE 4.9: Torsional members. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 318-89 (Revised 92) and ACI 318R-89 (Revised 92), Detroit, MI. With permission.) in Figure 4.9. The stiffness Kt of the torsional member is calculated by the following expression: Kt =
X
9Ecs C 3 l2 1 − cl22
(4.113)
where Ecs is the modulus of elasticity of the slab concrete and torsional constant C may be evaluated by dividing the cross-section into separate rectangular parts and carrying out the following summation: C=
X
1 − 0.63
x y
x3y 3
(4.114)
where x and y are the shorter and longer dimension, respectively, of each rectangular part. Where beams frame into columns in the direction of the span, the increased torsional stiffness Kta is obtained by multiplying the value Kt obtained from Equation 4.113 by the ratio of (a) moment inertia of slab with such beam, to (b) moment of inertia of slab without such beam. Various ways have been suggested for incorporating torsional members into a computer model of an equivalent frame. The model implied by the ACI Code is one that has the slab beam connected to the torsional members, which are projected out of the plane of the columns. Others have suggested that the torsional 1999 by CRC Press LLC
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members be replaced by rotational springs at column ends or, alternatively, at the slab-beam ends. Or, instead of rotational springs, columns may be modeled with an equivalent value of the moment of inertia modified by the equivalent column stiffness Kec given in the commentary of the code. Using Figure 4.7, Kec is computed as Kct + Kcb (4.115) Kec = Kct + Kcb 1+ Kta + Kta where Kct and Kcb are the top and bottom flexural stiffnesses of the column.
4.7.6
Detailing
The ACI Code specifies that reinforcement in two-way slabs without beams have minimum extensions as prescribed in Figure 4.10. Where adjacent spans are unequal, extensions of negative moment
FIGURE 4.10: Minimum extensions for reinforcement in two-way slabs without beams. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 318-89 (Revised 92) and ACI 318R-89 (Revised 92), Detroit, MI. With permission.)
reinforcement shall be based on the longer span. Bent bars may be used only when the depth-span ratio permits use of bends 45 degrees or less. And at least two of the column strip bottom bars in each direction shall be continuous or spliced at the support with Class A splices or anchored within support. These bars must pass through the column and be placed within the column core. The purpose of this “integrity steel” is to give the slab some residual capacity following a single punching shear failure. The ACI Code requires drop panels to extend in each direction from centerline of support a distance not less than one-sixth the span length, and the drop panel must project below the slab at 1999 by CRC Press LLC
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least one-quarter of the slab thickness. The effective support area of a column capital is defined by the intersection of the bottom surface of the slab with the largest right circular cone whose surfaces are located within the column and capital and are oriented no greater than 45 degrees to the axis of the column.
4.8
Frames
A structural frame is a three-dimensional structural system consisting of straight members that are built monolithically and have rigid joints. The frame may be one bay long and one story high—such as portal frames and gable frames—or it may consist of multiple bays and stories. All members of frame are considered continuous in the three directions, and the columns participate with the beams in resisting external loads. Consideration of the behavior of reinforced concrete frames at and near the ultimate load is necessary to determine the possible distributions of bending moment, shear force, and axial force that could be used in design. It is possible to use a distribution of moments and forces different from that given by linear elastic structural analysis if the critical sections have sufficient ductility to allow redistribution of actions to occur as the ultimate load is approached. Also, in countries that experience earthquakes, a further important design is the ductility of the structure when subjected to seismic-type loading, since present seismic design philosophy relies on energy dissipation by inelastic deformations in the event of major earthquakes.
4.8.1
Analysis of Frames
A number of methods have been developed over the years for the analysis of continuous beams and frames. The so-called classical methods—such as application of the theorem of three moments, the method of least work, and the general method of consistent deformation—have proved useful mainly in the analysis of continuous beams having few spans or of very simple frames. For the more complicated cases usually met in practice, such methods prove to be exceedingly tedious, and alternative approaches are preferred. For many years the closely related methods of slope deflection and moment distribution provided the basic analytical tools for the analysis of indeterminate concrete beams and frames. In offices with access to high-speed digital computers, these have been supplanted largely by matrix methods of analysis. Where computer facilities are not available, moment distribution is still the most common method. Approximate methods of analysis, based either on an assumed shape of the deformed structure or on moment coefficients, provide a means for rapid estimation of internal forces and moments. Such estimates are useful in preliminary design and in checking more exact solutions, and in structures of minor importance may serve as the basis for final design. Slope Deflection
The method of slope deflection entails writing two equations for each member of a continuous frame, one at each end, expressing the end moment as the sum of four contributions: (1) the restraining moment associated with an assumed fixed-end condition for the loaded span, (2) the moment associated with rotation of the tangent to the elastic curve at the near end of the member, (3) the moment associated with rotation of the tangent at the far end of the member, and (4) the moment associated with translation of one end of the member with respect to the other. These equations are related through application of requirements of equilibrium and compatibility at the joints. A set of simultaneous, linear algebraic equations results for the entire structure, in which the structural displacements are unknowns. Solution for these displacements permits the calculation of all internal forces and moments. This method is well suited to solving continuous beams, provided there are not very many spans. 1999 by CRC Press LLC
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Its usefulness is extended through modifications that take advantage of symmetry and antisymmetry, and of hinge-end support conditions where they exist. However, for multistory and multibay frames in which there are a large number of members and joints, and which will, in general, involve translation as well as rotation of these joints, the effort required to solve the correspondingly large number of simultaneous equations is prohibitive. Other methods of analysis are more attractive. Moment Distribution
The method of moment distribution was developed to solve problems in frame analysis that involve many unknown joint displacements. This method can be regarded as an iterative solution of the slope-deflection equations. Starting with fixed-end moments for each member, these are modified in a series of cycles, each converging on the precise final result, to account for rotation and translation of the joints. The resulting series can be terminated whenever one reaches the degree of accuracy required. After obtaining member-end moments, all member stress resultants can be obtained by use of the laws of statics. Matrix Analysis
Use of matrix theory makes it possible to reduce the detailed numerical operations required in the analysis of an indeterminate structure to systematic processes of matrix manipulation which can be performed automatically and rapidly by computer. Such methods permit the rapid solution of problems involving large numbers of unknowns. As a consequence, less reliance is placed on special techniques limited to certain types of problems; powerful methods of general applicability have emerged, such as the matrix displacement method. Account can be taken of such factors as rotational restraint provided by members perpendicular to the plane of a frame. A large number of alternative loadings may be considered. Provided that computer facilities are available, highly precise analyses are possible at lower cost than for approximate analyses previously employed. Approximate Analysis In spite of the development of refined methods for the analysis of beams and frames, increasing attention is being paid to various approximate methods of analysis. There are several reasons for this. Prior to performing a complete analysis of an indeterminate structure, it is necessary to estimate the proportions of its members in order to know their relative stiffness upon which the analysis depends. These dimensions can be obtained using approximate analysis. Also, even with the availability of computers, most engineers find it desirable to make a rough check of results—using approximate means—to detect gross errors. Further, for structures of minor importance, it is often satisfactory to design on the basis of results obtained by rough calculation. Provided that points of inflection (locations in members at which the bending moment is zero and there is a reversal of curvature of the elastic curve) can be located accurately, the stress resultants for a framed structure can usually be found on the basis of static equilibrium alone. Each portion of the structure must be in equilibrium under the application of its external loads and the internal stress resultants. The use of approximate analysis in determining stress resultants in frames is illustrated using a simple rigid frame in Figure 4.11. ACI Moment Coefficients
The ACI Code [1] includes moment and shear coefficients that can be used for the analysis of buildings of usual types of construction, span, and story heights. They are given in ACI Code Sec. 8.3.3. The ACI coefficients were derived with due consideration of several factors: a maximum allowable ratio of live to dead load (3:1); a maximum allowable span difference (the larger of two adjacent spans not exceed the shorter by more than 20%); the fact that reinforced concrete beams are never simply supported but either rest on supports of considerable width, such as walls, or are 1999 by CRC Press LLC
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FIGURE 4.11: Approximate analysis of rigid frame.
built monolithically like columns; and other factors. Since all these influences are considered, the ACI coefficients are necessarily quite conservative, so that actual moments in any particular design are likely to be considerably smaller than indicated. Consequently, in many reinforced concrete structures, significant economy can be effected by making a more precise analysis. Limit Analysis
Limit analysis in reinforced concrete refers to the redistribution of moments that occurs throughout a structure as the steel reinforcement at a critical section reaches its yield strength. Under working loads, the distribution of moments in a statically indeterminate structure is based on elastic theory, and the whole structure remains in the elastic range. In limit design, where factored loads are used, the distribution of moments at failure when a mechanism is reached is different from that distribution based on elastic theory. The ultimate strength of the structure can be increased as more sections reach their ultimate capacity. Although the yielding of the reinforcement introduces large deflections, which should be avoided under service, a statically indeterminate structure does not collapse when the reinforcement of the first section yields. Furthermore, a large reserve of strength is present between the initial yielding and the collapse of the structure. In steel design the term plastic design is used to indicate the change in the distribution of moments in the structure as the steel fibers, at a critical section, are stressed to their yield strength. Limit analysis of reinforced concrete developed as a result of earlier research on steel structures. Several studies had been performed on the principles of limit design and the rotation capacity of reinforced concrete plastic hinges. Full utilization of the plastic capacity of reinforced concrete beams and frames requires an extensive analysis of all possible mechanisms and an investigation of rotation requirements and capacities at 1999 by CRC Press LLC
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all proposed hinge locations. The increase of design time may not be justified by the limited gains obtained. On the other hand, a restricted amount of redistribution of elastic moments can safely be made without complete analysis and may be sufficient to obtain most of the advantages of limit analysis. A limited amount of redistribution is permitted under the ACI Code, depending upon a rough measure of available ductility; without explicit calculation of rotation requirements and capacities. The ratio ρ/ρb —or in the case of doubly reinforced members, (ρ − ρ 0 )/ρb —is used as an indicator of rotation capacity, where ρb is the balanced steel ratio. For singly reinforced members with ρ = ρb , experiments indicate almost no rotation capacity, since the concrete strain is nearly equal to εcu when steel yielding is initiated. Similarly, in a doubly reinforced member, when ρ − ρ 0 = ρb , very little rotation will occur after yielding before the concrete crushes. However, when ρ or ρ − ρ 0 is low, extensive rotation is usually possible. Accordingly, ACI Code Sec. 8.3 provides as follows: Except where approximate values for moments are used, it is permitted to increase or decrease negative moments calculated by elastic theory at supports of continuous flexural members for any assumed loading arrangement by not more than 20 [1 − (ρ − ρ 0 )/ρb ] percent. The modified negative moments shall be used for calculating moments at sections within the spans. Redistribution of negative moments shall be made only when the section at which moment is reduced is so designed that ρ or ρ − ρ 0 is not greater than 0.5 ρb [1992].
4.8.2
Design for Seismic Loading
The ACI Code contains provisions that are currently considered to be the minimum requirements for producing a monolithic concrete structure with adequate proportions and details to enable the structure to sustain a series of oscillations into the inelastic range of response without critical decay in strength. The provisions are intended to apply to reinforced concrete structures located in a seismic zone where major damage to construction has a high possibility of occurrence, and are designed with a substantial reduction in total lateral seismic forces due to the use of lateral load-resisting systems consisting of ductile moment-resisting frames. The provisions for frames are divided into sections on flexural members, columns, and joints of frames. Some of the important points stated are summarized below. Flexural Members
Members having a factored axial force not exceeding Ag fc0 /10, where Ag is gross section of area (in.2 ), are regarded as flexural members. An upper limit is placed on the flexural steel ratio ρ. The maximum value of ρ should not exceed 0.025. Provision is also made to ensure that a minimum quantity of top and bottom reinforcement is always present. Both the top and the bottom steel are to have a steel ratio of a least 200/fy , with the steel yield strength fy in psi throughout the length of the member. Recommendations are also made to ensure that sufficient steel is present to allow for unforeseen shifts in the points of contraflexure. At column connections, the positive moment capacity should be at least 50% of the negative moment capacity, and the reinforcement should be continuous through columns where possible. At external columns, beam reinforcement should be terminated in the far face of the column using a hook plus any additional extension necessary for anchorage. The design shear force Ve should be determined from consideration of the static forces on the portion of the member between faces of the joints. It should be assumed that moments of opposite sign corresponding to probable strength Mpr act at the joint faces and that the member is loaded with the factored tributary gravity load along its span. Figure 4.12 illustrates the calculation. Minimum web reinforcement is provided throughout the length of the member, and spacing should not exceed 1999 by CRC Press LLC
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FIGURE 4.12: Design shears for girders and columns. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 318-89 (Revised 92) and ACI 318R-89 (Revised 92), Detroit, MI. With permission.) d/4 in plastic hinge zones and d/2 elsewhere, where d is effective depth of member. The stirrups should be closed around bars required to act as compression reinforcement and in plastic hinge regions, and the spacing should not exceed specified values. Columns
Members having a factored axial force exceeding Ag fc0 /10 are regarded as columns of frames serving to resist earthquake forces. These members should satisfy the conditions that the shortest cross-sectional dimension—measured on a straight line passing through the geometric centroid— should not be less than 12 in. and that the ratio of the shortest cross-sectional dimension to the perpendicular dimension should not be less than 0.4. The flexural strengths of the columns should satisfy X X Mg (4.116) Me ≥ (6/5) P where Me is sum of moments, at the center of the joint, P corresponding to the design flexural strength of the columns framing into that joint and where Mg is sum of moments, at the center of the joint, corresponding to the design flexural strengths of the girders framing into that joint. Flexural strengths should be summed such that the column moments oppose the beam moments. 1999 by CRC Press LLC
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Equation 4.116 should be satisfied for beam moments acting in both directions in the vertical plane of the frame considered. The requirement is intended to ensure that plastic hinges form in the girders rather than the columns. The longitudinal reinforcement ratio is limited to the range of 0.01 to 0.06. The lower bound to the reinforcement ratio refers to the traditional concern for the effects of time-dependent deformations of the concrete and the desire to have a sizable difference between the cracking and yielding moments. The upper bound reflects concern for steel congestion, load transfer from floor elements to column in low-rise construction, and the development of large shear stresses. Lap splices are permitted only within the center half of the member length and should be proportioned as tension splices. Welded splices and mechanical connections are allowed for splicing the reinforcement at any section, provided not more than alternate longitudinal bars are spliced at a section and the distance between splices is 24 in. or more along the longitudinal axis of the reinforcement. If Equation 4.116 is not satisfied at a joint, columns supporting reactions from that joint should be provided with transverse reinforcement over their full height to confine the concrete and provide lateral support to the reinforcement. Where a spiral is used, the ratio of volume of spiral reinforcement to the core volume confined by the spiral reinforcement, ρs , should be at least that given by f 0 Ag −1 (4.117) ρs = 0.45 c fy Ac but not less than 0.12 fc0 /fyh , where Ac is the area of core of spirally reinforced compression member measured to outside diameter of spiral in in.2 and fyh is the specified yield strength of transverse reinforcement in psi. When rectangular reinforcement hoop is used, the total cross-sectional area of rectangular hoop reinforcement should not be less than that given by (4.118) Ash = 0.3 shc fc0 /fyh Ag /Ach − 1 0 Ash = 0.09shc fc /fyh (4.119) where s is the spacing of transverse reinforcement measured along the longitudinal axis of column, hc is the cross-sectional dimension of column core measured center-to-center of confining reinforcement, and Ash is the total cross-sectional area of transverse reinforcement (including crossties) within spacing s and perpendicular to dimension hc . Supplementary crossties, if used, should be of the same diameter as the hoop bar and should engage the hoop with a hook. Special transverse confining steel is required for the full height of columns that support discontinuous shear walls. The design shear force Ve should be determined from consideration of the maximum forces that can be generated at the faces of the joints at each end of the column. These joint forces should be determined using the maximum probable moment strength Mpr of the column associated with the range of factored axial loads on the column. The column shears need not exceed those determined from joint strengths based on the probable moment strength Mpr , of the transverse members framing into the joint. In no case should Ve be less than the factored shear determined by analysis of the structure Figure 4.12. Joints of Frames
Development of inelastic rotations at the faces of joints of reinforced concrete frames is associated with strains in the flexural reinforcement well in excess of the yield strain. Consequently, joint shear force generated by the flexural reinforcement is calculated for a stress of 1.25 fy in the reinforcement. Within the depth of the shallowed framing member, transverse reinforcement equal to at least onehalf the amount required for the column reinforcement should be provided where members frame into all four sides of the joint and where each member width is at least three-fourths the column width. Transverse reinforcement as required for the column reinforcement should be provided through the 1999 by CRC Press LLC
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joint to provide confinement for longitudinal beam reinforcement outside the column core if such confinement is not provided by a beam framing into the joint. The nominal shear strength of the joint should not be taken greater than the forces specified below for normal weight aggregate concrete: p 20 fc0 Aj
for joints confined on all four faces
p 15 fc0 Aj
for joints confined on three faces or on two opposite faces
p 12 fc0 Aj
for others
where Aj is the effective cross-sectional area within a joint in a plane parallel to plane of reinforcement generating shear in the joint (Figure 4.13). A member that frames into a face is considered to provide
FIGURE 4.13: Effective area of joint. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 318-89 (Revised 92) and ACI 318R-89 (Revised 92), Detroit, MI. With permission.)
confinement to the joint if at least three-quarters of the face of the joint is covered by the framing member. A joint is considered to be confined if such confining members frame into all faces of the joint. For lightweight-aggregate concrete, the nominal shear strength of the joint should not exceed three-quarters of the limits given above. Details of minimum development length for deformed bars with standard hooks embedded in normal and lightweight concrete and for straight bars are contained in ACI Code Sec. 21.6.4.
1999 by CRC Press LLC
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4.9
Brackets and Corbels
Brackets and corbels are cantilevers having shear span depth ratio, a/d, not greater than unity. The shear span a is the distance from the point of load to the face of support, and the distance d shall be measured at face of support (see Figure 4.14).
FIGURE 4.14: Structural action of a corbel. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 318-89 (Revised 92) and ACI 318R-89 (Revised 92), Detroit, MI. With permission.)
The corbel shown in Figure 4.14 may fail by shearing along the interface between the column and the corbel by yielding of the tension tie, by crushing or splitting of the compression strut, or by localized bearing or shearing failure under the loading plate. The depth of a bracket or corbel at its outer edge should be less than one-half of the required depth d at the support. Reinforcement should consist of main tension bars with area As and shear reinforcement with area Ah (see Figure 4.15 for notation). The area of primary tension reinforcement
FIGURE 4.15: Notation used. (From ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 318-89 (Revised 92) and ACI 318R-89 (Revised 92), Detroit, MI. With permission.) 1999 by CRC Press LLC
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As should be made equal to the greater of (Af + An ) or (2Avf /3 − An ), where Af is the flexural reinforcement required to resist moment [Vu a + Nuc (h − d)], An is the reinforcement required to resist tensile force Nuc , and Avf is the shear-friction reinforcement required to resist shear Vu : Af
=
An
=
Avf
=
Mu Vu a + Nuc (h − d) = φfy j d φfy j d Nuc φfy Vu φfy µ
(4.120) (4.121) (4.122)
In the above equations, fy is the reinforcement yield strength; φ is 0.9 for Equation 4.120 and 0.85 for Equations 4.121 and 4.122. In Equation 4.120, the lever arm j d can be approximated for all practical purposes in most cases as 0.85d. Tensile force Nuc in Equation 4.121 should not be taken less than 0.2 Vu unless special provisions are made to avoid tensile forces. Tensile force Nuc should be regarded as a live load even when tension results from creep, shrinkage, or temperature change. In Equation 4.122, Vu /φ(= Vn ) should not be taken greater than 0.2 fc0 bw d nor 800bw d in pounds in normal-weight concrete. For “all-lightweight” or “sand-lightweight” concrete, shear strength Vn should not be taken greater than (0.2 − 0.07a/d)fc0 bw d nor (800 − 280a/d)bw d in pounds. The coefficient of friction µ in Equation 4.122 should be 1.4λ for concrete placed monolithically, 1.0λ for concrete placed against hardened concrete with surface intentionally roughened, 0.6λ for concrete placed against hardened concrete not intentionally roughened, and 0.7λ for concrete anchored to as-rolled structural steel by headed studs or by reinforcing bars, where λ is 1.0 for normal weight concrete, 0.85 for “sandlightweight” concrete, and 0.75 for “all-lightweight” concrete. Linear interpolation of λ is permitted when partial sand replacement is used. The total area of closed stirrups or ties Ah parallel to As should not be less than 0.5(As − An ) and should be uniformly distributed within two-thirds of the depth of the bracket adjacent to As . At front face of bracket or corbel, primary tension reinforcement As should be anchored in one of the following ways: (a) by a structural weld to a transverse bar of at least equal size; weld to be designed to develop specified yield strength fy of As bars; (b) by bending primary tension bars As back to form a horizontal loop, or (c) by some other means of positive anchorage. Also, to ensure development of the yield strength of the reinforcement As near the load, bearing area of load on bracket or corbel should not project beyond straight portion of primary tension bars As , nor project beyond interior face of transverse anchor bar (if one is provided). When corbels are designed to resist horizontal forces, the bearing plate should be welded to the tension reinforcement As .
4.10
Footings
Footings are structural members used to support columns and walls and to transmit and distribute their loads to the soil in such a way that (a) the load bearing capacity of the soil is not exceeded, (b) excessive settlement, differential settlement, and rotations are prevented, and (c) adequate safety against overturning or sliding is maintained. When a column load is transmitted to the soil by the footing, the soil becomes compressed. The amount of settlement depends on many factors, such as the type of soil, the load intensity, the depth below ground level, and the type of footing. If different footings of the same structure have different settlements, new stresses develop in the structure. Excessive differential settlement may lead to the damage of nonstructural members in the buildings, even failure of the affected parts. Vertical loads are usually applied at the centroid of the footing. If the resultant of the applied loads does not coincide with the centroid of the bearing area, a bending moment develops. In this case, 1999 by CRC Press LLC
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the pressure on one side of the footing will be greater than the pressure on the other side, causing higher settlement on one side and a possible rotation of the footing. If the bearing soil capacity is different under different footings—for example, if the footings of a building are partly on soil and partly on rock—a differential settlement will occur. It is customary in such cases to provide a joint between the two parts to separate them, allowing for independent settlement.
4.10.1
Types of Footings
Different types of footings may be used to support building columns or walls. The most commonly used ones are illustrated in Figure 4.16(a–g). A simple file footing is shown in Figure 4.16(h).
FIGURE 4.16: Common types of footings for walls and columns. (From ACI Committee 340. 1990. Design Handbook in Accordance with the Strength Design Method of ACI 318-89. Volume 2, SP-17. With permission.)
For walls, a spread footing is a slab wider than the wall and extending the length of the wall [Figure 4.16(a)]. Square or rectangular slabs are used under single columns [Figure 4.16(b–d)]. When two columns are so close that their footings would merge or nearly touch, a combined footing [Figure 4.16(e)] extending under the two should be constructed. When a column footing cannot project in one direction, perhaps because of the proximity of a property line, the footing may be helped out by an adjacent footing with more space; either a combined footing or a strap (cantilever) footing [Figure 4.16(f)] may be used under the two. For structures with heavy loads relative to soil capacity, a mat or raft foundation [Figure 4.16(g)] may prove economical. A simple form is a thick, two-way-reinforced-concrete slab extending under 1999 by CRC Press LLC
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the entire structure. In effect, it enables the structure to float on the soil, and because of its rigidity it permits negligible differential settlement. Even greater rigidity can be obtained by building the raft foundation as an inverted beam-and-girder floor, with the girders supporting the columns. Sometimes, also, inverted flat slabs are used as mat foundations.
4.10.2
Design Considerations
Footings must be designed to carry the column loads and transmit them to the soil safely while satisfying code limitations. The design procedure must take the following strength requirements into consideration: • The area of the footing based on the allowable bearing soil capacity • Two-way shear or punching shear • One-way shear • Bending moment and steel reinforcement required • Dowel requirements • Development length of bars • Differential settlement These strength requirements will be explained in the following sections. Size of Footings The required area of concentrically loaded footings is determined from
Areq =
D+L qa
(4.123)
where qa is allowable bearing pressure and D and L are, respectively, unfactored dead and live loads. Allowable bearing pressures are established from principles of soil mechanics on the basis of load tests and other experimental determinations. Allowable bearing pressures qa under service loads are usually based on a safety factor of 2.5 to 3.0 against exceeding the ultimate bearing capacity of the particular soil and to keep settlements within tolerable limits. The required area of footings under the effects of wind W or earthquake E is determined from the following: Areq =
D+L+W 1.33qa
or
D+L+E 1.33qa
(4.124)
It should be noted that footing sizes are determined for unfactored service loads and soil pressures, in contrast to the strength design of reinforced concrete members, which utilizes factored loads and factored nominal strengths. A footing is eccentrically loaded if the supported column is not concentric with the footing area or if the column transmits—at its juncture with the footing—not only a vertical load but also a bending moment. In either case, the load effects at the footing base can be represented by the vertical load P and a bending moment M. The resulting bearing pressures are again assumed to be linearly distributed. As long as the resulting eccentricity e = M/P does not exceed the kern distance k of the footing area, the usual flexure formula qmax, min = 1999 by CRC Press LLC
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Mc P + A I
(4.125)
FIGURE 4.17: Assumed bearing pressures under eccentric footings. permits the determination of the bearing pressures at the two extreme edges, as shown in Figure 4.17(a). The footing area is found by trial and error from the condition qmax ≤ qa . If the eccentricity falls outside the kern, Equation 4.125 gives a negative value for q along one edge of the footing. Because no tension can be transmitted at the contact area between soil and footing, Equation 4.125 is no longer valid and bearing pressures are distributed as in Figure 4.17(b). Once the required footing area has been determined, the footing must then be designed to develop the necessary strength to resist all moments, shears, and other internal actions caused by the applied loads. For this purpose, the load factors of the ACI Code apply to footings as to all other structural components. Depth of footing above bottom reinforcement should not be less than 6 in. for footings on soil, nor less than 12 in. for footings on piles. Two-Way Shear (Punching Shear) ACI Code Sec. 11.12.2 allows a shear strength Vc of footings without shear reinforcement for two-way shear action as follows: p 4 p 0 fc bo d ≤ 4 fc0 bo d (4.126) Vc = 2 + βc
where βc is the ratio of long side to short side of rectangular area, bo is the perimeter of the critical section taken at d/2 from the loaded area (column section), and d is the effective depth of footing. This shear is a measure of the diagonal tension caused by the effect of the column load on the footing. Inclined cracks may occur in the footing at a distance d/2 from the face of the column on all sides. The footing will fail as the column tries to punch out part of the footing, as shown in Figure 4.18. One-Way Shear
For footings with bending action in one direction, the critical section is located at a distance d from the face of the column. The diagonal tension at section m-m in Figure 4.19 can be checked as is done in beams. The allowable shear in this case is equal to p (4.127) φVc = 2φ fc0 bd where b is the width of section m-m. The ultimate shearing force at section m-m can be calculated as follows: L c − −d (4.128) Vu = qu b 2 2 where b is the side of footing parallel to section m-m. 1999 by CRC Press LLC
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FIGURE 4.18: Punching shear (two-way).
FIGURE 4.19: One-way shear. Flexural Reinforcement and Footing Reinforcement
The theoretical sections for moment occur at face of the column (section n-n, Figure 4.20). The bending moment in each direction of the footing must be checked and the appropriate reinforcement must be provided. In square footings the bending moments in both directions are equal. To determine the reinforcement required, the depth of the footing in each direction may be used. As the bars in one direction rest on top of the bars in the other direction, the effective depth d varies with the diameter of the bars used. The value of dmin may be adopted. The depth of footing is often controlled by the shear, which requires a depth greater than that required by the bending moment. The steel reinforcement in each direction can be calculated in the case of flexural members as follows: As =
Mu φfy (d − a/2)
(4.129)
The minimum steel percentage requirement in flexural member is equal to 200/fy . However, ACI Code Sec. 10.5.3 indicates that for structural slabs of uniform thickness, the minimum area 1999 by CRC Press LLC
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FIGURE 4.20: Critical section of bending moment. and maximum spacing of steel in the direction of bending should be as required for shrinkage and temperature reinforcement. This last minimum steel reinforcement is very small and a higher minimum reinforcement ratio is recommended, but not greater than 200/fy . The reinforcement in one-way footings and two-way footings must be distributed across the entire width of the footing. In the case of two-way rectangular footings, ACI Code Sec 15.4.4 specifies that in the long direction the total reinforcement must be placed uniformly within a band width equal to the length of the short side of the footing according to 2 Reinforcement band width = Total reinforcement in short direction β +1
(4.130)
where β is the ratio of the long side to the short side of the footing. The band width must be centered on the centerline of the column (Figure 4.21). The remaining reinforcement in the short direction must be uniformly distributed outside the band width. This remaining reinforcement percentage should not be less than required for shrinkage and temperature.
FIGURE 4.21: Band width for reinforcement distribution. When structural steel columns or masonry walls are used, the critical sections for moments in footing are taken at halfway between the middle and the edge of masonry walls, and halfway between the face of the column and the edge of the steel base place (ACI Code Sec. 15.4.2). 1999 by CRC Press LLC
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Bending Capacity of Column at Base
The loads from the column act on the footing at the base of the column, on an area equal to the area of the column cross-section. Compressive forces are transferred to the footing directly by bearing on the concrete. Tensile forces must be resisted by reinforcement, neglecting any contribution by concrete. Forces acting on the concrete at the base of the column must not exceed the bearing strength of concrete as specified by the ACI Code Sec.10.15: (4.131) N = φ 0.85fc0 A1 where φ is 0.7 and A1 is the bearing area of the column. The value of the bearing strength given √ in Equation 4.131 may be multiplied by a factor A2 /A1 ≤ 2.0 for bearing on footings when the supporting surface is wider on all sides other than the loaded area. Here A2 is the area of the part of the supporting footing that is geometrically similar to and concentric with the load area √ (Figure 4.22). Since A2 > A1 , the factor A2 /A1 is greater than unity, indicating that the allowable
FIGURE 4.22: Bearing areas on footings. A1 = c2 , A2 = b2 .
bearing strength is increased because of the lateral support from the footing area surrounding the √ column base. If the calculated bearing force is greater than N or the modified one with r A2 /A1 , reinforcement must be provided to transfer the excess force. This is achieved by providing dowels or extending the column bars √ into the footing. If the calculated bearing force is less than either N or the modified one with r A2 /A1 , then minimum reinforcement must be provided. ACI Code Sec. 15.8.2 indicates that the minimum area of the dowel reinforcement is at least 0.005Ag but not less than 4 bars, where Ag is the gross area of the column section of the supported member. The minimum reinforcement requirements apply √ to the case in which the calculated bearing forces are greater than N or the modified one with r A2 /A1 . Dowels on Footings
It was explained earlier that dowels are required in any case, even if the bearing strength is adequate. The ACI Code specifies a minimum steel ratio ρ = 0.005 of the column section as compared to ρ = 0.01 as minimum reinforcement for the column itself. The minimum number of dowel bars needed is four; these may be placed at the four corners of the column. The dowel bars are usually extended into the footing, bent at their ends, and tied to the main footing reinforcement. ACI Code Sec. 15.8.2 indicates that #14 and #18 longitudinal bars, in compression only, may be lap-spliced with dowels. Dowels should not be larger than #11 bar and should extend (1) into supported member a distance not less than the development length of #14 or 18” bars or the splice length of the dowels—whichever is greater, and (2) into the footing a distance not less than the development length of the dowels. 1999 by CRC Press LLC
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Development Length of the Reinforcing Bars
The critical sections for checking the development length of reinforcing bars are the same as those for bending moments. Calculated tension or compression in reinforcement at each section should be developed on each side of that section by embedment length, hook (tension only) or mechanical device, or a combination thereof. The development length for compression bar is p (4.132) ld = 0.02fy db fc0 but not less than 0.0003fy db ≥ 8 in. For other values, refer to ACI Code, Chapter 12. Dowel bars must also be checked for proper development length. Differential Settlement
Footings usually support the following loads: • Dead loads from the substructure and superstructure • Live loads resulting from materials or occupancy • Weight of materials used in backfilling • Wind loads Each footing in a building is designed to support the maximum load that may occur on any column due to the critical combination of loadings, using the allowable soil pressure. The dead load, and maybe a small portion of the live load, may act continuously on the structure. The rest of the live load may occur at intervals and on some parts of the structure only, causing different loadings on columns. Consequently, the pressure on the soil under different loadings will vary according to the loads on the different columns, and differential settlement will occur under the various footings of one structure. Since partial settlement is inevitable, the problem is defined by the amount of differential settlement that the structure can tolerate. The amount of differential settlement depends on the variation in the compressibility of the soils, the thickness of compressible material below foundation level, and the stiffness of the combined footing and superstructure. Excessive differential settlement results in cracking of concrete and damage to claddings, partitions, ceilings, and finishes. For practical purposes it can be assumed that the soil pressure under the effect of sustained loadings is the same for all footings, thus causing equal settlements. The sustained load (or the usual load) can be assumed equal to the dead load plus a percentage of the live load, which occurs very frequently on the structure. Footings then are proportioned for these sustained loads to produce the same soil pressure under all footings. In no case is the allowable soil bearing capacity to be exceeded under the dead load plus the maximum live load for each footing.
4.10.3
Wall Footings
The spread footing under a wall [Figure 4.16(a)] distributes the wall load horizontally to preclude excessive settlement. The wall should be so located on the footings as to produce uniform bearing pressure on the soil (Figure 4.23), ignoring the variation due to bending of the footing. The pressure is determined by dividing the load per foot by the footing width. The footing acts as a cantilever on opposite sides of the wall under downward wall loads and upward soil pressure. For footings supporting concrete walls, the critical section for bending moment is at the face of the wall; for footings under masonry walls, halfway between the middle and edge of the 1999 by CRC Press LLC
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FIGURE 4.23: Reinforced-concrete wall footing. wall. Hence, for a one-foot-long strip of symmetrical concrete-wall footing, symmetrically loaded, the maximum moment, ft-lb, is 1 (4.133) Mu = qu (L − a)2 8 where qu is the uniform pressure on soil (lb/ft2 ), L is the width of footing (ft), and a is wall thickness (ft). For determining shear stresses, the vertical shear force is computed on the section located at a distance d from the face of the wall. Thus, L−a (4.134) −L Vu = qu 2 The calculation of development length is based on the section of maximum moment.
4.10.4
Single-Column Spread Footings
The spread footing under a column [Figure 4.16(b–d)] distributes the column load horizontally to prevent excessive total and differential settlement. The column should be located on the footing so as to produce uniform bearing pressure on the soil, ignoring the variation due to bending of the footing. The pressure equals the load divided by the footing area. In plan, single-column footings are usually square. Rectangular footings are used if space restrictions dictate this choice or if the supported columns are of strongly elongated rectangular crosssection. In the simplest form, they consist of a single slab [Figure 4.16(b)]. Another type is that of Figure 4.16(c), where a pedestal or cap is interposed between the column and the footing slab; the pedestal provides for a more favorable transfer of load and in many cases is required in order to provide the necessary development length for dowels. This form is also known as a stepped footing. All parts of a stepped footing must be poured in a single pour in order to provide monolithic action. Sometimes sloped footings like those in Figure 4.16(d) are used. They requires less concrete than stepped footings, but the additional labor necessary to produce the sloping surfaces (formwork, etc.) usually makes stepped footings more economical. In general, single-slab footings [Figure 4.16(b)] are most economical for thicknesses up to 3 ft. The required bearing area is obtained by dividing the total load, including the weight of the footing, by the selected bearing pressure. Weights of footings, at this stage, must be estimated and usually amount to 4 to 8% of the column load, the former value applying to the stronger types of soils. Once the required footing area has been established, the thickness h of the footing must be determined. In single footings the effective depth d is mostly governed by shear. Two different types of 1999 by CRC Press LLC
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shear strength are distinguished in single footings: two-way (or punching) shear and one-way (or beam) shear. Based on the Equations 4.126 and 4.127 for punching and one-way shear strength, the required effective depth of footing d is calculated. Single-column footings represent, as it were, cantilevers projecting out from the column in both directions and loaded upward by the soil pressure. Corresponding tension stresses are caused in both these directions at the bottom surface. Such footings are therefore reinforced by two layers of steel, perpendicular to each other and parallel to the edge. The steel reinforcement in each direction can be calculated using Equation 4.129. The critical sections for development length of footing bars are the same as those for bending. Development length may also have to be checked at all vertical planes in which changes of section or of reinforcement occur, as at the edges of pedestals or where part of the reinforcement may be terminated. When a column rests on a footing or pedestal, it transfers its load to only a part of the total area of the supporting member. The adjacent footing concrete provides lateral support to the directly loaded part of the concrete. This causes triaxial compression stresses that increase the strength of the concrete, which is loaded directly under the column. The design bearing strength of concrete must not exceed the one given √ in Equation 4.131 for forces acting on the concrete at the base of column and the modified one with r A2 /A1 for supporting area wider than the loaded area. If the calculated bearing force is greater than the design bearing strength, reinforcement must be provided to transfer the excess force. This is done either by extending the column bars into the footing or by providing dowels, which are embedded in the footing and project above it.
4.10.5
Combined Footings
Spread footings that support more than one column or wall are known as combined footings. They can be divided into two categories: those that support two columns, and those that support more than two (generally large numbers of) columns. In buildings where the allowable soil pressure is large enough for single footings to be adequate for most columns, two-column footings are seen to become necessary in two situations: (1) if columns are so close to the property line that single-column footings cannot be made without projecting beyond that line, and (2) if some adjacent columns are so close to each other that their footings would merge. When the bearing capacity of the subsoil is low so that large bearing areas become necessary, individual footings are replaced by continuous strip footings, which support more than two columns and usually all columns in a row. Mostly, such strips are arranged in both directions, in which case a grid foundation is obtained, as shown in Figure 4.24. Such a grid foundation can be done by single footings because the individual strips of the grid foundation represent continuous beams whose moments are much smaller than the cantilever moments in large single footings that project far out from the column in all four directions. For still lower bearing capacities, the strips are made to merge, resulting in a mat foundation, as shown in Figure 4.25. That is, the foundation consists of a solid reinforced concrete slab under the entire building. In structural action such a mat is very similar to a flat slab or a flat plate, upside down—that is, loaded upward by the bearing pressure and downward by the concentrated column reactions. The mat foundation evidently develops the maximum available bearing area under the building. If the soil’s capacity is so low that even this large bearing area is insufficient, some form of deep foundation, such as piles or caissons, must be used. Grid and mat foundations may be designed with the column pedestals—as shown in Figures 4.24 and 4.25—or without them, depending on whether or not they are necessary for shear strength and the development length of dowels. Apart from developing large bearing areas, another advantage of grid and mat foundations is that their continuity and rigidity help in reducing differential settlements of individual columns relative to each other, which may otherwise be caused by local variations in the 1999 by CRC Press LLC
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FIGURE 4.24: Grid foundation.
FIGURE 4.25: Mat foundation.
quality of subsoil, or other causes. For this purpose, continuous spread foundations are frequently used in situations where the superstructure or the type of occupancy provides unusual sensitivity to differential settlement.
4.10.6
Two-Column Footings
The ACI Codes does not provide a detailed approach for the design of combined footings. The design, in general, is based on an empirical approach. It is desirable to design combined footings so that the centroid of the footing area coincides with the resultant of the two column loads. This produces uniform bearing pressure over the entire area and forestalls a tendency for the footings to tilt. In plan, such footings are rectangular, trapezoidal, or T shaped, the details of the shape being arranged to produce coincidence of centroid and resultant. The simple relationships of Figure 4.26 facilitate the determination of the shapes of the bearing area [7]. In general, the distances m and n are given, the former being the distance from the center of the exterior column to the property line and the latter the distance from that column to the resultant of both column loads. Another expedient, which is used if a single footing cannot be centered under an exterior column, is to place the exterior column footing eccentrically and to connect it with the nearest interior column 1999 by CRC Press LLC
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FIGURE 4.26: Two-column footings. (From Fintel, M. 1985. Handbook of Concrete Engineering, 2nd ed., Van Nostrand Reinhold, New York. With permission.)
by a beam or strap. This strap, being counterweighted by the interior column load, resists the tilting tendency of the eccentric exterior footings and equalizes the pressure under it. Such foundations are known as strap, cantilever, or connected footings. The strap may be designed as a rectangular beam spacing between the columns. The loads on it include its own weight (when it does not rest on the soil) and the upward pressure from the footings. Width of the strap usually is selected arbitrarily as equal to that of the largest column plus 4 to 8 inches so that column forms can be supported on top of the strap. Depth is determined by the maximum bending moment. The main reinforcing in the strap is placed near the top. Some of the steel can be cut off where not needed. For diagonal tension, stirrups normally will be needed near the columns (Figure 4.27). In addition, longitudinal placement steel is set near the bottom of the strap, plus reinforcement to guard against settlement stresses. The footing under the exterior column may be designed as a wall footing. The portions on opposite sides of the strap act as cantilevers under the constant upward pressure of the soil. The interior footing should be designed as a single-column footing. The critical section for punching shear, however, differs from that for a conventional footing. This shear should be computed on a section at a distance d/2 from the sides and extending around the column at a distance d/2 from its faces, where d is the effective depth of the footing.
1999 by CRC Press LLC
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FIGURE 4.27: Strap (cantilever) footing. (From Fintel, M. 1985. Handbook of Concrete Engineering, 2nd ed., Van Nostrand Reinhold, New York. With permission.)
4.10.7
Strip, Grid, and Mat Foundations
In the case of heavily loaded columns, particularly if they are to be supported on relatively weak or uneven soils, continuous foundations may be necessary. They may consist of a continuous strip footing supporting all columns in a given row or, more often, of two sets of such strip footings intersecting at right angles so that they form one continuous grid foundation (Figure 4.24). For even larger loads or weaker soils the strips are made to merge, resulting in a mat foundation (Figure 4.25). For the design of such continuous foundations it is essential that reasonably realistic assumptions be made regarding the distribution of bearing pressures, which act as upward loads on the foundation. For compressible soils it can be assumed in first approximation that the deformation or settlement of the soil at a given location and the bearing pressure at that location are proportional to each other. If columns are spaced at moderate distances and if the strip, grid, or mat foundation is very rigid, the settlements in all portions of the foundation will be substantially the same. This means that the bearing pressure, also known as subgrade reaction, will be the same provided that the centroid of the foundation coincides with the resultant of the loads. If they do not coincide, then for such rigid foundations the subgrade reaction can be assumed as linear and determined from statics in the same manner as discussed for single footings. In this case, all loads—the downward column loads as well as the upward-bearing pressures—are known. Hence, moments and shear forces in the foundation can be found by statics alone. Once these are determined, the design of strip and grid foundations is similar to that of inverted continuous beams, and design of mat foundations is similar to that of inverted flat slabs or plates. 1999 by CRC Press LLC
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On the other hand, if the foundation is relatively flexible and the column spacing large, settlements will no longer be uniform or linear. For one thing, the more heavily loaded columns will cause larger settlements, and thereby larger subgrade reactions, than the lighter ones. Also, since the continuous strip or slab midway between columns will deflect upward relative to the nearby columns, soil settlement—and thereby the subgrade reaction—will be smaller midway between columns than directly at the columns. This is shown schematically in Figure 4.28. In this case the subgrade reaction can no longer be assumed as uniform. A reasonably accurate but fairly complex analysis can then be made using the theory of beams on elastic foundations.
FIGURE 4.28: Strip footing. (From Fintel, M. 1985. Handbook of Concrete Engineering, 2nd ed., Van Nostrand Reinhold, New York. With permission.)
A simplified approach has been developed that covers the most frequent situations of strip and grid foundations [4]. The method first defines the conditions under which a foundation can be regarded as rigid so that uniform or overall linear distribution of subgrade reactions can be assumed. This is the case when the average of two adjacent span lengths in a continuous strip does not exceed 1.75/λ, provided also that the adjacent span and column loads do not differ by more than 20% of the larger value. Here, s ks b (4.135) λ=4 3Ec I where ks = Sks0 ks0 = coefficient of subgrade reaction as defined in soils mechanics, basically force per unit area required to produce unit settlement, kips/ft3 b = width of footing, ft Ec = modulus of elasticity of concrete, kips/ft2 I = moment of inertia of footing, ft4 S = shape factor, being [(b + 1)/2b]2 for granular soils and (n + 0.5)/1.5n for cohesive soils, where n is the ratio of longer to shorter side of strip If the average of two adjacent spans exceeds 1.75/λ, the foundation is regarded as flexible. Provided that adjacent spans and column loads differ by no more than 20%, the complex curvelinear distribution of subgrade reaction can be replaced by a set of equivalent trapezoidal reactions, which are also shown in Figure 4.28. The report of ACI Committee 436 contains fairly simple equations 1999 by CRC Press LLC
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for determining the intensities of the equivalent pressures under the columns and at the middle of the spans and also gives equations for the positive and negative moments caused by these equivalent subgrade reactions. With this information, the design of continuous strip and grid footings proceeds similarly to that of footings under two columns. Mat foundations likewise require different approaches, depending on whether they can be classified as rigid or flexible. As in strip footings, if the column spacing is less than 1/λ, the structure may be regarded as rigid, soil pressure can be assumed as uniformly or linearly distributed, and the design is based on statics. On the other hand, when the foundation is considered flexible as defined above, and if the variation of adjacent column loads and spans is not greater than 20%, the same simplified procedure as for strip and grid foundations can be applied to mat foundations. The mat is divided into two sets of mutually perpendicular strip footings of width equal to the distance between midspans, and the distribution of bearing pressures and bending moments is carried out for each strip. Once moments are determined, the mat is in essence treated the same as a flat slab or plate, with the reinforcement allocated between column and middle strips as in these slab structures. This approach is feasible only when columns are located in a regular rectangular grid pattern. When a mat that can be regarded as rigid supports columns at random locations, the subgrade reactions can still be taken as uniform or as linearly distributed and the mat analyzed by statics. If it is a flexible mat that supports such randomly located columns, the design is based on the theory of plates on elastic foundation.
4.10.8
Footings on Piles
If the bearing capacity of the upper soil layers is insufficient for a spread foundation, but firmer strata are available at greater depth, piles are used to transfer the loads to these deeper strata. Piles are generally arranged in groups or clusters, one under each column. The group is capped by a spread footing or cap that distributes the column load to all piles in the group. Reactions on caps act as concentrated loads at the individual piles, rather than as distributed pressures. If the total of all pile reactions in a cluster is divided by area of the footing to obtain an equivalent uniform pressure, it is found that this equivalent pressure is considerably higher in pile caps than for spread footings. Thus, it is in any event advisable to provide ample rigidity—that is, depth for pile caps—in order to spread the load evenly to all piles. As in single-column spread footings, the effective portion of allowable bearing capacities of piles, Ra , available to resist the unfactored column loads is the allowable pile reaction less the weight of footing, backfill, and surcharge per pile. That is, Re = Ra − Wf
(4.136)
where Wf is the total weight of footing, fill, and surcharge divided by the number of piles. Once the available or effective pile reaction Re is determined, the number of piles in a concentrically loaded cluster is the integer next larger than n=
D+L Re
(4.137)
The effects of wind and earthquake moments at the foot of the columns generally produce an eccentrically loaded pile cluster in which different piles carry different loads. The number and location of piles in such a cluster is determined by successive approximation from the condition that the load on the most heavily loaded pile should not exceed the allowable pile reaction Ra . Assuming a linear distribution of pile loads due to bending, the maximum pile reaction is Rmax = 1999 by CRC Press LLC
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P M + n Ipg /c
(4.138)
where P is the maximum load (including weight of cap, backfill, etc.), M is the moment to be resisted by the pile group, both referred to the bottom of the cap, Ipg is the moment of inertia of the entire pile group about the centroidal axis about which bendings occurs, and c is the distance from that axis to the extreme pile. Piles are generally arranged in tight patterns, which minimizes the cost of the caps, but they cannot be placed closer than conditions of deriving and of undisturbed carrying capacity will permit. AASHTO requires that piles be spaced at least 2 ft 6 in. center to center and that the distance from the side of a pile to the nearest edge of the footing be 9 in. or more. The design of footings on piles is similar to that of single-column spread footings. One approach is to design the cap for the pile reactions calculated for the factored column loads. For a concentrically loaded cluster this would give Ru = (1.4D + 1.7L)/n. However, since the number of piles was taken as the next larger integer according to Equation 4.138, determining Ru in this manner can lead to a design where the strength of the cap is less than the capacity of the pile group. It is therefore recommended that the pile reaction for strength design be taken as Ru = Re × Average load factor
(4.139)
where the average load factor is (1.4D + 1.7L)/(D + L). In this manner the cap is designed to be capable of developing the full allowable capacity of the pile group. As in single-column spread footings, the depth of the pile cap is usually governed by shear. In this regard both punching and one-way shear need to be considered. The critical sections are the same as explained earlier under “Two-Way Shear (Punching Shear)” and “One-Way Shear.” The difference is that shears on caps are caused by concentrated pile reactions rather than by distributed bearing pressures. This poses the question of how to calculate shear if the critical section intersects the circumference of one or more piles. For this case the ACI Code accounts for the fact that pile reaction is not really a point load, but is distributed over the pile-bearing area. Correspondingly, for piles with diameters dp , it stipulates as follows: Computation of shear on any section through a footing on piles shall be in accordance with the following: (a) The entire reaction from any pile whose center is located dp /2 or more outside this section shall be considered as producing shear on that section. (b) The reaction from any pile whose center is located dp /2 or more inside the section shall be considered as producing no shear on that section. (c) For intermediate portions of the pile center, the portion of the pile reaction to be considered as producing shear on the section shall be based on straight-line interpolation between the full value at dp /2 outside the section and zero at dp /2 inside the section [1]. In addition to checking punching and one-way shear, punching shear must be investigated for the individual pile. Particularly in caps on a small number of heavily loaded piles, it is this possibility of a pile punching upward through the cap which may govern the required depth. The critical perimeter for this action, again, is located at a distance d/2 outside the upper edge of the pile. However, for relatively deep caps and closely spaced piles, critical perimeters around adjacent piles may overlap. In this case, fracture, if any, would undoubtedly occur along an outward-slanting surface around both adjacent piles. For such situations the critical perimeter is so located that its length is a minimum, as shown for two adjacent piles in Figure 4.29. 1999 by CRC Press LLC
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FIGURE 4.29: Modified critical section for shear with overlapping critical perimeters.
4.11
Walls
4.11.1
Panel, Curtain, and Bearing Walls
As a general rule, the exterior walls of a reinforced concrete building are supported at each floor by the skeleton framework, their only function being to enclose the building. Such walls are called panel walls. They may be made of concrete (often precast), cinder concrete block, brick, tile blocks, or insulated metal panels. The thickness of each of these types of panel walls will vary according to the material, type of construction, climatological conditions, and the building requirements governing the particular locality in which the construction takes place. The pressure of the wind is usually the only load that is considered in determining the structural thickness of a wall panel, although in some cases exterior walls are used as diaphragms to transmit forces caused by horizontal loads down to the building foundations. Curtain walls are similar to panel walls except that they are not supported at each story by the frame of the building; rather, they are self supporting. However, they are often anchored to the building frame at each floor to provide lateral support. A bearing wall may be defined as one that carries any vertical load in addition to its own weight. Such walls may be constructed of stone masonry, brick, concrete block, or reinforced concrete. Occasional projections or pilasters add to the strength of the wall and are often used at points of load concentration. Bearing walls may be of either single or double thickness, the advantage of the latter type being that the air space between the walls renders the interior of the building less liable to temperature variation and makes the wall itself more nearly moistureproof. On account of the greater gross thickness of the double wall, such construction reduces the available floor space. According to ACI Code Sec. 14.5.2 the load capacity of a wall is given by " # klc 2 0 (4.140) φPnw = 0.55φfc Ag 1 − 32h where φPnw = design axial load strength = gross area of section, in.2 Ag lc = vertical distance between supports, in. h = thickness of wall, in. φ = 0.7 and where the effective length factor k is taken as 0.8 for walls restrained against rotation at top or bottom or both, 1.0 for walls unrestrained against rotation at both ends, and 2.0 for walls not braced against lateral translation. 1999 by CRC Press LLC
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In the case of concentrated loads, the length of the wall to be considered as effective for each should not exceed the center-to-center distance between loads; nor should it exceed the width of the bearing plus 4 times the wall thickness. Reinforced concrete bearing walls should have a thickness of a least 1/25 times the unsupported height or width, whichever is shorter. Reinforced concrete bearing walls of buildings should be not less than 4 in. thick. Minimum ratio of horizontal reinforcement area to gross concrete area should be 0.0020 for deformed bars not larger than #5—with specified yield strength not less than 60,000 psi or 0.0025 for other deformed bars—or 0.0025 for welded wire fabric not larger than W31 or D31. Minimum ratio of vertical reinforcement area to gross concrete area should be 0.0012 for deformed bars not larger than #5—with specified yield strength not less than 60,000 psi or 0.0015 for other deformed bars—or 0.0012 for welded wire fabric not larger than W31 or D31. In addition to the minimum reinforcement, not less than two #5 bars shall be provided around all window and door openings. Such bars shall be extended to develop the bar beyond the corners of the openings but not less than 24 in. Walls more than 10 in. thick should have reinforcement for each direction placed in two layers parallel with faces of wall. Vertical and horizontal reinforcement should not be spaced further apart than three times the wall thickness, or 18 in. Vertical reinforcement need not be enclosed by lateral ties if vertical reinforcement area is not greater than 0.01 times gross concrete area, or where vertical reinforcement is not required as compression reinforcement. Quantity of reinforcement and limits of thickness mentioned above are waived where structural analysis shows adequate strength and stability. Walls should be anchored to intersecting elements such as floors, roofs, or to columns, pilasters, buttresses, and intersecting walls, and footings.
4.11.2
Basement Walls
In determining the thickness of basement walls, the lateral pressure of the earth, if any, must be considered in addition to other structural features. If it is part of a bearing wall, the lower portion may be designed either as a slab supported by the basement and floors or as a retaining wall, depending upon the type of construction. If columns and wall beams are available for support, each basement wall panel of reinforced concrete may be designed to resist the earth pressure as a simple slab reinforced in either one or two directions. A minimum thickness of 7.5 in. is specified for reinforced concrete basement walls. In wet ground a minimum thickness of 12 in. should be used. In any case, the thickness cannot be less than that of the wall above. Care should be taken to brace a basement wall thoroughly from the inside (1) if the earth is backfilled before the wall has obtained sufficient strength to resist the lateral pressure without such assistance, or (2) if it is placed before the first-floor slab is in position.
4.11.3
Partition Walls
Interior walls used for the purpose of subdividing the floor area may be made of cinder block, brick, precast concrete, metal lath and plaster, clay tile, or metal. The type of wall selected will depend upon the fire resistance required; flexibility of rearrangement; ease with which electrical conduits, plumbing, etc. can be accommodated; and architectural requirements.
4.11.4
Shears Walls
Horizontal forces acting on buildings—for example, those due to wind or seismic action—can be resisted by a variety of means. Rigid-frame resistance of the structure, augmented by the contribution of ordinary masonry walls and partitions, can provide for wind loads in many cases. However, when heavy horizontal loading is likely—such as would result from an earthquake—reinforced concrete 1999 by CRC Press LLC
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shear walls are used. These may be added solely to resist horizontal forces; alternatively, concrete walls enclosing stairways or elevator shafts may also serve as shear walls. Figure 4.30 shows a building with wind or seismic forces represented by arrows acting on the edge of each floor or roof. The horizontal surfaces act as deep beams to transmit loads to vertical resisting
FIGURE 4.30: Building with shear walls subject to horizontal loads: (a) typical floor; (b) front elevation; (c) end elevation.
elements A and B. These shear walls, in turn, act as cantilever beams fixed at their base to carry loads down to the foundation. They are subjected to (1) a variable shear, which reaches maximum at the base, (2) a bending moment, which tends to cause vertical tension near the loaded edge and compression at the far edge, and (3) a vertical compression due to ordinary gravity loading from the structure. For the building shown, additional shear walls C and D are provided to resist loads acting in the log direction of the structure. The design basis for shear walls, according to the ACI Code, is of the same general form as that used for ordinary beams: Vu Vn
≤ φVn = Vc + Vs
(4.141) (4.142)
Shear p strength Vn at any horizontal section for shear in plane of wall should not be taken greater than 10 fc0 hd. In this and all other equations pertaining to the design of shear walls, the distance of d may be taken equal to 0.8lw . A larger value of d, equal to the distance from the extreme compression face to the center of force of all reinforcement in tension, may be used when determined by a strain compatibility analysis. The value of Vc , the nominal shear strength provided by the concrete, may be based on the usual equations for beams, according to ACI Code. For walls subjected to vertical compression, p (4.143) Vc = 2 fc0 hd and for walls subjected to vertical tension Nu , Vc = 2 1 + 1999 by CRC Press LLC
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Nu 500Ag
p fc0 hd
(4.144)
where Nu is the factored axial load in pounds, taken negative for tension, and Ag is the gross area of horizontal concrete section in square inches. Alternatively, the value of Vc may be based on a more detailed calculation, as the lesser of p Nu d Vc = 3.3 fc0 hd + 4lw or
p # 0 + 0.2N / l h p f l 1.25 w u w c hd Vc = 0.6 fc0 + Mu /Vu − lw /2
(4.145)
"
(4.146)
p Equation 4.145 corresponds to the occurrence of a principal tensile stress of approximately 4 fc0 at the centroid of the shear-wallpsection. Equation 4.146 corresponds approximately to the occurrence of a flexural tensile stress of 6 fc0 at a section lw /2 above the section being investigated. Thus the two equations predict, respectively, web-shear cracking and flexure-shear cracking. When the quantity Mu /Vu − lw /2 is negative, Equation 4.146 is inapplicable. According to the ACI Code, horizontal sections located closer to the wall base than a distance lw /2 or hw /2, whichever less, may be designed for the same Vc as that computed at a distance lw /2 or hw /2. When the factored shear force Vu does not exceed φVc /2, a wall may be reinforced according to the minimum requirements given in Sec. 12.1. When Vu exceeds φVc /2, reinforcement for shear is to be provided according to the following requirements. The nominal shear strength Vs provided by the horizontal wall steel is determined on the same basis as for ordinary beams: Av fy d (4.147) Vs = s2 where Av is the area of horizontal shear reinforcement within vertical distance s2 , (in.2 ), s2 is the vertical distance between horizontal reinforcement, (in.), and fy is the yield strength of reinforcement, psi. Substituting Equation 4.147 into Equation 4.142, then combining with Equation 4.141, one obtains the equation for the required area of horizontal shear reinforcement within a distance s2 : Av =
(Vu − φVc ) s2 φfy d
(4.148)
The minimum permitted ratio of horizontal shear steel to gross concrete area of vertical section, ρn , is 0.0025 and the maximum spacing s2 is not exceed lw /5, 3h, or 18 in. Test results indicate that for low shear walls, vertical distributed reinforcement is needed as well as horizontal reinforcement. Code provisions require vertical steel of area Ah within a spacing s1 , such that the ratio of vertical steel to gross concrete area of horizontal section will not be less than hw (4.149) ρn = 0.0025 + 0.5 2.5 − (ρh − 0.0025) lw nor less than 0.0025. However, the vertical steel ratio need not be greater than the required horizontal steel ratio. The spacing of the vertical bars is not to exceed lw /3, 3h, or 18 in. Walls may be subjected to flexural tension due to overturning moment, even when the vertical compression from gravity loads is superimposed. In many but not all cases, vertical steel is provided, concentrated near the wall edges, as in Figure 4.31. The required steel area can be found by the usual methods for beams. The ACI Code contains requirements for the dimensions and details of structural walls serving as part of the earthquake-force resisting systems. The reinforcement ratio, ρv (= Asv /Acv ; where Acv is the net area of concrete section bounded by web thickness and length of section in the direction of 1999 by CRC Press LLC
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FIGURE 4.31: Geometry and reinforcement of typical shear wall: (a) cross section; (b) elevation. shear force considered, and Asv is the projection on Acv of area of distributed shear reinforcement crossing the plane of Acv ), for structural walls should not be less than 0.0025 along the longitudinal and transverse axes. Reinforcement provided for shear strength should be continuous p and should be distributed across the shear plane. If the design shear force does not exceed Acv fc0 , the shear reinforcement may conform to the reinforcement ratio given in Sec. 12.1. At least two curtains of reinforcement p should be used in a wall if the in-plane factored shear force assigned to the wall exceeds 2Acv fc0 . All continuous reinforcement in structural walls should be anchored or spliced in accordance with the provisions for reinforcement in tension for seismic design. Proportioning and details of structural walls that resist shear forces caused by earthquake motion is contained in the ACI Code Sec. 21.7.3.
4.12
Defining Terms
The terms common in concrete engineering as defined in and selected from the Cement and Concrete Terminology Report of ACI Committee 116 are given below [1, Further Reading]. Allowable stress: Maximum permissible stress used in design of members of a structure and based on a factor of safety against yielding or failure of any type. Allowable stress design (ASD): Design principle according to which stresses resulting from service or working loads are not allowed to exceed specified allowable values. Balanced load: Combination of axial force and bending moment that causes simultaneous crushing of concrete and yielding of tension steel. Balanced reinforcement: An amount and distribution of flexural reinforcement such that the tensile reinforcement reaches its specified yield strength simultaneously with the concrete in compression reaching its assumed ultimate strain of 0.003. Beam: A structural member subjected primarily to flexure; depth-to-span ratio is limited to 2/5 for continuous spans, or 4/5 for simple spans, otherwise the member is to be treated as a deep beam. 1999 by CRC Press LLC
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Beam-column: A structural member that is subjected simultaneously to bending and substantial axial forces. Bond: Adhesion and grip of concrete or mortar to reinforcement or to other surfaces against which it is placed; to enhance bond strength, ribs or other deformations are added to reinforcing bars. Camber: A deflection that is intentionally built into a structural element or form to improve appearance or to offset the deflection of the element under the effects of loads, shrinkage, and creep. Cast-in-place concrete: Concrete poured in its final or permanent location; also called in situ concrete; opposite of precast concrete. Column: A member used to support primarily axial compression loads with a height of at least three times its least lateral dimensions; the capacity of short columns is controlled by strength; the capacity of long columns is limited by buckling. Column strip: The portion of a flat slab over a row of columns consisting of the two adjacent quarter panels on each side of the column centerline. Composition construction: A type of construction using members made of different materials (e.g., concrete and structural steel), or combining members made of cast-in-place concrete and precast concrete such that the combined components act together as a single member; strictly speaking, reinforced concrete is also composite construction. Compression member: A member subjected primarily to longitudinal compression; often synonymous with “column”. Compressive strength: Strength typically measured on a standard 6 × 12 in. cylinder of concrete in an axial compression test, 28 d after casting. Concrete: A composite material that consists essentially of a binding medium within which are embedded particles or fragments of aggregate; in portland cement concrete, the binder is a mixture of portland cement and water. Confined concrete: Concrete enclosed by closely spaced transverse reinforcement, which restrains the concrete expansion in directions perpendicular to the applied stresses. Construction joint: The surface where two successive placements of concrete meet, across which it may be desirable to achieve bond, and through which reinforcement may be continuous. Continuous beam or slab: A beam or slab that extends as a unit over three or more supports in a given direction and is provided with the necessary reinforcement to develop the negative moments over the interior supports; a redundant structure that requires a statically indeterminant analysis (opposite of simple supported beam or slab). Cover: In reinforced concrete, the shortest distance between the surface of the reinforcement and the outer surface of the concrete; minimum values are specified to protect the reinforcement against corrosion and to assure sufficient bond strength. Cracks: Results of stresses exceeding concrete’s tensile strength capacity; cracks are ubiquitous in reinforced concrete and needed to develop the strength of the reinforcement, but a design goal is to keep their widths small (hairline cracks). 1999 by CRC Press LLC
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Cracked section: A section designed or analyzed on the assumption that concrete has no resistance to tensile stress. Cracking load: The load that causes tensile stress in a member to be equal to the tensile strength of the concrete. Deformed bar: Reinforcing bar with a manufactured pattern of surface ridges intended to prevent slip when the bar is embedded in concrete. Design strength: Ultimate load-bearing capacity of a member multiplied by a strength reduction factor. Development length: The length of embedded reinforcement to develop the design strength of the reinforcement; a function of bond strength. Diagonal crack: An inclined crack caused by a diagonal tension, usually at about 45 degrees to the neutral axis of a concrete member. Diagonal tension: The principal tensile stress resulting from the combination of normal and shear stresses acting upon a structural element. Drop panel: The portion of a flat slab in the area surrounding a column, column capital, or bracket which is thickened in order to reduce the intensity of stresses. Ductility: Capability of a material or structural member to undergo large inelastic deformations without distress; opposite of brittleness; very important material property, especially for earthquake-resistant design; steel is naturally ductile, concrete is brittle but it can be made ductile if well confined. Durability: The ability of concrete to maintain its qualities over long time spans while exposed to weather, freeze-thaw cycles, chemical attack, abrasion, and other service load conditions. Effective depth: Depth of a beam or slab section measured from the compression face to the centroid of the tensile reinforcement. Effective flange width: Width of slab adjoining a beam stem assumed to function as the flange of a T-section. Effective prestress: The stress remaining in the prestressing steel or in the concrete due to prestressing after all losses have occurred. Effective span: The lesser of the distance between centers of supports and the clear distance between supports plus the effective depth of the beam or slab. Flat slab: A concrete slab reinforced in two or more directions, generally without beams or girders to transfer the loads to supporting members, but with drop panels or column capitals or both. High-early strength cement: Cement producing strength in mortar or concrete earlier than regular cement. Hoop: A one-piece closed reinforcing tie or continuously wound tie that encloses the longitudinal reinforcement. Interaction diagram: Failure curve for a member subjected to both axial force and the bending moment, indicating the moment capacity for a given axial load and vice versa; used to develop design charts for reinforced concrete compression members. 1999 by CRC Press LLC
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Lightweight concrete: Concrete of substantially lower unit weight than that made using normalweight gravel or crushed stone aggregate. Limit analysis: See Plastic analysis. Limit design: A method of proportioning structural members based on satisfying certain strength and serviceability limit states. Load and resistance factor design (LRFD): See Ultimate strength design. Load factor: A factor by which a service load is multiplied to determine the factored load used in ultimate strength design. Modulus of elasticity: The ratio of normal stress to corresponding strain for tensile of compressive stresses below the proportional limit of the material; for steel, Es = 29,000 ksi; for concrete 0 it is highly variable with stress level and the strength p fc ; for normal-weight concrete and low 0 stresses, a common approximation is Ec = 57,000 fc . Modulus of rupture: The tensile strength of concrete as measured in a flexural test of a small prismatic specimen of plain concrete. Mortar: A mixture of cement paste and fine aggregate; in fresh concrete, the material occupying the interstices among particles of coarse aggregate. Nominal strength: The strength of a structural member based on its assumed material properties and sectional dimensions, before application of any strength reduction factor. Plastic analysis: A method of structural analysis to determine the intensity of a specified load distribution at which the structure forms a collapse mechanism. Plastic hinge: Region of flexural member where the ultimate moment capacity can be developed and maintained with corresponding significant inelastic rotation, as main tensile steel is stressed beyond the yield point. Post-tensioning: A method of prestressing reinforced concrete in which the tendons are tensioned after the concrete has hardened (opposite of pretensioning). Precast concrete: Concrete cast elsewhere than its final position, usually in factories or factory-like shop sites near the final site (opposite of cast-in-place concrete). Prestressed concrete: Concrete in which internal stresses of such magnitude and distribution are introduced that the tensile stresses resulting from the service loads are counteracted to a desired degree; in reinforced concrete the prestress is commonly introduced by tensioning embedded tendons. Prestressing steel: High strength steel used to prestress concrete, commonly seven-wire strands, single wires, bars, rods, or groups of wires or strands. Pretensioning: A method of prestressing reinforced concrete in which the tendons are tensioned before the concrete has hardened (opposite of post-tensioning). Ready-mixed concrete: Concrete manufactured for delivery to a purchaser in a plastic and unhardened state; usually delivered by truck. Rebar: Short for reinforcing bar; see Reinforcement. 1999 by CRC Press LLC
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Reinforced concrete: Concrete containing adequate reinforcement (prestressed or not) and designed on the assumption that the two materials act together in resisting forces. Reinforcement: Bars, wires, strands, and other slender members that are embedded in concrete in such a manner that the reinforcement and the concrete act together in resisting forces. Safety factor: The ratio of a load producing an undesirable state (such as collapse) and an expected or service load. Service loads: Loads on a structure with high probability of occurrence, such as dead weight supported by a member or the live loads specified in building codes and bridge specifications. Shear key: A recess or groove in a joint between successive lifts or placements of concrete, which is filled with concrete of the adjacent lift, giving shear strength to the joint. Shear span: The distance from a support of a simply supported beam to the nearest concentrated load. Shear wall: See Structural wall. Shotcrete: Mortar or concrete pneumatically projected at high velocity onto a surface. Silica fume: Very fine noncrystalline silica produced in electric arc furnaces as a by-product of the production of metallic silicon and various silicon alloys (also know as condensed silica fume); used as a mineral admixture in concrete. Slab: A flat, horizontal (or neatly so) molded layer of plain or reinforced concrete, usually of uniform thickness, either on the ground or supported by beams, columns, walls, or other frame work. See also Flat slab. Slump: A measure of consistency of freshly mixed concrete equal to the subsidence of the molded specimen immediately after removal of the slump cone, expressed in inches. Splice: Connection of one reinforcing bar to another by lapping, welding, mechanical couplers, or other means. Split cylinder test: Test for tensile strength of concrete in which a standard cylinder is loaded to failure in diametral compression applied along the entire length (also called Brazilian test). Standard cylinder: Cylindric specimen of 12-in. height and 6-in. diameter, used to determine standard compressive strength and splitting tensile strength of concrete. Stiffness coefficient: The coefficient kij of stiffness matrix K for a multi-degree of freedom structure is the force needed to hold the ith degree of freedom in place, if the jth degree of freedom undergoes a unit of displacement, while all others are locked in place. Stirrup: A reinforcement used to resist shear and diagonal tension stresses in a structural member; typically a steel bar bent into a U or rectangular shape and installed perpendicular to or at an angle to the longitudinal reinforcement, and properly anchored; the term “stirrup” is usually applied to lateral reinforcement in flexural members and the term “tie” to lateral reinforcement in compression members. See Tie. Strength design: See Ultimate strength design. 1999 by CRC Press LLC
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Strength reduction factor: Capacity reduction factor (typically designated as φ) by which the nominal strength of a member is to be multiplied to obtain the design strength; specified by the ACI Code for different types of members. Structural concrete: Concrete used to carry load or to form an integral part of a structure (opposite of, for example, insulating concrete). T-beam: A beam composed of a stem and a flange in the form of a “T”, with the flange usually provided by a slab. Tension stiffening effect: The added stiffness of a single reinforcing bar due to the surrounding uncracked concrete between bond cracks. Tie: Reinforcing bar bent into a loop to enclose the longitudinal steel in columns; tensile bar to hold a form in place while resisting the lateral pressure of unhardened concrete. Ultimate strength design (USD): Design principle such that the actual (ultimate) strength of a member or structure, multiplied by a strength factor, is no less than the effects of all service load combinations, multiplied by respective overload factors. Unbonded tendon: A tendon that is not bonded to the concrete. Under-reinforced beam: A beam with less than balanced reinforcement such that the reinforcement yields before the concrete crushes in compression. Water-cement ratio: Ratio by weight of water to cement in a mixture; inversely proportional to concrete strength. Water-reducing admixture: An admixture capable of lowering the mix viscosity, thereby allowing a reduction of water (and increase in strength) without lowering the workability (also called superplasticizer). Whitney stress block: A rectangular area of uniform stress intensity 0.85fc0 , whose area and centroid are similar to that of the actual stress distribution in a flexural member near failure. Workability: General property of freshly mixed concrete that defines the ease with which it can be placed into forms without honeycombs; closely related to slump. Working stress design: See Allowable stress design. Yield-line theory: Method of structural analysis of plate structures at the verge of collapse under factored loads.
References [1] ACI Committee 318. 1992. Building Code Requirements for Reinforced Concrete and Commentary, ACI 318-89 (Revised 92) and ACI 318R-89 (Revised 92) (347pp.). Detroit, MI. [2] ACI Committee 340. 1990. Design Handbook in Accordance with the Strength Design Method of ACI 318-89. Volume 2, SP-17 (222 pp.). [3] ACI Committee 363. 1984. State-of-the-art report on high strength concrete. ACI J. Proc. 81(4):364411. [4] ACI Committee 436. 1996. Suggested design procedures for combined footings and mats. J. ACI. 63:1041-1057. [5] Breen, J.E. 1991. Why structural concrete? IASE Colloq. Struct. Concr. Stuttgart, pp.15-26. 1999 by CRC Press LLC
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[6] Collins, M.P. and Mitchell, D. 1991. Prestressed Concrete Structures, 1st ed., Prentice Hall, Englewood Cliffs, N.J. [7] Fintel, M. 1985. Handbook of Concrete Engineering. 2nd ed., Van Nostrand Reinhold, New York. [8] MacGregor, J.G. 1992. Reinforced Concrete Mechanics and Design, 2nd ed., Prentice Hall, Englewood Cliffs, N.J. [9] Nilson, A.H. and Winter, G. 1992. Design of Concrete Structures, 11th ed., McGraw-Hill, New York. [10] Standard Handbook for Civil Engineers, 2nd ed., McGraw-Hill, New York. [11] Wang, C.-K. and Salmon, C. G. 1985. Reinforced Concrete Design, 4th ed., Harper Row, New York.
Further Reading [1] ACI Committee 116. 1990. Cement and Concrete Terminology, Report 116R-90, American Concrete Institute, Detroit, MI. [2] Ferguson, P.M., Breen, J.E., and Jirsa, J.O. 1988. Reinforced Concrete Fundamentals, 5th ed., John Wiley & Sons, New York. [3] Lin, T-Y. and Burns, N.H. 1981. Design of Prestressed Concrete Structures, 3rd ed., John Wiley & Sons, New York. [4] Meyer, C. 1996. Design of Concrete Structures, Prentice-Hall, Upper Saddle River, NJ.
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Scawthorn, C. “Earthquake Engineering” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Earthquake Engineering 5.1 5.2
5.3 5.4
Charles Scawthorn EQE International, San Francisco, California and Tokyo, Japan
5.1
Introduction Earthquakes
Causes of Earthquakes and Faulting • Distribution of Seismicity • Measurement of Earthquakes • Strong Motion Attenuation and Duration • Seismic Hazard and Design Earthquake • Effect of Soils on Ground Motion • Liquefaction and Liquefaction-Related Permanent Ground Displacement
Seismic Design Codes
Purpose of Codes • Historical Development of Seismic Codes • Selected Seismic Codes
Earthquake Effects and Design of Structures
Buildings • Non-Building Structures
5.5 Defining Terms References Further Reading
Introduction
Earthquakes are naturally occurring broad-banded vibratory ground motions, caused by a number of phenomena including tectonic ground motions, volcanism, landslides, rockbursts, and humanmade explosions. Of these various causes, tectonic-related earthquakes are the largest and most important. These are caused by the fracture and sliding of rock along faults within the Earth’s crust. A fault is a zone of the earth’s crust within which the two sides have moved — faults may be hundreds of miles long, from 1 to over 100 miles deep, and not readily apparent on the ground surface. Earthquakes initiate a number of phenomena or agents, termed seismic hazards, which can cause significant damage to the built environment — these include fault rupture, vibratory ground motion (i.e., shaking), inundation (e.g., tsunami, seiche, dam failure), various kinds of permanent ground failure (e.g., liquefaction), fire or hazardous materials release. For a given earthquake, any particular hazard can dominate, and historically each has caused major damage and great loss of life in specific earthquakes. The expected damage given a specified value of a hazard parameter is termed vulnerability, and the product of the hazard and the vulnerability (i.e., the expected damage) is termed the seismic risk. This is often formulated as Z E(D | H )p(H )dH (5.1) E(D) = H
where H p(·) D
= hazard = refers to probability = damage
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E(D|H ) = vulnerability E(·) = the expected value operator Note that damage can refer to various parameters of interest, such as casualties, economic loss, or temporal duration of disruption. It is the goal of the earthquake specialist to reduce seismic risk. The probability of having a specific level of damage (i.e., p(D) = d) is termed the fragility. For most earthquakes, shaking is the dominant and most widespread agent of damage. Shaking near the actual earthquake rupture lasts only during the time when the fault ruptures, a process that takes seconds or at most a few minutes. The seismic waves generated by the rupture propagate long after the movement on the fault has stopped, however, spanning the globe in about 20 minutes. Typically earthquake ground motions are powerful enough to cause damage only in the near field (i.e., within a few tens of kilometers from the causative fault). However, in a few instances, long period motions have caused significant damage at great distances to selected lightly damped structures. A prime example of this was the 1985 Mexico City earthquake, where numerous collapses of mid- and high-rise buildings were due to a Magnitude 8.1 earthquake occurring at a distance of approximately 400 km from Mexico City. Ground motions due to an earthquake will vibrate the base of a structure such as a building. These motions are, in general, three-dimensional, both lateral and vertical. The structure’s mass has inertia which tends to remain at rest as the structure’s base is vibrated, resulting in deformation of the structure. The structure’s load carrying members will try to restore the structure to its initial, undeformed, configuration. As the structure rapidly deforms, energy is absorbed in the process of material deformation. These characteristics can be effectively modeled for a single degree of freedom (SDOF) mass as shown in Figure 5.1 where m represents the mass of the structure, the elastic spring (of stiffness k = force / displacement) represents the restorative force of the structure, and the dashpot damping device (damping coefficient c = force/velocity) represents the force or energy lost in the process of material deformation. From the equilibrium of forces on mass m due to the spring and
FIGURE 5.1: Single degree of freedom (SDOF) system. dashpot damper and an applied load p(t), we find: mu¨ + cu˙ + ku = p(t)
(5.2)
the solution of which [32] provides relations between circular frequency of vibration ω, the natural frequency f , and the natural period T : $2 = f = 1999 by CRC Press LLC
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1 T
=
$ 2π
k m
=
1 2π
q
(5.3) k m
(5.4)
Damping tends to reduce the amplitude of vibrations. Critical damping refers to the value of damping such that free vibration of a structure will cease after one cycle (ccrit = 2mω). Damping is conventionally expressed as a percent of critical damping and, for most buildings and engineering structures, ranges from 0.5 to 10 or 20% of critical damping (increasing with displacement amplitude). Note that damping in this range will not appreciably affect the natural period or frequency of vibration, but does affect the amplitude of motion experienced.
5.2 5.2.1
Earthquakes Causes of Earthquakes and Faulting
In a global sense, tectonic earthquakes result from motion between a number of large plates comprising the earth’s crust or lithosphere (about 15 in total), (see Figure 5.2). These plates are driven by the convective motion of the material in the earth’s mantle, which in turn is driven by heat generated at the earth’s core. Relative plate motion at the fault interface is constrained by friction and/or asperities (areas of interlocking due to protrusions in the fault surfaces). However, strain energy accumulates in the plates, eventually overcomes any resistance, and causes slip between the two sides of the fault. This sudden slip, termed elastic rebound by Reid [101] based on his studies of regional deformation following the 1906 San Francisco earthquake, releases large amounts of energy, which constitutes the earthquake. The location of initial radiation of seismic waves (i.e., the first location of dynamic rupture) is termed the hypocenter, while the projection on the surface of the earth directly above the hypocenter is termed the epicenter. Other terminology includes near-field (within one source dimension of the epicenter, where source dimension refers to the length or width of faulting, whichever is less), far-field (beyond near-field), and meizoseismal (the area of strong shaking and damage). Energy is radiated over a broad spectrum of frequencies through the earth, in body waves and surface waves [16]. Body waves are of two types: P waves (transmitting energy via push-pull motion), and slower S waves (transmitting energy via shear action at right angles to the direction of motion). Surface waves are also of two types: horizontally oscillating Love waves (analogous to S body waves) and vertically oscillating Rayleigh waves. While the accumulation of strain energy within the plate can cause motion (and consequent release of energy) at faults at any location, earthquakes occur with greatest frequency at the boundaries of the tectonic plates. The boundary of the Pacific plate is the source of nearly half of the world’s great earthquakes. Stretching 40,000 km (24,000 miles) around the circumference of the Pacific Ocean, it includes Japan, the west coast of North America, and other highly populated areas, and is aptly termed the Ring of Fire. The interiors of plates, such as ocean basins and continental shields, are areas of low seismicity but are not inactive — the largest earthquakes known to have occurred in North America, for example, occurred in the New Madrid area, far from a plate boundary. Tectonic plates move very slowly and irregularly, with occasional earthquakes. Forces may build up for decades or centuries at plate interfaces until a large movement occurs all at once. These sudden, violent motions produce the shaking that is felt as an earthquake. The shaking can cause direct damage to buildings, roads, bridges, and other human-made structures as well as triggering fires, landslides, tidal waves (tsunamis), and other damaging phenomena. Faults are the physical expression of the boundaries between adjacent tectonic plates and thus may be hundreds of miles long. In addition, there may be thousands of shorter faults parallel to or branching out from a main fault zone. Generally, the longer a fault the larger the earthquake it can generate. Beyond the main tectonic plates, there are many smaller sub-plates (“platelets”) and simple blocks of crust that occasionally move and shift due to the “jostling” of their neighbors and/or the major plates. The existence of these many sub-plates means that smaller but still damaging earthquakes are possible almost anywhere, although often with less likelihood. 1999 by CRC Press LLC
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FIGURE 5.2: Global seismicity and major tectonic plate boundaries.
Faults are typically classified according to their sense of motion (Figure 5.3). Basic terms include
FIGURE 5.3: Fault types.
transform or strike slip (relative fault motion occurs in the horizontal plane, parallel to the strike of the fault), dip-slip (motion at right angles to the strike, up- or down-slip), normal (dip-slip motion, two sides in tension, move away from each other), reverse (dip-slip, two sides in compression, move towards each other), and thrust (low-angle reverse faulting). Generally, earthquakes will be concentrated in the vicinity of faults. Faults that are moving more rapidly than others will tend to have higher rates of seismicity, and larger faults are more likely than others to produce a large event. Many faults are identified on regional geological maps, and useful information on fault location and displacement history is available from local and national geological surveys in areas of high seismicity. Considering this information, areas of an expected large earthquake in the near future (usually measured in years or decades) can be and have been identified. However, earthquakes continue to occur on “unknown” or “inactive” faults. An important development has been the growing recognition of blind thrust faults, which emerged as a result of several earthquakes in the 1980s, none of which were accompanied by surface faulting [120]. Blind thrust faults are faults at depth occurring under anticlinal folds — since they have only subtle surface expression, their seismogenic potential can be evaluated by indirect means only [46]. Blind thrust faults are particularly worrisome because they are hidden, are associated with folded topography in general, including areas of lower and infrequent seismicity, and therefore result in a situation where the potential for an earthquake exists in any area of anticlinal geology, even if there are few or no earthquakes in the historic record. Recent major earthquakes of this type have included the 1980 Mw 7.3 El- Asnam (Algeria), 1988 Mw 6.8 Spitak (Armenia), and 1994 Mw 6.7 Northridge (California) events. Probabilistic methods can be usefully employed to quantify the likelihood of an earthquake’s occurrence, and typically form the basis for determining the design basis earthquake. However, the earthquake generating process is not understood well enough to reliably predict the times, sizes, and 1999 by CRC Press LLC
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locations of earthquakes with precision. In general, therefore, communities must be prepared for an earthquake to occur at any time.
5.2.2
Distribution of Seismicity
This section discusses and characterizes the distribution of seismicity for the U.S. and selected areas. Global
It is evident from Figure 5.2 that some parts of the globe experience more and larger earthquakes than others. The two major regions of seismicity are the circum-Pacific Ring of Fire and the TransAlpide belt, extending from the western Mediterranean through the Middle East and the northern India sub-continent to Indonesia. The Pacific plate is created at its South Pacific extensional boundary — its motion is generally northwestward, resulting in relative strike-slip motion in California and New Zealand (with, however, a compressive component), and major compression and subduction in Alaska, the Aleutians, Kuriles, and northern Japan. Subduction refers to the plunging of one plate (i.e., the Pacific) beneath another, into the mantle, due to convergent motion, as shown in Figure 5.4.
FIGURE 5.4: Schematic diagram of subduction zone, typical of west coast of South America, Pacific Northwest of U.S., or Japan.
Subduction zones are typically characterized by volcanism, as a portion of the plate (melting in the lower mantle) re-emerges as volcanic lava. Subduction also occurs along the west coast of South America at the boundary of the Nazca and South American plate, in Central America (boundary of the Cocos and Caribbean plates), in Taiwan and Japan (boundary of the Philippine and Eurasian plates), and in the North American Pacific Northwest (boundary of the Juan de Fuca and North American 1999 by CRC Press LLC
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plates). The Trans-Alpide seismic belt is basically due to the relative motions of the African and Australian plates colliding and subducting with the Eurasian plate. U.S.
Table 5.1 provides a list of selected U.S. earthquakes. The San Andreas fault system in California and the Aleutian Trench off the coast of Alaska are part of the boundary between the North American and Pacific tectonic plates, and are associated with the majority of U.S. seismicity (Figure 5.5 and Table 5.1). There are many other smaller fault zones throughout the western U.S. that are also helping to release the stress that is built up as the tectonic plates move past one another, (Figure 5.6). While California has had numerous destructive earthquakes, there is also clear evidence that the potential exists for great earthquakes in the Pacific Northwest [11].
FIGURE 5.5: U.S. seismicity. (From Algermissen, S. T., An Introduction to the Seismicity of the United States, Earthquake Engineering Research Institute, Oakland, CA, 1983. With permission. Also after Coffman, J. L., von Hake, C. A., and Stover, C. W., Earthquake History of the United States, U.S. Department of Commerce, NOAA, Pub. 41-1, Washington, 1980.)
On the east coast of the U.S., the cause of earthquakes is less well understood. There is no plate boundary and very few locations of active faults are known so that it is more difficult to assess where earthquakes are most likely to occur. Several significant historical earthquakes have occurred, such as in Charleston, South Carolina, in 1886, and New Madrid, Missouri, in 1811 and 1812, indicating that there is potential for very large and destructive earthquakes [56, 131]. However, most earthquakes in the eastern U.S. are smaller magnitude events. Because of regional geologic differences, eastern and central U.S. earthquakes are felt at much greater distances than those in the western U.S., sometimes up to a thousand miles away [58]. 1999 by CRC Press LLC
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TABLE 5.1
Selected U.S. Earthquakes
Yr
M
D
Lat.
Long.
M
MMI
1755
11
18
8
1774 1791
2 5
21 16
7 8
1811 1812 1812 1817 1836 1838 1857 1865 1868 1868 1872 1886 1892 1892 1892 1897
12 1 2 10 6 6 1 10 4 10 3 9 2 4 5 5
16 23 7 5 10 0 9 8 3 21 26 1 24 19 16 31
36 36.6 36.6
N N N
90 89.6 89.6
W W W
8.6 8.4 8.7
38 37.5 35 37 19 37.5 36.5 32.9 31.5 38.5 14
N N N N N N N N N N N
122 123 119 122 156 122 118 80 117 123 143
W W W W W W W W W W E
8.3 6.8 8.5 7.7 5.8
12 12 8 10 10 7 9 10 10 10 9 10 9 8
1899 1906
9 4
4 18
60 38
N N
142 123
W W
8.3 8.3
11
1915 1925 1927 1933 1934 1935 1940 1944 1949 1951 1952 1954 1957 1958 1959 1962 1964 1965 1971 1975 1975 1975 1980 1980 1980 1980 1983 1983 1983 1984 1986 1987 1987 1989 1989 1990 1992 1992 1992 1992 1992 1993 1993 1994 1994 1994 1995
10 6 11 3 12 10 5 9 4 8 7 12 3 7 8 8 3 4 2 3 8 11 1 5 7 11 5 10 11 4 7 10 11 6 10 2 4 4 6 6 6 3 9 1 1 2 10
3 29 4 11 31 19 19 5 13 21 21 16 9 10 18 30 28 29 9 28 1 29 24 25 27 8 2 28 16 24 8 1 24 26 18 28 23 25 28 28 29 25 21 16 17 3 6
40.5 34.3 34.5 33.6 31.8 46.6 32.7 44.7 47.1 19.7 35 39.3 51.3 58.6 44.8 41.8 61 47.4 34.4 42.1 39.4 19.3 37.8 37.6 38.2 41.2 36.2 43.9 19.5 37.3 34 34.1 33.2 19.4 37.1 34.1 34 40.4 34.2 34.2 36.7 45 42.3 40.3 34.2 42.8 65.2
N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N
118 120 121 118 116 112 116 74.7 123 156 119 118 176 137 111 112 148 122 118 113 122 155 122 119 83.9 124 120 114 155 122 117 118 116 155 122 118 116 124 117 116 116 123 122 76 119 111 149
W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W
7.8 6.2 7.5 6.3 7.1 6.2 7.1 5.6 7 6.9 7.7 7 8.6 7.9 7.7 5.8 8.3 6.5 6.7 6.2 6.1 7.2 5.9 6.4 5.2 7 6.5 7.3 6.6 6.2 6.1 6 6.3 6.1 7.1 5.5 6.3 7.1 6.7 7.6 5.6 5.6 5.9 4.6 6.8 6 6.4
9 10 10 8 11 10 7 11 8 9 7 7 7 8 8 7 7 8 6 6 9 7 7 8 8 9 7 7 5 9 7 -
Fat.
USD mills
81 3 50 60 -
5
700?
400
13
8
115
40
2 9 8
19 6 2 25
13
60 3
5 131 7 65 2 1 5 2 8 2
2 540 13 553 1 6 4 4 2 1 3 31 13 7 8 5 358 -
62 -
6,000 13
3
92
2
-
57
30,000
66
Locale Nr Cape Ann, MA (MMI from STA) Eastern VA (MMI from STA) E. Haddam, CT (MMI from STA) New Madrid, MO New Madrid, MO New Madrid, MO Woburn, MA (MMI from STA) California California San Francisco, CA San Jose, Santa Cruz, CA Hawaii Hayward, CA Owens Valley, CA Charleston, SC, Ms from STA San Diego County, CA Vacaville, Winters, CA Agana, Guam Giles County, VA (mb from STA) Cape Yakataga, AK San Francisco, CA (deaths more?) Pleasant Valley, NV Santa Barbara, CA Lompoc, Port San Luis, CA Long Beach, CA Baja, Imperial Valley, CA Helena, MT SE of Elcentro, CA Massena, NY Olympia, WA Hawaii Central, Kern County, CA Dixie Valley, NV Alaska Lituyabay, AK—Landslide Hebgen Lake, MT Utah Alaska Seattle, WA San Fernando, CA Pocatello Valley, ID Oroville Reservoir, CA Hawaii Livermore, CA Mammoth Lakes, CA Maysville, KY N Coast, CA Central, Coalinga, CA Borah Peak, ID Kapapala, HI Central Morgan Hill, CA Palm Springs, CA Whittier, CA Superstition Hills, CA Hawaii Loma Prieta, CA Claremont, Covina, CA Joshua Tree, CA Humboldt, Ferndale, CA Big Bear Lake, Big Bear, CA Landers, Yucca, CA Border of NV and CA Washington-Oregon Klamath Falls, OR PA, Felt, Canada Northridge, CA Afton, WY AK (Oil pipeline damaged)
Note: STA refers to [3]. From NEIC, Database of Significant Earthquakes Contained in Seismicity Catalogs, National Earthquake Information Center, Goldon, CO, 1996. With permission.
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FIGURE 5.6: Seismicity for California and Nevada, 1980 to 1986. M >1.5 (Courtesy of Jennings, C. W., Fault Activity Map of California and Adjacent Areas, Department of Conservation, Division of Mines and Geology, Sacramento, CA, 1994.)
Other Areas
Table 5.2 provides a list of selected 20th-century earthquakes with fatalities of approximately 10,000 or more. All the earthquakes are in the Trans-Alpide belt or the circum-Pacific ring of fire, and the great loss of life is almost invariably due to low-strength masonry buildings and dwellings. Exceptions to this rule are the 1923 Kanto (Japan) earthquake, where most of the approximately 140,000 fatalities were due to fire; the 1970 Peru earthquake, where large landslides destroyed whole towns; and the 1988 Armenian earthquake, where large numbers were killed in Spitak and Leninakan due to poor quality pre-cast concrete construction. The Trans-Alpide belt includes the Mediterranean, which has very significant seismicity in North Africa, Italy, Greece, and Turkey due to the Africa plate’s motion relative to the Eurasian plate; the Caucasus (e.g., Armenia) and the Middle East (Iran, Afghanistan), due to the Arabian plate being forced northeastward into the Eurasian plate by the African plate; and the Indian sub-continent (Pakistan, northern India), and the subduction boundary along the southwestern side of Sumatra and Java, which are all part of the Indian-Australian 1999 by CRC Press LLC
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plate. Seismicity also extends northward through Burma and into western China. The Philippines, Taiwan, and Japan are all on the western boundary of the Philippines sea plate, which is part of the circum-Pacific ring of fire. Japan is an island archipelago with a long history of damaging earthquakes [128] due to the interaction of four tectonic plates (Pacific, Eurasian, North American, and Philippine) which all converge near Tokyo. Figure 5.7 indicates the pattern of Japanese seismicity, which is seen to be higher in the north of Japan. However, central Japan is still an area of major seismic risk, particularly Tokyo,
FIGURE 5.7: Japanese seismicity (1960 to 1965).
which has sustained a number of damaging earthquakes in history. The Great Kanto earthquake of 1923 (M7.9, about 140,000 fatalities) was a great subduction earthquake, and the 1855 event (M6.9) had its epicenter in the center of present-day Tokyo. Most recently, the 1995 MW 6.9 Hanshin (Kobe) earthquake caused approximately 6,000 fatalities and severely damaged some modern structures as well as many structures built prior to the last major updating of the Japanese seismic codes (ca. 1981). 1999 by CRC Press LLC
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The predominant seismicity in the Kuriles, Kamchatka, the Aleutians, and Alaska is due to subduction of the Pacific Plate beneath the North American plate (which includes the Aleutians and extends down through northern Japan to Tokyo). The predominant seismicity along the western boundary of North American is due to transform faults (i.e., strike-slip) as the Pacific Plate displaces northwestward relative to the North American plate, although the smaller Juan de Fuca plate offshore Washington and Oregon, and the still smaller Gorda plate offshore northern California, are driven into subduction beneath North American by the Pacific Plate. Further south, the Cocos plate is similarly subducting beneath Mexico and Central America due to the Pacific Plate, while the Nazca Plate lies offshore South America. Lesser but still significant seismicity occurs in the Caribbean, primarily along a series of trenches north of Puerto Rico and the Windward islands. However, the southern boundary of the Caribbean plate passes through Venezuela, and was the source of a major earthquake in Caracas in 1967. New Zealand’s seismicity is due to a major plate boundary (Pacific with Indian-Australian plates), which transitions from thrust to transform from the South to the North Island [108]. Lesser but still significant seismicity exists in Iceland where it is accompanied by volcanism due to a spreading boundary between the North American and Eurasian plates, and through Fenno-Scandia, due to tectonics as well as glacial rebound. This very brief tour of the major seismic belts of the globe is not meant to indicate that damaging earthquakes cannot occur elsewhere — earthquakes can and have occurred far from major plate boundaries (e.g., the 1811-1812 New Madrid intraplate events, with several being greater than magnitude 8), and their potential should always be a consideration in the design of a structure. TABLE 5.2
Selected 20th Century Earthquakes with Fatalities Greater than 10,000
Yr
M
D
Lat.
Long.
M
MMI
Deaths
1976 1920 1923 1908 1932 1970 1990 1927 1915 1935 1939 1939 1978 1988 1976 1974 1948 1905 1917 1968 1962 1907 1960 1980 1934 1918 1933 1975
7 12 9 2 12 5 6 5 1 5 12 1 9 12 2 5 10 4 1 8 9 10 2 10 1 2 8 2
27 16 1 0 25 31 20 22 13 30 26 25 16 7 4 10 5 4 21 31 1 21 29 10 15 13 25 4
39.5 N 36.5 N 35.3 N 38.2 N 39.2 N 9.1 S 37 N 37.6 N 41.9 N 29.5 N 39.5 N 36.2 S 33.4 N 41 N 15.3 N 28.2 N 37.9 N 33 N 8S 33.9 N 35.6 N 38.5 N 30.4 N 36.1 N 26.5 N 23.5 N 32 N 40.6 N
118 E 106 E 140 E 15.6 E 96.5 E 78.8 W 49.4 E 103 E 13.6 E 66.8 E 38.5 E 72.2 W 57.5 E 44.2 E 89.2 W 104 E 58.6 E 76 E 115 E 59 E 49.9 E 67.9 E 9.6 W 1.4 E 86.5 E 117 E 104 E 123 E
8 8.5 8.2 7.5 7.6 7.8 7.7 8 7 7.5 7.9 8.3 7.4 6.8 7.5 6.8 7.2 8.6 — 7.3 7.3 7.8 5.9 7.7 8.4 7.3 7.4 7.4
10 — — — — 9 7 — 11 10 12 — — 10 9 — — — — — — 9 — — — — — 10
655,237 200,000 142,807 75,000 70,000 67,000 50,000 40,912 35,000 30,000 30,000 28,000 25,000 25,000 22,400 20,000 19,800 19,000 15,000 15,000 12,225 12,000 12,000 11,000 10,700 10,000 10,000 10,000
Damage USD millions $2,000 $2,800 $500
$100 $11 $16,200 $6,000
Locale China: NE: Tangshan China: Gansu and Shanxi Japan: Toyko, Yokohama, Tsunami Italy: Sicily China: Gansu Province Peru Iran: Manjil China: Gansu Province Italy: Abruzzi, Avezzano Pakistan: Quetta Turkey: Erzincan Chile: Chillan Iran: Tabas CIS: Armenia Guatemala: Tsunami China: Yunnan and Sichuan CIS: Turkmenistan: Aschabad India: Kangra Indonesia: Bali, Tsunami Iran Iran: NW CIS: Uzbekistan: SE Morocco: Agadir Algeria: Elasnam Nepal-India China: Guangdong Province China: Sichuan Province China: NE: Yingtao
From NEIC, Database of Significant Earthquakes Contained in Seismicity Catalogs, National Earthquake Information Center, Goldon, CO, 1996.
1999 by CRC Press LLC
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5.2.3
Measurement of Earthquakes
Earthquakes are complex multi-dimensional phenomena, the scientific analysis of which requires measurement. Prior to the invention of modern scientific instruments, earthquakes were qualitatively measured by their effect or intensity, which differed from point-to-point. With the deployment of seismometers, an instrumental quantification of the entire earthquake event — the unique magnitude of the event — became possible. These are still the two most widely used measures of an earthquake, and a number of different scales for each have been developed, which are sometimes confused.1 Engineering design, however, requires measurement of earthquake phenomena in units such as force or displacement. This section defines and discusses each of these measures. Magnitude
An individual earthquake is a unique release of strain energy. Quantification of this energy has formed the basis for measuring the earthquake event. Richter [103] was the first to define earthquake magnitude as (5.5) ML = log A − log Ao where ML is local magnitude (which Richter only defined for Southern California), A is the maximum trace amplitude in microns recorded on a standard Wood-Anderson short-period torsion seismometer,2 at a site 100 km from the epicenter, log Ao is a standard value as a function of distance, for instruments located at distances other than 100 km and less than 600 km. Subsequently, a number of other magnitudes have been defined, the most important of which are surface wave magnitude MS , body wave magnitude mb , and moment magnitude MW . Due to the fact that ML was only locally defined for California (i.e., for events within about 600 km of the observing stations), surface wave magnitude MS was defined analogously to ML using teleseismic observations of surface waves of 20-s period [103]. Magnitude, which is defined on the basis of the amplitude of ground displacements, can be related to the total energy in the expanding wave front generated by an earthquake, and thus to the total energy release. An empirical relation by Richter is log10 Es = 11.8 + 1.5Ms
(5.6)
where Es is the total energy in ergs.3 Note that 101.5 = 31.6, so that an increase of one magnitude unit is equivalent to 31.6 times more energy release, two magnitude units increase is equivalent to 1000 times more energy, etc. Subsequently, due to the observation that deep-focus earthquakes commonly do not register measurable surface waves with periods near 20 s, a body wave magnitude mb was defined [49], which can be related to Ms [38]: mb = 2.5 + 0.63Ms
(5.7)
Body wave magnitudes are more commonly used in eastern North America, due to the deeper earthquakes there. A number of other magnitude scales have been developed, most of which tend to saturate — that is, asymptote to an upper bound due to larger earthquakes radiating significant amounts of energy at periods longer than used for determining the magnitude (e.g., for Ms , defined by
1 Earthquake magnitude and intensity are analogous to a lightbulb and the light it emits. A particular lightbulb has only one energy level, or wattage (e.g., 100 watts, analogous to an earthquake’s magnitude). Near the lightbulb, the light intensity is very bright (perhaps 100 ft-candles, analogous to MMI IX), while farther away the intensity decreases (e.g., 10 ft-candles, MMI V). A particular earthquake has only one magnitude value, whereas it has many intensity values. 2 The instrument has a natural period of 0.8 s, critical damping ration 0.8, magnification 2,800. 3 Richter [104] gives 11.4 for the constant term, rather than 11.8, which is based on subsequent work. The uncertainty in the data make this difference, equivalent to an energy factor = 2.5 or 0.27 magnitude units, inconsequential.
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measuring 20 s surface waves, saturation occurs at about Ms > 7.5). More recently, seismic moment has been employed to define a moment magnitude Mw ( [53]; also denoted as bold-face M) which is finding increased and widespread use: log Mo = 1.5Mw + 16.0
(5.8)
where seismic moment Mo (dyne-cm) is defined as [74] Mo = µAu¯
(5.9)
where µ is the material shear modulus, A is the area of fault plane rupture, and u¯ is the mean relative displacement between the two sides of the fault (the averaged fault slip). Comparatively, Mw and Ms are numerically almost identical up to magnitude 7.5. Figure 5.8 indicates the relationship between moment magnitude and various magnitude scales.
FIGURE 5.8: Relationship between moment magnitude and various magnitude scales. (From Campbell, K. W., Strong Ground Motion Attenuation Relations: A Ten-Year Perspective, Earthquake Spectra, 1(4), 759-804, 1985. With permission.)
For lay communications, it is sometimes customary to speak of great earthquakes, large earthquakes, etc. There is no standard definition for these, but the following is an approximate categorization: Earthquake Magnitude∗
Micro Not felt
Small 8
∗ Not specifically defined.
From the foregoing discussion, it can be seen that magnitude and energy are related to fault rupture length and slip. Slemmons [114] and Bonilla et al. [17] have determined statistical relations between these parameters, for worldwide and regional data sets, segregated by type of faulting (normal, reverse, strike-slip). The worldwide results of Bonilla et al. for all types of faults are Ms = 6.04 + 0.708 log10 L s = .306 log10 L = −2.77 + 0.619Ms s = .286 1999 by CRC Press LLC
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(5.10) (5.11)
Ms = log10 d =
6.95 + 0.723 log10 d −3.58 + 0.550Ms
s = .323 s = .282
(5.12) (5.13)
which indicates, for example that, for Ms = 7, the average fault rupture length is about 36 km (and the average displacement is about 1.86 m). Conversely, a fault of 100 km length is capable of about a Ms = 7.54 event. More recently, Wells and Coppersmith [130] have performed an extensive analysis of a dataset of 421 earthquakes. Their results are presented in Table 5.3a and b. Intensity
In general, seismic intensity is a measure of the effect, or the strength, of an earthquake hazard at a specific location. While the term can be applied generically to engineering measures such as peak ground acceleration, it is usually reserved for qualitative measures of location-specific earthquake effects, based on observed human behavior and structural damage. Numerous intensity scales were developed in pre-instrumental times. The most common in use today are the Modified Mercalli Intensity (MMI) [134], Rossi-Forel (R-F), Medvedev-Sponheur-Karnik (MSK) [80], and the Japan Meteorological Agency (JMA) [69] scales. MMI is a subjective scale defining the level of shaking at specific sites on a scale of I to XII. (MMI is expressed in Roman numerals to connote its approximate nature). For example, moderate shaking that causes few instances of fallen plaster or cracks in chimneys constitutes MMI VI. It is difficult to find a reliable relationship between magnitude, which is a description of the earthquake’s total energy level, and intensity, which is a subjective description of the level of shaking of the earthquake at specific sites, because shaking severity can vary with building type, design and construction practices, soil type, and distance from the event. Note that MMI X is the maximum considered physically possible due to “mere” shaking, and that MMI XI and XII are considered due more to permanent ground deformations and other geologic effects than to shaking. Other intensity scales are defined analogously (see Table 5.5, which also contains an approximate conversion from MMI to acceleration a [PGA, in cm/s2 , or gals]). The conversion is due to Richter [103] (other conversions are also available [84]. log a = MMI/3 − 1/2
(5.14)
Intensity maps are produced as a result of detailed investigation of the type of effects tabulated in Table 5.4, as shown in Figure 5.9 for the 1994 MW 6.7 Northridge earthquake. Correlations have been developed between the area of various MMIs and earthquake magnitude, which are of value for seismological and planning purposes. Figure 10 correlates Af elt vs. MW . For pre-instrumental historical earthquakes, Af elt can be estimated from newspapers and other reports, which then can be used to estimate the event magnitude, thus supplementing the seismicity catalog. This technique has been especially useful in regions with a long historical record [4, 133]. Time History
Sensitive strong motion seismometers have been available since the 1930s, and they record actual ground motions specific to their location (Figure 5.11). Typically, the ground motion records, termed seismographs or time histories, have recorded acceleration (these records are termed accelerograms),
4 Note that L = g(M ) should not be inverted to solve for M = f (L), as a regression for y = f (x) is different than a s s
regression for x = g(y).
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1999 by CRC Press LLC
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Table 5.3a Regressions of Rupture Length, Rupture Width, Rupture Area and Moment Magnitude Equationa M = a + b ∗ log(SRL)
log(SRL) = a + b∗ M
M = a + b ∗ log(RLD)
log(RLD) = a + b∗ M
M = a + b ∗ log(RW )
log(RW ) = a + b∗ M
M = a + b ∗ log(RA)
log(RA) = a + b∗ M
Slip typeb
Number of events
SS R N All SS R N All SS R N All SS R N All SS R N All SS R N All SS R N All SS R N All
43 19 15 77 43 19 15 77 93 50 24 167 93 50 24 167 87 43 23 153 87 43 23 153 83 43 22 148 83 43 22 148
Coefficients and standard errors a (sa) b(sb) 5.16(0.13) 5.00(0.22) 4.86(0.34) 5.08(0.10) −3.55(0.37) −2.86(0.55) −2.01(0.65) −3.22(0.27) 4.33(0.06) 4.49(0.11) 4.34(0.23) 4.38(0.06) −2.57(0.12) −2.42(0.21) −1.88(0.37) −2.44(0.11) 3.80(0.17) 4.37(0.16) 4.04(0.29) 4.06(0.11) −0.76(0.12) −1.61(0.20) −1.14(0.28) −1.01(0.10) 3.98(0.07) 4.33(0.12) 3.93(0.23) 4.07(0.06) −3.42(0.18) −3.99(0.36) −2.87(0.50) −3.49(0.16)
1.12(0.08) 1.22(0.16) 1.32(0.26) 1.16(0.07) 0.74(0.05) 0.63(0.08) 0.50(0.10) 0.69(0.04) 1.49(0.05) 1.49(0.09) 1.54(0.18) 1.49(0.04) 0.62(0.02) 0.58(0.03) 0.50(0.06) 0.59(0.02) 2.59(0.18) 1.95(0.15) 2.11(0.28) 2.25(0.12) 0.27(0.02) 0.41(0.03) 0.35(0.05) 0.32(0.02) 1.02(0.03) 0.90(0.05) 1.02(0.10) 0.98(0.03) 0.90(0.03) 0.98(0.06) 0.82(0.08) 0.91(0.03)
Standard deviation s
Correlation coefficient r
Magnitude range
0.28 0.28 0.34 0.28 0.23 0.20 0.21 0.22 0.24 0.26 0.31 0.26 0.15 0.16 0.17 0.16 0.45 0.32 0.31 0.41 0.14 0.15 0.12 0.15 0.23 0.25 0.25 0.24 0.22 0.26 0.22 0.24
0.91 0.88 0.81 0.89 0.91 0.88 0.81 0.89 0.96 0.93 0.88 0.94 0.96 0.93 0.88 0.94 0.84 0.90 0.86 0.84 0.84 0.90 0.86 0.84 0.96 0.94 0.92 0.95 0.96 0.94 0.92 0.95
5.6 to 8.1 5.4 to 7.4 5.2 to 7.3 5.2 to 8.1 5.6 to 8.1 5.4 to 7.4 5.2 to 7.3 5.2 to 8.1 4.8 to 8.1 4.8 to 7.6 5.2 to 7.3 4.8 to 8.1 4.8 to 8.1 4.8 to 7.6 5.2 to 7.3 4.8 to 8.1 4.8 to 8.1 4.8 to 7.6 5.2 to 7.3 4.8 to 8.1 4.8 to 8.1 4.8 to 7.6 5.2 to 7.3 4.8 to 8.1 4.8 to 7.9 4.8 to 7.6 5.2 to 7.3 4.8 to 7.9 4.8 to 7.9 4.8 to 7.6 5.2 to 7.3 4.8 to 7.9
Length/width range (km) 1.3 to 432 3.3 to 85 2.5 to 41 1.3 to 432 1.3 to 432 3.3 to 85 2.5 to 41 1.3 to 432 1.5 to 350 1.1 to 80 3.8 to 63 1.1 to 350 1.5 to 350 1.1 to 80 3.8 to 63 1.1 to 350 1.5 to 350 1.1 to 80 3.8 to 63 1.1 to 350 1.5 to 350 1.1 to 80 3.8 to 63 1.1 to 350 3 to 5,184 2.2 to 2,400 19 to 900 2.2 to 5,184 3 to 5,184 2.2 to 2,400 19 to 900 2.2 to 5,184
a SRL—surface rupture length (km); RLD —subsurface rupture length (km); RW —downdip rupture width (km); RA—rupture area (km2 ). b SS—strike slip; R—reverse; N—normal.
From Wells, D. L. and Coopersmith, K. J., Empirical Relationships Among Magnitude, Rupture Length, Rupture Width, Rupture Area and Surface Displacements, Bull. Seis. Soc. Am., 84(4), 974-1002, 1994. With permission.
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Table 5.3b Regressions of Displacement and Moment Magnitude Equationa M = a + b ∗ log(MD)
log(MD) = a + b∗ M
M = a + b ∗ log(AD)
log(AD) = a + b∗ M
Slip typeb
Number of events
SS { Rc N All SS {R N All SS {R N All SS {R N All
43 21 16 80 43 21 16 80 29 15 12 56 29 15 12 56
Coefficients and standard errors a (sa) b(sb) 6.81(0.05) 6.52(0.11) 6.61(0.09) 6.69(0.04) −7.03(0.55) − 1.84(1.14) −5.90(1.18) −5.46(0.51) 7.04(0.05) 6.64(0.16) 6.78(0.12) 6.93(0.05) −6.32(0.61) − 0.74(1.40) −4.45(1.59) −4.80(0.57)
0.78(0.06) 0.44(0.26) 0.71(0.15) 0.74(0.07) 1.03(0.08) 0.29(0.17) 0.89(0.18) 0.82(0.08) 0.89(0.09) 0.13(0.36) 0.65(0.25) 0.82(0.10) 0.90(0.09) 0.08(0.21) 0.63(0.24) 0.69(0.08)
Standard deviation s
Correlation coefficient r
Magnitude range
0.29 0.52 0.34 0.40 0.34 0.42 0.38 0.42 0.28 0.50 0.33 0.39 0.28 0.38 0.33 0.36
0.90 0.36 0.80 0.78 0.90 0.36 0.80 0.78 0.89 0.10 0.64 0.75 0.89 0.10 0.64 0.75
5.6 to 8.1 5.4 to 7.4 5.2 to 7.3 5.2 to 8.1 5.6 to 8.1 5.4 to 7.4 5.2 to 7.3 5.2 to 8.1 5.6 to 8.1 5.8 to 7.4 6.0 to 7.3 5.6 to 8.1 5.6 to 8.1 5.8 to 7.4 6.0 to 7.3 5.6 to 8.1
Displacement range (km) 0.01 to 14.6 0.11 to 6.5 } 0.06 to 6.1 0.01 to 14.6 0.01 to 14.6 0.11 to 6.5 } 0.06 to 6.1 0.01 to 14.6 0.05 to 8.0 0.06 to 1.5 } 0.08 to 2.1 0.05 to 8.0 0.05 to 8.0 0.06 to 1.5 } 0.08 to 2.1 0.05 to 8.0
a MD —maximum displacement (m); AD —average displacement (M).
b SS—strike slip; R—reverse; N—normal. c Regressions for reverse-slip relationships shown in italics and brackets are not significant at a 95% probability level.
From Wells, D. L. and Coopersmith, K. J., Empirical Relationships Among Magnitude, Rupture Length, Rupture Width, Rupture Area and Surface Displacements, Bull. Seis. Soc. Am., 84(4), 974-1002, 1994. With permission.
TABLE 5.4 I II III IV V VI VII VIII
IX X XI XII
Modified Mercalli Intensity Scale of 1931
Not felt except by a very few under especially favorable circumstances. Felt only by a few persons at rest, especially on upper floors of buildings. Delicately suspended objects may swing. Felt quite noticeably indoors, especially on upper floors of buildings, but many people do not recognize it as an earthquake. Standing motor cars may rock slightly. Vibration like passing truck. Duration estimated. During the day felt indoors by many, outdoors by few. At night some awakened. Dishes, windows, and doors disturbed; walls make creaking sound. Sensation like heavy truck striking building. Standing motor cars rock noticeably. Felt by nearly everyone; many awakened. Some dishes, windows, etc. broken; a few instances of cracked plaster; unstable objects overturned. Disturbance of trees, poles, and other tall objects sometimes noticed. Pendulum clocks may stop. Felt by all; many frightened and run outdoors. Some heavy furniture moved; a few instances of fallen plaster or damaged chimneys. Damage slight. Everybody runs outdoors. Damage negligible in buildings of good design and construction slight to moderate in well built ordinary structures; considerable in poorly built or badly designed structures. Some chimneys broken. Noticed by persons driving motor cars. Damage slight in specially designed structures; considerable in ordinary substantial buildings, with partial collapse; great in poorly built structures. Panel walls thrown out of frame structures. Fall of chimneys, factory stacks, columns, monuments, walls. Heavy furniture overturned. Sand and mud ejected in small amounts. Changes in well water. Persons driving motor cars disturbed. Damage considerable in specially designed structures; well-designed frame structures thrown out of plumb; great in substantial buildings, with partial collapse. Buildings shifted off foundations. Ground cracked conspicuously. Underground pipes broken. Some well-built wooden structures destroyed; most masonry and frame structures destroyed with foundations; ground badly cracked. Rails bent. Landslides considerable from river banks and steep slopes. Shifted sand and mud. Water splashed over banks. Few, if any (masonry), structures remain standing. Bridges destroyed. Broad fissures in ground. Underground pipelines completely out of service. Earth slumps and land slips in soft ground. Rails bent greatly. Damage total. Waves seen on ground surfaces. Lines of sight and level distorted. Objects thrown upward into the air.
After Wood, H. O. and Neumann, Fr., Modified Mercalli Intensity Scale of 1931, Bull. Seis. Soc. Am., 21, 277-283, 1931.
TABLE 5.5 Comparison of Modified Mercalli (MMI) and Other Intensity Scales aa
MMIb
R-Fc
MSKd
JMAe
0.7 1.5 3 7 15 32 68 147 316 681 (1468)f (3162)f
I II III IV V VI VII VIII IX X XI XII
I I to II III IV to V V to VI VI to VII VIIIVIII+ to IX− IX+ X — —
I II III IV V VI VII VIII IX X XI XII
0 I II II to III III IV IV to V V V to VI VI VII
a gals b Modified Mercalli Intensity c Rossi-Forel d Medvedev-Sponheur-Karnik e Japan Meteorological Agency
f a values provided for reference only. MMI > X are due more
to geologic effects.
for many years in analog form on photographic film and, more recently, digitally. Analog records required considerable effort for correction due to instrumental drift, before they could be used. Time histories theoretically contain complete information about the motion at the instrumental location, recording three traces or orthogonal records (two horizontal and one vertical). Time histories (i.e., the earthquake motion at the site) can differ dramatically in duration, frequency content, and amplitude. The maximum amplitude of recorded acceleration is termed the peak ground acceleration, PGA (also termed the ZPA, or zero period acceleration). Peak ground velocity (PGV) and peak ground displacement (PGD) are the maximum respective amplitudes of velocity and displacement. Acceleration is normally recorded, with velocity and displacement being determined by numerical integration; however, velocity and displacement meters are also deployed, to a lesser extent. Accel1999 by CRC Press LLC
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FIGURE 5.9: MMI maps, 1994 MW 6.7 Northridge Earthquake. (1) Far-field isoseismal map. Roman numerals give average MMI for the regions between isoseismals; arabic numerals represent intensities in individual communities. Squares denote towns labeled in the figure. Box labeled “FIG. 2” identifies boundaries of that figure. (2) Distribution of MMI in the epicentral region. (Courtesy of Dewey, J.W. et al., Spacial Variations of Intensity in the Northridge Earthquake, in Woods, M.C. and Seiple, W.R., Eds., The Northridge California Earthquake of 17 January 1994, California Department of Conservation, Division of Mines and Geology, Special Publication 116, 39-46, 1995.) eration can be expressed in units of cm/s2 (termed gals), but is often also expressed in terms of the fraction or percent of the acceleration of gravity (980.66 gals, termed 1g). Velocity is expressed in cm/s (termed kine). Recent earthquakes (1994 Northridge, Mw 6.7 and 1995 Hanshin [Kobe] Mw 6.9) have recorded PGA’s of about 0.8g and PGV’s of about 100 kine — almost 2g was recorded in the 1992 Cape Mendocino earthquake. Elastic Response Spectra
If the SDOF mass in Figure 5.1 is subjected to a time history of ground (i.e., base) motion similar to that shown in Figure 5.11, the elastic structural response can be readily calculated as a function of time, generating a structural response time history, as shown in Figure 5.12 for several oscillators with differing natural periods. The response time history can be calculated by direct integration of Equation 5.1 in the time domain, or by solution of the Duhamel integral [32]. However, this is time-consuming, and the elastic response is more typically calculated in the frequency domain 1 v(t) = 2π
Z
∞
$ =−∞
H ($ )c($ ) exp(i$ t)d$
(5.15)
where v(t) = the elastic structural displacement response time history $ = frequency 1 is the complex frequency response function H ($ ) = −$ 2 m+ic+k R∞ c($ ) = $ =−∞ p(t) exp(−i$ t)dt is the Fourier transform of the input motion (i.e., the Fourier transform of the ground motion time history) which takes advantage of computational efficiency using the Fast Fourier Transform. 1999 by CRC Press LLC
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FIGURE 5.10: log Afelt (km2 ) vs. MW . Solid circles denote ENA events and open squares denote California earthquakes. The dashed curve is the MW − Afelt relationship of an earlier study, whereas the solid line is the fit determined by Hanks and Johnston, for California data. (Courtesy of Hanks J. W. and Johnston A. C., Common Features of the Excitation and Propagation of Strong Ground Motion for North American Earthquakes, Bull. Seis. Soc. Am., 82(1), 1-23, 1992.)
FIGURE 5.11: Typical earthquake accelerograms. (Courtesy of Darragh, R. B., Huang, M. J., and Shakal, A. F., Earthquake Engineering Aspects of Strong Motion Data from Recent California Earthquakes, Proc. Fifth U.S. Natl. Conf. Earthquake Eng., 3, 99-108, 1994, Earthquake Engineering Research Institute. Oakland, CA.)
For design purposes, it is often sufficient to know only the maximum amplitude of the response time history. If the natural period of the SDOF is varied across a spectrum of engineering interest (typically, for natural periods from .03 to 3 or more seconds, or frequencies of 0.3 to 30+ Hz), then the plot of these maximum amplitudes is termed a response spectrum. Figure 5.12 illustrates this process, resulting in Sd , the displacement response spectrum, while Figure 5.13 shows (a) the Sd , 1999 by CRC Press LLC
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FIGURE 5.12: Computation of deformation (or displacement) response spectrum. (From Chopra, A. K., Dynamics of Structures, A Primer, Earthquake Engineering Research Institute, Oakland, CA, 1981. With permission.) displacement response spectrum, (b) Sv , the velocity response spectrum (also denoted PSV, the pseudo spectral velocity, pseudo to emphasize that this spectrum is not exactly the same as the relative velocity response spectrum [63], and (c) Sa , the acceleration response spectrum. Note that Sv =
2π Sd = $ Sd T
and Sa =
2π Sv = $ Sv = T
2π T
(5.16)
2 Sd = $ 2 Sd
(5.17)
Response spectra form the basis for much modern earthquake engineering structural analysis and design. They are readily calculated if the ground motion is known. For design purposes, however, response spectra must be estimated. This process is discussed below. Response spectra may be plotted in any of several ways, as shown in Figure 5.13 with arithmetic axes, and in Figure 5.14 where the 1999 by CRC Press LLC
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FIGURE 5.13: Response spectra spectrum. (From Chopra, A. K., Dynamics of Structures, A Primer, Earthquake Engineering Research Institute, Oakland, CA, 1981. With permission.)
velocity response spectrum is plotted on tripartite logarithmic axes, which equally enables reading of displacement and acceleration response. Response spectra are most normally presented for 5% of critical damping. While actual response spectra are irregular in shape, they generally have a concave-down arch or trapezoidal shape, when plotted on tripartite log paper. Newmark observed that response spectra tend to be characterized by three regions: (1) a region of constant acceleration, in the high frequency portion of the spectra; (2) constant displacement, at low frequencies; and (3) constant velocity, at intermediate frequencies, as shown in Figure 5.15. If a spectrum amplification factor is defined as the ratio of the spectral parameter to the ground motion parameter (where parameter indicates acceleration, velocity or displacement), then response spectra can be estimated from the data in Table 5.6, provided estimates of the ground motion parameters are available. An example spectra using these data is given in Figure 5.15. A standardized response spectra is provided in the Uniform Building Code [126] for three soil types. The spectra is a smoothed average of normalized 5% damped spectra obtained from actual ground 1999 by CRC Press LLC
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FIGURE 5.14: Response spectra, tri-partite plot (El Centro S 0◦ E component). (From Chopra, A. K., Dynamics of Structures, A Primer, Earthquake Engineering Research Institute, Oakland, CA, 1981. With permission.)
motion records grouped by subsurface soil conditions at the location of the recording instrument, and are applicable for earthquakes characteristic of those that occur in California [111]. If an estimate of ZPA is available, these normalized shapes may be employed to determine a response spectra, appropriate for the soil conditions. Note that the maximum amplification factor is 2.5, over a period range approximately 0.15 s to 0.4 - 0.9 s, depending on the soil conditions. Other methods for estimation of response spectra are discussed below. 1999 by CRC Press LLC
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FIGURE 5.15: Idealized elastic design spectrum, horizontal motion (ZPA = 0.5g, 5% damping, one sigma cumulative probability. (From Newmark, N. M. and Hall, W. J., Earthquake Spectra and Design, Earthquake Engineering Research Institute, Oakland, CA, 1982. With permission.) TABLE 5.6 Spectrum Amplification Factors for Horizontal Elastic Response Damping,
One sigma (84.1%)
Median (50%)
% Critical
A
V
D
A
V
D
0.5 1 2 3 5 7 10 20
5.10 4.38 3.66 3.24 2.71 2.36 1.99 1.26
3.84 3.38 2.92 2.64 2.30 2.08 1.84 1.37
3.04 2.73 2.42 2.24 2.01 1.85 1.69 1.38
3.68 3.21 2.74 2.46 2.12 1.89 1.64 1.17
2.59 2.31 2.03 1.86 1.65 1.51 1.37 1.08
2.01 1.82 1.63 1.52 1.39 1.29 1.20 1.01
From Newmark, N. M. and Hall, W. J., Earthquake Spectra and Design, Earthquake Engineering Research Institute, Oakland, CA, 1982. With permission.
Inelastic Response Spectra
While the foregoing discussion has been for elastic response spectra, most structures are not expected, or even designed, to remain elastic under strong ground motions. Rather, structures are expected to enter the inelastic region — the extent to which they behave inelastically can be defined by the ductility factor, µ µ= 1999 by CRC Press LLC
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um uy
(5.18)
FIGURE 5.16: Normalized response spectra shapes. (From Uniform Building Code, Structural Engineering Design Provisions, vol. 2, Intl. Conf. Building Officials, Whittier, 1994. With permission.)
where um is the maximum displacement of the mass under actual ground motions, and uy is the displacement at yield (i.e., that displacement which defines the extreme of elastic behavior). Inelastic response spectra can be calculated in the time domain by direct integration, analogous to elastic response spectra but with the structural stiffness as a non-linear function of displacement, k = k(u). If elastoplastic behavior is assumed, then elastic response spectra can be readily modified to reflect inelastic behavior [90] on the basis that (a) at low frequencies (0.3 Hz 33 Hz), accelerations are equal; and (c) at intermediate frequencies, the absorbed energy is preserved. Actual construction of inelastic response spectra on this basis is shown in Figure 5.17, where DV AAo is the elastic spectrum, which is reduced to D 0 and V 0 by the ratio of 1/µ for frequencies less than 2 Hz, and by the ratio of 1/(2µ − 1)1/2 between 2 and 8 Hz. Above 33 Hz there is no reduction. The result is the inelastic acceleration spectrum (D 0 V 0 A0 Ao ), while A00 A0o is the inelastic displacement spectrum. A specific example, for ZPA = 0.16g, damping = 5% of critical, and µ = 3 is shown in Figure 5.18. Response Spectrum Intensity and Other Measures
While the elastic response spectrum cannot directly define damage to a structure (which is essentially inelastic deformation), it captures in one curve the amount of elastic deformation for a wide variety of structural periods, and therefore may be a good overall measure of ground motion intensity. On this basis, Housner defined a response spectrum intensity as the integral of the elastic response spectrum velocity over the period range 0.1 to 2.5 s. Z SI (h) = 1999 by CRC Press LLC
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2.5
T =0.1
Sv(h, T )dT
(5.19)
FIGURE 5.17: Inelastic response spectra for earthquakes. (After Newmark, N. M. and Hall, W. J., Earthquake Spectra and Design, Earthquake Engineering Research Institute, Oakland, CA, 1982.)
where h = damping (as a percentage of ccrit ). A number of other measures exist, including Fourier amplitude spectrum [32] and Arias Intensity [8]: π IA = g
Z
t
a 2 (t)dt
(5.20)
0
Engineering Intensity Scale
Lastly, Blume [14] defined a measure of earthquake intensity, the Engineering Intensity Scale (EIS), which has been relatively underutilized but is worth noting as it attempts to combine the engineering benefits of response spectra with the simplicity of qualitative intensity scales, such as MMI. The EIS is simply a 10x9 matrix which characterizes a 5% damped elastic response spectra (Figure 5.19). Nine period bands (0.01-.1, -.2, -.4, -.6, -1.0, -2.0, - 4.0, -7.0, -10,0 s), and ten Sv levels (0.01-0.1, -1.0, -4.0, -10.0, -30.0, -60.0, -100., -300., -1000. kine) are defined. As can be seen, since the response spectrum for the example ground motion in period band II (0.1-0.2 s) is predominantly in Sv level 5 (10-30 kine), it is assigned EIS 5 (X is assigned where the response spectra does not cross a period band). In this manner, a nine-digit EIS can be assigned to a ground motion (in the example, it is X56,777,76X), which can be reduced to three digits (5,7,6) by averaging, or even to one digit (6, for this example). Numerically, single digit EIS values tend to be a unit or so lower than the equivalent MMI intensity value. 1999 by CRC Press LLC
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FIGURE 5.18: Example inelastic response spectra. (From Newmark, N. M. and Hall, W. J., Earthquake Spectra and Design, Earthquake Engineering Research Institute, Oakland, CA, 1982. With permission.)
5.2.4
Strong Motion Attenuation and Duration
The rate at which earthquake ground motion decreases with distance, termed attenuation, is a function of the regional geology and inherent characteristics of the earthquake and its source. Three major factors affect the severity of ground shaking at a site: (1) source — the size and type of the earthquake, (2) path — the distance from the source of the earthquake to the site and the geologic characteristics of the media earthquake waves pass through, and (3) site-specific effects — type of soil at the site. In the simplest of models, if the seismogenic source is regarded as a point, then from considering the relation of energy and earthquake magnitude and the fact that the volume of a hemisphere is proportion to R 3 (where R represents radius), it can be seen that energy per unit volume is proportional to C10aM R −3 , where C is a constant or constants dependent on the earth’s crustal properties. The constant C will vary regionally — for example, it has long been observed that attenuation in eastern North America (ENA) varies significantly from that in western North America (WNA) — earthquakes in ENA are felt at far greater distances. Therefore, attenuation relations are regionally dependent. Another regional aspect of attenuation is the definition of terms, especially magnitude, where various relations are developed using magnitudes defined by local observatories. A very important aspect of attenuation is the definition of the distance parameter; because attenuation is the change of ground motion with location, this is clearly important. Many investigators use differing definitions; as study has progressed, several definitions have emerged: (1) hypocentral distance (i.e., straight line distance from point of interest to hypocenter, where hypocentral distance 1999 by CRC Press LLC
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FIGURE 5.19: Engineering intensity scale (EIS) matrix with example. (From Blume, J. A., An Engineering Intensity Scale for Earthquakes and Other Ground Motions, Bull. Seis. Soc. Am., 60(1), 217-229, 1970. With permission.)
may be arbitrary or based on regression rather than observation), (2) epicentral distance, (3) closest distance to the causative fault, and (4) closest horizontal distance from the station to the point on the earth’s surface that lies directly above the seismogenic source. In using attenuation relations, it is critical that the correct definition of distance is consistently employed. Methods for estimating ground motion may be grouped into two major categories: empirical and methods based on seismological models. Empirical methods are more mature than methods based on seismological models, but the latter are advantageous in explicitly accounting for source and path, therefore having explanatory value. They are also flexible, they can be extrapolated with more confidence, and they can be easily modified for additional factors. Most seismological model-based methods are stochastic in nature — Hanks and McGuire’s [54] seminal paper has formed the basis 1999 by CRC Press LLC
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for many of these models, which “assume that ground acceleration is a finite-duration segment of a stationary random process, completely characterized by the assumption that acceleration follows Brune’s [23] source spectrum (for California data, typically about 100 bars), and that the duration of strong shaking is equal to reciprocal of the source corner frequency” fo (the frequency above which earthquake radiation spectra vary with $ −3 - below fo , the spectra are proportional to seismic moment [108]). Since there is substantial ground motion data in WNA, seismological model-based relations have had more value in ENA, where few records exist. The Hanks-McGuire method has, therefore, been usefully applied in ENA [123] where Boore and Atkinson [18] found, for hard-rock sites, the relation: (5.21) log y = c0 + c1 r − log r where y = a ground motion parameter (PSV, unless ci coefficients for amax are used) r = hypocentral distance (km) P = ξoi + ξni (MW − 6)n I = 0, 1 summation for n = 1, 2, 3 (see Table 5.7) ci TABLE 5.7
Eastern North America Hard-Rock Attenuation Coefficientsa
Response frequency (Hz)
ξ0
ξ1
ξ2
ξ3 −5.364E − 02
0.2
c0 : c1 :
1.743E + 00 −3.130E − 04
1.064E + 00 1.415E − 03
−4.293E − 02 −1.028E − 03
0.5
c0 : c1 :
2.141E + 00 −2.504E − 04
8.521E − 01
−1.670E − 01 −2.612E − 04
1.0
c0 : c1 :
2.300E + 00 −1.024E − 03
6.655E − 01 −1.144E − 04
−1.538E − 01 1.109E − 04
2.0
c0 : c1 :
2.317E + 00 −1.683E − 03
5.070E − 01 1.492E − 04
−9.317E − 02 1.203E − 04
5.0
c0 : c1 :
2.239E + 00 −2.537E − 03
3.976E − 01 5.468E − 04
−4.564E − 02 7.091E − 05
10.0
c0 : c1 :
2.144E + 00 −3.094E − 03
3.617E − 01 7.640E − 04
−3.163E − 02
20.0
c0 : c1 :
2.032E + 00 −3.672E − 03
3.438E − 01 8.956E − 04
−2.559E − 02 −4.219E − 05
amax
c0 : c1 :
3.763E + 00 −3.885E − 03
3.354E − 01 1.042E − 03
−2.473E − 02 −9.169E − 05
a See Equation 5.21. From Boore, D.M. and Atkinson, G.M., Stochastic Prediction of Ground Motion and Spectral Response Parameters at Hard-Rock Sites in Eastern North America, Bull. Seis. Soc. Am., 77, 440-487, 1987. With permission.
Similarly, Toro and McGuire [123] furnish the following relation for rock sites in ENA: ln Y = c0 + c1 M + c2 ln(R) + c3 R
(5.22)
where the c0 - c3 coefficients are provided in Table 5.8, M represents mLg , and R is the closest distance between the site and the causative fault at a minimum depth of 5 km. These results are valid for hypocentral distances of 10 to 100 km, and mLg 4 to 7. More recently, Boore and Joyner [19] have extended their hard-rock relations to deep soil sites in ENA: (5.23) log y = a 00 + b(m − 6) + c(m − 6)2 + d(m − 6)3 − log r + kr where a 00 and other coefficients are given in Table 5.9, m is moment magnitude (MW ), and r is hypocentral distance (km) although the authors suggest that, close to long faults, the distance should 1999 by CRC Press LLC
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TABLE 5.8
ENA Rock Attenuation Coefficientsa
Y
c0
c1
c2
c3
PSRV (1 Hz) PSRV (5 Hz) PSRV (10 Hz) PGA (cm/s2 )
−9.283 −2.757 −1.717 2.424
2.289 1.265 1.069 0.982
−1.000 −1.000 −1.000 −1.004
−.00183 −.00310 −.00391 −.00468
a See Equation 5.22. Spectral velocities are given in cm/s; peak acceleration is given in cm/s2 From Toro, G.R. and McGuire, R.K., An Investigation Into Earthquake Ground Motion Characteristics in Eastern North America, Bull. Seis. Soc. Am., 77, 468-489, 1987. With permission.
be the nearest distance to seismogenic rupture. The coefficients in Table 5.9 should not be used outside the ranges 10 < r < 400 km, and 5.0 < MW < 8.5. TABLE 5.9 Coefficients for Ground-Motion Estimation at Deep-Soil Sites a in Eastern North America in Terms of MW a0
a 00
b
c
d
k
M at maxb
SV 0.05 0.10 0.15 0.20 0.30 0.40 0.50 0.75 1.00 1.50 2.00 3.00 4.00
0.020 0.040 0.015 0.015 0.010 0.015 0.010 0.000 0.000 0.000 0.000 0.000 0.000
1.946 2.267 2.377 2.461 2.543 2.575 2.588 2.586 2.567 2.511 2.432 2.258 2.059
0.431 0.429 0.437 0.447 0.472 0.499 0.526 0.592 0.655 0.763 0.851 0.973 1.039
− 0.028 − 0.026 − 0.031 − 0.037 − 0.051 − 0.066 − 0.080 − 0.111 − 0.135 − 0.165 − 0.180 − 0.176 − 0.145
− 0.018 − 0.018 − 0.017 − 0.016 − 0.012 − 0.009 − 0.007 − 0.001 0.002 0.004 0.002 − 0.008 − 0.022
− 0.00350 − 0.00240 − 0.00190 − 0.00168 − 0.00140 − 0.00110 − 0.00095 − 0.00072 − 0.00058 − 0.00050 − 0.00039 − 0.00027 − 0.00020
8.35 8.38 8.38 8.38 8.47 8.50 8.48 8.58 8.57 8.55 8.47 8.38 8.34
amax SV max SA max
0.030 0.020 0.040
3.663 2.596 4.042
0.448 0.608 0.433
− 0.037 − 0.038 − 0.029
− 0.016 − 0.022 − 0.017
− 0.00220 − 0.00055 − 0.00180
8.38 8.51 8.40
T (sec)
a The distance used is generally the hypocentral distance; we suggest that, close to long faults,
the distance should be the nearest distance to seismogenic rupture. The response spectra are for random horizontal components and 5% damping. The units of amax and SA are cm/s2 ; the units of SV are cm/s. The coefficients in this table should not be used outside the ranges < < < < 10= r = 400 km and 5.0 = M = 8.5. See also Equation 5.23. b “M at max” is the magnitude at which the cubic equation attains its maximum value; for larger magnitudes, we recommend that the motions be equated to those for “M at max”. From Boore, D.M. and Joyner, W.B., Estimation of Ground Motion at Deep-Soil Sites in Eastern North America, Bull. Seis. Soc. Am., 81(6), 2167-2185, 1991. With permission.
In WNA, due to more data, empirical methods based on regression of the ground motion parameter vs. magnitude and distance have been more widely employed, and Campbell [28] offers an excellent review of North American relations up to 1985. Initial relationships were for PGA, but regression of the amplitudes of response spectra at various periods is now common, including consideration of fault type and effects of soil. Some current favored relationships are: Campbell and Bozorgnia [29] (PGA - Worldwide Data) ln(P GA)
=
−3.512 + 0.904M − 1.328 ln
q {Rs2 + [0.149 exp(0.647M)]2 }
+ [1.125 − 0.112 ln(Rs ) − 0.0957M]F 1999 by CRC Press LLC
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+ [0.440 − 0.171 ln(Rs )]Ssr + [0.405 − 0.222 ln(Rs )]Shr + ε where P GA M Rs F Ssr Shr Ssr = Shr ε
(5.24)
= the geometric mean of the two horizontal components of peak ground acceleration (g) = moment magnitude (MW ) = the closest distance to seismogenic rupture on the fault (km) = 0 for strike-slip and normal faulting earthquakes, and 1 for reverse, reverse-oblique, and thrust faulting earthquakes = 1 for soft-rock sites = 1 for hard-rock sites = 0 for alluvium sites = a random error term with zero mean and standard deviation equal to σln (P GA), the standard error of estimate of ln(P GA)
This relation is intended for meizoseismal applications, and should not be used to estimate PGA at distances greater than about 60 km (the limit of the data employed for the regression). The relation is based on 645 near-source recordings from 47 worldwide earthquakes (33 of the 47 are California records — among the other 14 are the 1985 MW 8.0 Chile, 1988 MW 6.8 Armenia, and 1990 MW 7.4 Manjil Iran events). Rs should not be assigned a value less than the depth of the top of the seismogenic crust, or 3 km. Regarding the uncertainty, ε was estimated as: 0.55 if P GA < 0.068 if 0.068 ≤ P GA ≤ 0.21 σln (P GA) = 0.173 − 0.140 ln(P GA) 0.39 if P GA > 0.21 Figure 5.20 indicates, for alluvium, median values of the attenuation of peak horizontal acceleration with magnitude and style of faulting. Joyner and Boore (PSV - WNA Data) [20, 67] Similar to the above but using a two-step regression technique in which the ground motion parameter is first regressed against distance and then amplitudes regressed against magnitude, Boore, Joyner, and Fumal [20] have used WNA data to develop relations for PGA and PSV of the form: log Y where Y M r
= b1 + b2 (M − 6) + b3 (M − 6)2 + b4 r + b5 log10 r + b6 GB + b7 GC + εr + εe
(5.25)
= the ground motion parameter (in cm/s for PSV, and g for PGA) = moment magnitude (MW ) = (d 2 + h2 )(1/2) = distance (km), where h is a fictitious depth determined by regression, and d is the closest horizontal distance from the station to the point on the earth’s surface that lies directly above the rupture GB , GC = site classification indices (GB = 1 for class B site, GC =1 for class C site, both zero otherwise), where Site Class A has shear wave velocities (averaged over the upper 30 m) > 750 m/s, Site Class B is 360 to 750 m/s, and Site Class C is 180 to 360 m/s (class D sites, < 180 m/s, were not included). In effect, class A are rock, B are firm soil sites, C are deep alluvium/soft soils, and D would be very soft sites εr + εe = independent random variable measures of uncertainty, where εr takes on a specific value for each record, and εe for each earthquake = coefficients (see Table 5.10 and Table 5.11) bi , h The relation is valid for magnitudes between 5 and 7.7, and for distances (d) ≤ 100 km. The coefficients in Equation 5.25 are for 5% damped response spectra — Boore et al. [20] also provide similar coefficients for 2%, 10%, and 20% damped spectra, as well as for the random horizontal 1999 by CRC Press LLC
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FIGURE 5.20: Campbell and Bozorgnia worldwide attenuation relationship showing (for alluvium) the scaling of peak horizontal acceleration with magnitude and style of faulting. (From Campbell, K.W. and Bozorgnia, Y., Near-Source Attenuation of Peak Horizontal Acceleration from Worldwide Accelerograms Recorded from 1957 to 1993, Proc. Fifth U.S. National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, Oakland, CA, 1994. With permission.)
coefficient (i.e., both horizontal coefficients, not just the larger, are considered). Figure 5.21 presents curves of attenuation of PGA and PSV for Site Class C, using these relations, while Figure 5.22 presents a comparison of this, the Campbell and Bozorgnia [29] and Sadigh et al. [105] attenuation relations, for two magnitude events on alluvium. The foregoing has presented attenuation relations for PGA (Worldwide) and response spectra (ENA and WNA). While there is some evidence [136] that meizoseismal strong ground motion may not differ as much regionally as previously believed, regional attenuation in the far-field differs significantly (e.g., ENA vs. WNA). One regime that has been treated in a special class has been large subduction zone events, such as those that occur in the North American Pacific Northwest (PNW), in Alaska, off the west coast of Central and South America, off-shore Japan, etc. This is due to the very large earthquakes that are generated in these zones, with long duration and a significantly different path. A number of relations have been developed for these events [10, 37, 81, 115, 138] which should be employed in those regions. A number of other investigators have developed attenuation relations for other regions, such as China, Japan, New Zealand, the Trans-Alpide areas, etc., which should be reviewed when working in those areas (see the References). In addition to the seismologically based and empirical models, there is another method for attenuation or ground motion modeling, which may be termed semi-empirical methods (Figure 5.23) [129]. The approach discretizes the earthquake fault into a number of subfault elements, finite rupture on each of which is modeled with radiation therefrom modeled via Green’s functions. The resulting wave-trains are combined with empirical modeling of scattering and other factors to generate time-histories of ground motions for a specific site. The approach utilizes a rational framework with powerful explanatory features, and offers useful application in the very near-field of large earthquakes, where it is increasingly being employed. The foregoing has also dealt exclusively with horizontal ground motions, yet vertical ground motions can be very significant. The common practice for many years has been to take the ratio (V /H ) 1999 by CRC Press LLC
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TABLE 5.10 Coefficients for 5% Damped PSV, for the Larger Horizontal Component T(s)
B1
B2
B3
B4
B5
B6
B7
H
.10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .22 .24 .26 .28 .30 .32 .34 .36 .38 .40 .42 .44 .46 .48 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00
1.700 1.777 1.837 1.886 1.925 1.956 1.982 2.002 2.019 2.032 2.042 2.056 2.064 2.067 2.066 2.063 2.058 2.052 2.045 2.038 2.029 2.021 2.013 2.004 1.996 1.988 1.968 1.949 1.932 1.917 1.903 1.891 1.881 1.872 1.864 1.858 1.849 1.844 1.842 1.844 1.849 1.857 1.866 1.878 1.891 1.905
.321 .320 .320 .321 .322 .323 .325 .326 .328 .330 .332 .336 .341 .345 .349 .354 .358 .362 .366 .369 .373 .377 .380 .383 .386 .390 .397 .404 .410 .416 .422 .427 .432 .436 .440 .444 .452 .458 .464 .469 .474 .478 .482 .485 .488 .491
−.104 −.110 −.113 −.116 −.117 −.117 −.117 −.117 −.115 −.114 −.112 −.109 −.105 −.101 −.096 −.092 −.088 −.083 −.079 −.076 −.072 −.068 −.065 −.061 −.058 −.055 −.048 −.042 −.037 −.033 −.029 −.025 −.022 −.020 −.018 −.016 −.014 −.013 −.012 −.013 −.014 −.016 −.019 −.022 −.025 −.028
.00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000
−.921 −.929 −.934 −.938 −.939 −.939 −.939 −.938 −.936 −.934 −.931 −.926 −.920 −.914 −.908 −.902 −.897 −.891 −.886 −.881 −.876 −.871 −.867 −.863 −.859 −.856 −.848 −.842 −.837 −.833 −.830 −.827 −.826 −.825 −.825 −.825 −.828 −.832 −.837 −.843 −.851 −.859 −.868 −.878 −.888 −.898
.039 .065 .087 .106 .123 .137 .149 .159 .169 .177 .185 .198 .208 .217 .224 .231 .236 .241 .245 .249 .252 .255 .258 .261 .263 .265 .270 .275 .279 .283 .287 .290 .294 .297 .301 .305 .312 .319 .326 .334 .341 .349 .357 .365 .373 .381
.128 .150 .169 .187 .203 .217 .230 .242 .254 .264 .274 .291 .306 .320 .333 .344 .354 .363 .372 .380 .388 .395 .401 .407 .413 .418 .430 .441 .451 .459 .467 .474 .481 .486 .492 .497 .506 .514 .521 .527 .533 .538 .543 .547 .551 .554
6.18 6.57 6.82 6.99 7.09 7.13 7.13 7.10 7.05 6.98 6.90 6.70 6.48 6.25 6.02 5.79 5.57 5.35 5.14 4.94 4.75 4.58 4.41 4.26 4.16 3.97 3.67 3.43 3.23 3.08 2.97 2.89 2.85 2.83 2.84 2.87 3.00 3.19 3.44 3.74 4.08 4.46 4.86 5.29 5.74 6.21
The equations are to be used for 5.0 1 are peculiar to shells in that they produce undulating deformations around the cross-section with no net translation. The relatively large Fourier coefficients associated with n = 2,3,4,5 indicate that a significant portion of the loading will cause shell deformations in these modes. In turn, the corresponding local forces are significantly higher than a beam-like response would produce. To account for the internal conditions in the tower during operation, it is common practice to add an axisymmetric internal suction coefficient H = 0.5 to the external pressure coefficients H (θ ). In terms of the Fourier series representation, this would increase A0 to −0.8922. The dynamic amplification of the effective velocity pressure is represented by the parameter g in Equation 14.5. This parameter reflects the resonant part of the response of the structure and may be as much as 0.2 depending on the dynamic characteristics of the structure. However, when the basis 1999 by CRC Press LLC
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FIGURE 14.11: Surface roughness k/a and maximum side-suction. of q(z) includes some dynamic portion, such as the fastest-mile-of-wind, (1 + g) is commonly taken as 1.0. Cooling towers are often constructed in groups and close to other structures, such as chimneys or boiler houses, which may be higher than the tower itself. When the spacing of towers is closer than 1.5 times the base diameter or 2 times the throat diameter, or when other tall structures are nearby, the wind pressure on any single tower may be altered in shape and intensity. Such effects should be studied carefully in boundary-layer wind tunnels in order not to overlook dramatic increases in the wind loading. Earthquake loading on hyperbolic cooling towers is produced by ground motions transmitted from the foundation through the supporting columns and the lintel into the shell. If the base motion is assumed to be uniform vertically and horizontally, the circumferential effects are axisymmetrical (n = 0) and antisymmetrical (n = 1), respectively (see Figure 14.12). In the meridional direction, the magnitude and distribution of the earthquake-induced forces is a function of the mass of the tower and the dynamic properties of the structure (natural frequencies and damping) as well as the acceleration produced by the earthquake at the base of the structure. The most appropriate technique for determining the loads applied by a design earthquake to the shell and components is the response spectrum method which, in turn, requires a free vibration analysis to evaluate the natural frequencies [2, 3, 4]. It is common to use elastic spectra with 5% of critical damping. The supporting columns and foundation are critical for this loading condition and should be modeled in appropriate detail [3, 4]. Temperature variations on cooling towers arise from two sources: operating conditions and sunshine on one side. Typical operating conditions are an external temperature of −15◦ C and internal temperature of +30◦ C. This is an axisymmetrical effect, n = 0 on Figure 14.12. For sunshine, a 1999 by CRC Press LLC
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FIGURE 14.12: Harmonic components of the radial displacement. temperature gradient of 25◦ C constant over the height and distributed as a half-wave around one half of the circumference is appropriate. This loading would require a Fourier expansion in the form of Equation 14.6 and higher harmonic components, n >1, to be considered. Construction loads are generally caused by the fixing devices of climbing formwork, by tower crane anchors, and by attachments for material transport equipment as shown in Figure 14.13. These loads must be considered on the portion of the shell extant at the phase of construction. Non-uniform settlement due to varying subsoil stiffness may be a consideration. Such effects should be modeled considering the interaction of the foundation and the soil.
14.6
Methods of Analysis
Thin shells may resist external loading through forces acting parallel to the shell surface, forces acting perpendicular to the shell surface, and moments. While the analysis of such shells may be formulated within the three-dimensional theory of elasticity, there are reduced theories which are two-dimensional and are expressed in terms of force and moment intensities. These intensities are traditionally based on a reference surface, generally the middle surface, and are forces and moments per unit length of the middle surface element upon which they act. They are called stress resultants and stress couples, respectively, and are associated with the three directions: circumferential, θ 1 ; meridional, θ 2 ; and normal, θ 3 . In Figure 14.14, the extensional stress resultants, n11 and n22 , the in-plane shearing stress resultants, n12 = n21 , and the transverse shear stress resultants, q12 = q21 , are shown in the left diagram along with the components of the applied loading in the circumferential, meridional, and normal directions, p1 , p2 , and p3 , respectively. The bending stress couples, m11 and m22 , and the twisting stress couples, m12 = m21 , are shown in the right diagram along with the displacements v1 , v2 , and v3 in the respective directions. Historically, doubly curved thin shells have been designed to resist applied loading primarily through the extensional and shearing forces in the “plane” of the shell surface, as opposed to the 1999 by CRC Press LLC
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FIGURE 14.13: Attachments on shell wall.
transverse shears and bending and twisting moments which predominate in flat plates loaded normally to their surface. This is known as membrane action, as opposed to bending action, and is consistent with an accompanying theory and calculation methodology which has the advantage of being statically determinate. This methodology was well-suited for the pre-computer age and enabled many large thin shells, including cooling towers, to be rationally designed and economically constructed [9]. Because the conditions that must be provided at the shell boundaries in order to insure membrane action are not always achievable, shell bending should be taken into account even for shells designed by membrane theory. Remarkably, the accompanying bending often is confined to narrow regions in the vicinity of the boundaries and other discontinuities and may have only a minor effect on the shell design, such as local thickening and/or additional reinforcement. Many clever and insightful techniques have been developed over the years to approximate the effects of local bending in shells designed by the membrane theory. As we have passed into and advanced in the computer age, it is no longer appropriate to use the membrane theory to analyze such extraordinary thin shells, except perhaps for preliminary design purposes. The finite element method is widely accepted as the standard contemporary technique and the attention shifts to the level of sophistication to be used in the finite element model. As is often the case, the greater the level of sophistication specified, the more data required. Consequently, 1999 by CRC Press LLC
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FIGURE 14.14: Surface loads, stress resultants, stress couples, and displacements.
a model may evolve through several stages, starting with a relatively simple version that enables the structure to be sized, to the most sophisticated version that may depict such phenomena as the sequence of progressive collapse of the as-built shell under various static and dynamic loading scenarios, the incremental effects of the progressive stages of construction, the influence of the operating environment, aging and deterioration on the structure, etc. The techniques described in the following paragraphs form a hierarchical progression from the relatively simple to the very complex, depending on the objective of the analysis. In modeling cooling tower shells using the finite element method, there are a number of options. For the shell wall, ring elements, triangular elements, or quadrilateral elements have been used. Earlier, flat elements adapted from the two-dimensional elasticity and plate formulations were used to approximate the doubly curved surface. Such elements present a number of theoretical and computational problems and are not recommended for the analysis of shells. Currently, shell elements degenerated from three-dimensional solid elements are very popular. These elements have been utilized in both the ring and quadrilateral form. The column region at the base of the shell presents a special modeling challenge. For static analysis, the lower boundary is often idealized as a uniform support at the lintel level. Then, a portion of the lower shell and the columns is considered in a subsequent analysis to account for the concentrated actions of the columns, which may penetrate only a relatively short distance into the shell wall. For dynamic analysis, it is important to include the column region along with the veil in the model. An equivalent shell element has proved useful in this regard if ring elements are used to model the shell [3, 4]. It may also be desirable to include some of the foundation elements, such as a ring beam at the base and even the supporting piles in a dynamic or settlement model. The linear static analysis method is based on the classical bending theory of thin shells. While this theory has been formulated for many years, solutions for doubly curved shells have not been readily achievable until the development of computer-based numerical methods, most notably the finite element method. The outputs of such an analysis are the stress resultants and couples, defined on Figure 14.14, over the entire shell surface and the accompanying displacements. The analysis is based on the initial geometry, linear elastic material behavior, and a linear kinematic law. Some representative results of such analyses for a large cooling tower (Figure 14.15) are shown in Figures 14.16 through 14.24 for some of the important loading conditions discussed in the preceding section. The finite element model used considers the shell to be fixed at the top of the columns and, thus, does not account for the effect of the concentrated column reactions. Also, in considering the analyses under the individual loading conditions, it should be remembered that the effects are to be factored and combined to produce design values. 1999 by CRC Press LLC
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FIGURE 14.15: Design project for a 200-m high cooling tower: geometry.
FIGURE 14.16: Circumferential forces n11G under deadweight. 1999 by CRC Press LLC
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The dead load analysis results in Figures 14.16 and 14.17 indicate that the shell is always under compression in both directions, except for a small circumferential tension near the top. This is a very desirable feature of this geometrical form. In Figures 14.18 through 14.20, the results of an analysis for a quasistatic wind load using the K1.0 distribution from Figure 14.10 are shown. Large tensions in both the meridional and circumferential directions are present. The regions of tension may extend a considerable distance along the circumference from the windward meridian, and the magnitude of the forces is strongly dependent on the distribution selected. In contrast to bluff bodies, where the magnitude of the extensional force along the meridian would be essentially a function of the overturning moment, the cylindrical-type body is also strongly influenced by the circumferential distribution of the applied pressure, a function of the surface roughness. The major effect of the shearing forces is at the level of the lintel where they are transferred into the columns. The internal suction effects (Figures 14.21 and 14.22) are significant only in the circumferential direction. For the service temperature case shown in Figures 14.23 and 14.24, the main effects are bending in the lower region of the shell wall. The analysis of hyperbolic cooling towers for instability or buckling is a subject that has been investigated for several decades [1]. Shell buckling is a complex topic to treat analytically in any case due to the influence of imperfections; for reinforced concrete, it is even more difficult. While the governing equations may be generalized to treat instability by using nonlinear strain-displacement relations and thereby introducing the geometric stiffness matrix, the correlation between the resulting analytical solutions and the possible failure of a reinforced concrete cooling tower is questionable. Nevertheless, it has been common to analyze cooling tower shells under an unfactored combination of dead load plus wind load plus internal suction. The corresponding buckling pattern is shown in Figure 14.25. Interaction diagrams calibrated from experimental studies based on bifurcation buckling are also available [9, 12, 13]. Additionally, there are empirical methods based on wind tunnel tests that consider a snap-through buckle at the upper edge at each stage of construction [13]. These formulas are proportional to h/R and are convenient for establishing an appropriate shell thickness. If buckling safety is evaluated based on such a linear buckling analysis or an experimental investigation, the buckling safety factor for realistic material parameters should exceed 5.0. Presently, however, the use of bifurcation buckling analyses should be confined to preliminary proportioning since more rational procedures based on nonlinear analysis have been developed to predict the collapse of reinforced concrete shells, as discussed in the following paragraphs. Advances in the analyses of reinforced concrete have produced the capability to analyze shells taking into account the layered composition of the cross-section as shown in Figure 14.26. Using realistic material properties for steel and for concrete, including tension stiffening in the form shown in Figures 14.27 through 14.29, load-deflection relationships may be constructed for appropriate load combinations. These relationships progress from the linear elastic phase to initial cracking of the concrete through spreading of the cracks until collapse. Results from a nonlinear study are presented in Figures 14.30 through 14.33. The geometry of the shell is given in Figure 14.30, the wind load factor λ is plotted against the maximum lateral displacement at the top of the shell in Figure 14.31, and the deformed shape for the collapse load is shown in Figure 14.32. Also, the pattern of cracking corresponding to the initial yielding of the reinforcement is indicated in Figure 14.33. For reinforced concrete shells, this type of analysis represents the state-of-the-art and provides a realistic evaluation of the capacity of such shells against extreme loading [8]. Also durability assessments can be performed by this concept, from which particularly weak and crack-endangered regions of the shell can be identified and further reinforced [10]. It is possible to obtain an estimate of the wind load factor, λ, from the results of a linear elastic analysis, even from a calculation based on membrane theory. This estimate is computed as the cracking load for the shell under a combination of D + λW and is predicated on the notion that the 1999 by CRC Press LLC
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FIGURE 14.17: Meridional forces n22G under deadweight.
FIGURE 14.18: Circumferential forces n11W under wind load.
1999 by CRC Press LLC
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FIGURE 14.19: Meridional forces n22W under wind load.
FIGURE 14.20: Shear forces n12W under wind load.
1999 by CRC Press LLC
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FIGURE 14.21: Circumferential forces n11S under internal suction. reinforcement may add only a modest amount of capacity to the tower beyond the cracking load [6]. The amount of reinforcement in the wall is often controlled by a specified minimum percentage augmented by that required to resist the net tension due to the factored load combinations. The steel provided is often less than the capacity of the concrete in tension, which is presumed to be lost when the concrete cracks. Therefore, the cracking load represents most of the ultimate capacity of the tower. The maximum meridional tension location under the wind loading is identified, for example, as the value of n22 = 863 kN/m in Figure 14.19. The dead load at this location is obtained from Figure 14.17 as −701 kN/m. Taking the concrete tensile capacity as 2,400 kN/m2 and the wall thickness as 16 cm, the tensile strength is 384 kN/m. Therefore, we have − 701 + λ863 = 384
(14.7)
giving λ = 1.26 as the lower bound on the ultimate strength of the tower. Note that the tower used for the linear elastic analysis is much taller than the one shown in Figure 14.30. The dynamic analysis of cooling towers is usually associated with design for earthquake-induced forces. The most efficient approach is the response spectrum method, but a time history analysis may be appropriate if nonlinearities are to be included [2, 7]. For large shells the dynamic response due to wind is often investigated, at least to determine the positions of the nodal lines and areas of particularly intensive vibrations. In any case the first step is to carry out a free vibration analysis. This analysis represents the modes of free vibration associated with each natural frequency, f , or its inverse the natural period T, as the product of a circumferential mode proportional to sin nθ or cos nθ and a longitudinal mode along the z axis [3, 4]. Some representative results are shown on Figures 14.34 and 14.35, as discussed below. As an illustration, the cooling tower from Figure 14.4 is again considered. Some key circumferential and longitudinal modes for a fixed-base boundary condition are shown in Figure 14.35. Also, the effects of different cornice stiffnesses are demonstrated. This model may be regarded as preliminary 1999 by CRC Press LLC
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FIGURE 14.22: Meridional forces n22S under internal suction. in that the relatively soft column supports are not properly represented, but it illustrates the salient characteristics of the modes of vibration. Most interesting are the frequency curves on Figure 14.34 for the first 10 harmonics, also demonstrating the influence of different cornice stiffnesses. Note that the natural frequencies decrease with increasing n until a minimum is reached whereupon they increase, a very typical behavior for cylindrical-type shells. Also, the stiffening of the cornice tends to raise the minimum frequency, which is desirable for resistance to dynamic wind. Longitudinally, the cornice stiffness effect is significant for odd modes only. Specifically for earthquake effects, only the first mode participates in a linear analysis for uniform horizontal base motion and the respective values for n = 1 should be entered into the design response spectrum. Results from a seismic analysis of a cooling tower are presented in Figures 14.36 to 14.39. The cooling tower of Figure 14.4 is subjected to a horizontal base excitation based on Figure 14.36, leading to a first circumferential mode (n = 1) participation. A response spectrum analysis provides the lateral displacements w of the tower axis, the meridional forces n22 , and the shear forces n12 as shown on the indicated figures. In general, cooling tower shells have proven to be reasonably resistant against seismic excitations, but obviously the most critical region is the connection between the columns and the lintel as portrayed in Figure 14.40.
14.7
Design and Detailing of Components
The structural elements of the tower should be constructed with a suitable grade of concrete following the provisions of applicable codes and standards. The design of the mixture should reflect the conditions for placement of the concrete and the external and internal environment of the tower. The shell wall should be of a thickness which will permit two layers of reinforcement in two perpendicular directions to be covered by a minimum of 3 cm of concrete, and should be no less than 16 cm thick [7, 13]. The buckling considerations mentioned in the previous section have proven to 1999 by CRC Press LLC
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FIGURE 14.23: Circumferential bending moments m11T under service temperature.
FIGURE 14.24: Meridional bending moments m22T under service temperature.
1999 by CRC Press LLC
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FIGURE 14.25: Buckling pattern of tower shell with upper ring beam: D + W + S.
FIGURE 14.26: Layered model for reinforced concrete shell.
be a convenient and evidently acceptable criteria for setting the minimum wall thickness, subject to a nonlinear analysis. The formula qc = 0.052E(h/R)2.3
(14.8)
where E = modulus of elasticity, has been used to estimate the critical shell buckling pressure qc [1, 13]. Then, h(z) is selected to provide a factor of safety of at least 5.0 with respect to the maximum velocity pressure along the windward meridian, q(z)(1 + g). Also, the cornice should have a minimum stiffness of (14.9) Ix /dH = 0.0015m3 where Ix is the moment of inertia of the uncracked cross-section about the vertical axis [13]. Some typical forms of the cornice cross-section are shown in Figure 14.41. 1999 by CRC Press LLC
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FIGURE 14.27: Elasto-plastic material law for steel.
FIGURE 14.28: Biaxial failure envelope of Kupfter/Hilsdorf/R¨usch.
The elements of the cooling tower should be reinforced with deformed steel bars so as to provide for the tensile forces and moments arising from the controlling combination of factored loading cases. The shell walls may be proportioned as rectangular cross-sections subjected to axial forces and bending. As mentioned above, a mesh of two orthogonal layers of reinforcement should be provided in the shell walls, generally in the meridional and circumferential directions [2]. In each direction, the inner and outer layers should generally be the same, except near the edges where the bending may require an unsymmetrical mesh. It is preferable to locate the circumferential reinforcement outside of the meridional reinforcement except near the lintel, where the meridional reinforcement should be on the outside to stabilize the circumferential bars [13]. A typical heavily reinforced segment of the lintel, also showing the anchorage of the column reinforcement into the shell, is depicted in Figure 14.42. A summary of the most important minimum construction values for the shell wall is given in Figure 14.43 [13]. The bars should not be smaller than 8 mm diameter and, for meridional bars, not 1999 by CRC Press LLC
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FIGURE 14.29: Additional modulus of elasticity due to tension stiffening.
FIGURE 14.30: Shell geometry and wall thickness.
smaller than 10 mm. Further, a minimum of 0.35 to 0.45%, depending on the admissible cracking, should be used in each direction. The minimum cover, as mentioned above, should be 3 cm, the maximum spacing of the bars should be 20 cm, and the splices should be staggered as specified for the construction of walls in the applicable codes or standards. Particular attention should be given to splices in tensile zones. The supporting columns should ideally be proportioned for the forces and moments computed from an analysis in which they are represented as discrete members, using the appropriate factored loading combinations [3]. If the column region has not been modeled discretely, but rather by a 1999 by CRC Press LLC
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FIGURE 14.31: Load-displacement diagram for load combination D + λ • W .
FIGURE 14.32: Displacement plot of the shell for load combination D + λ • W : load factor = 1.83.
continuum approximation, the columns may be proportioned to resist the tributary factored forces and moments at the interface with the lintel, as computed from the shell analysis. The effective length may be taken as unity. Particular attention should be directed toward splices of the column bars when net tension is present. Since large bars will be involved, welded splices are recommended in such regions. It is possible to add discrete circumferential stiffeners to the shell to increase the stability or to restore capacity that may have been lost due to cracking or other deterioration [5] (see Figure 14.9). Such stiffeners can generally be included in a finite element model of the shell wall and should be proportioned for the forces computed from such an analysis. The eccentricity of the stiffeners with respect to the circumferential axis should be considered when the stiffeners are only on one side of the shell. The foundations should be proportioned for the factored forces induced by the column reactions, or from the computed forces if the foundation is included in the model with the shell and columns. 1999 by CRC Press LLC
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FIGURE 14.33: Crack pattern of the outer face of the shell for load combination D + λ • W : load factor = 1.54.
FIGURE 14.34: Natural frequencies for different cornice stiffnesses for the cooling tower on Figure 14.4. Reinforcement detailing and cover should be in accordance with the applicable codes or standards. Several improved forms for cooling tower foundations have been suggested. Figure 14.44 shows a flat ring footing suitable for uniform soil conditions, while Figure 14.45 portrays a stiff ring beam foundation appropriate for soil conditions that are non-uniform around the circumference. An example of an individual pier on bedrock is given in Figure 14.46.
1999 by CRC Press LLC
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FIGURE 14.35: Normalized natural vibration modes for the cooling tower on Figure 14.4.
FIGURE 14.36: Seismic response spectrum.
14.8
Construction
Tolerances for tall concrete cooling tower shells have been debated for many years and reasonable values should take into consideration what is achievable and what is measurable. It should be noted that state-of-the-art finite element models are capable of analyzing the as-built shell as well as the design configuration, so that the effects of those irregularities arising during construction, or even those discovered later, may be quantitatively studied and sometimes corrected. It is recommended that the actual wall thickness be no less than the design thickness and exceed this thickness by not more than 10%. The imperfections of the shell wall middle surface should not exceed one-half of the wall thickness or 10 cm. Deviations from the design geometry occurring 1999 by CRC Press LLC
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FIGURE 14.37: First axial mode seismic response.
FIGURE 14.38: Second axial mode seismic response.
during the construction should be corrected gradually, limiting the angular change in either direction to 1.5%. The column heads should be within 0.005 times the column height or ±6.0 cm of the design position, and foundation structures should also be within ±6.0 cm of the design location [13]. Formwork and scaffolding systems are generally proprietary and are provided by the constructor. Nevertheless, their influence on the shell quality is of utmost importance and diligent attention of the engineer is required. In general, the system should be designed to provide safety to operating personnel and to produce a sound structure. The working platforms should be designed for realistic loading, and scaffolding systems used for continuous material transport should be designed and built taking into account the resulting loads. The connections and joints between individual scaffolding units should be designed and built to act independently in case of collapse, so that the loss of one unit would not affect the adjacent units. Furthermore, at least two independent safety devices should be in place to prevent collapse. 1999 by CRC Press LLC
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FIGURE 14.39: SRSS superposition of first and second axial modes.
FIGURE 14.40: Column to lintel connection. 1999 by CRC Press LLC
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FIGURE 14.41: Suitable forms of the cornice.
FIGURE 14.42: Lintel reinforcement.
1999 by CRC Press LLCFIGURE 14.43:
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Important minimum construction values.
FIGURE 14.44: Flat ring foundation.
FIGURE 14.45: Ring beam foundation.
FIGURE 14.46: Individual reinforced concrete foundation on concrete base. 1999 by CRC Press LLC
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The shell wall should be designed to resist the anchor loads of the scaffolding, based on the strength of the concrete which is expected to be available when the anchors are loaded. Continuous monitoring of the concrete strength during the climbing process is essential. Cooling tower shells are subjected to a relatively severe environment over their lifetime, which may span several decades, and special care must be taken in order to provide a durable structure. The tower is subjected to the physical loads produced by wind, temperature, and moisture acting on concrete which may still be drying and hardening. Over the lifetime of the structure, it may be exposed to severe frost action in a saturated state, chemical attacks due to noxious substances in the atmosphere and in the water and water vapor, biological attacks due to microorganisms, and possibly additional chemical attacks due to reintroduced cleaned flue gases. The concrete should be of high-quality approved materials including fly-ash. It should have the following properties: • • • •
High resistance against chemical attacks High early strength High structural density High resistance against frost
The surface finish should be of high quality and the surface should be smooth and essentially free of shrink holes. Air bubbles deeper than 4 mm and unintended surface irregularities at joints should be avoided. The shell should be coated with a curing agent providing a high blocking effect and long durability. Several single component (acrylate or polyurethene-based) or double component (epoxy resinbased) coating systems are approved worldwide and are in a process of continual improvement. Of utmost importance for any coating is the homogeneity of the applied film between ≥ 200µm for single and ≥ 300µ for double component systems, since the durability of the complete coating is determined by the thinnest film spots.
References [1] Abel, J.F. and Gould, P.L. 1981. Buckling of Concrete Cooling Towers Shells, ACI SP-67, American Concrete Institute, Detroit, Michigan, pp. 135-160. [2] ACI-ASCE Committee 334. 1977. Recommended Practice for the Design and Construction of Reinforced Concrete Cooling Towers, ACI J., 74(1), 22-31. [3] Gould, P.L. 1985. Finite Element Analysis of Shells of Revolution, Pitman. [4] Gould, P.L., Suryoutomo, H., and Sen, S.K. 1974. Dynamic analysis of column-supported hyperboloidal shells, Earth. Eng. Struct. Dyn., 2, 269-279. [5] Gould, P.L. and Guedelhoefer, O.C. 1988. Repair and Completion of Damaged Cooling Tower, J. Struct. Eng., 115(3), 576-593. [6] Hayashi, K. and Gould, P.L. 1983. Cracking load for a wind-loaded reinforced concrete cooling tower, ACI J., 80(4), 318-325. [7] IASS-Recommendations for the Design of Hyperbolic or Other Similarly Shaped Cooling Towers. 1977. Intern. Assoc. for Shell and Space Structures, Working Group No. 3, Brussels. [8] Kr¨atzig, W.B. and Zhuang, Y. 1992. Collapse simulation of reinforced natural draught cooling towers, Eng. Struct., 14(5), 291-299. [9] Kr¨atzig, W.B. and Meskouris, K. 1993. Natural draught cooling towers: An increasing need for structural research, Bull. IASS, 34(1), 37-51. [10] Kr¨atzig, W.B. and Gruber, K.P. 1996. Life-Cycle Damage Simulations of Natural Draught Cooling Towers in Natural Draught Cooling Towers, Wittek, U. and Kr¨atzig, W., Eds., A.A. Balkema, Rotterdam, 151-158. 1999 by CRC Press LLC
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[11] Minimum Design Loads for Buildings and Other Structures. 1994. ASCE Standard 7-93, ASCE, New York. [12] Mungan, I. 1976. Buckling stress states of hyperboloidal shells, J. Struct. Div., ASCE, 102, 2005-2020. [13] VGB Guideline. 1990. Structural Design of Cooling Towers, VGB-Technical Committee, “Civil Engineering Problems of Cooling Towers”, Essen, Germany.
Further Reading [1] Proceedings (First) International Symposium on Very Tall Reinforced Concrete Cooling Towers. 1978. I.A.S.S., E.D.F., Paris, France, November. [2] Gould, P.L., Kr¨atzig, W.B., Mungan, I., and Wittek, U., Eds. 1984. Natural Draught Cooling Towers. Proceedings of the 2nd International Symposium on Natural Draught Cooling Towers, Springer-Verlag, Heidelberg. [3] Proceedings Third International Symposium on Natural Draught Cooling Towers. 1989. I.A.S.S., E.D.F., Paris, France, April. [4] Wittek, U. and Kr¨atzig, W.B., Eds. 1996. Natural Draught Cooling Towers. Proceedings of the 4th International Symposium on Natural Draught Cooling Towers, A.A. Balkema, Rotterdam. [5] British Standard Institution. 1996. BS 4485, Part 4: British Standard for Water Cooling Towers. Document 96/17117 DC 22. [6] Syndicat National du B´eton Arm´e et des Techniques Industrialis´ees. 1996. Regles de conception et de realisation des refrigerants atmospheriques en beton arm´e, Paris.
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Fang, S.J.; Roy, S. and Kramer, J. “Transmission Structures” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Transmission Structures 15.1 Introduction and Application
Application • Structure Configuration and Material • Constructibility • Maintenance Considerations • Structure Families • State of the Art Review
15.2 Loads on Transmission Structures
General • Calculation of Loads Using NESC Code • Calculation of Loads Using the ASCE Guide • Special Loads • Security Loads • Construction and Maintenance Loads • Loads on Structure • Vertical Loads • Transverse Loads • Longitudinal Loading
15.3 Design of Steel Lattice Tower
Tower Geometry • Analysis and Design Methodology • Allowable Stresses • Connections • Detailing Considerations • Tower Testing
15.4 Transmission Poles
General • Stress Analysis • Tubular Steel Poles • Wood Poles • Concrete Poles • Guyed Poles
15.5 Transmission Tower Foundations
Shu-jin Fang, Subir Roy, and Jacob Kramer Sargent & Lundy, Chicago, IL
15.1
Geotechnical Parameters • Foundation Types—Selection and Design • Anchorage • Construction and Other Considerations • Safety Margins for Foundation Design • Foundation Movements • Foundation Testing • Design Examples
15.6 Defining Terms References
Introduction and Application
Transmission structures support the phase conductors and shield wires of a transmission line. The structures commonly used on transmission lines are either lattice type or pole type and are shown in Figure 15.1. Lattice structures are usually composed of steel angle sections. Poles can be wood, steel, or concrete. Each structure type can also be self-supporting or guyed. Structures may have one of the three basic configurations: horizontal, vertical, or delta, depending on the arrangement of the phase conductors.
15.1.1 Application Pole type structures are generally used for voltages of 345-kV or less, while lattice steel structures can be used for the highest of voltage levels. Wood pole structures can be economically used for relatively shorter spans and lower voltages. In areas with severe climatic loads and/or on higher voltage lines with multiple subconductors per phase, designing wood or concrete structures to meet the large 1999 by CRC Press LLC
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FIGURE 15.1: Transmission line structures.
loads can be uneconomical. In such cases, steel structures become the cost-effective option. Also, if greater longitudinal loads are included in the design criteria to cover various unbalanced loading contingencies, H-frame structures are less efficient at withstanding these loads. Steel lattice towers can be designed efficiently for any magnitude or orientation of load. The greater complexity of these towers typically requires that full-scale load tests be performed on new tower types and at least the 1999 by CRC Press LLC
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tangent tower to ensure that all members and connections have been properly designed and detailed. For guyed structures, it may be necessary to proof-test all anchors during construction to ensure that they meet the required holding capacity.
15.1.2
Structure Configuration and Material
Structure cost usually accounts for 30 to 40% of the total cost of a transmission line. Therefore, selecting an optimum structure becomes an integral part of a cost-effective transmission line design. A structure study usually is performed to determine the most suitable structure configuration and material based on cost, construction, and maintenance considerations and electric and magnetic field effects. Some key factors to consider when evaluating the structure configuration are: • A horizontal phase configuration usually results in the lowest structure cost. • If right-of-way costs are high, or the width of the right-of-way is restricted or the line closely parallels other lines, a vertical configuration may be lower in total cost. • In addition to a wider right-of-way, horizontal configurations generally require more tree clearing than vertical configurations. • Although vertical configurations are narrower than horizontal configurations, they are also taller, which may be objectionable from an aesthetic point of view. • Where electric and magnetic field strength is a concern, the phase configuration is considered as a means of reducing these fields. In general, vertical configurations will have lower field strengths at the edge of the right-of-way than horizontal configurations, and delta configurations will have the lowest single-circuit field strengths and a double-circuit with reverse or low-reactance phasing will have the lowest possible field strength. Selection of the structure type and material depends on the design loads. For a single circuit 230-kV line, costs were estimated for single-pole and H-frame structures in wood, steel, and concrete over a range of design span lengths. For this example, wood H-frames were found to have the lowest installed cost, and a design span of 1000 ft resulted in the lowest cost per mile. As design loads and other parameters change, the relative costs of the various structure types and materials change.
15.1.3
Constructibility
Accessibility for construction of the line should be considered when evaluating structure types. Mountainous terrain or swampy conditions can make access difficult and use of helicopter may become necessary. If permanent access roads are to be built to all structure locations for future maintenance purposes, all sites will be accessible for construction. To minimize environmental impacts, some lines are constructed without building permanent access roads. Most construction equipment can traverse moderately swampy terrain by use of widetrack vehicles or temporary mats. Transporting concrete for foundations to remote sites, however, increases construction costs. Steel lattice towers, which are typically set on concrete shaft foundations, would require the most concrete at each tower site. Grillage foundations can also be used for these towers. However, the cost of excavation, backfill and compaction for these foundations is often higher than the cost of a drilled shaft. Unless subsurface conditions are poor, most pole structures can be directly embedded. However, if unguyed pole structures are used at medium to large line angles, it may be necessary to use drilled shaft foundations. Guyed structures can also create construction difficulties in that a wider area must be accessed at each structure site to install the guys and anchors. Also, careful coordination is required to ensure that all guys are tensioned equally and that the structure is plumb. 1999 by CRC Press LLC
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Hauling the structure materials to the site must also be considered in evaluating constructibility. Transporting concrete structures, which weigh at least five times as much as other types of structures, will be difficult and will increase the construction cost of the line. Heavier equipment, more trips to transport materials, and more matting or temporary roadwork will be required to handle these heavy poles.
15.1.4
Maintenance Considerations
Maintenance of the line is generally a function of the structure material. Steel and concrete structures should require very little maintenance, although the maintenance requirements for steel structures depends on the type of finish applied. Tubular steel structures are usually galvanized or made of weathering steel. Lattice structures are galvanized. Galvanized or painted structures require periodic inspection and touch-up or reapplication of the finish while weathering steel structures should have relatively low maintenance. Wood structures, however, require more frequent and thorough inspections to evaluate the condition of the poles. Wood structures would also generally require more frequent repair and/or replacement than steel or concrete structures. If the line is in a remote location and lacks permanent access roads, this can be an important consideration in selecting structure material.
15.1.5
Structure Families
Once the basic structure type has been established, a family of structures is designed, based on the line route and the type of terrain it crosses, to accommodate the various loading conditions as economically as possible. The structures consist of tangent, angle, and deadend structures. Tangent structures are used when the line is straight or has a very small line angle, usually not exceeding 3◦ . The line angle is defined as the deflection angle of the line into adjacent spans. Usually one tangent type design is sufficient where terrain is flat and the span lengths are approximately equal. However, in rolling and mountainous terrain, spans can vary greatly. Some spans, for example, across a long valley, may be considerably larger than the normal span. In such cases, a second tangent design for long spans may prove to be more economical. Tangent structures usually comprise 80 to 90% of the structures in a transmission line. Angle towers are used where the line changes direction. The point at which the direction change occurs is generally referred to as the point of intersection (P.I.) location. Angle towers are placed at the P.I. locations such that the transverse axis of the cross arm bisects the angle formed by the conductor, thus equalizing the longitudinal pulls of the conductors in the adjacent spans. On lines where large numbers of P.I. locations occur with varying degrees of line angles, it may prove economical to have more than one angle structure design: one for smaller angles and the other for larger angles. When the line angle exceeds 30◦ , the usual practice is to use a deadend type design. Deadend structures are designed to resist wire pulls on one side. In addition to their use for large angles, the deadend structures are used as terminal structures or for sectionalizing a long line consisting of tangent structures. Sectionalizing provides a longitudinal strength to the line and is generally recommended every 10 miles. Deadend structures may also be used for resisting uplift loads. Alternately, a separate strain structure design with deadend insulator assemblies may prove to be more economical when there is a large number of structures with small line angle subjected to uplift. These structures are not required to resist the deadend wire pull on one side.
15.1.6
State of the Art Review
A major development in the last 20 years has been in the area of new analysis and design tools. These include software packages and design guidelines [12, 6, 3, 21, 17, 14, 9, 8], which have greatly 1999 by CRC Press LLC
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improved design efficiency and have resulted in more economical structures. A number of these tools have been developed based on test results, and many new tests are ongoing in an effort to refine the current procedures. Another area is the development of the reliability based design concept [6]. This methodology offers a uniform procedure in the industry for calculation of structure loads and strength, and provides a quantified measure of reliability for the design of various transmission line components. Aside from continued refinements in design and analysis, significant progress has been made in the manufacturing technology in the last two decades. The advance in this area has led to the increasing usage of cold formed shapes, structures with mixed construction such as steel poles with lattice arms or steel towers with FRP components, and prestressed concrete poles [7].
15.2
Loads on Transmission Structures
15.2.1
General
Prevailing practice and most state laws require that transmission lines be designed, as a minimum, to meet the requirements of the current edition of the National Electrical Safety Code (NESC) [5]. NESC’s rules for the selection of loads and overload capacity factors are specified to establish a minimum acceptable level of safety. The ASCE Guide for Electrical Transmission Line Structural Loading (ASCE Guide) [6] provides loading guidelines for extreme ice and wind loads as well as security and safety loads. These guidelines use reliability based procedures and allow the design of transmission line structures to incorporate specified levels of reliability depending on the importance of the structure.
15.2.2
Calculation of Loads Using NESC Code
NESC code [5] recognizes three loading districts for ice and wind loads which are designated as heavy, medium, and light loading. The radial thickness of ice and the wind pressures specified for the loading districts are shown in Table 15.1. Ice build-up is considered only on conductors and shield wires, and is usually ignored on the structure. Ice is assumed to weigh 57 lb/ft3 . The wind pressure applies to cylindrical surfaces such as conductors. On the flat surface of a lattice tower member, the wind pressure values are multiplied by a force coefficient of 1.6. Wind force is applied on both the windward and leeward faces of a lattice tower. TABLE 15.1
Ice, Wind, and Temperature Loading districts
Radial thickness of ice (in.) Horizontal wind pressure (lb/ft2 ) Temperature (◦ F)
Heavy
Medium
Light
0.50
0.25
0
4
4
9
0
+15
+30
NESC also requires structures to be designed for extreme wind loading corresponding to 50 year fastest mile wind speed with no ice loads considered. This provision applies to all structures without conductors, and structures over 60 ft supporting conductors. The extreme wind speed varies from a basic speed of 70 mph to 110 mph in the coastal areas. In addition, NESC requires that the basic loads be multiplied by overload capacity factors to 1999 by CRC Press LLC
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determine the design loads on structures. Overload capacity factors make it possible to assign relative importance to the loads instead of using various allowable stresses for different load conditions. Overload capacity factors specified in NESC have a larger value for wood structures than those for steel and prestressed concrete structures. This is due to the wide variation found in wood strengths and the aging effect of wood caused by decay and insect damage. In the 1990 edition, NESC introduced an alternative method, where the same overload factors are used for all the materials but a strength reduction factor is used for wood.
15.2.3
Calculation of Loads Using the ASCE Guide
The ASCE Guide [6] specifies extreme ice and extreme wind loads, based on a 50-year return period, which are assigned a reliability factor of 1. These loads can be increased if an engineer wants to use a higher reliability factor for an important line, for example a long line, or a line which provides the only source of load. The load factors used to increase the ASCE loads for different reliability factors are given in Table 15.2. TABLE 15.2
Load Factor to Adjust Line Reliability
Line reliability factor, LRF Load return period, RP Corresponding load factor, a˜
1 50 1.0
2 100 1.15
4 200 1.3
8 400 1.4
In calculating wind loads, the effects of terrain, structure height, wind gust, and structure shape are included. These effects are explained in detail in the ASCE Guide. ASCE also recommends that the ice loads be combined with a wind load equal to 40% of the extreme wind load.
15.2.4
Special Loads
In addition to the weather related loads, transmission line structures are designed for special loads that consider security and safety aspects of the line. These include security loads for preventing cascading type failures of the structures and construction and maintenance loads that are related to personnel safety.
15.2.5
Security Loads
Longitudinal loads may occur on the structures due to accidental events such as broken conductors, broken insulators, or collapse of an adjacent structure in the line due to an environmental event such as a tornado. Regardless of the triggering event, it is important that a line support structure be designed for a suitable longitudinal loading condition to provide adequate resistance against cascading type failures in which a larger number of structures fail sequentially in the longitudinal direction or parallel to the line. For this reason, longitudinal loadings are sometimes referred to as “anticascading”, “failure containment”, or “security loads”. There are two basic methods for reducing the risk of cascading failures, depending on the type of structure, and on local conditions and practices. These methods are: (1) design all structures for broken wire loads and (2) install stop structures or guys at specified intervals. Design for Broken Conductors
Certain types of structures such as square-based lattice towers, 4-guyed structures, and single shaft steel poles have inherent longitudinal strength. For lines using these types of structures, the 1999 by CRC Press LLC
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recommended practice is to design every structure for one broken conductor. This provides the additional longitudinal strength for preventing cascading failures at a relatively low cost. Anchor Structures
When single pole wood structures or H-frame structures having low longitudinal strength are used on a line, designing every structure for longitudinal strength can be very expensive. In such cases, stop or anchor structures with adequate longitudinal strength are provided at specific intervals to limit the cascading effect. The Rural Electrification Administration [19] recommends a maximum interval of 5 to 10 miles between structures with adequate longitudinal capacity.
15.2.6
Construction and Maintenance Loads
Construction and maintenance (C&M) loads are, to a large extent, controllable and are directly related to construction and maintenance methods. A detailed discussion on these types of loads is included in the ASCE Loading Guide, and Occupation Safety and Health Act (OSHA) documents. It should be emphasized, however, that workers can be seriously injured as a result of structure overstress during C&M operations; therefore, personnel safety should be a paramount factor when establishing C&M loads. Accordingly, the ASCE Loading Guide recommends that the specified C&M loads be multiplied by a minimum load factor of 1.5 in cases where the loads are “static” and well defined; and by a load factor of 2.0 when the loads are “dynamic”, such as those associated with moving wires during stringing operations.
15.2.7
Loads on Structure
Loads are calculated on the structures in three directions: vertical, transverse, and longitudinal. The transverse load is perpendicular to the line and the longitudinal loads act parallel to the line.
15.2.8
Vertical Loads
The vertical load on supporting structures consists of the weight of the structure plus the superimposed weight, including all wires, ice coated where specified. Vertical load of wire Vw in. (lb/ft) is given by the following equations: Vw = wt. of bare wire (lb/f t) + 1.24(d + I )I
(15.1)
where d = diameter of wire (in.) I = ice thickness (in.) Vertical wire load on structure (lb) = V w × vertical design span × load factor
(15.2)
Vertical design span is the distance between low points of adjacent spans and is indicated in Figure 15.2.
15.2.9
Transverse Loads
Transverse loads are caused by wind pressure on wires and structure, and the transverse component of the line tension at angles. 1999 by CRC Press LLC
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FIGURE 15.2: Vertical and horizontal design spans. Wind Load on Wires
The transverse load due to wind on the wire is given by the following equations: Wh
= =
p × d/12 × Horizontal Span × OCF (without ice) p × (d + 2I )/12 × Horizontal Span × OCF (with ice)
(15.3) (15.4)
where = transverse wind load on wire in lb Wh p = wind pressure in lb/ft2 d = diameter of wire in in. I = radial thickness of ice in in. OCF = Overload Capacity Factor Horizontal span is the distance between midpoints of adjacent spans and is shown in Figure 15.2. Transverse Load Due to Line Angle
Where a line changes direction, the total transverse load on the structure is the sum of the transverse wind load and the transverse component of the wire tension. The transverse component of the tension may be of significant magnitude, especially for large angle structures. To calculate the total load, a wind direction should be used which will give the maximum resultant load considering the effects on the wires and structure. The transverse component of wire tension on the structure is given by the following equation: H = 2T sin θ/2
(15.5)
where H = transverse load due to wire tension in pounds T = wire tension in pounds θ = Line angle in degrees Wind Load on Structures
In addition to the wire load, structures are subjected to wind loads acting on the exposed areas of the structure. The wind force coefficients on lattice towers depend on shapes of member sections, solidity ratio, angle of incidence of wind (face-on wind or diagonal wind), and shielding. Methods 1999 by CRC Press LLC
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for calculating wind loads on transmission structures are given in the ASCE Guide as well the NESC code.
15.2.10
Longitudinal Loading
There are several conditions under which a structure is subjected to longitudinal loading: Deadend Structures—These structures are capable of withstanding the full tension of the conductors and shield wires or combinations thereof, on one side of the structure. Stringing— Longitudinal load may occur at any one phase or shield wire due to a hang-up in the blocks during stringing. The longitudinal load is taken as the stringing tension for the complete phase (i.e., all subconductors strung simultaneously) or a shield wire. In order to avoid any prestressing of the conductors, stringing tension is typically limited to the minimum tension required to keep the conductor from touching the ground or any obstructions. Based on common practice and according to the IEEE “Guide to the Installation of Overhead Transmission Line Conductors” [4], stringing tension is generally about one-half of the sagging tension. Therefore, the longitudinal stringing load is equal to 50% of the initial, unloaded tension at 60◦ F. Longitudinal Unbalanced Load—Longitudinal unbalanced forces can develop at the structures due to various conditions on the line. In rugged terrain, large differentials in adjacent span lengths, combined with inclined spans, could result in significant longitudinal unbalanced load under ice and wind conditions. Non-uniform loading of adjacent spans can also produce longitudinal unbalanced loads. This load is based on an ice shedding condition where ice is dropped from one span and not the adjacent spans. Reference [12] includes a software that is commonly used for calculating unbalanced loads on the structure.
EXAMPLE 15.1: Problem
Determine the wire loads on a small angle structure in accordance with the data given below. Use NESC medium district loading and assume all intact conditions. Given Data: Conductor: 954 kcm 45/7 ACSR Diameter = 1.165 in. Weight = 1.075 lb/ft Wire tension for NESC medium loading = 8020 lb Shield Wire: 3 No.6 Alumoweld Diameter = 0.349 in. Weight = 0.1781 lb/ft Wire tension for NESC medium loading = 2400 lb Wind Span = 1500 ft Weight Span = 1800 ft Line angle = 5◦ Insulator weight = 170 lb 1999 by CRC Press LLC
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Solution NESC Medium District Loading
4 psf wind, 1/4-in. ice Ground Wire Iced Diameter = 0.349 + 2 × 0.25 = 0.849 in. Conductor Ice Diameter = 1.165 + 2 × 0.25 = 1.665 in. Overload Capacity Factors for Steel Transverse Wind = 2.5 Wire Tension = 1.65 Vertical = 1.5 Conductor Loads On Tower Transverse Wind = 4 psf × 1.665"/12 × 1500 × 2.5 = 2080 lb Line Angle = 2 × 8020 × sin 2.5◦ × 1.65 = 1150 lb Total = 3230 lb Vertical Bare Wire = 1.075 × 1800 × 1.5 = 2910 lb Ice = {1.24(d + I )I }1800 × 1.5 = 1.24(1.165 + .25).25 × 1800 × 1.5 = 1185 lb Insulator = 170 × 1.5 = 255 lb Total = 4350 lb Ground Wire Loads on Tower Transverse Wind = 4 psf × 0.849/12 × 1500 × 2.5 = 1060 lb Line Angle = 2 × 2400 × sin 2.5 × 1.65 = 350 lb Total = 1410 lb
15.3
Design of Steel Lattice Tower
15.3.1
Tower Geometry
A typical single circuit, horizontal configuration, self-supported lattice tower is shown in Figure 15.3. The design of a steel lattice tower begins with the development of a conceptual design, which establishes the geometry of the structure. In developing the geometry, structure dimensions are established for the tower window, crossarms and bridge, shield wire peak, bracing panels, and the slope of the 1999 by CRC Press LLC
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FIGURE 15.3: Single circuit lattice tower.
tower leg below the waist. The most important criteria for determining structure geometry are the minimum phase to phase and phase to steel clearance requirements, which are functions of the line voltage. Spacing of phase conductors may sometimes be dictated by conductor galloping considerations. Height of the tower peak above the crossarm is based on shielding considerations for lightning protection. The width of the tower base depends on the slope of the tower leg below the waist . The overall structure height is governed by the span length of the conductors between structures. The lattice tower is made up of a basic body, body extension, and leg extensions. Standard designs are developed for these components for a given tower type. The basic body is used for all the towers regardless of the height. Body and leg extensions are added to the basic body to achieve the desired tower height. The primary members of a tower are the leg and the bracing members which carry the vertical and shear loads on the tower and transfer them to the foundation. Secondary or redundant bracing members are used to provide intermediate support to the primary members to reduce their unbraced length and increase their load carrying capacity. The slope of the tower leg from the waist down has a significant influence on the tower weight and should be optimized to achieve an economical tower 1999 by CRC Press LLC
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design. A flatter slope results in a wider tower base which reduces the leg size and the foundation size, but will increase the size of the bracing. Typical leg slopes used for towers range from 3/4 in. 12 for light tangent towers to 2 1/2 in. 12 for heavy deadend towers. The minimum included angle ∞ between two intersecting members is an important factor for proper force distribution. Reference [3] recommends a minimum included angle of 15◦ , intended to develop a truss action for load transfer and to minimize moment in the member. However, as the tower loads increase, the preferred practice is to increase the included angle to 20◦ for angle towers and 25◦ for deadend towers [23]. Bracing members below the waist can be designed as a tension only or tension compression system as shown in Figure 15.4. In a tension only system shown in (a), the bracing members are designed
FIGURE 15.4: Bracing systems.
to carry tension forces only, the compression forces being carried by the horizontal strut. In a tension/compression system shown in (b) and (c), the braces are designed to carry both tension and compression. A tension only system may prove to be economical for lighter tangent towers. But for heavier towers, a tension/compression system is recommended as it distributes the load equally to the tower legs. A staggered bracing pattern is sometimes used on the adjacent faces of a tower for ease of connections and to reduce the number of bolt holes at a section. Tests [23] have shown that staggering of main bracing members may produce significant moment in the members especially for heavily loaded towers. For heavily loaded towers, the preferred method is to stagger redundant bracing members and connect the main bracing members on the adjacent faces at a common panel point.
15.3.2
Analysis and Design Methodology
The ASCE Guide for Design of Steel Transmission Towers [3] is the industry document governing the analysis and design of lattice steel towers. A lattice tower is analyzed as a space truss. Each member of the tower is assumed pin-connected at its joints carrying only axial load and no moment. Today, finite element computer programs [12, 21, 17] are the typical tools for the analysis of towers for ultimate design loads. In the analytical model the tower geometry is broken down into a discrete number of joints (nodes) and members (elements). User input consists of nodal coordinates, member end incidences and properties, and the tower loads. For symmetric towers, most programs can generate the complete geometry from a part of the input. Loads applied on the tower are ultimate loads which include overload capacity factors discussed in Section 15.2. Tower members are then designed to 1999 by CRC Press LLC
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the yield strength or the buckling strength of the member. Tower members typically consist of steel angle sections, which allow ease of connection. Both single- and double-angle sections are used. Aluminum towers are seldom used today due to the high cost of aluminum. Steel types commonly used on towers are ASTM A-36 (Fy = 36 ksi) or A-572 (F y = 50 ksi). The most common finish for steel towers is hot-dipped galvanizing. Self-weathering steel is no longer used for towers due to the “pack-out” problems experienced in the past resulting in damaged connections. Tower members are designed to carry axial compressive and tensile forces. Allowable stress in compression is usually governed by buckling, which causes the member to fail at a stress well below the yield strength of the material. Buckling of a member occurs about its weakest axis, which for a single angle section is at an inclination to the geometric axes. As the unsupported length of the member increases, the allowable stress in buckling is reduced. Allowable stress in a tension member is the full yield stress of the material and does not depend on the member length. The stress is resisted by a net cross-section, the area of which is the gross area minus the area of the bolt holes at a given section. Tension capacity of an angle member may be affected by the type of end connection [3]. For example, when one leg of the angle is connected, the tension capacity is reduced by 10%. A further reduction takes place when only the short leg of an unequal angle is connected.
15.3.3
Allowable Stresses
Compression Member
The allowable compressive stress in buckling on the gross cross-sectional area of axially loaded compression members is given by the following equations [3]: i 1 − (KL/R)2 /(2Cc2 ) F y
h
if KL/R = Cc or less
Fa
=
Fa Cc
= 286000/(kl/r)2 if KL/R > Cc 1/2 = (3.14)(2E/Fy)
(15.6) (15.7) (15.8)
where Fa = allowable compressive stress (ksi) Fy = yield strength (ksi) E = modulus of elasticity (ksi) L/R = maximum slenderness ratio = unbraced length /radius of gyration K = effective length co-efficient The angle member must also be checked for local buckling considerations. If the ratio of the angle effective width to angle thickness (w/t) exceeds 80/(F y)1/2 , the value of F a will be reduced in accordance with the provisions of Reference [3]. The above formulas indicate that the allowable buckling stress is largely dependent on the effective slenderness ratio (kl/r) and the material yield strength (F y). It may be noted, however, that Fy influences the buckling capacity for short members only (kl/r < Cc). For long members (kl/r > Cc), the allowable buckling stress is unaffected by the material strength. The slenderness ratio is calculated for different axes of buckling and the maximum value is used for the calculation of allowable buckling stress. In some cases, a compression member may have an intermediate lateral support in one plane only. This support prevents weak axis and in-plane buckling but not the out-of-plane buckling. In such cases, the slenderness ratio in the member geometric axis will be greater than in the member weak axis, and will control the design of the member. The effective length coefficient K adjusts the member slenderness ratio for different conditions of framing eccentricity and the restraint against rotation provided at the connection. Values of K 1999 by CRC Press LLC
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for six different end conditions, curves one through six, have been defined in Reference [3]. This reference also specifies maximum slenderness ratios of tower members, which are as follows: Type of Member Leg Bracing Redundant
Maximum KL/R 150 200 250
Tests have shown that members with very low L/R are subjected to substantial bending moment in addition to axial load. This is especially true for heavily loaded towers where members are relatively stiff and multiple bolted rigid joints are used [22]. A minimum L/R of 50 is recommended for compression members. Tension Members
The allowable tensile force on the net cross-sectional area of a member is given by the following equation [3]: Pt = F y · An · K
(15.9)
where Pt = allowable tensile force (kips) F y = yield strength of the material (ksi) An = net cross-sectional area of the angle after deducting for bolt holes (in.2 ). For unequal angles, if the short leg is connected, An is calculated by considering the unconnected leg to be the same size as the connected leg K = 1.0 if both legs of the angle connected = 0.9 if one leg connected The allowable tensile force must also meet the block shear criteria at the connection in accordance with the provisions of Reference [3]. Although the allowable force in a tension member does not depend on the member length, Reference [3] specifies a maximum L/R of 375 for these members. This limit minimizes member vibration under everyday steady state wind, and reduces the risk of fatigue in the connection.
15.3.4
Connections
Transmission towers typically use bearing type bolted connections. Commonly used bolt sizes are 5/8", 3/4", and 7/8" in diameter. Bolts are tightened to a snug tight condition with torque values ranging from 80 to 120 ft-lb. These torques are much smaller than the torque used in friction type connections in steel buildings. The snug tight torque ensures that the bolts will not slip back and forth under everyday wind loads thus minimizing the risk of fatigue in the connection. Under full design loads, the bolts would slip adding flexibility to the joint, which is consistent with the truss assumption. Load carrying capacity of the bolted connections depends on the shear strength of the bolt and the bearing strength of the connected plate. The most commonly used bolt for transmission towers is A-394, Type 0 bolt with an allowable shear stress of 55.2 ksi across the threaded part. The maximum allowable stress in bearing is 1.5 times the minimum tensile strength of the connected part or the bolt. Use of the maximum bearing stress requires that the edge distance from the center of the bolt hole to the edge of the connected part be checked in accordance with the provisions of Reference [3]. 1999 by CRC Press LLC
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15.3.5
Detailing Considerations
Bolted connections are detailed to minimize eccentricity as much as possible. Eccentric connections give rise to a bending moment causing additional shear force in the bolts. Sometimes small eccentricities may be unavoidable and should be accounted for in the design. The detailing specification should clearly specify the acceptable conditions of eccentricity. Figure 15.5 shows two connections, one with no eccentricity and the second with a small eccentricity. In the first case the lines of force passing through the center of gravity (c.g.) of the members
FIGURE 15.5: Brace details. intersect at a common point. This is the most desired condition producing no eccentricity. In the second case, the lines of force of the two bracing members do not intersect with that of the leg member thus producing an eccentricity in the connection. It is common practice to accept a small eccentricity as long as the intersection of the lines of force of the bracing members does not fall outside the width of the leg member. In some cases it may be necessary to add gusset plates to avoid large eccentricities. In detailing double angle members, care should be taken to avoid a large gap between the angles that are typically attached together by stitch bolts at specified intervals. Tests [23] have shown that a double angle member with a large gap between the angles does not act as a composite member. This results in one of the two angles carrying significantly more load than the other angle. It is recommended that the gap between the two angles of a double angle member be limited to 1/2 in. The minimum size of a member is sometimes dictated by the size of the bolt on the connected leg. The minimum width of members that can accommodate a single row of bolts is as follows: Bolt diameter
Minimum width of member
5/8" 3/4" 7/8"
1 3/4" 2" 2 1/2"
Tension members are detailed with draw to facilitate erection. Members 15 ft in length, or less, are detailed 1/8 in. short, plus 1/16 in. for each additional 10 ft. Tension members should have at least two bolts on one end to facilitate the draw.
15.3.6
Tower Testing
Full scale load tests are conducted on new tower designs and at least the tangent tower to verify the adequacy of the tower members and connections to withstand the design loads specified for that structure. Towers are required to pass the tests at 100% of the ultimate design loads. Tower tests 1999 by CRC Press LLC
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also provide insight into actual stress distribution in members, fit-up verification and action of the structure in deflected positions. Detailed procedures of tower testing are given in Reference [3].
EXAMPLE 15.2:
Description Check the adequacy of the following tower components shown in Figure 15.3. Member 1 (compressive leg of the leg extension) Member force = 132 kips (compression) Angle size = L5 × 5 × 3/8" Fy = 50 ksi Member 2 (tension member) Tensile force = 22 kips Angle size = L2 1/2 × 2 × 3/16 (long leg connected) Fy = 36 ksi Bolts at the splice connection of Member 1 Number of 5/8" bolts = 6 (Butt Splice) Type of bolt = A-394, Type O Solution Member 1
Member force = 132 kips (compression) Angle size = L5 × 5 × 3/8" Fy = 50 ksi Find maximum L/R Properties of L 5 × 5 × 3/8" Area = 3.61 in.2 rx = ry = 1.56 in. rz = 0.99 in. Member 1 has the same bracing pattern in adjacent planes. Thus, the unsupported length is the same in the weak (z − z) axis and the geometric axes (x − x and y − y). lz = lx = ly = 61" Maximum L/R = 61/0.99 = 61.6 Allowable Compressive Stress: Using Curve 1 for leg member (no framing eccentricity), per Reference [3], k = 1.0 KL/R 1999 by CRC Press LLC
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=
L/R = 61.6
Cc
Fa
= (3.14)(2E/Fy)1/2 = (3.14)(2 × 29000/50)1/2 = 107.0 which is > KL/R i h = 1 − (KL/R)2 /(2Cc2 ) F y i h = 1 − (61.6)2 /(2 × 107.02 ) 50.0 =
41.7 ksi
Allowable compressive load = 41.7 ksi × 3.61 in. = 150.6 kips > 132 kips → O.K. Check local buckling: w/t 80/(Fy)1/2
= (5.0 − 7/8)/(3/8) = 11.0 = 80/(50)1/2 = 11.3 > 11.0 O.K.
Member 2 Tensile force = 22 kips Angle size = L 2 − 1/2 × 2 × 3/16 Area = 0.81 in.2 Fy = 36 ksi Find tension capacity Pt = Fy · An · K Diameter of bolt hole = 5/8" + 1/16" = 11/16" Assuming one bolt hole deduction in 2 − 1/2" leg width, Area of bolt hole
= =
angle th. × hole diam. (3/16)(11/16) = 0.128 in.2
An
=
gross area − bolt hole area
K Pt
= = =
0.81 − 0.128 = 0.68 in.2 0.9, since member end is connected by one leg (36)(0.68)(0.9) = 22.1 kips > 22.0 kips, O.K.
Bolts for Member 1 Number of 5/8" bolts = 6 (Butt Splice) Type of bolt = A-394, Type O Shear Strength F v = 55.2 ksi Root area thru threads = 0.202 in.2 1999 by CRC Press LLC
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Shear capacity of bolts: Bolts act in double shear at butt splice Shear capacity of 6 bolts in double shear = 2 × (Root area) × 55.2 ksi × 6 = 133.8 kips > 132 kips ⇒ O.K. Bearing capacity of connected part: Thickness of connected angle = 3/8" Fy of angle = 50 ksi Capacity of bolt in bearing = 1.5 × F u × th. of angle × dia. of bolt F u of 50 ksi material = 65 ksi Capacity of 6 bolts in bearing = 1.5 × 65 × 3/8 × 5/8 × 6 = 137.1 kips > 132 kips, O.K.
15.4
Transmission Poles
15.4.1
General
Transmission poles made of wood, steel, or concrete are used on transmission lines at voltages up to 345-kv. Wood poles can be economically used for relatively shorter spans and lower voltages whereas steel poles and concrete poles have greater strength and are used for higher voltages. For areas where severe climatic loads are encountered, steel poles are often the most cost-effective choice. Pole structures have two basic configurations: single pole and H-frame (Figure 15.1). Single pole structures are used for lower voltages and shorter spans. H-frame structures consist of two poles connected by a framing comprised of the cross arm, the V-braces, and the X-braces. The use of X-braces significantly increases the load carrying capacity of H-frame structures. At line angles or deadend conditions, guying is used to decrease pole deflections and to increase their transverse or longitudinal structural strength. Guys also help prevent uplift on H-frame structures. Large deflections would be a hindrance in stringing operations.
15.4.2
Stress Analysis
Transmission poles are flexible structures and may undergo relatively large lateral deflections under design loads. A secondary moment (or P − 1 effect) will develop in the poles due to the lateral deflections at the load points. This secondary moment can be a significant percent of the total moment. In addition, large deflections of poles can affect the magnitude and direction of loads caused by the line tension and stringing operations. Therefore, the effects of pole deflections should be included in the analysis and design of single and multi-pole transmission structures. To properly analyze and design transmission structures, the standard industry practice today is to use nonlinear finite element computer programs. These computer programs allow efficient evaluation of pole structures considering geometric and/or material nonlinearities. For wood poles, there are several popular computer software programs available from EPRI [15]. They are specially developed for design and analysis of wood pole structures. Other general purpose commercial programs auch as SAP-90 and STAAD [20, 10] are available for performing small displacement P − 1 analysis. 1999 by CRC Press LLC
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15.4.3
Tubular Steel Poles
Steel transmission poles are fabricated from uniformly tapered hollow steel sections. The crosssections of the poles vary from round to 16-sided polygonal with the 12-sided dodecagonal as the most common shape. The poles are formed into design cross-sections by braking, rolling, or stretch bending. For these structures the usual industry practice is that the analysis, design, and detailing are performed by the steel pole supplier. This facilitates the design to be more compatible with fabrication practice and available equipment. Design of tubular steel poles is governed by the ASCE Manual # 72 [9]. The Manual provides detailed design criteria including allowable stresses for pole masts and connections and stability considerations for global and local buckling. It also defines the requirements for fabrication, erection, load testing, and quality assurance. It should be noted that steel transmission pole structures have several unique design features as compared to other tubular steel structures. First, they are designed for ultimate, or maximum anticipated loads. Thus, stress limits of the Manual #72 are not established for working loads but for ultimate loads. Second, Manual #72 requires that stability be provided for the structure as a whole and for each structural element. In other words, the effects of deflected structural shape on structural stability should be considered in the evaluation of the whole structure as well as the individual element. It relies on the use of the large displacement nonlinear computer analysis to account for the P −1 effect and check for stability. To prevent excessive deflection effects, the lateral deflection under factored loads is usually limited to 5 to 10% of the pole height. Pre-cambering of poles may be used to help meet the imposed deflection limitation on angle structures. Lastly, due to its polygonal cross-sections combined with thin material, special considerations must be given to calculation of member section properties and assessment of local buckling. To ensure a polygonal tubular member can reach yielding on its extreme fibers under combined axial and bending compression, local buckling must be prevented. This can be met by limiting the width to thickness ratio, w/t, to 240/(F y)1/2 for tubes with 12 or fewer sides and 215/(F y)1/2 for hexdecagonal tubes. If the axial stress is 1 ksi or less, the w/t limit may be increased to 260/(F y)1/2 for tubes with 8 or fewer sides [9]. Special considerations should be given in the selection of the pole materials where poles are to be subjected to subzero temperatures. To mitigate potential brittle fracture, use of steel with good impact toughness in the longitudinal direction of the pole is necessary. Since the majority of pole structures are manufactured from steels of a yield strength of 50 to 65 ksi (i.e., ASTM A871 and A572), it is advantageous to specify a minimum Charpy-V-notch impact energy of 15 ft-lb at 0◦ F for plate thickness of 1/2 in. or less and 15 ft-lb at −20◦ F for thicker plates. Likewise, high strength anchor bolts made of ASTM A615-87 Gr.75 steel should have a minimum Charpy V-notch of 15 ft-lbs at −20◦ F. Corrosion protection must be considered for steel poles. Selection of a specific coating or use of weathering steel depends on weather exposure, past experience, appearance, and economics. Weathering steel is best suited for environments involving proper wetting and drying cycles. Surfaces that are wet for prolonged periods will corrode at a rapid rate. A protective coating is required when such conditions exist. When weathering steel is used, poles should also be detailed to provide good drainage and avoid water retention. Also, poles should either be sealed or well ventilated to assure the proper protection of the interior surface of the pole. Hot-dip galvanizing is an excellent alternate means for corrosion protection of steel poles above grade. Galvanized coating should comply with ASTM A123 for its overall quality and for weight/thickness requirements. Pole sections are normally joined by telescoping or slip splices to transfer shears and moments. They are detailed to have a lap length no less than 1.5 times the largest inside diameter. It is important 1999 by CRC Press LLC
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to have a tight fit in slip joint to allow load transfer by friction between sections. Locking devices or flanged joints will be needed if the splice is subjected to uplift forces.
15.4.4
Wood Poles
Wood poles are available in different species. Most commonly used are Douglas Fir and Southern Yellow Pine, with a rupture bending stress of 8000 psi, and Western Red Cedar with a rupture bending stress of 6000 psi. The poles are usually treated with a preservative (pentachlorophenol or creosote). Framing materials for crossarm and braces are usually made of Douglas Fir or Southern Yellow Pine. Crossarms are typically designed for a rupture bending stress of 7400 psi. Wood poles are grouped into a wide range of classes and heights. The classification is based on minimum circumference requirements specified by the American National Standard (ANSI) specification 05.1 for each species, each class, and each height [2]. The most commonly used pole classes are class 1, 2, 3, and H-1. Table 15.3 lists the moment capacities at groundline for these common classes of wood poles. Poles of the same class and length have approximately the same capacity regardless of the species. TABLE 15.3 Moment Capacity at Ground Line for 8000 psi Douglas Fir and Southern Pine Poles Class Minimum circumference at top (in.) Length of pole (ft)
Ground line distance from butt (ft)
50 55 60 65 70 75 80 85 90 95 100 105 110
7 7.5 8 8.5 9 9.5 10 10.5 11 11 11 12 12
H-1 29
1 27
2 25
3 23
Ultimate moment capacity, ft-lb 220.3 246.4 266.8 288.4 311.2 335.3 360.6 387.2 405.2 438.0 461.5 461.5 514.2
187.2 204.2 222.3 241.5 261.9 283.4 306.2 321.5 337.5 357.3 387.3 387.3 424.1
152.1 167.1 183.0 200.0 218.1 230.3 250.2 263.7 285.5 303.2 321.5 321.5 354.1
121.7 134.7 148.7 163.5 179.4 190.2 201.5 213.3 225.5 —
The basic design principle for wood poles, as in steel poles, is to assure that the applied loads with appropriate overload capacity factors do not exceed the specified stress limits. In the design of a single unguyed wood pole structure, the governing criteria is to keep the applied moments below the moment capacity of wood poles, which are assumed to have round solid sections. Theoretically the maximum stress for single unguyed poles under lateral load does not always occur at the ground line. Because all data have been adjusted to the ground line per ANSI 05.1 pole dimensions, only the stress or moment at the ground line need to be checked against the moment capacity. The total ground line moment is the sum of the moment due to transverse wire loads, the moment due to wind on pole, and the secondary moment. The moment due to the eccentric vertical load should also be included if the conductors are not symmetrically arranged. Design guidelines for wood pole structures are given in the REA (Rural Electrification Administration) Bulletin 62-1 [18] and IEEE Wood Transmission Structural Design Guide [15]. Because of the use of high overload factors, the REA and NESC do not require the consideration of secondary moments in the design of wood poles unless the pole is very flexible. It also permits the use of rupture stress. In contrast, IEEE requires the secondary moments be included in the design and recommends 1999 by CRC Press LLC
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lower overload factors and use of reduction factors for computing allowable stresses. Designers can use either of the two standards to evaluate the allowable horizontal span for a given wood pole. Conversely, a wood pole can be selected for a given span and pole configuration. For H-frames with X-braces, maximum moments may not occur at ground line. Sections at braced location of poles should also be checked for combined moments and axial loads.
15.4.5
Concrete Poles
Prestressed concrete poles are more durable than wood or steel poles and they are aesthetically pleasing. The reinforcing of poles consists of a spiral wire cage to prevent longitudinal cracks and high strength longitudinal strands for prestressing. The pole is spinned to achieve adequate concrete compaction and a dense smooth finish. The concrete pole typically utilizes a high strength concrete (around 12000 psi) and 270 ksi prestressing strands. Concrete poles are normally designed by pole manufacturers. The guideline for design of concrete poles is given in Reference [8]. Standard concrete poles are limited by their ground line moment capacity. Concrete poles are, however, much heavier than steel or wood poles. Their greater weight increases transportation and handling costs. Thus, concrete poles are used most cost-effectively when there is a manufacturing plant near the project site.
15.4.6
Guyed Poles
At line angles and deadends, single poles and H-frames are guyed in order to carry large transverse loads or longitudinal loads. It is a common practice to use bisector guys for line angles up to 30◦ and in-line guys for structures at deadends or larger angles. The large guy tension and weight of conductors and insulators can exert significant vertical compression force on poles. Stability is therefore a main design consideration for guyed pole structures. Structural Stability
The overall stability of guyed poles under combined axial compression and bending can be assessed by either a large displacement nonlinear finite element stress analysis or by the use of simplified approximate methods. The rigorous stability analysis is commonly used by steel and concrete pole designers. The computer programs used are capable of assessing the structural stability of the guyed poles considering the effects of the stress-dependent structural stiffness and large displacements. But, in most cases, guys are modeled as tension-only truss elements instead of geometrically nonlinear cable elements. The effect of initial tension in guys is neglected in the analysis. The simplified stability method is typically used in the design of guyed wood poles. The pole is treated as a strut carrying axial loads only and guys are to carry the lateral loads. The critical buckling load for a tapered guyed pole may be estimated by the Gere and Carter method [13]. P cr = P (Dg/Da)e
(15.10)
where P is the Euler buckling load for a pole with a constant diameter of Da at guy attachment and is equal to 9.87 EI /(kl)2 ; Dg is the pole diameter at groundline; kl is the effective column length depending on end condition; e is an exponent constant equal to 2.7 for fixed-free ends and 2.0 for other end conditions. It should be noted that the exact end condition at the guyed attachment is difficult to evaluate. Common practice is to assume a hinged-hinged condition with k equal to 1.0. A higher k value should be chosen when there is only a single back guy. For a pole guyed at multiple levels, the column stability may be checked as follows by comparing the maximum axial compression against the critical buckling load, P cr, at the lowest braced location 1999 by CRC Press LLC
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of the pole [15]: [P 1 + P 2 + P 3 + · · ·] /P cr < 1/OCF
(15.11)
where OCF is the overload capacity factor and P 1, P 2, and P 3 are axial loads at various guy levels. Design of Guys
Guys are made of strands of cable attached to the pole and anchor by shackles, thimbles, clips, or other fittings. In the tall microwave towers, initial tension in the guys is normally set between 8 to 15% of the rated breaking strength (RBS) of the cable. However, there is no standard initial tension specified for guyed transmission poles. Guys are installed before conductors and ground wires are strung and should be tightened to remove slack without causing noticeable pole deflections. Initial tension in guys are normally in the range of 5 to 10% of RBS. For design of guys, the maximum tension under factored loads per NESC shall not exceed 90% of the cable breaking strength. Note that for failure containment (broken conductors) the guy tension may be limited to 0.85 RBS. A lower allowable of 65% of RBS would be needed if a linear load-deformation behavior of guyed poles is desired for extreme wind and ice conditions per ASCE Manual #72. Considerations should be given to the range of ambient temperatures at the site. A large temperature drop may induce a significant increase of guy tension. Guys with an initial tension greater than 15% of RBS of the guy strand may be subjected to aeolian vibrations.
EXAMPLE 15.3:
Description Select a Douglas Fir pole unguyed tangent structure shown below to withstand the NESC heavy district loads. Use an OCF of 2.5 for wind and 1.5 for vertical loads and a strength reduction factor of 0.65. Horizontal load span is 400 ft and vertical load span is 500 ft. Examine both cases with and without the P − 1 effect. The NESC heavy loading is 0.5 in. ice, 4 psf wind, and 0◦ F.
1999 by CRC Press LLC
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Ground Wire Loads H 1 = 0.453#/ft V 1 = 0.807#/ft Conductor Loads H 2 = H 3 = H 4 = 0.732#/ft V 2 = V 3 = V 4 = 2.284#/ft Horizontal Span = 400 ft Vertical Span = 500 ft Line Angle = 0◦ Solution A 75-ft class 1 pole is selected as the first trial. The pole will have a length of 9.5 ft buried below the groundline. The diameter of the pole is 9.59 in. at the top (Dt) and 16.3 in. at the groudline (Dg). Moment at groundline due to transverse wind on wire loads is
Mh = (0.732)(2.5)(400)(58 + 53.5 + 49) + (0.453)(2.5)(400)(65) = 146930 ft-lbs Moment at groundline due to vertical wire loads Mv = (2.284)(1.5)(500)(8 + 7 − 7) = 13700 ft-lbs Moment due to 4 psf wind on pole Mw
= (wind pressure) (OCF )H 2 (Dg + 2Dt)/72 = (4)(2.5)(65.5)2 (16.3 + 9.59 × 2)/72 = 21140 ft-lbs
The total moment at groundline Mt = 146930 + 13700 + 21140 = 181770 ft-lbs or 181.7 ft-kips This moment is less than the moment capacity of the 75-ft class 1 pole, 184.2 ft-kips ( i.e., 0.65 × 283.4, refer to Table 15.3). Thus, the 75-ft class 1 pole is adequate if the P − 1 effect is ignored. To include the effect of the pole displacement, the same pole was modeled on the SAP-90 computer program using a modulus of elasticity of 1920 ksi. Under the factored NESC loading, the maximum displacement at the top of the pole is 67.9 in. The associated secondary moment at the groundline is 28.5 ft-kips, which is approximately 15.7% of the primary moment. As a result, a 75-ft class H1 Douglas Fir pole with an allowable moment of 217.9 ft-kips is needed when the P − 1 effect is considered.
15.5
Transmission Tower Foundations
Tower foundation design requires competent engineering judgement. Soil data interpretation is critical as soil and rock properties can vary significantly along a transmission line. In addition, construction procedures and backfill compaction greatly influence foundation performance. Foundations can be designed for site specific loads or for a standard maximum load design. The best approach is to use both a site specific and standardized design. The selection should be based on the number of sites that will have a geotechnical investigation, inspection, and verification of soil conditions. 1999 by CRC Press LLC
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15.5.1
Geotechnical Parameters
To select and design the most economical type of foundation for a specific location, soil conditions at the site should be known through existing site knowledge or new explorations. Inspection should also be considered to verify that the selected soil parameters are within the design limits. The subsurface investigation program should be consistent with foundation loads, experience in the right-of-way conditions, variability of soil conditions, and the desired level of reliability. In designing transmission structure foundations, considerations must be given to frost penetration, expansive or shrinking soils, collapsing soils, black shales, sinkholes, and permafrost. Soil investigation should consider the unit weight, angle of internal friction, cohesion, blow counts, and modulus of deformation. The blow count values are correlated empirically to the soil value. Lab tests can measure the soil properties more accurately especially in clays.
15.5.2
Foundation Types—Selection and Design
There are many suitable types of tower foundations such as steel grillages, pressed plates, concrete footings, precast concrete, rock foundations, drilled shafts with or without bells, direct embedment, pile foundations, and anchors. These foundations are commonly used as support for lattice, poles, and guyed towers. The selected type depends on the cost and availability [14, 24]. Steel Grillages
These foundations consist entirely of steel members and should be designed in accordance with Reference [3]. The surrounding soil should not be considered as bracing the leg. There are pyramid arrangements that transfer the horizontal shear to the base through truss action. Other types transfer the shear through shear members that engage the lateral resistance of the compacted backfill. The steel can be purchased with the tower steel and concrete is not required at the site. Cast in Place Concrete
Cast in place concrete foundation consists of a base mat and a square of cylindrical pier. Most piers are kept in vertical position. However, the pier may be battered to allow the axial loads in the tower legs to intersect the mat centroid. Thus, the horizontal shear loads are greatly reduced for deadends and large line angles. Either stub angles or anchor bolts are embedded in the top of the pier so that the upper tower section can be spliced directly to the foundation. Bolted clip angles, welded stud shear connectors, or bottom plates are added to the stub angle. This type can also be precast elsewhere and delivered to the site. The design is accomplished by Reference [1]. Drilled Concrete Shafts
The drilled concrete shaft is the most common type of foundation now being used to support transmission structures. The shafts are constructed by power auguring a circular excavation, placing the reinforcing steel and anchor, and pouring concrete. Tubular steel poles are attached to the shafts using base plates welded to the pole with anchor bolts embedded in the foundation (Figure 15.6a). Lattice towers are attached through the use of stub angles or base plates with anchor bolts. Loose granular soil may require a casing or a slurry. If there is a water level, tremi concrete is required. The casing, if used, should be pulled as the concrete is poured to allow friction along the sides. A minimum 4" slump should allow good concrete flow. Belled shafts should not be attempted in granular soil. If conditions are right, this foundation type is the fastest and most economical to install as there is no backfilling required with dependency on compaction. Lateral procedures for design of drilled shafts under lateral and uplift loads are given in References [14] and [25]. 1999 by CRC Press LLC
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FIGURE 15.6: Direct embedment.
Rock Foundations
If bedrock is close to the surface, a rock foundation can be installed. The rock quality designation (RQD) is useful in evaluating rock. Uplift capacity can be increased with drilled anchor rods or by shaping the rock. Blasting may cause shatter or fracture to rock. Drilling or power hammers are therefore preferred. It is also helpful to wet the hole before placing concrete to ensure a good bond. Direct Embedment
Direct embedment of structures is the oldest form of foundation as it has been used on wood pole transmission lines since early times. Direct embedment consists of digging a hole in the ground, inserting the structure into the hole, and backfilling. Thus, the structure acts as its own foundation 1999 by CRC Press LLC
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transferring loads to the in situ soil via the backfill. The backfill can be a stone mix, stone-cement mix, excavated material, polyurethane foam, or concrete (see Figure 15.6b and c). The disadvantage of direct embedment is the dependency on the quality of backfill material. To accurately get deflection and rotation of direct embedded structures, the stiffness of the embedment must be considered. Rigid caisson analysis will not give accurate results. The performance criteria for deflection should be for the combined pole and foundation. Instability of the augured hole and the presence of water may require a liner or double liners (see Figure 15.6d). The design procedure for direct embedment is similar to drilled shafts [14, 25, 16]. Vibratory Shells
Steel shells are installed by using a vibratory hammer. The top 6 or 8 ft (similar to slip joint requirements) of soil inside the shell is excavated and the pole is inserted. The annulus is then filled with a high strength non-shrink grout. The pole can also be attached through a flange connection which eliminates excavating and grouting. The shell design is similar to drilled shafts. Piles
Piles are used to transmit loads through soft soil layers to stiffer soils or rock. The piles can be of wood, prestressed concrete, cast in place concrete, concrete filled shells, steel H piles, steel pipes filled with concrete, and prestressed concrete cylinder piles. The pipe selection depends on the loads, materials, and cost. Pile foundations are normally used more often for lattice towers than for H-framed structures or poles because piles have high axial load capacity and relatively low shear and bending capacity. Besides the external loading, piles can be subjected to the handling, drying, and soil stresses. If piles are not tested, the design should be conservative. Reference [14] should be consulted for bearing, uplift, lateral capacity, and settlement. Driving formulas can be used to estimate dynamic capacity of the pile or group. Timber piles are susceptible to deterioration and should be treated with a preservative. Anchors
Anchors are usually used to support guyed structures. The uplift capacity of rock anchors depends on the quality of the rock, the bond of the grout and rock with steel, and the steel strength. The uplift capacity of soil anchors depends on the resistance between grout and soil and end bearing if applicable. Multi-belled anchors in cohesive soil depend on the number of bells. The capacity of Helix anchors can be determined by the installation torque developed by the manufacturer. Spread anchor plate anchors depend on the soil weight plus the soil resistance. Anchors provide resistance to upward forces. They may be prestressed or deadman anchors. Deadmen anchors are not loaded until the structure is loaded, while prestressed anchors are loaded when installed or proof loaded. Helix soil anchors have deformed plates installed by rotating the anchor into the ground with a truck-mounted power auger. The capacity of the anchor is correlated to the amount of torque. Anchors are typically designed in accordance with the procedure given in Reference [14].
15.5.3
Anchorage
Anchorage of the transmission tower can consist of anchor bolts, stub angles with clip angles, or shear connectors and designed by Reference [3]. The anchor bolts can be smooth bars with a nut or head at the bottom, or deformed reinforcing bars with the embedment determined by Reference [1]. If the anchor bolt base plate is in contact with the foundation, the lateral or shear load is transferred to the foundation by shear friction. If there is no contact between the base plate and the concrete (anchor 1999 by CRC Press LLC
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bolts with leveling nuts), the lateral load is transferred to the concrete by the side bearing of the anchor bolt. Thus, anchor bolts should be designed for a combination of tension (or compression), shear, and bending by linear interaction.
15.5.4
Construction and Other Considerations
Backfill
Excavated foundations require a high level of compaction that should be inspected and tested. During the original design the degree of compaction that may actually be obtained should be considered. This construction procedure of excavation and compaction increases the foundation costs. Corrosion
The type of soil, moisture, and stray electric currents could cause corrosion of metals placed below the ground. Obtaining resistivity measurements would determine if a problem exists. Consideration could then be given to increasing the steel thickness, a heavier galvanizing coat, a bituminous coat, or in extreme cases a cathodic protection system. Hard epoxy coatings can be applied to steel piles. In addition, concrete can deteriorate in acidic or high sulfate soils.
15.5.5
Safety Margins for Foundation Design
The NESC requires the foundation design loads to be taken the same as NESC load cases used for design of the transmission structures. The engineer must use judgement in determining safety factors depending on the soil conditions, importance of the structures, and reliability of the transmission line. Unlike structural steel or concrete, soil does not have well-defined properties. Large variations exist in the geotechnical parameters and construction techniques. Larger safety margins should be provided where soil conditions are less uniform and less defined. Although foundation design is based on ultimate strength design, there is no industry standard on strength reduction factors at present. The latest research [11] shows that uplift test results differed significantly from analytical predictions and uplift capacity. Based on a statistical analysis of 48 uplift tests on drilled piers and 37 tests on grillages and plates, the coefficients of variation were found to be approximately 30%. To achieve a 95% reliability, which is a 5% exclusion limit, an uplift strength reduction factor of 0.8 to 0.9 is recommended for drilled shafts and 0.7 to 0.8 for backfilled types of foundations.
15.5.6
Foundation Movements
Foundation movements may change the structural configuration and cause load redistribution in lattice structures and framed structures. For pole structures a small foundation movement can induce a large displacement at the top of the pole which will reduce ground clearance or cause problems in wire stringing. The amount of tolerable foundation settlements depends on the structure type and load conditions. However, there is no industry standard at the present time. For lattice structures, it is suggested that the maximum vertical foundation movement be limited to 0.004 times the base dimensions. If larger movements are expected, foundations can be designed to limit their movements or the structures can be designed to withstand the specified foundation movements.
15.5.7
Foundation Testing
Transmission line foundations are load tested to verify the foundation design for specific soils, adequacy of the foundation, research investigation, and to determine strength reduction factors. The 1999 by CRC Press LLC
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load tests will refine foundation selection and verify the soil conditions and construction techniques. The load tests may be in uplift, download, lateral loads, overturning moment, or any necessary combination. There should also be a geotechnical investigation at the test site to correlate the soil data with other locations. There are various test set-ups, depending on what type of loading is to be applied and what type of foundation is to be tested. The results should compare the analytical methods used to actual behaviors. The load vs. the foundation movements should be plotted in order to evaluate the foundation performance.
15.5.8
Design Examples
EXAMPLE 15.4: Spread Footing
Problem—Determine the size of a square spread footing for a combined moment (175 ft-k) and axial load (74 kips) using two alternate methods. In the first method, the minimum factor of safety against overturning is 1.7 and the maximum soil pressure is kept below an allowable soil bearing of 4000 psf. In the second method, no factor of safety against overturning is specified. Instead, the spread footing is designed so that the resultant reaction is within the middle third. This example shows that keeping the resultant in the middle third is a conservative design. Solution Method 1
Try a 8 ft x 8 ft footing
1999 by CRC Press LLC
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P
=
74 kips
Mo
=
175 kip-ft
P increase for footing size increase = 0.3 kips/ft2 e = 175 k-ft/74 kips = 2.4 ft > 8 ft/6 = 1.33 ft Therefore, resultant is outside the middle third of the mat. (40 − 2.40 ) × 3 = 4.8 ft S.P. = (74 k)(2)/(4.8 ft)(8 ft) = 3850 psf < 4000 psf MR = (74 k)(4 ft) = 296 k-ft MR /Mo = 296/175 = 1.7 FOS against overturning, O.K. Method 2 (increase mat size to keep the resultant in the middle third) Try a 11.3 ft x 11.3 ft mat
h i P increase = (11.3 ft)2 − (8 ft)2 × 0.3 k/ft2 = 19.1 kips e = 175 k-ft/(74 + 19.1) kips = 1.88 ft = 11.3 ft/6 Resultant is within middle third. S.P. = (93.1 k)(2)/(11.3)2 = 1460 lbs/ft2 < 4000 lbs/ft2 Therefore, O.K. Increase in mat size = (11.3/8)2 = 1.99 Therefore, mat size has doubled, assuming that the mat thickness remains the same. 1999 by CRC Press LLC
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EXAMPLE 15.5: Design of a Drilled Shaft
Problem—Determine the depth of a 5-ft diameter drilled shaft in cohesive soil with a cohesion of 1.25 ksf by both Broms and modified Broms methods. The foundation is subjected to a combined moment of 2000 ft-k and a shear of 20 kips under extreme wind loading. Manual calculation by Broms method is shown herein while the modified Broms method is made by the use of a computer program (CADPRO) [25], which determines the depth required, lateral displacement, and rotation of the foundation top. Calculations are made for various factors of safety (or strength reduction factor). The equations used in this example are based on Reference [25]. Foundation in Cohesive Soil: M
=
2000 ft-kips
V
=
20 kips
Cohesion: C D
Solution 1. Use Broms Method [14] M H
=
2000 + 20 × 1
=
2020 ft-k
=
M/V = 2020/20 = 101
1999 by CRC Press LLC
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= 1.25 ksf = 5"
q
=
V /(9 CD) = 20/(9)(1.25)(5) = 0.356 i h L = 1.5D + q 1 + (2 + (4H + 6D)/q)0.5 i h = (1.5)(5) + .356 1 + (2 + (4)(101) + (6)(5))/0.356)0.5 = 20.3 ft 2. Comparison of Results of Broms Method and Modified Broms Method. Depth from
where F OS 8 1 θ
= = = =
Modified Brom
F OS
8
C used
broms (ft)
D (ft)
1
θ
1.0 1.33 1.5 1.75 2.0
1.0 0.75 0.667 0.575 0.5
1.25 .9375 .833 .714 .625
20.3 22.3 23.2 24.6 25.8
19 19.5 20.5 23.0 24.0
.935 .89 .81 .653 .603
.457 .474 .366 .262 .23
factor of safety strength reduction factor displacement, in. rotation, degrees
3. Conclusions This example demonstrates that the modified Broms method provides a more economical design than the Broms method. It also shows that as the depth increases by 26%, the factor of safety increased from 1.0 to 2.0. The cost will also increase proportionally.
15.6
Defining Terms
Bearing connection: Shear resistance is provided by bearing of bolt against the connected part. Block shear: A combination of shear and tensile failure through the end connection of a member. Buckling: Mode of failure of a member under compression at stresses below the material yield stress. Cascading effect: Progressive failure of structures due to an accident event. Circuit: A system of usually three phase conductors. Eccentric connection: Lines of force in intersecting members do not pass through a common work point, thus producing moment in the connection. Galloping: High amplitude, low frequency oscillation of snow covered conductors due to wind on uneven snow formation. Horizontal span: The horizontal distance between the midspan points of adjacent spans. Leg and bracing members: Tension or compression members which carry the loads on the structure to the foundation. Line angle: Denotes the change in the direction of a transmission line. Line tension: The longitudinal tension in a conductor or shield wire. Longitudinal load: Load on the supporting structure in a direction parallel to the line. Overload capacity factor: A multiplier used with the unfactored load to establish the design factored load. Phase conductors: Wires or cables intended to carry electric currents, extending along the route of the transmission line, supported by transmission structures. 1999 by CRC Press LLC
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Redundant member: Members that reduce the enbraced length of leg or brace members by providing intermediate support. Sag: The distance measured vertically from a conductor to the straight line joining its two points of support. Self supported structure: Unguyed structure supported on its own foundation. Shear friction: A mechanism to transfer the shear force at anchor bolts to the concrete through wedge action and tension of anchor. Shield wires: Wires installed on transmission structures intended to protect phase conductors against lightning strokes. Slenderness ratio: Ratio of the member unsupported length to its least radius of gyration. Span length: The horizontal distance between two adjacent supporting structures. Staggered bracing: Brace members on adjoining faces of a lattice tower are not connected to a common point on the leg. Stringing: Installation of conductor or shield wire on the structure. Transverse load: Load on the supporting structure in a direction perpendicular to the line. Uplift load: Vertically upward load at the wire attachment to the structure. Vertical span: The horizontal distance between the maximum sag points of adjacent spans. Voltage: The effective potential difference between any two conductors or between a conductor and ground.
References [1] ACI Committee 318, 1995, Building Code Requirements for Reinforced Concrete with Commentary, American Concrete Institute (ACI), Detroit, MI. [2] ANSI, 1979, Specification and Dimensions for Wood Poles, ANSI 05.1, American National Standard Institute, New York. [3] ANSI/ASCE, 1991, Design of Steel Latticed Transmission Structures, Standard 10-90, American National Standard Institute and American Society of Civil Engineers, New York. (Former ASCE Manual No. 52). [4] ANSI/IEEE, 1992, IEEE Guide to the Installation of Overhead Transmission Line Conductors, Standard 524, American National Structure Institute and Institute of Electrical and Electronic Engineers, New York. [5] ANSI/IEEE, 1993, National Electrical Safety Code, Standard C2, American National Standard Institute and Institute of Electrical and Electronic Engineers, New York. [6] ASCE, 1984, Guideline for Transmission Line Structural Loading, Committee on Electrical Transmission Structures, American Society of Civil Engineers, New York. [7] ASCE, 1986, Innovations in the Design of Electrical Transmission Structures, Proc. Conf. Struct. Div. Am. Soc. Civil Eng., New York. [8] ASCE, 1987, Guide for the Design and Use of Concrete Pole, American Society of Civil Engineers, New York. [9] ASCE, 1990, Design of Steel Transmission Pole Structures, ASCE Manual No. 72, Second ed., American Society of Civil Engineers, New York. [10] CSI, 1992, SAP90—A Series of Computer Programs for the Finite Element Analysis of Structures—Structural Analysis User’s Manual, Computer and Structures, Berkeley, CA. [11] EPRI, 1983, Transmission Line Structure Foundations for Uplift-Compression Loading: Load Test Summaries, EPRI Report EL-3160, Electric Power Research Institute, Palo Alto, CA. 1999 by CRC Press LLC
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[12] EPRI, 1990, T.L. Workstation Code, EPRI (Electric Power Research Institute), Report EL-6420, Vol. 1-23, Palo Alto, CA. [13] Gere, J.M. and Carter, W.O., 1962, Critical Buckling Loads for Tapered Columns, J. Struct. Div., ASCE, 88(ST1), 112. [14] IEEE, 1985, IEEE Trial-Use Guide for Transmission Structure Foundation Design, Standard 891, Institute of Electrical and Electronics Engineers, New York. [15] IEEE, 1991, IEEE Trial-Use Guide for Wood Transmission Structures, IEEE Standard 751, Institute of Electrical and Electronic Engineers, New York. [16] Kramer, J. M., 1978, Direct Embedment of Transmission Structures, Sargent & Lundy Transmission and Substation Conference, Chicago, IL. [17] Peyrot, A.H., 1985, Microcomputer Based Nonlinear Structural Analysis of Transmission Line Systems, IEEE Trans. Power Apparatus and Systems, PAS-104 (11). [18] REA, 1980, Design Manual for High Voltage Transmission Lines, Rural Electrification Administration (REA) Bulletin 62-1. [19] REA, 1992, Design Manual for High Voltage Transmission Lines, Rural Electrification Administration (REA), Bulletin 1724E-200. [20] REI, 1993, Program STAAD-III—Structural Analysis and Design—User’s Manual, Research Engineers, Orange, CA. [21] Rossow, E.C., Lo, D., and Chu, S.L, 1975, Efficient Design-Analysis of Physically Nonlinear Trusses, J. Struct. Div., 839-853, ASCE, New York. [22] Roy, S., Fang, S., and Rossow, E.C., 1982, Secondary Effects of Large Defection in Transmission Tower Structures, J. Energy Eng., ASCE, 110-2, 157-172. [23] Roy, S. and Fang, S., 1993, Designing and Testing Heavy Dead-End Towers, Proc. Am. Power Conf., 55-I, 839-853, ASCE, New York. [24] Simpson, K.D. and Yanaga, C.Y., 1982, Foundation Design Considerations for Transmission Structure, Sargent & Lundy Transmission and Distribution Conference, Chicago, IL. [25] Simpson, K.D., Strains, T.R., et. al., 1992, Transmission Line Computer Software: The New Generation of Design Tool, Sargent & Lundy Transmission and Distribution Conference, Chicago, IL.
1999 by CRC Press LLC
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Duan, L. and Reno, M. “Performance-Based Seismic Design Criteria For Bridges” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Performance-Based Seismic Design Criteria For Bridges Notations 16.2 Introduction
Damage to Bridges in Recent Earthquakes • No-CollapseBased Design Criteria • Performance-Based Design Criteria • Background of Criteria Development
16.3 Performance Requirements
General • Safety Evaluation Earthquake • Functionality Evaluation Earthquake • Objectives of Seismic Design
16.4 Loads and Load Combinations
Load Factors and Combinations • Earthquake Load Load • Buoyancy and Hydrodynamic Mass
•
Wind
16.5 Structural Materials
Existing Materials • New Materials
16.6 Determination of Demands
Analysis Methods • Modeling Considerations
16.7 Determination of Capacities
Limit States and Resistance Factors • Effective Length of Compression Members • Nominal Strength of Steel Structures • Nominal Strength of Concrete Structures • Structural Deformation Capacity • Seismic Response Modification Devices
16.8 Performance Acceptance Criteria
General • Structural Component Classifications • Steel Structures • Concrete Structures • Seismic Response Modification Devices
Lian Duan and Mark Reno Division of Structures, California Department of Transportation, Sacramento, CA
Defining Terms Acknowledgments References Further Reading Appendix A 16.A.1 Section Properties for Latticed Members 16.A.2 Buckling Mode Interaction For Compression Built-up members 16.A.3 Acceptable Force D/C Ratios and Limiting Values 16.A.4 Inelastic Analysis Considerations
Notations The following symbols are used in this chapter. The section number in parentheses after definition of a symbol refers to the section where the symbol first appears or is defined. A = cross-sectional area (Figure 16.9) 1999 by CRC Press LLC
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Ab Aclose Ad Ae Aequiv
= = = = =
Af Ag Agt Agv Ai Ant Anv Ap Ar As Aw A∗i A∗equiv
= = = = = = = = = = = = =
B C Cb Cw D1
= = = = =
DCaccept E Ec Es Et (EI )eff FL Fr
= = = = = = = =
Fu Fumax Fy Fyf Fymax Fyw G Ib If Ii Is Ix−x Iy−y
= = = = = = = = = = = = =
cross-sectional area of batten plate (Section 17.A.1) area enclosed within mean dimension for a box (Section 17.A.1) cross-sectional area of all diagonal lacings in one panel (Section 17.A.1) effective net area (Figure 16.9) cross-sectional area of a thin-walled plate equivalent to lacing bars considering shear transferring capacity (Section 17.A.1) flange area (Section 17.A.1) gross section area (Section 16.7.3) gross area subject to tension (Figure 16.9) gross area subject to shear (Figure 16.9) cross-sectional area of individual component i (Section 17.A.1) net area subject to tension (Figure 16.9) net area subject to shear (Figure 16.9) cross-sectional area of pipe (Section 16.7.3) nominal area of rivet (Section 16.7.3) cross-sectional area of steel members (Figure 16.8) cross-sectional area of web (Figure 16.12) cross-sectional area above or below plastic neutral axis (Section 17.A.1) cross-sectional area of a thin-walled plate equivalent to lacing bars or battens assuming full section integrity (Section 17.A.1) ratio of width to depth of steel box section with respect to bending axis (Section 17.A.4) distance from elastic neutral axis to extreme fiber (Section 17.A.1) bending coefficient dependent on moment gradient (Figure 16.10) warping constant, in.6 (Table 16.2) damage index defined as ratio of elastic displacement demand to ultimate displacement (Section 17.A.3) Acceptable force demand/capacity ratio (Section 16.8.1) modulus of elasticity of steel (Figure 16.8) modulus of elasticity of concrete (Section 16.5.2) modulus of elasticity of reinforcement (Section 16.5.2) tangent modulus (Section 17.A.4) effective flexural stiffness (Section 17.A.4) smaller of (Fyf − Fr ) or Fyw , ksi (Figure 16.10) compressive residual stress in flange; 10 ksi for rolled shapes, 16.5 ksi for welded shapes (Figure 16.10) specified minimum tensile strength of steel, ksi (Section 16.5.2) specified maximum tensile strength of steel, ksi (Section 16.5.2) specified minimum yield stress of steel, ksi (Section 16.5.2) specified minimum yield stress of the flange, ksi (Figure 16.10) specified maximum yield stress of steel, ksi (Section 16.5.2) specified minimum yield stress of the web, ksi (Figure 16.10) shear modulus of elasticity of steel (Table 16.2) moment of inertia of a batten plate (Section 17.A.1) moment of inertia of one solid flange about weak axis (Section 17.A.1) moment of inertia of individual component i (Section 17.A.1) moment of inertia of the stiffener about its own centroid (Section 16.7.3) moment of inertia of a section about x-x axis (Section 17.A.1) moment of inertia of a section about y-y axis considering shear transferring capacity (Section 17.A.1)
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Iy J Ka K L Lg M M1 M2 Mn MnFLB MnLTB MnWLB Mp Mr Mu My Mp−batten Mεc
= = = = = = = = = = = = = = = = = = =
Ns P Pcr
= = =
PL PG Pn Pu Py Pn∗ PnLG Pnb f Pn Pns y Pn comp Pn Pnten Q Qi Re Rn S Seff Sx Tn Vc Vn Vp Vs Vt
= = = = = = = = = = = = = = = = = = = = = = = = = =
moment of inertia about minor axis, in.4 (Table 16.2) torsional constant, in.4 (Figure 16.10) effective length factor of individual components between connectors (Figure 16.8) effective length factor of a compression member (Section 16.7.2) unsupported length of a member (Figure 16.8) free edge length of gusset plate (Section 16.7.3) bending moment (Figure 16.26) larger moment at end of unbraced length of beam (Table 16.2) smaller moment at end of unbraced length of beam (Table 16.2) nominal flexural strength (Figure 16.10) nominal flexural strength considering flange local buckling (Figure 16.10) nominal flexural strength considering lateral torsional buckling (Figure 16.10) nominal flexural strength considering web local buckling (Figure 16.10) plastic bending moment (Figure 16.10) elastic limiting buckling moment (Figure 16.10) factored bending moment demand (Section 16.7.3) yield moment (Figure 16.10) plastic moment of a batten plate about strong axis (Figure 16.12) moment at which compressive strain of concrete at extreme fiber equal to 0.003 (Section 16.7.4) number of shear planes per rivet (Section 16.7.3) axial force (Section 17.A.4) elastic buckling load of a built-up member considering buckling mode interaction (Section 17.A.2) elastic buckling load of an individual component (Section 17.A.2) elastic buckling load of a global member (Section 17.A.2) nominal axial strength (Figure 16.8) factored axial load demands (Figure 16.13) yield axial strength (Section 16.7.3) nominal compressive strength of column (Figure 16.8) nominal compressive strength considering buckling mode interaction (Figure 16.8) nominal tensile strength considering block shear rupture (Figure 16.9) nominal tensile strength considering fracture in net section (Figure 16.9) nominal compressive strength of a solid web member (Figure 16.8) nominal tensile strength considering yielding in gross section (Figure 16.9) nominal compressive strength of lacing bar (Figure 16.12) nominal tensile strength of lacing bar (Figure 16.12) full reduction factor for slender compression elements (Figure 16.8) force effect (Section 16.4.1) hybrid girder factor (Figure 16.10) nominal shear strength (Section 16.7.3) elastic section modulus (Figure 16.10) effective section modulus (Figure 16.10) elastic section modulus about major axis, in.3 (Figure 16.10) nominal tensile strength of a rivet (Section 16.7.3) nominal shear strength of concrete (Section 16.7.4) nominal shear strength (Figure 16.12) plastic shear strength (Section 16.7.3) nominal shear strength of transverse reinforcement (Section 16.7.4) shear strength carried bt truss mechanism (Section 16.7.4)
1999 by CRC Press LLC
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Vu X1 X2 Z a b bi d fc0 fcmin fr fyt h k kv l
= = = = = = = = = = = = = = = =
m
=
mbatten mlacing n nr
= = = =
r ri ry t ti tequiv tw vc x xi xi∗
= = = = = = = = = = =
y yi∗
= =
1ed 1u α αx αy β βm βt
= = = = = = = =
βx βy
= =
factored shear demand (Section 16.7.3) beam buckling factor defined by AISC-LRFD [4] (Figure 16.11) beam buckling factor defined by AISC-LRFD [4] (Figure 16.11) plastic section modulus (Figure 16.10) distance between two connectors along member axis (Figure 16.8) width of compression element (Figure 16.8) length of particular segment of (Section 17.A.1) effective depth of (Section 16.7.4) specified compressive strength of concrete (Section 16.7.5) specified minimum compressive strength of concrete (Section 16.5.2) modulus of rupture of concrete (Section 16.5.2) probable yield strength of transverse steel (Section 16.7.4) depth of web (Figure 16.8) or depth of member in lacing plane (Section 17.A.1) buckling coefficient (Table 16.3) web plate buckling coefficient (Figure 16.12) length from the last rivet (or bolt) line on a member to first rivet (or bolt) line on a member measured along the centerline of member (Section 16.7.3) number of panels between point of maximum moment to point of zero moment to either side [as an approximation, half of member length (L/2) may be used] (Section 17.A.1) number of batten planes (Figure 16.12) number of lacing planes (Figure 16.12) number of equally spaced longitudinal compression flange stiffeners (Table 16.3) number of rivets connecting lacing bar and main component at one joint (Figure 16.12) radius of gyration, in. (Figure 16.8) radius of gyration of local member, in. (Figure 16.8) radius of gyration about minor axis, in. (Figure 16.10) thickness of unstiffened element (Figure 16.8) average thickness of segment bi (Section 17.A.1) thickness of equivalent thin-walled plate (Section 17.A.1) thickness of the web (Figure 16.10) permissible shear stress carried by concrete (Section 16.7.4) subscript relating symbol to strong axis or x-x axis (Figure 16.13) distance between y-y axis and center of individual component i (Section 17.A.1) distance between center of gravity of a section A∗i and plastic neutral y-y axis (Section 17.A.1) subscript relating symbol to strong axis or y-y axis (Figure 16.13) distance between center of gravity of a section A∗i and plastic neutral x-x axis (Section 17.A.1) elastic displacement demand (Section 17.A.3) ultimate displacement (Section 17.A.3) separation ratio (Section 17.A.2) parameter related to biaxial loading behavior for x-x axis (Section 17.A.4) parameter related to biaxial loading behavior for y-y axis (Section 17.A.4) 0.8, reduction factor for connection (Section 16.7.3) reduction factor for moment of inertia specified by Equation 16.28 (Section 17.A.1) reduction factor for torsion constant may be determined Equation 16.38 (Section 17.A.1) parameter related to uniaxial loading behavior for x-x axis (Section 17.A.4) parameter related to uniaxial loading behavior for y-y axis (Section 17.A.4)
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δo γLG λ λb λbp λbr λbpr λc λcp λcpr λcr λp λpr λr λp−Seismic µ1 µφ ρ 00 φ φ φb φbs φc φt φtf φty comp σc σcten σs τu εs εsh comp εc γi η
= imperfection (out-of-straightness) of individual component (Section 17.A.2) = buckling mode interaction factor to account for buckling model interaction (Figure 16.8) = width-thickness ratio of compression element (Figure 16.8) = rLy (slenderness parameter of flexural moment dominant members) (Figure 16.10) = limiting beam slenderness parameter for plastic moment for seismic design (Figure 16.10) = limiting beam slenderness parameter for elastic lateral torsional buckling (Figure 16.10) = limiting q beam slenderness parameter determined by Equation 16.25 (Table 16.2) Fy = KL rπ E (slenderness parameter of axial load dominant members) (Figure 16.8) = 0.5 (limiting column slenderness parameter for 90% of the axial yield load based on AISC-LRFD [4] column curve) (Table 16.2) = limiting column slenderness parameter determined by Equation 16.24 (Table 16.2) = limiting column slenderness parameter for elastic buckling (Table 16.2) = limiting width-thickness ratio for plasticity development specified in Table 16.3 (Figure 16.10) = limiting width-thickness ratio determined by Equation 16.23 (Table 16.2) = limiting width-thickness ratio (Figure 16.8) = limiting width-thickness ratio for seismic design (Table 16.2) = displacement ductility, ratio of ultimate displacement to yield displacement (Section 16.7.4) = curvature ductility, ratio of ultimate curvature to yield curvature (Section 17.A.3) = ratio of transverse reinforcement volume to volume of confined core (Section 16.7.4) = resistance factor (Section 16.7.1) = angle between diagonal lacing bar and the axis perpendicular to the member axis (Figure 16.12) = resistance factor for flexure (Figure 16.13) = resistance factor for block shear (Section 16.7.1) = resistance factor for compression (Figure 16.13) = resistance factor for tension (Figure 16.9) = resistance factor for tension fracture in net (section 16.7.1) = resistance factor for tension yield (Figure 16.9) = maximum concrete stress under uniaxial compression (Section 16.7.5) = maximum concrete stress under uniaxial tension (Section 16.7.5) = maximum steel stress under uniaxial tension (Section 16.7.5) = shear strength of a rivet (Section 16.7.3) = maximum steel strain under uniaxial tension (Section 16.7.5) = strain hardening strain of steel (Section 16.5.2) = maximum concrete strain under uniaxial compression (Section 16.7.5) = load factor corresponding to Qi (Section 16.4.1) = a factor relating to ductility, redundancy, and operational importance (Section 16.4.1)
16.2
Introduction
16.2.1
Damage to Bridges in Recent Earthquakes
Since the beginning of civilization, earthquake disasters have caused both death and destruction — the structural collapse of homes, buildings, and bridges. About 20 years ago, the 1976 Tangshan earthquake in China resulted in the tragic death of 242,000 people, while 164,000 people were severely 1999 by CRC Press LLC
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injured, not to mention the entire collapse of the industrial city of Tangshan [39]. More recently, the 1989 Loma Prieta and the 1994 Northridge earthquakes in California [27, 28] and the 1995 Kobe earthquake in Japan [29] have exacted their tolls in the terms of deaths, injuries, and the collapse of the infrastructure systems which can in turn have detrimental effects on the economies. The damage and collapse of bridge structures tend to have a more lasting image on the public. Figure 16.1 shows the collapsed elevated steel conveyor at Lujiatuo Mine following the 1976 Tangshan earthquake in China. Figures 16.2 and 16.3 show damage from the 1989 Loma Prieta earthquake: the San Francisco-Oakland Bay Bridge east span drop off and the collapsed double deck portion of the Cypress freeway, respectively. Figure 16.4 shows a portion of the R-14/I-5 interchange following the 1994 Northridge earthquake, which also collapsed following the 1971 San Fernando earthquake in California while it was under construction. Figure 16.5 shows a collapsed 500-m section of the elevated Hanshin Expressway during the 1995 Kobe earthquake in Japan. These examples of bridge damage, though tragic, have served as full-scale laboratory tests and have forced bridge engineers to reconsider their design principles and philosophies. Since the 1971 San Fernando earthquake, it has been a continuing challenge for bridge engineers to develop a safe seismic design procedure so that the structures are able to withstand the sometimes unpredictable devastating earthquakes.
FIGURE 16.1: Collapsed elevated steel conveyor at Lujiatuo Mine following the 1976 Tangshan earthquake in China. (From California Institute of Technology, The Greater Tangshan Earthquake, California, 1996. With permission.)
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FIGURE 16.2: Aerial view of collapsed upper and lower decks of the San Francisco-Oakland Bay Bridge (I-80) following the 1989 Loma Prieta earthquake in California. (Photo by California Department of Transportation. With permission.)
16.2.2 No-Collapse-Based Design Criteria For seismic design and retrofit of ordinary bridges, the primary philosophy is to prevent collapse during severe earthquakes [13, 24, 25]. The structural survival without collapse has been a basis of seismic design and retrofit for many years [13]. To prevent the collapse of bridges, two alternative design approaches are commonly in use. First is the conventional force-based approach where the adjustment factor Z for ductility and risk assessment [12], or the response modification factor R [1], is applied to elastic member force levels obtained by acceleration spectra analysis. The second approach is the newer displacement-based design approach [13] where displacements are a major consideration in design. For more detailed information, reference is made to a comprehensive and state-of-the-art book by Prietley et al. [35]. Much of the information in this book is backed by California Department of Transportation (Caltrans)-supported research, directed at the seismic performance of bridge structures.
16.2.3
Performance-Based Design Criteria
Following the 1989 Loma Prieta earthquake, bridge engineers recognized the need for site-specific and project-specific design criteria for important bridges. A bridge is defined as “important” when one of the following criteria is met: • The bridge is required to provide secondary life safety. • Time for restoration of functionality after closure creates a major economic impact. • The bridge is formally designated as critical by a local emergency plan. 1999 by CRC Press LLC
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FIGURE 16.3: Collapsed Cypress Viaduct (I-880) following the 1989 Loma Prieta earthquake in California.
FIGURE 16.4: Collapsed SR-14/I-5 south connector overhead following the 1994 Northridge earthquake in California. (Photo by James MacIntyre. With permission.)
Caltrans, in cooperation with various emergency agencies, has designated and defined the various important routes throughout the state of California. For important bridges, such as I-880 replacement [23] and R-14/I-5 interchange replacement projects, the design criteria [10, 11] including site-specific Acceleration Response Spectrum (ARS) curves and specific design procedures to reflect the desired performance of these structures were developed. 1999 by CRC Press LLC
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FIGURE 16.5: Collapsed Hanshin Expressway following the 1995 Kobe earthquake in Japan. (Photo by Mark Yashinsky. With permission.)
In 1995, Caltrans, in cooperation with engineering consulting firms, began the task of seismic retrofit design for the seven major toll bridges including the San Francisco-Oakland Bay Bridge (SFOBB) in California. Since the traditional seismic design procedures could not be directly applied to these toll bridges, various analysis and design concepts and strategies have been developed [7]. These differences can be attributed to the different post-earthquake performance requirements. As shown in Figure 16.6, the performance requirements for a specific project or bridge must be the first item to be established. Loads, materials, analysis methods and approaches, and detailed acceptance criteria are then developed to achieve the expected performance. The no-collapse-based design criteria shall be used unless performance-based design criteria is required.
16.2.4
Background of Criteria Development
It is the purpose of this chapter to present performance-based criteria that may be used as a guideline for seismic design and retrofit of important bridges. More importantly, this chapter provides concepts for the general development of performance-based criteria. The appendices, as an integral part of the criteria, are provided for background and information of criteria development. However, it must be recognized that the desired performance of the structure during various earthquakes ultimately defines the design procedures. Much of this chapter was primarily based on the Seismic Retrofit Design Criteria (Criteria) which was developed for the SFOBB West Spans [17]. The SFOBB Criteria was developed and based on past successful experience, various codes, specifications, and state-of-the-art knowledge. The SFOBB, one of the national engineering wonders, provides the only direct highway link between San Francisco and the East Bay Communities. SFOBB (Figure 16.7) carries Interstate Highway 80 approximately 8-1/4 miles across San Francisco Bay since it first opened to traffic in 1936. The west spans of SFOBB, consisting of twin, end-to-end double-deck suspension bridges and a three-span double-deck continuous truss, crosses the San Francisco Bay from the city of San Francisco to Yerba Buena Island. The seismic retrofit design of SFOBB West Spans, as the top priority project of the California Department of Transportation, is a challenge to bridge engineers. A performance-based design Criteria [17] was, therefore, developed for SFOBB West Spans. 1999 by CRC Press LLC
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FIGURE 16.6: Development procedure of performance-based seismic design criteria for important bridges.
16.3
Performance Requirements
16.3.1
General
The seismic design and retrofit of important bridges shall be performed by considering both the higher level Safety Evaluation Earthquake (SEE), which has a mean return period in the range of 1000 to 2000 years, and the lower level Functionality Evaluation Earthquake (FEE), which has a mean return period of 300 years with a 40% probability of exceedance during the expected life of the bridge. It is important to note that the return periods of both the SEE and FEE are dictated by the engineers and seismologists.
16.3.2
Safety Evaluation Earthquake
The bridge shall remain serviceable after a SEE. Serviceable is defined as sustaining repairable damage with minimum impact to functionality of the bridge structure. In addition, the bridge will be open to emergency vehicles immediately following the event, provided bridge management personnel can provide access. 1999 by CRC Press LLC
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(a) West crossing spans.
(b) East crossing spans. FIGURE 16.7: San Francisco-Oakland Bay Bridge. (Photo by California Department of Transportation. With permission.)
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16.3.3
Functionality Evaluation Earthquake
The bridge shall remain fully operational after a FEE. Fully operational is defined as full accessibility to the bridge by current normal daily traffic. The structure may suffer repairable damage, but repair operations may not impede traffic in excess of what is currently required for normal daily maintenance.
16.3.4
Objectives of Seismic Design
The objectives of seismic design are as follows: 1. To keep the Critical structural components in the essentially elastic range during the SEE. 2. To achieve safety, reliability, serviceability, constructibility, and maintainability when the Seismic Response Modification Devices (SRMDs), i.e., energy dissipation and isolation devices, are installed in bridges. 3. To devise expansion joint assemblies between bridge frames that either retain traffic support or, with the installation of deck plates, are able to carry the designated traffic after being subjected to SEE displacements. 4. To provide ductile load paths and detailing to ensure bridge safety in the event that future demands might exceed those demands resulting from current SEE ground motions.
16.4
Loads and Load Combinations
16.4.1
Load Factors and Combinations
New and retrofitted bridge components shall be designed for the applicable load combinations in accordance with the requirements of AASHTO-LRFD [1]. The load effect shall be obtained by Load effect = η
X
γi Qi
(16.1)
where Qi = force effect η = a factor relating to ductility, redundancy, and operational importance = load factor corresponding to Qi γi . The AASHTO-LRFD load factors or load factors η = 1.0 and γi = 1.0 may be used for seismic design. The live load on the bridge shall be determined by ADTT (Average Daily Truck Traffic) value for the project. The bridge shall be analyzed for the worst case with or without live load. The mass of the live loads shall not be included in the dynamic calculations. The intent of the live load combination is to include the weight effect of the vehicles only.
16.4.2
Earthquake Load
The earthquake load – ground motions and response spectra shall be considered at two levels: SEE and FEE. The ground motions and response spectra may be generated in accordance with Caltrans Guidelines [14, 15]. 1999 by CRC Press LLC
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16.4.3
Wind Load
1. Wind load on structures — Wind loads shall be applied as a static equivalent load in accordance with AASHTO-LRFD [1] . 2. Wind load on live load — Wind pressure on vehicles shall be represented by a uniform load of 0.100 kips/ft (1.46 kN/m) applied at right angles to the longitudinal axis of the structure and 6.0 ft (1.85 m) above the deck to represent the wind load on vehicles. 3. Wind load dynamics — The expansion joints, SRMDs, and wind locks (tongues) shall be evaluated for the dynamic effects of wind loads.
16.4.4
Buoyancy and Hydrodynamic Mass
The buoyancy shall be considered to be an uplift force acting on all components below design water level. Hydrodynamic mass effects [26] shall be considered for bridges over water.
16.5
Structural Materials
16.5.1
Existing Materials
For seismic retrofit design, aged concrete with specified strength of 3250 psi (22.4 MPa) can be considered to have a compressive strength of 5000 psi (34.5 MPa). If possible, cores of existing concrete should be taken. Behavior of structural steel and reinforcement shall be based on mill certificate or tensile test results. If they are not available in bridge archives, a nominal strength of 1.1 times specified yield strength may be used [13].
16.5.2
New Materials
Structural Steel
New structural steel used shall be AASHTO designation M270 (ASTM designation A709) Grade 36 and Grade 50. Welds shall be as specified in the Bridge Welding Code ANSI/AASHTO/AWS D1.5-95 [8]. Partial penetration welds shall not be used in regions of structural components subjected to possible inelastic deformation. High strength bolts conforming to ASTM designation A325 shall be used for all new connections and for upgrading strengths of existing riveted connections. New bolted connections shall be designed as bearing-type for seismic loads and shall be slip-critical for all other load cases. All bolts with a required length under the head greater than 8 in. shall be designated as ASTM A449 threaded rods (requiring nuts at each end) unless a verified source of longer bolts can be identified. New anchor bolts shall be designated as ASTM A449 threaded rods. Structural Concrete
All concrete shall be normal weight concrete with the following properties: Specified compressive strength: Modulus of elasticity: Modulus of rupture:
fcmin = 4, 000 p Ec = 57,000 fc0 p fr = 5 fc0
psi (27.6MPa) psi psi
Reinforcement
All reinforcement shall use ASTM A706 (Grade 60) with the following specified properties: 1999 by CRC Press LLC
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Specified minimum yield stress: Specified minimum tensile strength: Specified maximum yield stress: Specified maximum tensile strength: Modulus of elasticity: Strain hardening strain: εsh =
16.6
Determination of Demands
16.6.1
Analysis Methods
Fy = 60 ksi Fu = 90 ksi Fymax = 78 ksi Fumax = 107 ksi Es = 29,000 ksi 0.0150 0.0125 0.0100 0.0075 0.0050
(414 MPa) (621 MPa) (538 MPa) (738 MPa) (200,000 MPa)
for #8 and smallers bars for #9 for #10 and #11 for #14 for #18
Static Linear Analysis
Static linear analysis shall be used to determine member forces due to self weight, wind, water currents, temperature, and live load. Dynamic Response Spectrum Analysis
1. Dynamic response spectrum analysis shall be used for the local and regional stand alone models and the simplified global model described in Section 16.6.2 to determine mode shapes, structure periods, and initial estimates of seismic force and displacement demands. 2. Dynamic response spectrum analysis may be used on global models prior to time history analysis to verify model behavior and eliminate modeling errors. 3. Dynamic response spectrum analysis may be used to identify initial regions or members of likely inelastic behavior which need further refined analysis using inelastic nonlinear elements. 4. Site specific ARS curves shall be used, with 5% damping. 5. Modal responses shall be combined using the Complete Quadratic Combination (CQC) method and the resulting orthogonal responses shall be combined using either the Square Root of the Sum of the Squares (SRSS) method or the “30%” rule, e.g., RH = Max(Rx + 0.3Ry , Ry + 0.3Rx ) [13]. 6. Due to the expected levels of inelastic structural response in some members and regions, dynamic response spectrum analysis shall not be used to determine final design demand values or to assess the performance of the retrofitted structures. Dynamic Time History Analysis
Site specific multi-support dynamic time histories shall be used in a dynamic time history analysis. All analyses incorporating significant nonlinear behavior shall be conducted using nonlinear inelastic dynamic time history procedures. 1. Linear elastic dynamic time history analysis — Linear elastic dynamic time history analysis is defined as dynamic time history analysis with considerations of geometrical linearity 1999 by CRC Press LLC
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(small displacement), linear boundary conditions, and elastic members. It shall only be used to check regional and global models. 2. Nonlinear elastic dynamic time history analysis — Nonlinear elastic time history analysis is defined as dynamic time history analysis with considerations of geometrical nonlinearity, linear boundary conditions, and elastic members. It shall be used to determine areas of inelastic behavior prior to incorporating inelasticity into the regional and global models. 3. Nonlinear inelastic dynamic time history analysis – Level I — Nonlinear inelastic dynamic time history analysis – Level I is defined as dynamic time history analysis with considerations of geometrical nonlinearity, nonlinear boundary conditions, other inelastic elements (for example, dampers) and elastic members. It shall be used for the final determination of force and displacement demands for existing structures in combination with static gravity, wind, thermal, water current, and live load as specified in Section 16.4. 4. Nonlinear inelastic dynamic time history analysis – Level II — Nonlinear inelastic dynamic time history analysis – Level II is defined as dynamic time history analysis with considerations of geometrical nonlinearity, nonlinear boundary conditions, other inelastic elements (for example, dampers) and inelastic members. It shall be used for the final evaluation of response of the structures. Reduced material and section properties, and the yield surface equation suggested in the Appendix may be used for inelastic considerations.
16.6.2
Modeling Considerations
Global, Regional, and Local Models
The global models focus on the overall behavior and may include simplifications of complex structural elements. Regional models concentrate on regional behavior. Local models emphasize the localized behavior, especially complex inelastic and nonlinear behavior. In regional and global models where more than one foundation location is included in the model, multi-support time history analysis shall be used. Boundary Conditions
Appropriate boundary conditions shall be included in the regional models to represent the interaction between the regional model and the adjacent portion of the structure not explicitly included. The adjacent portion not specifically included may be modeled using simplified structural combinations of springs, dashpots, and lumped masses. Appropriate nonlinear elements such as gap elements, nonlinear springs, SRMDs, or specialized nonlinear finite elements shall be included where the behavior and response of the structure is determined to be sensitive to such elements. Soil-Foundation-Structure-Interaction
Soil-Foundation-Structure-Interaction may be considered using nonlinear or hysteretic springs in the global and regional models. Foundation springs at the base of the structure which reflect the dynamic properties of the supporting soil shall be included in both regional and global models. Section Properties of Latticed Members
For latticed members, the procedure proposed in the Appendix may be used for member characterization. 1999 by CRC Press LLC
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Damping
When nonlinear member properties are incorporated in the model, Rayleigh damping shall be reduced, for example by 20%, compared with analysis with elastic member properties. Seismic Response Modification Devices
The SRMDs, i.e., energy dissipation and isolation devices, shall be modeled explicitly using their hysteretic characteristics as determined by tests.
16.7
Determination of Capacities
16.7.1 Limit States and Resistance Factors Limit States
The limit states are defined as those conditions of a structure at which it ceases to satisfy the provisions for which it was designed. Two kinds of limit states corresponding to SEE and FEE specified in Section 16.3 apply for seismic design and retrofit. Resistance Factors
To account for unavoidable inaccuracies in the theory, variation in the material properties, workmanship, and dimensions, nominal strength of structural components should be modified by a resistance factor φ to obtain the design capacity or strength (resistance). The following resistance factors shall be used for seismic design: • • • •
16.7.2
For tension fracture in net section For block shear For bolts and welds For all other cases
φtf φbs φ φ
= 0.8 = 0.8 = 0.8 = 1.0
Effective Length of Compression Members
The effective length factor K for compression members shall be determined in accordance with Chapter 17 of this Handbook.
16.7.3
Nominal Strength of Steel Structures
Members
1. General — Steel members include rolled members and built-up members, such as latticed, battened, and perforated members. The design strength of those members shall be according to applicable provisions of AISC-LRFD [4]. Section properties of latticed members shall be determined in accordance with the Appendix. 2. Compression members — For compression members, the nominal strength shall be determined in accordance with Section E2 and Appendix B of AISC-LRFD [4]. For builtup members, effects of interaction of buckling modes shall be considered in accordance with the Appendix. A detailed procedure in a flowchart format is shown in Figure 16.8. 3. Tension Members — For tension members, the design strength shall be determined in accordance with Sections D1 and J4 of AISC-LRFD [4]. It is the smallest value obtained according to (i) yielding in gross section, (ii) fracture in net section, and (iii) block shear rupture. A detailed procedure is shown in Figure 16.9. 1999 by CRC Press LLC
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FIGURE 16.8: Evaluation procedure for nominal compressive strength of steel members.
4. Flexural members — For flexural members, the nominal flexural strength shall be determined in accordance with Section F1 and Appendices B, F, and G of AISC-LRFD [4]. • For critical members, the nominal flexural strength is the smallest value according to (i) initial yielding, (ii) lateral-torsional buckling, (iii) flange local buckling, and (iv) web local buckling. • For other members, the nominal flexural strength is the smallest value according to (i) plastic moment, (ii) lateral-torsional buckling, (iii) flange local buckling, and (iv) web local buckling.
1999 by CRC Press LLC
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FIGURE 16.9: Evaluation procedure for tensile strength of steel members.
Detailed procedures for flexural strength of box- and I-shaped members are shown in Figures 16.10 and 16.11, respectively. 5. Nominal shear strength — For solid-web steel members, the nominal shear strength shall be determined in accordance with Appendix F2 of AISC-LRFD [4]. For latticed members, the shear strength shall be based on shear-flow transfer-capacity of lacing bar, battens, and connectors as discussed in the Appendix. A detailed procedure for shear strength is shown in Figure 16.12. 6. Members subjected to bending and axial force — For members subjected to bending and axial force, the evaluation shall be according to Section H1 of AISC-LRFD [4], i.e., the bi-linear interaction equation shall be used. The recent study on “Cyclic Testing of Latticed Members for San Francisco-Oakland Bay Bridge” at UCSD [37] recommends that the AISC-LRFD interaction equation can be used directly for seismic evaluation of latticed members. A detailed procedure for steel beam-columns is shown in Figure 16.13. Gusset Plate Connections
1. General description — Gusset plates shall be evaluated for shear, bending, and axial forces according to Article 6.14.2.8 of AASHTO-LRFD [1]. The internal stresses in the gusset plate shall be determined according to Whitmore’s method in which the effective area is defined as the width bound by two 30◦ lines drawn from the first row of the bolt or rivet 1999 by CRC Press LLC
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FIGURE 16.10: Evaluation procedure for nominal flexural strength of box-shaped steel members.
group to the last bolt or rivet line. The stresses in the gusset plate may be determined by more rational methods or refined computer models. 2. Tension strength — The tension capacity of the gusset plates shall be calculated according to Article 6.13.5.2 of AASHTO-LRFD [1]. 3. Compressive strength — The compression capacity of the gusset plates shall be calculated according to Article 6.9.4.1 of AASHTO-LRFD [1]. In using the AASHTO-LRFD Equations (6.9.4.1-1) and (6.9.4.1-2), symbol l is the length from the last rivet (or bolt) line on a member to first rivet (or bolt) line on a chord measured along the centerline of the member; K is effective length factor = 0.65; As is average effective cross-section area defined by Whitmore’s method. 4. Limit of free edge to thickness ratio of gusset p plate — When the free edge length to thickness ratio of a gusset plate Lg /t > 1.6 E/Fy , the compression stress of a gusset plate shall be less than 0.8 Fy ; otherwise the plate shall be stiffened. The free edge length to thickness ratio of a gusset plate shall satisfy the following limit specified in Article 6.14.2.8 of AASHTO-LRFD [1]. s Lg E ≤ 2.06 (16.2) t Fy When the free edge is stiffened, the following requirements shall be satisfied: • The stiffener plus a width of 10t of gusset plate shall have an l/r ratio less than or equal to 40. 1999 by CRC Press LLC
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FIGURE 16.11: Evaluation procedure for nominal flexural strength of I-shaped steel members.
• The stiffener shall have an l/r ratio less than or equal to 40 between fasteners. • The stiffener moment of inertia shall satisfy [38]: ( Is ≥
q 1.83t 4 (b/t)2 − 144 9.2t 4
(16.3)
where Is = the moment of inertia of the stiffener about its own centroid b
= the width of the gusset plate perpendicular to the edge
t
= the thickness of the gusset plate
5. In-plane moment strength of gusset plate (strong axis) — The nominal moment strength of a gusset plate shall be calculated by the following equation in Article 6.14.2.8 of AASHTO-LRFD [1]: Mn = SFy
(16.4)
where S = elastic section modulus about the strong axis 6. In-plane shear strength for a gusset plate — The nominal shear strength of a gusset plate shall be calculated by the following equations: 1999 by CRC Press LLC
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FIGURE 16.12: Evaluation procedure for nominal shear strength of steel members.
Based on gross section: Vn = smaller
0.4Fy Agv 0.6Fy Agv
for flexural shear for uniform shear
(16.5)
0.4Fu Anv 0.6Fu Anv
for flexural shear for uniform shear
(16.6)
Based on net section: Vn = smaller
where Agv = gross area subject to shear Anv = net area subject to shear Fu = minimum tensile strength of the gusset plate 7. Initial yielding of gusset plate in combined in-plane moment, shear, and axial load — The initial yielding strength of a gusset plate subjected to a combined in-plane moment, shear, and axial load shall be determined by the following equations: Pu Mu + ≤1 Mn Py 1999 by CRC Press LLC
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(16.7)
FIGURE 16.13: Evaluation procedure for steel beam-columns.
or
Vu Vn
2 +
Pu Py
2 ≤1
(16.8)
where Vu = factored shear Mu = factored moment Pu = factored axial load Mn = nominal moment strength determined by Equation 16.4 Vn = nominal shear strength determined by Equation 16.5 Py = yield axial strength (Ag Fy ) Ag = gross section area of gusset plate 8. Full yielding of gusset plate in combined in-plane moment, shear, and axial load — Full yielding strength for a gusset plate subjected to combined in-plane moment, shear, and axial load has the form [6]: 1999 by CRC Press LLC
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Mu + Mp
Pu Py
2
+ 1−
4
Vu Vp
Pu Py
2 = 1
(16.9)
where Mp = plastic moment of pure bending (ZFy ) Vp = shear capacity of gusset plate (0.6Ag Fy ) Z = plastic section modulus 9. Block shear capacity — The block shear capacity shall be calculated according to Article 6.13.4 of AASHTO-LRFD [1]. 10. Out-of-plane moment and shear consideration — Moment will be resolved into a couple acting on the near and far side gusset plates. This will result in tension or compression on the respective plates. This force will produce weak axis bending of the gusset plate. Connections Splices
The splice section shall be evaluated for axial tension, flexure, and combined axial and flexural loading cases according to AISC-LRFD [4]. The member splice capacity shall be equal to or greater than the capacity of the smaller of the two members being spliced. Eyebars
The tensile capacity of the eyebars shall be calculated according to Article D3 of AISC-LRFD [4]. Anchor Bolts (Rods) and Anchorage Assemblies 1. Anchorage assemblies for nonrocking mechanisms shall be anchored with sufficient capacity to develop the lesser of the seismic force demand and plastic strength of the columns. Anchorage assemblies may be designed for rocking mechanisms where yield is permitted — at which point rocking commences. Shear keys shall be provided to prevent excess lateral movement. The nominal shear strength of pipe guided shear keys shall be calculated by:
Rn = 0.6Fy Ap
(16.10)
where Ap = cross-section area of pipe 2. Evaluation of anchorage assemblies shall be based on reinforced concrete structure behavior with bonded or unbonded anchor rods under combined axial load and bending moment. All anchor rods outside of the compressive region may be taken to full minimum tensile strength. 3. The nominal strength of anchor bolts (rods) for shear, tension, and combined shear and tension shall be calculated according to Article 6.13.2 of AASHTO-LRFD [1]. 4. Embedment length of anchor rods shall be such that p a ductile failure occurs. Concrete failure surfaces shall be based on a shear stress of 2 fc0 and account for edge distances and overlapping shear zones. In no case should edge distances or embedments be less than those shown in Table 8-26 of the AISC-LRFD Manual [3]. New anchor rods shall be threaded to assure development. 1999 by CRC Press LLC
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Rivets and Holes
1. The bearing capacity on rivet holes shall be calculated according to Article 6.13.2.9 of AASHTO-LRFD [1]. 2. Nominal shear strength of a rivet shall be calculated by the following formula: Rn = 0.75βFu Ar Ns
(16.11)
where β = 0.8, reduction factor for connections with more than two rivets and to account for deformation of connected material which causes nonuniform rivet shear force (see Article C6.13.2.7 of AASHTO-LRFD [1]) Fu = minimum tensile strength of the rivet Ar = the nominal area of the rivet (before driving) Ns = number of shear planes per rivet It should be pointed out that the 0.75 factor is the ratio of the shear strength τu to the tensile strength Fu of a rivet. The research work by Kulak et al. [31] found that this ratio is independent of the rivet grade, installation procedure, diameter, and grip length and is about 0.75. 3. Tension capacity of a rivet shall be calculated by the following formula: Tn = Ar Fu
(16.12)
4. Tensile capacity of a rivet subjected to combined tension and shear shall be calculated by the following formula: s Tn = Ar Fu 1 −
Vu Rn
(16.13)
where Vu = factored shear force Rn = nominal shear strength of a rivet determined by Equation 16.11
Bolts and Holes
1. The bearing capacity on bolt holes shall be calculated according to Article 6.13.2.9 of AASHTO-LRFD [1]. 2. The nominal strength of a bolt for shear, tension, and combined shear and tension shall be calculated according to Article 6.13.2 of AASHTO-LRFD [1].
Prying Action
Additional tension forces resulting from prying action must be accounted for in determining applied loads on rivets or bolts. The connected elements (primarily angles) must also be checked for adequate flexural strength. Prying action forces shall be determined from the equations presented in AISC-LRFD Manual Volume 2, Part 11 [3].
1999 by CRC Press LLC
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16.7.4
Nominal Strength of Concrete Structures
Nominal Moment Strength
The nominal moment strength Mn shall be calculated by considering combined biaxial bending and axial loads. It is defined as: My (16.14) Mn = smaller Mεc where My = moment corresponding to first steel yield Mεc = moment at which compressive strain of concrete at extreme fiber equal to 0.003 Nominal Shear Strength
The nominal shear strength Vn shall be calculated by the following equations [12, 13]. Vn or Vn
where Ag = As = Vt = D0 = Pu = d = s = fyt = µ1 =
16.7.5
Vc
=
νc
=
Factor 1
=
ρ 00
=
Vs
=
= =
Vc + Vs Vc + Vt
0.8νc Ag p p Pu 2 1 + 2,000A fc0 ≤ 3 fc0 g p larger p Factor 1 × 1 + Pu fc0 ≤ 4 fc0 2,000Ag ρ 00 fyt + 3.67 − µ1 ≤ 3.0 150 volume of transverse reinforcement volume of confined core A f d/s for rectangular sections ν yt A s fyt D 0 for circular sections 2s
(16.15a) (16.15b)
(16.16) (16.17)
(16.18)
(16.19)
gross section area of concrete member cross-sectional area of transverse reinforcement within space s shear strength carried by truss mechanism hoop or spiral diameter factored axial load associated with design shear Vu and Pu /Ag is in psi effective depth of section space of transverse reinforcement probable yield strength of transverse steel (psi) ductility demand ratio (1.0 will be used)
Structural Deformation Capacity
Steel Structures
Displacement capacity shall be evaluated by considering both material and geometrical nonlinearity. Proper boundary conditions for various structures shall be carefully adjusted. The ultimate 1999 by CRC Press LLC
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available displacement capacity is defined as the displacement corresponding to a load that drops a maximum of 20% from the peak load. Reinforced Concrete Structures
Displacement capacity shall be evaluated using stand-alone push-over analysis models. Both the geometrical and material nonlinearities, as well as the foundation (nonlinear soil springs) shall be taken into account. The ultimate available displacement capacity is defined as the displacement corresponding to a maximum of 20% load reduction from the peak load, or to a specified stress-strain failure limit (surface), whichever occurs first. The following parameters shall be used to define stress-strain failure limit (surface): comp
=
σcten
= = = =
σc
comp
εc
σs εs where comp σc σcten fc0 σs εs comp εc
= = = = = =
16.7.6
0.85fc0
p fr = 5 fc0 0.003 Fu 0.12
maximum concrete stress under uniaxial compression maximum concrete stress under uniaxial tension specified compressive concrete strength maximum steel stress under uniaxial tension maximum steel strain under uniaxial tension maximum concrete strain under uniaxial compression
Seismic Response Modification Devices
General
The SRMDs include the energy dissipation and seismic isolation devices. The basic purpose of energy dissipation devices is to increase the effective damping of the structure by adding dampers to the structure thereby reducing forces, deflections, and impact effects. The basic purpose of isolation devices is to change the fundamental mode of vibration so that the structure is subjected to lower earthquake forces. However, the reduction in force may be accompanied by an increase in displacement demand that shall be accommodated within the isolation system and any adjacent structures. Determination of SRMDs Properties
The properties of SRMDs shall be determined by the specified testing program. References are made to AASHTO-Guide [2], Caltrans [18], and JMC [30]. The following items shall be addressed rigorously in the testing specification: • • • •
Scales of specimens; at least two full-scale tests are required Loading (including lateral and vertical) history and rate Durability — design life Expected levels of strength and stiffness deterioration
1999 by CRC Press LLC
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16.8
Performance Acceptance Criteria
16.8.1
General
To achieve the performance objectives stated in Section 16.3, the various structural components shall satisfy the acceptable demand/capacity ratios, DCaccept , specified in this section. The general design format is given by the formula: Demand (16.20) ≤ DCaccept Capacity where demand, in terms of various factored forces (moment, shear, axial force, etc.), and deformations (displacement, rotation, etc.) shall be obtained by the nonlinear inelastic dynamic time history analysis – Level I defined in Section 16.6; and capacity, in terms of factored strength and deformations, shall be obtained according to the provisions set forth in Section 16.7. For members subjected to combined loadings, the definition of force D/C ratio:] D/C ratios is given in the Appendix.
16.8.2
Structural Component Classifications
Structural components are classified into two categories: critical and other. It is the aim that other components may be permitted to function as “fuses” so that the critical components of the bridge system can be protected during FEE and SEE. As an example, Table 16.1 shows structural component classifications and their definition for SFOBB West Span components. TABLE 16.1
Structural Component Classification
Component classification
Critical
Other
Definition
Example (SFOBB West Spans)
Components on a critical path that carry bridge gravity load directly. The loss of capacity of these components would have serious consequences on the structural integrity of the bridge
Suspension cables Continuous trusses Floor beams and stringers Tower legs Central anchorage A-Frame Piers W-1 and W2 Bents A and B Caisson foundations Anchorage housings Cable bents
All components other than critical
All other components
Note: Structural components include members and connections.
16.8.3
Steel Structures
General Design Procedure
Seismic design of steel members shall be in accordance with the procedure shown in Figure 16.14. Seismic retrofit design of steel members shall be in accordance with the procedure shown in Figure 16.15. Connections
Connections shall be evaluated over the length of the seismic event. For connecting members with force D/C ratios larger than one, 25% greater than the nominal capacities of the connecting members shall be used for connection design. 1999 by CRC Press LLC
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FIGURE 16.14: Steel member seismic design procedure.
General Limiting Slenderness Parameters and Width-Thickness Ratios
For all steel members regardless of their force D/C ratios, slenderness parameters λc for axial load dominant members, and λb for flexural dominant members shall not exceed the limiting values (0.9λcr or 0.9λbr for critical, λcr or λbr for other) shown in Table 16.2. Acceptable Force D/C Ratios and Limiting Values
Acceptable force D/C ratios, DCaccept and associated limiting slenderness parameters and width-thickness ratios for various members are specified in Table 16.2. For all members with D/C ratios larger than one, slenderness parameters and width- thickness ratios shall not exceed the limiting values specified in Table 16.2. For existing steel members with D/C ratios less than one, width-thickness ratios may exceed λr specified in Table 16.3 and AISC-LRFD [4]. The following symbols are used in Table 16.2: Mn = nominal moment strength of a member determined by Section 16.7 Pn = nominal axial strength of a member determined by Section 16.7 λ = width-thickness (b/t or h/tw ) ratio of compressive elements p λc = (KL/rπ) Fy /E, slenderness parameter of axial load dominant members λb = L/ry , slenderness parameter of flexural moment dominant members 1999 by CRC Press LLC
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FIGURE 16.15: Steel member seismic retrofit design procedure.
TABLE 16.2 Acceptable Force Demand/Capacity Ratios and Limiting Slenderness Parameters and Width/Thickness Ratios Limiting ratios Slenderness parameter (λc and λb )
Width/thickness λ (b/t or h/tw )
force D/C ratio D/Caccept
Axial load dominant Pu /Pn ≥ Mu /Mn Flexural moment dominant Mu /Mn > Pu /Pn
0.9 λcr λcpr λcp 0.9 λbr λbpr λbp
λr λpr λp λr λpr λp
DCr = 1.0 1.0 ∼ 1.2 DCp = 1.2 DCr = 1.0 1.2 ∼ 1.5 DCp = 1.5
Axial load dominant Pu /Pn ≥ Mu /Mn Flexural moment dominant Mu /Mn > Pu /Pn
λcr λcpr λcp λbr λbpr λbp
λr λpr
DCr = 1.0 1.0 ∼ 2.0 DCp = 2 DCr = 1.0 1.0 ∼ 2.5 DCp = 2.5
Member classification
Critical
Other
1999 by CRC Press LLC
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Acceptable
λp−Seismic λr λpr λp−Seismic
TABLE 16.3
Limiting Width-Thickness Ratio
1999 by CRC Press LLC
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λcp = 0.5, limiting column slenderness parameter for 90% of the axial yield load based on AISCLRFD [4] column curve λbp = limiting beam slenderness parameter for plastic moment for seismic design λcr = 1.5, limiting column slenderness parameter for elastic buckling based on AISC-LRFD [4] column curve λbr = limiting beam slenderness parameter for elastic lateral torsional buckling
λbr
=
Mr
=
X1
=
where A = L = J = r = ry = Fyw = Fyf = E = G = Sx = Iy = Cw =
√ 57,000 J A rMr X1 FL
1+
q
for solid rectangular bars and box sections 1 + X2 FL2
for doubly symmetric I-shaped members and channels
FL Sx for I-shaped member Fyf Sx for solid rectangular and box section r π EGJ A 4Cw Sx 2 Fyw ; X2 = ; FL = smaller F Sx 2 Iy GJ yf − Fr
cross-sectional area, in.2 unsupported length of a member torsional constant, in.4 radius of gyration, in. radius of gyration about minor axis, in. yield stress of web, ksi yield stress of flange, ksi modulus of elasticity of steel (29,000 ksi) shear modulus of elasticity of steel (11,200 ksi) section modulus about major axis, in.3 moment of inertia about minor axis, in.4 warping constant, in.6
For doubly symmetric and singly symmetric I-shaped members with compression flange equal to or larger than the tension flange, including hybrid members (strong axis bending): [3,600+2,200(M1 /M2 )] for other members Fy (16.21) λbp = for critical members √300 Fyf
in which = larger moment at end of unbraced length of beam M1 = smaller moment at end of unbraced length of beam M2 (M1 /M2 ) = positive when moments cause reverse curvature and negative for single curvature For solid rectangular bars and symmetric box beam (strong axis bending): ( λbp =
[5,000+3,000(M1 /M2 )] √ Fy 3,750 JA Mp
≥
in which Mp = plastic moment (Zx Fy ) Zx = plastic section modulus about major axis 1999 by CRC Press LLC
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3,000 Fy
for other members for critical members
(16.22)
FIGURE 16.16: Typical cross-sections for steel members (SFOBB west spans). λr , λp , λp−Seismic are limiting width thickness ratios specified by Table 16.3 h DCp −DCaccept i λp + λr − λp for critical members DCp −DCr i h λpr = DC −DC p accept λ for other members p−Seismic + λr − λp−Seismic DCp −DCr For axial load dominant members (Pu /Pn ≥ Mu /Mn ) DCp −DCaccept λcp + 0.9λcr − λcp for critical members DCp −DCr λcpr = λ + λ − λ DCp −DCaccept for other members cp cr cp DCp −DCr For flexural moment dominant members (Mu /Mn > Pu /Pn ) DCp −DCaccept λbp + 0.9λbr − λbp for critical members DCp −DCr λbpr = DCp −DCaccept λ + λ −λ for other members bp br bp DCp −DCr 1999 by CRC Press LLC
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(16.23)
(16.24)
(16.25)
16.8.4
Concrete Structures
General
For all concrete compression members regardless of their force D/C ratios, slenderness parameters KL/r shall not exceed 60. For critical components, force DCaccept = 1.2 and deformation DCaccept = 0.4. For other components, force DCaccept = 2.0 and deformation DCaccept = 0.67. Beam-Column (Bent Cap) Joints
For concrete box girder bridges, the beam-column (bent cap) joints shall be evaluated and designed in accordance with the following guidelines [16, 40]: 1. Effective Superstructure Width — The effective width of a superstructure (box girder) on either side of a column to resist longitudinal seismic moment at bent (support) shall not be taken as larger than the superstructure depth. • The immediately adjacent girder on either side of a column within the effective superstructure width is considered effective. • Additional girders may be considered effective if refined bent-cap torsional analysis indicates that the additional girders can be mobilized. 2. Minimum Bent-Cap Width — Minimum cap width outside the column shall not be less than D/4 (D is column diameter or width in that direction) or 2 ft (0.61 m). 3. Acceptable Joint Shear Stress • For p existing unconfined joints, acceptable principal tensile stress shall be taken as p 3.5 fc0 psi (0.29 fc0 MPa). If the principal tensile stress demand exceeds this value, the joint shear reinforcement specified in (4) shall be provided. p • For new joints, acceptable principal tensile stress shall be taken as 12 fc0 psi (1.0 p fc0 MPa). • For existing and new joints, acceptable principal compressive stress shall be taken as 0.25 fc0 . 4. Joint Shear Reinforcement • Typical flexure and shear reinforcement (see Figures 16.17 and 16.18) in bent caps shall be supplemented in the vicinity of columns to resist joint shear. All joint shear reinforcement shall be well distributed and provided within D/2 from the face of column. • Vertical reinforcement including cap stirrups and added bars shall be 20% of the column reinforcement anchored into the joint. Added bars shall be hooked around main longitudinal cap bars. Transverse reinforcement in the joint region shall consist of hoops with a minimum reinforcement ratio of 0.4 (column steel area)/(embedment length of column bar into the bent cap)2 . • Horizontal reinforcement shall be stitched across the cap in two or more intermediate layers. The reinforcement shall be shaped as hairpins, spaced vertically at not more than 18 in. (457 mm). The hairpins shall be 10% of column reinforcement. Spacing shall be denser outside the column than that used within the column. • Horizontal side face reinforcement shall be 10% of the main cap reinforcement including top and bottom steel. 1999 by CRC Press LLC
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FIGURE 16.17: Example cap joint shear reinforcement — skews 0◦ to 20◦ . • For bent caps skewed greater than 20◦ , the vertical J-bars hooked around longitudinal deck and bent cap steel shall be 8% of the column steel (see Figure 16.18). The J-bars shall be alternatively 24 in. (600 mm) and 30 in. (750 mm) long and placed within a width of the column dimension on either side of the column centerline. • All vertical column bars shall be extended as high as practically possible without interfering with the main cap bars.
16.8.5
Seismic Response Modification Devices
General
Analysis methods specified in Section 16.6 shall apply for determining seismic design forces and displacements on SRMDs. Properties or capacities of SRMDs shall be determined by specified tests. Acceptance Criteria
SRMDs shall be able to perform their intended function and maintain their design parameters for the design life (for example, 40 years) and for an ambient temperature range (for example from 30◦ 1999 by CRC Press LLC
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FIGURE 16.18: Example cap joint shear reinforcement — skews > 20◦ .
to 125◦ F). The devices shall have accessibility for periodic inspections, maintenance, and exchange. In general, the SRMDs shall satisfy at least the following requirements:
• To remain stable and provide increasing resistance with the increasing displacement. Stiffness degradation under repeated cyclic load is unacceptable. • To dissipate energy within the design displacement limits. For example: provisions may be made to limit the maximum total displacement imposed on the device to prevent device displacement failure, or the device shall have a displacement capacity 50% greater than the design displacement. • To withstand or dissipate the heat build-up during reasonable seismic displacement time history. • To survive for the number of cycles of displacement expected under wind excitation during the life of the device and to function at maximum wind force and displacement levels for at least, for example, five hours.
1999 by CRC Press LLC
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Defining Terms
Bridge: A structure that crosses over a river, bay, or other obstruction, permitting the smooth and safe passage of vehicles, trains, and pedestrians. Buckling model interaction: A behavior phenomenon of compression built-up member; that is, interaction between the individual (or local) buckling mode and the global buckling mode. Built-up member: A member made of structural metal elements that are welded, bolted, and/or riveted together. Capacity: factored strength and deformation capacity obtained according to specified provisions. Critical components: Structural components on a critical path that carry bridge gravity load directly. The loss of capacity of these components would have serious consequences on the structural integrity of the bridge. Damage index: A ratio of elastic displacement demand to ultimate displacement. D/C ratio: A ratio of demand to capacity. Demands: In terms of various forces (moment, shear, axial force, etc.) and deformation (displacement, rotation, etc.) obtained by structural analysis. Ductility: A nondimensional factor, i.e., ratio of ultimate deformation to yield deformation. Effective length factor K: A factor that when multiplied by actual length of the end-restrained column gives the length of an equivalent pin-ended column whose elastic buckling load is the same as that of the end-restrained column Functionality evaluation earthquake (FEE): An earthquake that has a mean return period of 300 years with a 40% probability of exceedance during the expected life of the bridge. Latticed member: A member made of metal elements that are connected by lacing bars and batten plates. LRFD (Load and Resistance Factor Design): A method of proportioning structural components (members, connectors, connecting elements, and assemblages) such that no applicable limit state is exceeded when the structure is subjected to all appropriate load combinations. Limit states: Those conditions of a structure at which it ceases to satisfy the provision for which it was designed. No-collapse-based design: Design that is based on survival limit state. The overall design concern is to prevent the bridge from catastrophic collapse and to save lives. Other components: All components other than critical. Performance-based seismic design: Design that is based on bridge performance requirements. The design philosophy is to accept some repairable earthquake damage and to keep bridge functional performance after earthquakes. Safety evaluation earthquake (SEE): An earthquake that has a mean return period in the range of 1000 to 2000 years. Seismic design: Design and analysis considering earthquake loads. Seismic response modification devices (SRMDs): Seismic isolation and energy dissipation devices including isolators, dampers, or isolation/dissipation (I/D) devices. Ultimate deformation: Deformation refers to a loading state at which structural system or a structural member can undergo change without losing significant load-carrying capacity. 1999 by CRC Press LLC
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The ultimate deformation is usually defined as the deformation corresponding to a load that drops a maximum of 20% from the peak load. Yield deformation: Deformation corresponds to the points beyond which the structure starts to respond inelastically.
Acknowledgments First, we gratefully acknowledge the support from Professor Wai-Fah Chen, Purdue University. Without his encouragement, drive, and review, this chapter would not have been done in this timely manner. Much of the material presented in this chapter was taken from the San Francisco – Oakland Bay Bridge West Spans Seismic Retrofit Design Criteria (Criteria) which was developed by Caltrans engineers. Substantial contribution to the Criteria came from the following: Lian Duan, Mark Reno, Martin Pohll, Kevin Harper, Rod Simmons, Susan Hida, Mohamed Akkari, and Brian Sutliff. We would like to acknowledge the careful review of the SFOBB Criteria by Caltrans engineers: Abbas Abghari, Steve Altman, John Fujimoto, Don Fukushima, Richard Heninger, Kevin Keady, John Kung, Mike Keever, Rick Land, Ron Larsen, Brian Maroney, Steve Mitchell, Ramin Rashedi, Jim Roberts, Bob Tanaka, Vinacs Vinayagamoorthy, Ray Wolfe, Ray Zelinski, and Gus Zuniga. The SFOBB Criteria was also reviewed by the Caltrans Peer Review Committee for the Seismic Safety Review of the Toll Bridge Retrofit Designs: Chuck Seim (Chairman), T.Y. Lin International; Professor Frieder Seible, University of California at San Diego; Professor Izzat M. Idriss, University of California at Davis; and Gerard Fox, Structural Consultant in New York. We are thankful for their input. We are also appreciative of the review and suggestions of I-Hong Chen, Purdue University; Professor Ahmad Itani, University of Nevada at Reno; Professor Dennis Mertz, University of Delaware; and Professor Chia-Ming Uang, University of California at San Diego. We express our sincere thanks to Enrico Montevirgen and Jerry Helm for their careful preparation of figures. Finally, we gratefully acknowledge the continuous support of the California Department of Transportation.
References [1] AASHTO. 1994. LRFD Bridge Design Specifications, 1st ed., American Association of State Highway and Transportation Officials, Washington, D.C. [2] AASHTO. 1997. Guide Specifications for Seismic Isolation Design, American Association of State Highway and Transportation Officials, Washington, D.C. [3] AISC. 1994. Manual of Steel Construction — Load and Resistance Factor Design, Vol. 1-2, 2nd ed., American Institute of Steel Construction, Chicago, IL. [4] AISC. 1993. Load and Resistance Factor Design Specification for Structural Steel Buildings, 2nd ed., American Institute of Steel Construction, Chicago, IL. [5] AISC. 1992. Seismic Provisions for Structural Steel Buildings, American Institute of Steel Construction, Chicago, IL. [6] ASCE. 1971. Plastic Design in Steel — A Guide and Commentary, 2nd ed., American Society of Civil Engineers, New York. [7] Astaneh-Asl, A. and Roberts, J. eds. 1996. Seismic Design, Evaluation and Retrofit of Steel Bridges, Proceedings of the Second U.S. Seminar, San Francisco, CA. 1999 by CRC Press LLC
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[8] AWS. 1995. Bridge Welding Code. (ANSI/AASHTO/AWS D1.5-95), American Welding Society, Miami, FL. [9] Bazant , Z. P. and Cedolin, L. 1991. Stability of Structures, Oxford University Press, New York. [10] Caltrans. 1993. Design Criteria for I-880 Replacement, California Department of Transportation, Sacramento, CA. [11] Caltrans. 1994. Design Criteria for SR-14/I-5 Replacement, California Department of Transportation, Sacramento, CA. [12] Caltrans. 1995. Bridge Design Specifications, California Department of Transportation, Sacramento, CA. [13] Caltrans. 1995. Bridge Memo to Designers (20-4), California Department of Transportation, Sacramento, CA. [14] Caltrans. 1996. Guidelines for Generation of Response-Spectrum-Compatible Rock Motion Time History for Application to Caltrans Toll Bridge Seismic Retrofit Projects, Caltrans Seismic Advisory Board, California Department of Transportation, Sacramento, CA. [15] Caltrans. 1996. Guidelines for Performing Site Response Analysis to Develop Seismic Ground Motions for Application to Caltrans Toll Bridge Seismic Retrofit Projects, Caltrans Seismic Advisory Board, California Department of Transportation, Sacramento, CA. [16] Caltrans. 1996. Seismic Design Criteria for retrofit of the West Approach to the San FranciscoOakland Bay Bridge, Prepared by Keever, M., California Department of Transportation, Sacramento, CA. [17] Caltrans. 1997. San Francisco – Oakland Bay Bridge West Spans Seismic Retrofit Design Criteria, Prepared by Reno, M. and Duan, L., edited by Duan, L., California Department of Transportation, Sacramento, CA. [18] Caltrans. 1997. Full Scale Isolation Bearing Testing Document (Draft), Prepared by Mellon, D., California Department of Transportation, Sacramento, CA. [19] Duan, L. and Chen, W.F. 1990. “A Yield Surface Equation for Doubly Symmetrical Section,” Structural Eng., 12(2), 114-119. CRC [20] Duan, L. and Cooper, T. R. 1995. “Displacement Ductility Capacity of Reinforced Concrete Columns,” ACI Concrete Int., 17(11). 61-65. [21] Duan, L. and Reno, M. 1995. “Section Properties of Latticed Members,” Research Report, California Department of Transportation, Sacramento, CA. [22] Duan, L., Reno, M., and Uang, C.M. 1997. “Buckling Model Interaction for Compression Built-up Members,” AISC Eng. J. (in press). [23] ENR. 1997. Seismic Superstar Billion Dollar California Freeway — Cover story: Rising form the Rubble, New Freeway Soars and Swirls Near Quake, Engineering News Records, Jan. 20, McGraw-Hill, New York. [24] FHWA. 1987. Seismic Design and Retrofit Manual for Highway Bridges, Report No. FHWAIP-87-6, Federal Highway Administration, Washington, D.C. [25] FHWA. 1995. Seismic Retrofitting Manual for Highway Bridges, Publication No. FHWA-RD94-052, Federal Highway Administration, Washington, D.C. [26] Goyal, A. and Chopra, A.K. 1989. Earthquake Analysis & Response of Intake-Outlet Towers, EERC Report No. UCB/EERC-89/04, University of California, Berkeley, CA. [27] Housner, G.W. 1990.Competing Against Time, Report to Governor George Deuknejian from The Governor’s Broad of Inquiry on the 1989 Loma Prieta Earthquake, Sacramento, CA. [28] Housner, G.W. 1994. The Continuing Challenge — The Northridge Earthquake of January 17, 1994, Report to Director, California Department of Transportation, Sacramento, CA. [29] Institute of Industrial Science (IIS). 1995. Incede Newsletter, Special Issue, International Center for Disaster-Mitigation Engineering, Institute of Industrial Science, The University of Tokyo, Japan. 1999 by CRC Press LLC
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[30] Japan Ministry of Construction (JMC). 1994. Manual of Menshin Design of Highway Bridges, English Version: EERC, Report 94/10, University of California, Berkeley, CA. [31] Kulak, G.L., Fisher, J.W., and Struik, J.H. 1987. Guide to the Design Criteria for Bolted and Riveted Joints, 2nd ed., John Wiley & Sons, New York. [32] Liew, J.Y.R. 1992. “Advanced Analysis for Frame Design,” Ph.D. Dissertation, Purdue University, West Lafayette, IN. [33] McCormac, J. C. 1989. Structural Steel Design, LRFD Method, Harper & Row, New York. [34] Park, R. and Paulay, T. 1975. Reinforced Concrete Structures, John Wiley & Sons, New York. [35] Priestley, M.J.N., Seible, F., and Calvi, G.M. 1996. Seismic Design and Retrofit of Bridges, John Wiley & Sons, New York. [36] Salmon, C. G. and Johnson, J. E. 1996. Steel Structures: Design and Behavior, Emphasizing Load and Resistance Factor Design, Fourth ed., HarperCollins College Publishers, New York. [37] Uang, C. M. and Kleiser, M. 1997. “Cyclic Testing of Latticed Members for San FranciscoOakland Bay Bridge,” Final Report, Division of Structural Engineering, University of California at San Diego, La Jolla, CA. [38] USS. 1968. Steel Design Manual, Brockenbrough, R.L. and Johnston, B.G., Eds., United States Steel Corporation, ADUSS 27-3400-02, Pittsburgh, PA. [39] Xie, L. L. and Housner, G. W. 1996. The Greater Tangshan Earthquake, Vol. I and IV, California Institute of Technology, Pasadena, CA. [40] Zelinski, R. 1994. Seismic Design Momo Various Topics Preliminary Guidelines, California Department of Transportation, Sacramento, CA.
Further Reading [1] Chen, W.F. and Duan, L. 1998. Handbook of Bridge Engineering, (in press) CRC Press, Boca Raton, FL. [2] Clough, R.W. and Penzien, J. 1993. Dynamics of Structures, 2nd ed., McGraw-Hill, New York. [3] Fukumoto, Y. and Lee, G. C. 1992. Stability and Ductility of Steel Structures under Cyclic Loading, CRC Press, Boca Raton, FL. [4] Gupta, A.K. 1992. Response Spectrum Methods in Seismic Analysis and Design of Structures, CRC Press, Boca Raton, FL.
Appendix A 16.A.1 Section Properties for Latticed Members This section presents practical formulas proposed by Duan and Reno [21] for calculating section properties for latticed members. Concept
It is generally assumed that section properties can be computed based on cross-sections of main components if the lacing bars and battens can assure integral action of the solid main components [33, 36]. To consider actual section integrity, reduction factors βm for moment of inertia, and βt for torsional constant are proposed depending on shear-flow transferring-capacity of lacing bars and connections. For clarity and simplicity, typical latticed members as shown in Figure 16.19 are discussed. 1999 by CRC Press LLC
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FIGURE 16.19: Typical latticed members. Section Properties
1. Cross-sectional area — The contribution of lacing bars is assumed negligible. The crosssectional area of latticed member is only based on main components. A=
X
Ai
(16.26)
where Ai is cross-sectional area of individual component i. 2. Moment of inertia — For lacing bars or battens within web plane (bending about y-y axis in Figure 16.19) Iy−y =
X
I(y−y)i + βm
X
Ai xi2
(16.27)
where Iy−y = moment of inertia of a section about y-y axis considering shear transferring capacity = moment of inertia of individual component i Ii = distance between y-y axis and center of individual component i xi βm = reduction factor for moment of inertia and may be determined by the following formula:
1999 by CRC Press LLC
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For laced member (Figure 16.19a): ( m sin φ × smaller of βm
comp
mlacing Pn
+ Pnten
mlacing nr Ar (0.6Fu ) Fyf Af
=
≤ 1.0
(16.28a)
For battened member (Figure 16.19b)
βm
=
mbatten Ab 0.6Fyw m × smaller of mbatten 2Mp−batten / h mbatten nr Ar (0.6Fu ) Fyf Af
≤ 1.0
(16.28b)
in which φ = the angle between the diagonal lacing bar and the axis perpendicular to the member axis (see Figure 16.19) = cross-sectional area of batten plate Ab = flange area Af Fyf = yield strength of flange = yield strength of web member (battens or lacing bars) Fyw = ultimate strength of rivets Fu m = number of panels between point of maximum moment to point of zero moment to either side (as an approximation, half of member length L/2 may be used) = number of batten planes mbatten = number of lacing planes mlacing = number of rivets of connecting lacing bar and main component at one nr joint h = depth of member in lacing plane = nominal area of rivet Ar Mp−batten = plastic moment of a batten plate about strong axis comp = nominal compressive strength of lacing bar and can be determined by Pn AISC-LRFD [4] column curve = nominal tensile strength of lacing bar and can be determined by AISCPnten LRFD [4] Since the section integrity mainly depends on the shear transference between various components, it is rational to introduce the βm factor in Equation 16.27. As seen in Equations 16.28a and 16.28b, βm is defined as the ratio of the shear capacity transferred by lacing bars/battens and connections to the shear-flow (Fyf Af ) required by the plastic bending moment of a fully integral section. For laced members, the shear transferring capacity is controlled by either lacing bars or connecting rivets, the smaller of the two values should be used in Equation 16.28a. For battened members, the shear transferring capacity is controlled by either pure shear strength of battens (0.6 Fyw Ab ), or flexural strength of battens or connecting rivets, the smaller of the three values should be used in Equation 16.28b. It is important to point out that the limiting value unity for βm implies a fully integral section when shear can be transferred fully by lacings and connections. For lacing bars or battens within flange plane (bending about x-x axis in Figure 16.19).
1999 by CRC Press LLC
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The contribution of lacing bars is assumed negligible and only the main components are considered. X X I(x−x)i + Ai yi2 (16.29) Ix−x = 3. Elastic section modulus S=
I C
(16.30)
where S = elastic section modulus of a section C = distance from elastic neutral axis to extreme fiber 4. Plastic section modulus For lacing bars or battens within flange plane (bending about x-x axis in Figure 16.19) Zx−x =
X
yi∗ A∗i
(16.31)
For lacing bars or battens within web plane (bending about y-y axis in Figure 16.19) Zy−y = βm where Z = xi∗ = yi∗ = A∗i =
X
xi∗ A∗i
(16.32)
plastic section modulus of a section about plastic neutral axis distance between center of gravity of a section A∗i and plastic neutral y-y axis distance between center of gravity of a section A∗i and plastic neutral x-x axis cross-section area above or below plastic neutral axis
It should be pointed out that the plastic neutral axis is generally different from the elastic neutral axis. The plastic neutral axis is defined by equal plastic compression and tension forces for this section. 5. Torsional constant For a box-shaped section J =
4 (Aclose )2 P bi
(16.33)
ti
For an open thin-walled section J =
X bi t 3 i
3
(16.34)
where Aclose = area enclosed within mean dimension for a box = length of a particular segment of the section bi = average thickness of segment bi ti For determination of torsional constant of a latticed member, it is proposed that the lacing bars or batten plates be replaced by reduced equivalent thin-walled plates defined as:
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Aequiv = βt A∗equiv
(16.35)
For laced member (Figure 16.19a) A∗equiv
=
3.12Ad sin φ cos2 φ
(16.36a)
For battened member (Figure 16.19b) A∗equiv
=
tequiv =
74.88 2ah Ib
+
(16.36b)
a2 If
Aequiv h
(16.37)
where a = distance between two battens along member axis Aequiv = cross-section area of a thin-walled plate equivalent to lacing bars considering shear transferring capacity A∗equiv = cross-section area of a thin-walled plate equivalent to lacing bars or battens assuming full section integrity tequiv = thickness of equivalent thin-walled plate = cross-sectional area of all diagonal lacings in one panel Ad = moment of inertia of a batten plate Ib If = moment of inertia of a side solid flange about the weak axis = reduction factor for torsion constant may be determined by the following βt formula: For laced member (Figure 16.19a) cos φ × smaller of βt
=
comp
Pn + Pnten nr Ar (0.6Fu )
0.6Fyw A∗equiv
≤ 1.0
(16.38a)
For battened member (Figure 16.19b)
βt
=
Ab 0.6Fyw h/a 2Mp−batten /a smaller of nr Ar (0.6Fu ) h/a 0.6Fyw A∗equiv
≤ 1.0
(16.38b)
The torsional integrity is from lacings and battens. A reduction factor βt , similar to that used for the moment of inertia, is introduced to consider section integrity when the lacing is weaker than the solid plate side of the section. βt factor is defined as the ratio of the shear capacity transferred by lacing bars and connections to the shear-flow (0.6Fyw A∗equiv ) required by the equivalent thin-walled plate. It is seen that the limiting value of unity for βt implies a fully integral section when shear in the equivalent thin-walled plate can be transferred fully by lacings and connections. 1999 by CRC Press LLC
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Based on the equal lateral stiffness principle, an equivalent thin-walled plate for a lacing plane, Equation 16.36a and 16.36b can be obtained by considering E/G = 2.6 for steel material and shape factor for shear n = 1.2 for a rectangular section. 6. Warping constant For a box-shaped section Cw ≈ 0
(16.39)
For an I-shaped section Cw =
If h2 2
(16.40)
where If = moment of inertia of one solid flange about the weak axis (perpendicular to the flange) of the cross-section h = distance between center of gravity of two flanges
16.A.2 Buckling Mode Interaction For Compression Built-up members An important phenomenon of the behavior of these compressive built-up members is the interaction of buckling modes [9]; i.e., interaction between the individual (or local) component buckling (Figure 16.20a) and the global member buckling (Figure 16.20b). This section presents the practical approach proposed by Duan, Reno, and Uang [22] that may be used to determine the effects of interaction of buckling modes for capacity assessment of existing built-up members. Buckling Mode Interaction Factor
To consider buckling mode interaction between the individual components and the global member, it is proposed that the usual effective length factor K of a built-up member be multiplied by a buckling mode interaction factor, γLG , that is, KLG = γLG K
(16.41)
where K = usual effective length factor of a built-up member KLG = effective length factor considering buckling model interaction Limiting Effective Slenderness Ratios
For practical design, the following two limiting effective slenderness ratios are suggested for consideration of buckling mode interaction: p (16.42) Ka a/rf = 1.1 E/F Ka a/rf = 0.75(KL/r) (16.43) where (Ka a/rf ) is the largest effective slenderness ratio of individual components between connectors and (KL/r) is governing effective slenderness ratio of a built-up member. The first limit Equation 16.42 is based on the argument that if an individual component is very short, the failure mode of the component would be material yielding (say 95% of the yield load), not member buckling. This implies that no interaction of buckling modes occurs when an individual component is very short. 1999 by CRC Press LLC
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FIGURE 16.20: Buckling modes of built-up members.
The second limit Equation 16.43 is set forth by the current design specifications. Both the AISCLRFD [4] and AASHTO-LRFD [1] imply that when the effective slenderness ratio (Ka a/ri ) of each individual component between the connectors does not exceed 75% of the governing effective slenderness ratio of the built-up member, no strength reduction due to interaction of buckling modes needs to be considered. The study reported by Duan, Reno, and Uang [22] has justified that the rule of (Ka a/ri ) < 0.75(KL/r) is consistent with the theory. Analytical Equation
The buckling mode interaction factor γLG is defined as: s s PG π 2 EI = γLG = Pcr (KL)2 Pcr where Pcr = elastic buckling load considering buckling mode interaction PG = elastic buckling load without considering buckling mode interaction L = unsupported member length I = moment of inertia of a built-up member γLG can be computed by the following equation: v u 1 + α2 u γLG = u u α2 u1 + u (δo /a)2 (Ka a/rf )2 u 1+ " #3 t (Ka a/r )2 2 1−
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f (γLG KL/r)2
(16.44)
(16.45)
where (δo /a) = imperfection (out-of-straightness) parameter of individual component (see Figure 16.21) α = separation as defined as: h α= 2
s
Af h = If 2rf
(16.46)
in which If = moment inertia of one side individual components (see Figure 16.21) Af = cross-section area of one side individual components (see Figure 16.21) h = depth of latticed member, distance between center of gravity of two flanges in lacing plane (see Figure 16.21) r = radius of gyration of a built-in member rf = radius of gyration of individual component
FIGURE 16.21: Typical cross-section and local components. For widely separated built-up members with α ≥ 2, the buckling mode interaction factor γLG can be accurately estimated by the following equation on the conservative side:
γLG
v u u (δo /a)2 (Ka a/rj )2 =u i t1 + h (K a/rj )2 3 2 1 − (γ aKL/r) 2
(16.47)
LG
Graphical Solution
Although γLG can be obtained by solving Equation 16.45, an iteration procedure must be used. For design purposes, solutions in chart forms are more desirable. Figures 16.22 to 16.24 provide engineers with alternative graphic solutions for widely separated built-up members with α ≥ 2. In these figures, the out-of-straightness ratios (δo /a) considered are 1/500, 1/1000, and 1/1500, and the effective slenderness ratios (KL/r) considered are 20, 40, 60, 100, and 140. In all these figures, the top line represents KL/r = 20 and the bottom line represents KL/r = 140. 1999 by CRC Press LLC
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FIGURE 16.22: Buckling mode interaction factor γLG for δ0 /a = 1/500.
FIGURE 16.23: Buckling mode interaction factor γLG for δ0 /a = 1/1000.
16.A.3 Acceptable Force D/C Ratios and Limiting Values Since it is uneconomical and impossible to design bridges to withstand seismic forces elastically, the non-linear inelastic responses of the bridges are expected. The performance-based criteria accepts certain seismic damage in some other components so that the critical components and the bridges will be kept essentially elastic and functional after the SEE and FEE. This section presents the concept of acceptable force D/C ratios, limiting member slenderness parameters and limiting width-thickness ratios, as well as expected ductility. 1999 by CRC Press LLC
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FIGURE 16.24: Buckling mode interaction factor γLG for δ0 /a = 1/1500.
FIGURE 16.25: Definition of force D/C ratios for combined loadings.
Definition of Force Demand/Capacity (D/C) Ratios
For members subjected to an individual load, force demand is defined as a factored individual force, such as factored moment, shear, axial force, etc. which shall be obtained by the nonlinear dynamic time history analysis – Level I specified in Section 16.6, and capacity is defined according to the provisions in Section 16.7. For members subjected to combined loads, force D/C ratio is based on the force interaction. For example, for a member subjected to combined axial load and bending moment (Figure 16.25), force demand D is defined as the distance from the origin point O(0, 0) to the factored force point d(Pu , Mu ), and capacity C is defined as the distance from the origin point O(0, 0) to the point c(P ∗ , M ∗ ) on the specified interaction surface or curve (failure surface or curve). 1999 by CRC Press LLC
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Ductility and Load-Deformation Curves
“Ductility” is usually defined as a nondimensional factor, i.e., ratio of ultimate deformation to yield deformation [20, 34]. It is normally expressed by two forms: 1. curvature ductility (µφ = φu /φy ) 2. displacement ductility (µ1 = 1u /1y ) Representing section flexural behavior, curvature ductility is dependent on the section shape and material properties. It is based on the moment-curvature curve. Indicating structural system or member behavior, displacement ductility is related to both the structural configuration and its section behavior. It is based on the load-displacement curve. A typical load-deformation curve including both the ascending and descending branches is shown in Figure 16.26. The yield deformation (1y or φy ) corresponds to a loading state beyond which the structure starts to respond inelastically. The ultimate deformation (1u or φu ) refers to the a loading state at which a structural system or a structural member can undergo without losing significant load-carrying capacity. It is proposed that the ultimate deformation (curvature or displacement) is defined as the deformation corresponding to a load that drops a maximum of 20% from the peak load.
FIGURE 16.26: Load-deformation curves.
Force D/C Ratios and Ductility
The following discussion will give engineers a direct measure of seismic damage of structural components during an earthquake. Figure 16.27 shows a typical load-response curve for a single degree of freedom system. Displacement ductility is: µ1 =
1u 1y
(16.48)
A new term, Damage Index, is defined herein as the ratio of elastic displacement demand to ultimate 1999 by CRC Press LLC
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FIGURE 16.27: Response of a single degree of freedom of system. displacement: D1 =
1ed 1u
(16.49)
When damage index D1 < 1/µ1 (1ed < 1y ), it implies that no damage occurs and the structure responds elastically; 1/µ1 < D1 < 1.0 indicates certain damages occur and the structure responds inelastically; D1 > 1.0, however, means that a structural system collapses. Based on the equal displacement principle, the following relationship is obtained as: 1ed Force Demand = µ1 D1 = Force Capacity 1y
(16.50)
It is seen from Equation 16.50 that the force D/C ratio is related to both the structural characters in terms of ductility µ1 and the degree of damage in terms of damage index D1 . Table 16.4 shows detailed data for this relationship. TABLE 16.4 Index Force D/C ratio 1.0 1.2 1.5 2.0 2.5
Force D/C Ratio and Damage
Damage index D1 No damage 0.4 0.5 0.67 0.83
Expected system displacement ductility µ1 No requirement 3.0 3.0 3.0 3.0
General Limiting Values
To ensure the important bridges have ductile load paths, general limiting slenderness parameters and width-thickness ratios are specified in Sections 16.8.3 and 16.8.4. For steel members, λcr is the limiting member slenderness parameter for column elastic buckling and is taken as 1.5 from AISC-LRFD [4] and λbr is the limiting member slenderness parameter for 1999 by CRC Press LLC
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beam elastic torsional buckling and is calculated by AISC-LRFD [4]. For a critical member, a more strict requirement, 90% of those elastic buckling limits is proposed. Regardless of the force D/C ratios, all steel members must not exceed these limiting values. For existing steel members with D/C ratios less than one, this limit may be relaxed. For concrete members, the general limiting parameter KL/r = 60 is proposed. Acceptable Force D/C Ratios DCaccept
Acceptable force D/C ratios (DCaccept ) depends on both the structural characteristics in terms of ductility µ1 and the degree of damage to the structure that can be accepted by practicing engineers in terms of damage index D1 . To ensure a steel member has enough inelastic deformation capacity during an earthquake, it is necessary to limit both the member slenderness parameters and the section width-thickness ratios within the specified ranges so that the acceptable D/C ratios and the energy dissipation can be achieved. Upper Bound Acceptable D/C Ratio DCp 1. For other members, the large acceptable force D/C ratios (DCp = 2 to 2.5) are proposed in Table 16.2. This implies that the damage index equals 0.67 ∼ 0.83 and more damage will occur at other members and large member ductility will be expected. To achieve this, • the limiting width-thickness ratio was taken as λp−Seismic from AISC-Seismic Provisions [5], which can provide flexural ductility 8 to 10. • the limiting slenderness parameters were taken as λbp for flexural moment dominant members from AISC-LRFD [4], which can provide flexural ductility 8 to 10. 2. For critical members, small acceptable force D/C ratios (DCp = 1.2 to 1.5) are proposed in Table 16.2, as the design purpose is to keep critical members essentially elastic and allow little damage (damage index equals to 0.4 ∼ 0.5). Thus, small member ductility is expected. To achieve this, • the limiting width-thickness ratio was taken as λp from AISC-LRFD [5], which can provide flexural ductility at least 4. • the limiting slenderness parameters were taken as λbp for flexural moment dominant members from AISC-LRFD [4], which can provide flexural ductility at least 4. 3. For axial load dominant members, the limiting slenderness parameters were taken as λcp = 0.5, corresponding to 90% of the axial yield load by the AISC-LRFD [4] column curve. This limit will provide the potential for axial load dominated members to develop expected inelastic deformation. Lower Bound Acceptable D/C Ratio DCr
The lower bound acceptable force D/C ratio DCrc = 1 is proposed in Table 16.2. For DCaccept =1, it is unnecessary to enforce more strict limiting values for members and sections. Therefore, the limiting slenderness parameters for elastic global buckling specified in Table 16.2 and the limiting width-thickness ratios specified in Table 16.3 for elastic local buckling are proposed. Acceptable D/C Ratios Between Upper and Lower Bounds DCr < DCaccept < DCp
When acceptable force D/C ratios are between the upper and the lower bounds, DCr < DCaccept < DCp , a linear interpolation as shown in Figure 16.28 is proposed to determine the limiting slenderness 1999 by CRC Press LLC
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parameters and width-thickness ratios. The following formulas can be used: h DCp −DCaccept i λp + λr − λp for critical members DCp −DCr i h λpr = DCp −DCaccept λ for other members p−Seismic + λr − λp−Seismic DCp −DCr
(16.51)
FIGURE 16.28: Acceptable D/C ratios and limiting slenderness parameters and width-thickness ratios. For axial load dominant members (Pu /Pn ≥ Mu /Mn ) DCp −DCaccept λcp + 0.9λcr − λcp for critical members DCp −DCr λcpr = λ + λ − λ DCp −DCaccept for other members cp cr cp DCp −DCr For flexural moment dominant members (Mu /Mn > Pu /Pn ) DCp −DCaccept λbp + 0.9λbr − λbp for critical members DCp −DCr λbpr = λ + λ − λ DCp −DCaccept for other members bp br bp DCp −DCr
(16.52)
(16.53)
where λr , λp , λp−Seismic are limiting width-thickness ratios specified by Table 16.3. Limiting Width-Thickness Ratios
The basic limiting width-thickness ratios λr , λp , λp−Seismic specified in Table 16.3 are proposed for important bridges.
16.A.4 Inelastic Analysis Considerations This section presents concepts and formulas of reduced material and section properties and yield surface for steel members for possible use in inelastic analysis. 1999 by CRC Press LLC
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Stiffness Reduction
Concepts of stiffness reduction — tangent modulus — have been used to calculate inelastic effective length factors by AISC-LRFD [4], and to account for both the effects of residual stresses and geometrical imperfection by Liew [32]. To consider inelasticity of a material, the tangent modulus of the material Et may be used in analysis. For practical application, stiffness reduction factor (SRF) = (Et /E) can be taken as the ratio of the inelastic to elastic buckling load of the column: SRF =
(Pcr )inelastic Et ≈ E (Pcr )elastic
(16.54)
where (Pcr )inelastic and (Pcr )elastic can be calculated by AISC-LRFD [4] column equations: (Pcr )inelastic
=
(Pcr )elastic
=
0.658λc As Fy 0.877 As Fy λ2c 2
(16.55) (16.56)
in which As is the gross section area of the member and λc is the slenderness parameter. By utilizing the calculated axial compression load P , the tangent modulus Et can be obtained as: E for P /Py ≤ 0.39 (16.57) Et = −3(P /Py ) ln(P /Py ) for P /Py > 0.39 Reduced Section Properties
In an initial structural analysis, the section properties based on a fully integral section of a latticed member may be used. If section forces obtained from this initial analysis are lower than the section strength controlled by the shear-flow transferring capacity, assumed fully integral section properties used in the analysis are rational. Otherwise, section properties considering a partially integral section, as discussed in Section 17.A.1, Iequiv and Jequiv , may be used in the further analysis. This concept is similar to “cracked section” analysis for reinforced concrete structures [13]. 1. Moment of inertia — latticed members (a) For lacing bars or battens within web plane (bending about y-y axis in Figure 16.19) The following assumptions are made: • Moment-curvature curve (Figure 16.29) behaves bi-linearly until the section reaches its ultimate moment capacity. • For moments less than Md = βm Mu , the moment at first stiffness degradation, the section can be considered fully integral. • For moments larger than Md and less than Mu , the ultimate moment capacity of the section, the section is considered as a partially integral one; that is, bending stiffness should be based on a reduced moment of inertia defined by Equation 16.27. An equivalent moment of inertia, Iequiv , based on the secant stiffness can be obtained as: Iequiv =
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∗ I Iy−y y−y
∗ βm Iy−y + (1 − βm )Iy−y
(16.58)
FIGURE 16.29: Idealized moment-curvature curve of a latticed member section.
where Iy−y = moment of inertia of a section about the y-y axis considering shear transferring capacity ∗ = moment of inertia of a section about the y-y axis assuming full section integrity Iy−y βm = reduction factor for the moment of inertia and may be determined by Equation 16.28 (b) For lacing bars or battens within flange plane (bending about x-x axis in Figure 16.19) Equation 16.29 is still valid for structural analysis. 2. Effective flexural stiffness — For steel members, when 8y < 8 < 8u , the further reduced section property, effective flexural stiffness (EI )eff may be used in the analysis. Mu Mu ≤ EIequiv ≤ (EI )eff = 8u 8
(16.59)
3. Torsional constant — latticed members Based on similar assumptions and principles used for moment of inertia, an equivalent torsional constant, Jequiv , is derived as follows: Jequiv =
J ∗J βt J + (1 − βt )J ∗
(16.60)
where J = torsional constant of a section considering shear transferring capacity (See Section 17.A.1) J ∗ = torsional constant of a section assuming full section integrity βt = reduction factor for torsional constant may be determined by Equation 16.38 1999 by CRC Press LLC
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Yield Surface Equation for Doubly Symmetrical Sections
The yield or failure surface concept has been conveniently used in inelastic analysis to describe the full plastification of steel sections under action of axial force combined with biaxial bending. A four parameter yield surface equation for doubly symmetrical steel sections (I, thin-walled circular tube, thin-walled box, solid rectangular and circular sections), developed by Duan and Chen [19], is presented in this section for possible use in a nonlinear analysis. The general shape of the yield surface for a doubly symmetrical steel section as shown in Figure 16.30 can be described approximately by the following general equation: My αy Mx αx + = 1.0 (16.61) Mpcx Mpcy where Mpcx and Mpcy are the moment capacities about the respective axes, reduced for the presence
FIGURE 16.30: Typical yield surface for doubly symmetrical sections.
of axial load; they can be obtained by the following formulas: " βx # P Mpcx = Mpx 1 − Py " βy # P Mpcy = Mpy 1 − Py
(16.62)
(16.63)
where P = axial force Mx = bending moment about the x-x principal axis My = bending moment about the y-y principal axis Mpx = plastic moment about x-x principal axis Mpy = plastic moment about y-y principal axis The four parameters αx , αy , βx , and βy are dependent on sectional shapes and area distribution. It is seen that αx and αy represent biaxial loading behavior, while βx and βy describe uniaxial loading behavior. They are listed in Table 16.5: Equation 16.61 represents a smooth and convex surface in the three-dimensional stress-resultant space. It meets all special conditions and is easy to implement in a computer-based structural analysis. 1999 by CRC Press LLC
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TABLE 16.5
Parameters for Doubly Symmetrical Sections αx
αy
βx
βy
Solid rectangular Solid circular I-shape
1.7 + 1.3 (P /Py ) 2.0 2.0
1.7 + 1.3 (P /Py ) 2.0 1.2 + 2 (P /Py )
2.0 2.1 1.3
2.0 2.1 2 + 1.2 (Aw /Af )
Thin-walled box Thin-walled circular
1.7 + 1.5 (P /Py ) 2.0
1.7 + 1.5 (P /Py ) 2.0
2 − 0.5 B ≥ 1.3 1.75
2 − 0.5 B ≥ 1.3 1.75
Section types
Note: B is the ratio of width-to-depth of the box section with respect to the bending axis.
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Duan, L. and Chen, W.F. “Effective Length Factors of Compression Members” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Effective Length Factors of Compression Members 17.1 17.2 17.3 17.4
Introduction Basic Concept Isolated Columns Framed Columns—Alignment Chart Method
Alignment Chart Method • Requirements for Braced Frames • Simplified Equations to Alignment Charts
17.5 Modifications to Alignment Charts
Different Restraining Girder End Conditions • Different Restraining Column End Conditions • Column Restrained by Tapered Rectangular Girders • Unsymmetrical Frames • Effects of Axial Forces in Restraining Members in Braced Frames • Consideration of Partial Column Base Fixity • Inelastic Kfactor
17.6 Framed Columns—Alternative Methods
LeMessurier Method • Lui Method • Remarks
17.7 Unbraced Frames With Leaning Columns Rigid Columns • Leaning Columns • Remarks
17.8 Cross Bracing Systems 17.9 Latticed and Built-Up Members
Lian Duan Division of Structures, California Department of Transportation, Sacramento, CA
W.F. Chen School of Civil Engineering, Purdue University, West Lafayette, IN
17.1
Laced Columns • Columns with Battens • Laced-Battened Columns • Columns with Perforated Cover Plates • Built-Up Members with Bolted and Welded Connectors
17.10Tapered Columns 17.11Crane Columns 17.12Columns in Gable Frames 17.13Summary 17.14Defining Terms References Further Reading .
Introduction
The concept of the effective length factors of columns has been well established and widely used by practicing engineers and plays an important role in compression member design. The most structural design codes and specifications have provisions concerning the effective length factor. The aim of this chapter is to present a state-of-the-art engineering practice of the effective length factor for the design of columns in structures. In the first part of this chapter, the basic concept of the effective length 1999 by CRC Press LLC
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factor is discussed. And then, the design implementation for isolated columns, framed columns, crossing bracing systems, latticed members, tapered columns, crane columns, as well as columns in gable frames is presented. The determination of whether a frame is braced or unbraced is also addressed. Several detailed examples are given to illustrate the determination of effective length factors for different cases of engineering applications.
17.2
Basic Concept
Mathematically, the effective length factor or the elastic K-factor is defined as: s K=
Pe = Pcr
s
π 2 EI L2 Pcr
(17.1)
where Pe is the Euler load, the elastic buckling load of a pin-ended column; Pcr is the elastic buckling load of an end-restrained framed column; E is the modulus of elasticity; I is the moment of inertia in the flexural buckling plane; and L is the unsupported length of column. Physically, the K-factor is a factor that when multiplied by actual length of the end-restrained column (Figure 17.1a) gives the length of an equivalent pin-ended column (Figure 17.1b) whose buckling load is the same as that of the end-restrained column. It follows that effective length, KL,
FIGURE 17.1: Isolated columns. 1999 by CRC Press LLC
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of an end-restrained column is the length between adjacent inflection points of its pure flexural buckling shape. Specifications provide the resistance equations for pin-ended columns, while the resistance of framed columns can be estimated through the K-factor to the pin-ended columns strength equation. Theoretical K-factor is determined from an elastic eigenvalue analysis of the entire structural system, while practical methods for the K-factor are based on an elastic eigenvalue analysis of selected subassemblages. The effective length concept is the only tool currently available for the design of compression members in engineering structures, and it is an essential part of analysis procedures.
17.3
Isolated Columns
From an eigenvalue analysis, the general K-factor equation of an end-restrained column as shown in Figure 17.1 is obtained as: C + RkA L S −(C + S) EI kB L −(C + S) S C + REI (17.2) det =0 3 −(C + S) −(C + S) 2(C + S) − π 2 + Tk L K
EI
where the stability functions C and S are defined as: C
=
(π/K) sin (π/K) − (π/K)2 cos (π/K) 2 − 2 cos (π/K) − (π/K) sin (π/K)
(17.3)
S
=
(π/K)2 − (π/K) sin (π/K) 2 − 2 cos (π/K) − (π/K) sin (π/K)
(17.4)
The largest value of K that satisfies Equation 17.2 gives the elastic buckling load of an end-restrained column. Figure 17.2 [1, 3, 4] summarizes the theoretical K-factors for columns with some idealized end conditions. The recommended K-factors are also shown in Figure 17.2 for practical design applications. Since actual column conditions seldom comply fully with idealized conditions used in buckling analysis, the recommended K-factors are always equal to or greater than their theoretical counterparts.
17.4
Framed Columns—Alignment Chart Method
In theory, the effective length factor K for any column in a framed structure can be determined from a stability analysis of the entire structural analysis—eigenvalue analysis. Methods available for stability analysis include the slope-deflection method [17, 35, 71], three-moment equation method [13], and energy methods [42]. In practice, however, such analysis is not practical, and simple models are often used to determine the effective length factors for framed columns [38, 47, 55, 72]. One such practical procedure that provides an approximate value of the elastic K-factor is the alignment chart method [46]. This procedure has been adopted by the AISC [3, 4], ACI 318-95 [2], and AASHTO [1] specifications, among others. At present, most engineers use the alignment chart method in lieu of an actual stability analysis.
17.4.1
Alignment Chart Method
The structural models employed for determination of K-factor for framed columns in the alignment chart method are shown in Figure 17.3. The assumptions used in these models are [4, 17]: 1999 by CRC Press LLC
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FIGURE 17.2: Theoretical and recommended K-factors for isolated columns with idealized end conditions. 1. 2. 3. 4.
All members have constant cross-section and behave elastically. Axial forces in the girders are negligible. All joints are rigid. For braced frames, the rotations at the near and far ends of the girders are equal in magnitude and opposite in direction (i.e., girders are bent in single curvature). 5. For unbraced frames, the rotations at the near and far ends of the girders are equal in magnitude and direction (i.e., girders are bent in double curvature). √ 6. The stiffness parameters, L P /EI , of all columns are equal. 7. All columns buckle simultaneously. Using the slope-deflection equation method and stability functions, the effective length factor equations of framed columns are obtained as follows. For columns in braced frames: π/K 2 tan(π/2K) GA + GB GA GB 1− + −1=0 (π/K)2 + 4 2 tan(π/K) π/K
(17.5)
For columns in unbraced frames: π/K GA GB (π/K)2 − 362 =0 − 6 (GA + GB ) tan (π/K)
(17.6)
where GA and GB are stiffness ratios of columns and girders at two end joints, A and B, of the column section being considered, respectively. They are defined by: P (Ec Ic /Lc ) (17.7) GA = P A A Eg Ig /Lg 1999 by CRC Press LLC
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FIGURE 17.3: Subassemblage models for K-factors of framed columns.
GB
=
P (Ec Ic /Lc ) PB B Eg Ig /Lg
(17.8)
P where indicates a summation of all members rigidly connected to the joint and lying in the plane in which buckling of column is being considered; subscripts c and g represent columns and girders, respectively. Equations 17.5 and 17.6 can be expressed in the form of alignment charts, as shown in Figure 17.4. It is noted that for columns in braced frames, the range of K is 0.5 ≤ K ≤ 1.0; for columns in unbraced frames, the range is 1.0 ≤ K ≤ ∞. For column ends supported by but not rigidly connected to a footing or foundations, G is theoretically infinity, but, unless actually designed as a true friction free pin, may be taken as 10 for practical design. If the column end is rigidly attached to a properly designed footing, G may be taken as 1.0 [4].
EXAMPLE 17.1:
Given: A two-story steel frame is shown in Figure 17.5. Using the alignment chart, determine the K-factor for the elastic column DE. E = 29,000 ksi (200 GPa) and F y = 36 ksi (248 MPa). Solution
1. For the given frame, section properties are 1999 by CRC Press LLC
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FIGURE 17.4: Alignment charts for effective length factors of framed columns.
Members
Section
Ix in.4 (mm4 × 108 )
L in. (mm)
Ix /L in.3 (mm3 )
AB and GH BC and HI DE EF BE EH CF FI
W 10x22 W10x22 W10x45 W10x45 W18x50 W18x86 W16x40 W16x67
118 (0.49) 118 (0.49) 248 (1.03) 248 (1.03) 800 (3.33) 1530 (6.37) 518 (2.16) 954 (3.97)
180 (4,572) 144 (3,658) 180 (4,572) 144 (3,658) 300 (7,620) 360 (9,144) 300 (7,620) 360 (9,144)
0.656(10,750) 0.819(13,412) 1.378(22,581) 1.722(28,219) 2.667(43,704) 4.250(69,645) 1.727(28,300) 2.650(43,426)
2. Calculate G-factor for column DE: GE
=
GD
=
P 1.378 + 1.722 (Ec Ic /Lc ) = = 0.448 PE 2.667 + 4.250 I /L E g g g E 10 (AISC-LRFD, 1993)
3. From the alignment chart in Figure 17.4b, K = 1.8 is obtained.
17.4.2
Requirements for Braced Frames
In stability design, one of the major decisions engineers have to make is the determination of whether a frame is braced or unbraced. The AISC-LRFD [4] states that a frame is braced when “lateral stability is provided by diagonal bracing, shear walls or equivalent means”. However, there is no specific provision for the “amount of stiffness required to prevent sidesway buckling” in the AISC, 1999 by CRC Press LLC
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FIGURE 17.5: An unbraced two-story frame. AASHTO, and other specifications. In actual structures, a completely braced frame seldom exists. But in practice, some structures can be analyzed as braced frames as long as the lateral stiffness provided by the bracing system is large enough. The following brief discussion may provide engineers with the tools to make engineering decisions regarding the basic requirements for a braced frame. 1. Lateral Stiffness Requirement Galambos [34] presented a simple conservative procedure to evaluate minimum lateral stiffness provided by a bracing system so that the frame is considered braced. P Pn (17.9) Required lateral stiffness, Tk = Lc P where represents the summation of all columns in one story, Pn is the nominal axial compression strength of a column using the effective length factor K = 1, and Lc is the unsupported length of a column. 2. Bracing Size Requirement Galambos [34] applied Equation 17.9 to a diagonal bracing (Figure 17.6) and obtained minimum requirements of diagonal bracing for a braced frame as Ab =
1 + (Lb /Lc )2
3/2 P
(Lb /Lc ) E 2
Pn
(17.10)
where Ab is the cross-sectional area of diagonal bracing and Lb is the span length of the beam. A recent study by Aristizabal-Ochoa [8] indicates that the size of the diagonal bracing required for a totally braced frame is about 4.9 and 5.1% of the column cross-section for 1999 by CRC Press LLC
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FIGURE 17.6: Diagonal cross bracing system. a “rigid frame” and “simple framing”, respectively, and increases with the moment inertia of the column, the beam span, and the beam-to-column span ratio, Lb /Lc .
17.4.3
Simplified Equations to Alignment Charts
1. ACI 318-95 Equations The ACI Building Code [2] recommends the use of alignment charts as the primary design aid for estimating K-factors, following two sets of simplified K-factor equations as an alternative: For braced frames [19]: K K
= 0.7 + 0.05 (GA + GB ) ≤ 1.0 = 0.85 + 0.05Gmin ≤ 1.0
(17.11) (17.12)
The smaller of the above two expressions provides an upper bound to the effective length factor for braced compression members. For unbraced frames [32]: For Gm < 2 K=
20 − Gm p 1 + Gm 20
(17.13)
For Gm ≥ 2 p K = 0.9 1 + Gm For columns hinged at one end 1999 by CRC Press LLC
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(17.14)
K = 2.0 + 0.3G
(17.15)
where Gm is the average of G values at the two ends of the columns. 2. Duan-King-Chen Equations A graphical alignment chart determination of the K-factor is easy to perform, while solving the chart Equations 17.5 and 17.6 always involves iteration. Although the ACI code provides simplified K-factor equations, generally, they may not lead to an economical design [40]. To achieve both accuracy and simplicity for design purposes, the following alternative K-factor equations were proposed by Duan, King and Chen [48]. For braced frames: 1 1 1 − − 5 + 9GA 5 + 9GB 10 + GA GB
(17.16)
1 1 1 − − 1 + 0.2GA 1 + 0.2GB 1 + 0.01GA GB
(17.17)
K =1− For unbraced frames: For K < 2 K =4− For K ≥ 2
K=
2π a √ 0.9 + 0.81 + 4ab
(17.18)
where
a
=
b
=
GA GB +3 GA + GB 36 +6 GA + GB
(17.19) (17.20)
3. French Equations For braced frames: K=
3GA GB + 1.4 (GA + GB ) + 0.64 3GA GB + 2.0 (GA + GB ) + 1.28
(17.21)
For unbraced frames: s K=
1.6GA GB + 4.0 (GA + GB ) + 7.5 GA + GB + 7.5
(17.22)
Equations 17.21 and 17.22 first appeared in the French Design Rules for Steel Structure [31] since 1966, and were later incorporated into the European Recommendation for Steel Construction [28]. They provide a good approximation to the alignment charts [26]. 1999 by CRC Press LLC
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17.5
Modifications to Alignment Charts
In using the alignment charts in Figure 17.4 and Equations 17.5 and 17.6, engineers must always be aware of the assumptions used in the development of these charts. When actual structural conditions differ from these assumptions, unrealistic design may result [4, 43, 53]. The SSRC (Structural Stability Research Council) guide [43] provides methods that enable engineers to make simple modifications of the charts for some special conditions, such as unsymmetrical frames, column base conditions, girder far end conditions, and flexible conditions. A procedure that can be used to account for far ends of restraining columns being hinged or fixed was proposed by Duan and Chen [21, 22]. Consideration of effects of material inelasticity on the K-factor was developed originally by Yura [73] and expanded by Disque [20]. LeMessurier [52] presented an overview of unbraced frames with or without leaning columns. An approximate procedure is also suggested by AISC-LRFD [4]. Special attention should also be paid to calculation of the proper G values [10, 49] when partially restrained (PR) connections are used in frames. Several commonly used modifications are summarized in this section.
17.5.1
Different Restraining Girder End Conditions
When the end conditions of restraining girders are not rigidly jointed to columns, the girder stiffness (Ig /Lg ) used in the calculation of GA and GB in Equations 17.7 and 17.8 should be multiplied by a modification factor, αk , as: P (Ec Ic /Lc ) (17.23) G= P αk Eg Ig /Lg where the modification factor, αk , for braced frames developed by Duan and Lu [25] and for unbraced frames proposed by Kishi, Chen, and Goto [49] are given in Table 17.1 and 17.2. In these tables, RkN and RkF are elastic spring constants at the near and far ends of a restraining girder, respectively. RkN and RkF are the tangent stiffness of a semi-rigid connection at buckling. TABLE 17.1 Modification Factor αk for Braced Frames with Semi-Rigid Connections End conditions of restraining girder Near end Far end Rigid Rigid
Rigid Hinged
Rigid
Semi-rigid
Rigid
Fixed
Semi-rigid
Rigid
Semi-rigid
Hinged
αk
1.0 1.5 6Eg Ig 4Eg Ig 1+ L R / 1+ L R g kF g kF 2.04E I 1/ 1 + L Rg g g kN
3E I 1.5/ 1 + L Rg g g kN
Semi-rigid Semi-rigid
Semi-rigid Fixed
6Eg Ig 1+ L R g kF
/R ∗
4E I 2/ 1 + L Rg g g kN
E I 2 4E I 4Eg Ig 4 1+ L R − Lg g Note: R ∗ = 1 + L Rg g RkN RkF g kN g kF g
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TABLE 17.2 Modification Factor, αk , for Unbraced Frames with Semi-Rigid Connections End conditions of restraining girder Near end Far end Rigid Rigid
Rigid Hinged
Rigid
Semi-rigid
Rigid
Fixed
Semi-rigid
Rigid
αk
1 0.5 2Eg Ig 4Eg Ig 1+ L R / 1+ L R g kF g kF
2/3 4E I 1/ 1 + L Rg g
Semi-rigid
Hinged
Semi-rigid
Semi-rigid
3E I 0.5/ 1 + L Rg g g kN 2Eg Ig 1+ L R /R ∗
Semi-rigid
g kN
g kF
4E I
(2/3)/ 1 + L Rg g g kN
Fixed
E I 2 4E I 4Eg Ig 4 Note:R ∗ = 1 + L Rg g 1+ L R − Lg g RkN RkF g kN g kF g
EXAMPLE 17.2:
Given: A steel frame is shown in Figure 17.5. Using the alignment chart with the necessary modifications, determine the K-factor for elastic column EF . E = 29,000 ksi (200 GPa) and Fy = 36 ksi (248 MPa). Solution 1. Calculate G-factor with modification for column EF . Since the far end of restraining girders are hinged, girder stiffness should be multiplied by 0.5 (see Table 17.2). Using section properties in Example 17.1, we obtain:
GF
P 1.722 (Ec Ic /Lc ) = = P = 0.787 0.5(1.727) + 0.5(2.650) αk Eg Ig /Lg = 0.448
GE
2. From the alignment chart in Figure 17.4b, K = 1.22 is obtained.
17.5.2
Different Restraining Column End Conditions
To consider different far end conditions of restraining columns, the general effective length factor equations for column C2 (Figure 17.3) were derived by Duan and Chen [21, 22, 23]. By assuming that the far ends of columns C1 and C3 are hinged and using the slope-deflection equation approach for the subassemblies shown in Figure 17.3, we obtain the following. 1. For a Braced Frame [21]: " C2
− S2
GAC1 + GBC3 + GAC2 GBC2 + + 2C
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1 GA
+
1 GB
2GBC3 2GAC1 GA + GB
+
C
4 GA GB
=0
2 − GAC1 GBC3 CS
#
(17.24)
where C and S are stability functions as defined by Equations 17.3 and 17.4; GA and GB are defined in Equations 17.7 and 17.8; GAC1 , GAC2 , GBC2 , and GBC3 are stiffness ratios of columns at A-th and B-th ends of the columns being considered, respectively. They are defined as: Eci Ici /Lci GCi = P (Eci Ici /Lci )
(17.25)
P where indicates a summation of all columns rigidly connected to the joint and lying in the plane in which buckling of the column is being considered. Although Equation 17.24 was derived for the special case in which the far ends of both columns C1 and C3 are hinged, this equation is also applicable if adjustment to GCi is made as follows: (1) if the far end of column Ci(C1 or C3) is fixed, then take GCi = 0 (except for GC2 ); (2) if the far end of the column Ci(C1 or C3) is rigidly connected, then take GCi = 0 and GC2 = 1.0. Therefore, Equation 17.24 can be specialized for the following conditions: (a) If the far ends of both columns C1 and C3 are fixed, we have GAC1 = GBC3 = 0 and Equation 17.24 reduces to C − S (GAC2 GBC2 ) + 2C 2
2
1 1 + GA GB
+
4 =0 GA GB
(17.26)
(b) If the far end of column C1 is rigidly connected and the far end of column C3 is fixed, we have GAC2 = 1.0 and GAC1 = GBC3 = 0, and Equation 17.24 reduces to C 2 − S 2 + GBC2 + 2C
1 1 + GA GB
+
4 =0 GA GB
(17.27)
(c) If the far end of column C1 is rigidly connected and the far end of column C3 is hinged, we have GAC1 = 0 and GAC2 = 1.0, and Equation 17.24 reduces to
C2
−
2GBC3 S 2 GBC3 + GBC2 + GA C 1 4 1 + =0 + + 2C GA GB GA GB
(17.28)
(d) If the far end of column C1 is hinged and the far end of column C3 is fixed, we have GBC3 = 0 and Equation 17.24 reduces to
C2
1999 by CRC Press LLC
c
−
2GAC1 S 2 GAC1 + GAC2 GBC2 + GB C 1 4 1 + =0 + + 2C GA GB GA GB
(17.29)
(e) If the far ends of both columns C1 and C3 are rigidly connected (i.e., assumptions used in developing the alignment chart), we have GC2 = 1.0 and GCi = 0, and Equation 17.24 reduces to C − S + 2C 2
2
1 1 + GA GB
+
4 =0 GA GB
(17.30)
which can be rewritten in the form of Equation 17.5. 2. For an Unbraced Frame [22, 23]: a11 det a21 a31
a12 a22 a32
a13 a23 a33
=0
(17.31)
or
a11 a22 a33 + a21 a32 a13 + a31 a23 a12 − a31 a22 a13 − a21 a12 a33 + a11 a23 a32 = 0
(17.32)
where
a11
=
a22
=
a33
=
a12 a21 a31
= = =
a13
=
a23
=
6 S2 − GAC1 GA C 6 S2 C+ − GBC3 G C B 1 π 2 −2 C + S − 2 K GAC2 S GBC2 S a32 = C + S S2 −(C + S) + GAC1 S + C S2 −(C + S) + GBC3 S + C C+
(17.33) (17.34) (17.35) (17.36) (17.37) (17.38) (17.39)
(17.40)
Although Equation 17.31 was derived for the special case in which the far ends of both columns C1 and C3 are hinged, it can be adjusted to account for the following cases: (1) if the far end of column Ci(C1 or C3) is fixed, then take GCi = 0 (except for GC2 ); (2) if the far end of column Ci(C1 or C3) is rigidly connected, then take GCi = 0 and GC2 = 1.0. Therefore, Equation 17.31 can be used for the following conditions: (a) If the far ends of both columns C1 and C3 are fixed, we take GC1 = GC3 = 0, and obtain from Equations 17.33, 17.34, 17.39, and 17.40, 1999 by CRC Press LLC
c
a11
=
a22
=
a13
=
6 GA 6 C+ GB a23 = −(C + S) C+
(17.41) (17.42) (17.43)
(b) If the far end of column C1 is rigidly connected and the far end of column C3 is fixed, we take GAC2 = 1.0 and GAC1 = GBC3 = 0, and obtain from Equations 17.33, 17.34, 17.36, 17.39, and 17.40, a11
=
a22
=
a12 a13
= =
6 GA 6 C+ GB S a23 = −(C + S) C+
(17.44) (17.45) (17.46) (17.47)
(c) If the far end of column C1 is rigidly connected and the far end of column C3 is hinged, we take GAC1 = 0 and GAC2 = 1.0, and obtain from Equations 17.33, 17.36, and 17.39, 6 GA
a11
=
C+
a12 a13
= =
S −(C + S)
(17.48) (17.49) (17.50)
(d) If the far end of column C1 is hinged and the far end of column C3 is fixed, we have GBC3 = 0.0, and obtain from Equations 17.34 and 17.40, a22
=
a23
=
6 GB −(C + S) C+
(17.51) (17.52)
(e) If the far ends of both columns C1 and C3 are rigidly connected (i.e., assumptions used in developing the alignment chart, that is θC = θB and θD = θA ), we take GC2 = 1.0 and GCi = 0, and obtain from Equations 17.33 to 17.40,
1999 by CRC Press LLC
c
a11
=
a22
=
a12 a13
= =
6 GA 6 C+ GB a21 = S a23 = −(C + S) C+
(17.53) (17.54) (17.55) (17.56)
Equation 17.31 is reduced to the form of Equation 17.6. The procedures to obtain the K-factor directly from the alignment charts without resorting to solve Equations 17.24 and 17.31 were also proposed by Duan and Chen [21, 22].
17.5.3
Column Restrained by Tapered Rectangular Girders
A modification factor, αT , was developed by King et al. [48] for those framed columns restrained by tapered rectangular girders with different far end conditions. The following modified G-factor is introduced in connection with the use of alignment charts: P (Ec Ic /Lc ) G= P αT Eg Ig /Lg
(17.57)
where Ig is the moment of inertia of the girder at near end. Both closed-form and approximate solutions for modification factor αT were derived. It is found that the following two-parameter power function can describe the closed-form solutions very well: αT = D (1 − r)β
(17.58)
in which the parameter D is a constant depending on the far end conditions, and β is a function of the far end conditions and tapering factor, α and r, as defined in Figure 17.7.
FIGURE 17.7: Tapered rectangular girders. (From King, W.S., Duan, L., et al., Eng. Struct., 15(5), 369, 1993. With kind permission from Elsevier Science, Ltd, The Boulevard, Langford Lane, Kidlington OX5 IGB, UK.) 1999 by CRC Press LLC
c
For a braced frame:
For an unbraced frame:
1.0 2.0 D= 1.5
rigid far end fixed far end hinged far end
1.0 2/3 D= 0.5
rigid far end fixed far end hinged far end
1. For a linearly tapered rectangular girder (Figure 17.7a) For a braced frame: rigid far end 0.02 + 0.4r 0.75 − 0.1r fixed far end β= 0.75 − 0.1r hinged far end
(17.59)
(17.60)
(17.61)
For an unbraced frame: 0.95 0.70 β= 0.70
rigid far end fixed far end hinged far end
2. For a symmetrically tapered rectangular girder (Figure 17.7b) For a braced frame: rigid far end 3 − 1.7a 2 − 2a β= fixed far end 3 + 2.5a 2 − 5.55a hinged far end 3 − a 2 − 2.7a
(17.62)
(17.63)
For an unbraced frame: 3 + 3.8a 2 − 6.5a β= 3 + 2.3a 2 − 5.45a 3 − 0.3a
rigid far end fixed far end hinged far end
(17.64)
EXAMPLE 17.3:
Given: A one-story frame with a symmetrically tapered rectangular girder is shown in Figure 17.8. Assuming r = 0.5, a = 0.2, and Ig = 2Ic = 2I , determine K-factor for column AB. Solution
1. Using the alignment chart with modification For joint A, since the far end of the girder is rigid, use Equations 17.64 and 17.58, 1999 by CRC Press LLC
c
FIGURE 17.8: A simple frame with rectangular sections. (From King, W.S., Duan, L., et al., Eng. Struct., 15(5), 369, 1993. With kind permission from Elsevier Science, Ltd, The Boulevard, Langford Lane, Kidlington OX5 IGB, UK.)
β αT GA GB
= 3 + 3.8 (0.2)2 − 6.5 (0.2) = 1.852 = (1 − 0.5)1.852 = 0.277 P Ec Ic /Lc EI /L = 3.61 = P = 0.277E(2I )/2L αT Eg Ig /Lg = 1.0 (AISC – LRFD 1993)
From the alignment chart in Figure 17.4b, K = 1.59 is obtained. 2. Using the alignment chart without modification A direct use of Equations 17.7 and 17.8 with an average section (0.75 h) results in:
Ig
=
GA
=
0.753 (2I ) = 0.844I EI /L = 2.37 GB = 1.0 0.844EI /2L
From the alignment chart in Figure 17.4b K = 1.49, or (1.49−1.59)/1.59 = −6 % in error on the less conservative side.
17.5.4
Unsymmetrical Frames
When the column sizes or column loads are not identical, adjustments to the alignment charts are necessary to obtain a correct K-factor. SSRC Guide [43] presents a set of curves as shown in Figure 17.9 for a modification factor, β, originally developed by Chu and Chow [18]. Kadjusted = βKalignment chart 1999 by CRC Press LLC
c
(17.65)
FIGURE 17.9: Chart for the modification factor β in an unsymmetrical frame. If the K-factor of the column under the load λP is desired, further modifications to K are necessary. Denoting K 0 as the effective length factor of the column with Ic0 = αIc subjected to the axial load, P 0 = λP , as shown in Figure 17.9, then we have: r L α 0 (17.66) K = Kadjusted 0 L λ Equation 17.66 can be used to determine K-factors for columns in adjacent stories with different heights, L0 .
17.5.5
Effects of Axial Forces in Restraining Members in Braced Frames
Bridge and Fraser [14] observed that K-factors of a column in a braced frame may be greater than unity due to “negative” restraining effects. Figure 17.10 shows the solutions obtained by considering both the “positive” and “negative” values of G-factors. The shaded portion of the graph corresponds to the alignment chart shown in Figure 17.4a when both GA and GB are positive. To account for the effect of axial forces in the restraining members, Bridge and Fraser [14] proposed a more general expression for G-factor: G 1999 by CRC Press LLC
c
=
(I /L) (I n /L)n γn mn
P
FIGURE 17.10: Effective length chart considering both positive and negative effects in braced frame. (From Bridge, R.Q. and Fraser, D.J., ASCE, J. Struct. Eng., 113(6), 1341, 1987. With permission.) =
stiffness of member i under investigation stiffness of all rigidly connected members
(17.67)
where γ is a function of the stability functions S and C (Equations 17.3 and 17.4); m is a factor to account for the end conditions of the restraining member (see Figure 17.11); and subscript n represents the other members rigidly connected to member i. The summation in the denominator is for all members meeting at the joint. Using Figures 17.10 and 17.11 and Equation 17.67, the effective length factor Ki for the i-th member can be determined by the following steps: 1. Sketch the buckled shape of the structure under consideration. 2. Assume a value of Ki for the member being investigated. 3. Calculate values of Kn for each of the other members that are rigidly connected to the i-th member using the equation Li Kn = Ki Ln 4. 5. 6. 7.
Pi Pn
In Ii
(17.68)
Calculate γ and obtain m from Figure 17.11 for each member. Calculate Gi for the i-th member using Equation 17.67. Obtain Ki from Figure 17.10 and compare with the assumed Ki at Step 2. Repeat the procedure by using the calculated Ki as the assumed Ki until Ki calculated at the end the cycle is approximately (say 10%) equal to the Ki at the beginning of the cycle.
1999 by CRC Press LLC
c
s
FIGURE 17.11: Values of γ and m to account for the effect of axial forces in the restraining members.
8. Repeat steps 2 to 7 for other members of the frame. 9. The largest set of K values obtained is then used for design. The above procedure has been illustrated [14] and verified [50] to provide a good elastic K-factor of columns in braced frames.
EXAMPLE 17.4:
Given: A braced column is shown in Figure 17.12. Consider axial force effects to determine K-factors for columns AB and BC. Solutions 1. Sketch the buckled shape as shown in Figure 17.12b 2. Assume KAB = 0.94. 1999 by CRC Press LLC
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FIGURE 17.12: Braced columns. 3. Calculate KBC by Equation 17.68.
KBC
=
LAB KAB + LBC
=
1.22
s
PAB PBC
IBC IAB
L = 0.94 L
s
2P I P (1.2I )
4. Calculate γ and obtain m from Figure 17.11 for member BC. Since KBC > 1.0 γBC = 1 −
1 1 =1− = 0.33 2 1.222 KBC
Far end is pinned, mBC = 1.5 5. Calculate G-factor for the member AB using Equation 17.67. (I /L) (1.2I /L) = 2.42 = (I /L) γ m (I /L)(0.33)(1.5) n n n n
GB
=
P
GA
=
∞
6. From Figure 17.10, KAB = 0.93. Comparing with the assumed KAB = 0.94 O.K. 7. Repeat the above procedure for member BC. 1999 by CRC Press LLC
c
Assume KBC = 1.2 Calculate KAB by Equation 17.68 s r PBC IAB L P (1.2I ) LBC = 0.93 = 1.2 KAB = KBC LAB PAB IBC L 2P I Calculate γ and obtain m from Figure 17.11 for member AB Since KAB < 1.0 γAB = 2 −
2 2 =2− = −0.312 2 0.932 KAB
Far end is pinned, mAB = 1.5 Calculate G-factor for the member BC using Equation 17.67. (I /L) (I /L) = = −1.78 (I /L) γ m (1.2I /L)(−0.312)(1.5) n n n n
GB
=
P
GA
=
∞
Read Figure 17.10, KBC = 1.18 Comparing with the assumed KAB = 1.20 O.K. 8. It is seen that the largest set of K-factors is KAB = 1.22 and KBC = 0.93
17.5.6
Consideration of Partial Column Base Fixity
In computing the effective length factor for monolithic connections, it is important to properly evaluate the degree of fixity in foundation. The following two approaches can be used to account for foundation fixity. 1. Fictitious Restraining Beam Approach Galambos [33] proposed that the effect of partial base fixity can be modelled as a fictitious beam. The approximate expression for the stiffness of the fictitious beam accounting for rotation of foundation in the soil has the form: qBH 3 Is = LB 72Esteel
(17.69)
where q is the modulus of subgrade reaction (varies from 50 to 400 lb/in.3 , 0.014 to 0.109 N/mm3 ); B and H are the width and length (in bending plane) of the foundation; and Esteel is the modulus of elasticity of steel. Based on studies by Salmon, Schenker, and Johnston [65], the approximate expression for the stiffness of the fictitious beam accounting for the rotations between column ends and footing due to deformation of base plate, anchor bolts, and concrete can be written as: bd 2 Is = LB 72Esteel /Econcrete 1999 by CRC Press LLC
c
(17.70)
where b and d are the width and length of the base plate, and subscripts concrete and steel represent concrete and steel, respectively. Galambos [33] suggested that the smaller of the stiffness calculated by Equations 17.69 and 17.70 be used in determining K-factors. 2. AASHTO-LRFD Approach The following values are suggested by AASHTO-LRFD [1]: G = 1.5 G = 3.0 G = 5.0 G = 1.0
17.5.7
footing anchored on rock footing not anchored on rock footing on soil footing on multiple rows of end bearing piles
Inelastic K-factor
The effect of material inelasticity and end restrain on the K-factors has been studied during the last two decades [12, 15, 20, 44, 45, 58, 64, 67, 68, 69, 73] The inelastic K-factor developed originally by Yura [73] and expanded by Disque [20] makes use of the alignment charts with simple modifications. To consider inelasticity of material, the G values as defined by Equations 17.7 and 17.8 are replaced by G∗ [20] as follows: Et (17.71) G G∗ = SRF (G) = E in which Et is the tangent modulus of the material. For practical application, stiffness reduction factor (SRF ) = (Et /E) can be taken as the ratio of the inelastic to elastic buckling stress of the column Pu /Ag Et (Fcr )inelastic ≈ ≈ (17.72) SRF = E (Fcr )elastic (Fcr )elastic where Pu is the factored axial load and Ag is the cross-sectional area of the member. (Fcr )inelastic and (Fcr )elastic can be calculated by AISC-LRFD [4] column equations: (Fcr )inelastic
=
(Fcr )elastic
=
λc
=
(0.658)λc Fy 0.877 Fy λ2c r KL Fy rπ E 2
(17.73) (17.74) (17.75)
in which K is the elastic effective length factor and r is the radius of gyration about the plane of buckling. Table 17.3 gives SRF values for different stress levels and slenderness parameters.
EXAMPLE 17.5:
Given: A two-story steel frame is shown in Figure 17.5. Use the alignment chart to determine K-factor for inelastic column DE. E = 29,000 ksi (200 GPa) and Fy = 36 ksi (248 MPa). Solution 1. Calculate the axial stress ratio: 300 Pu = 0.63 = Ag Fy 13.3(36) 2. Obtain SRF = 0.793 from Table 17.3 1999 by CRC Press LLC
c
Stiffness Reduction Factor (SRF) for G
TABLE 17.3 values
Pu Ag Fy
KL r elastic
36 ksi (248 MPa)
50 ksi (345 MPa)
λc
SRF (Eq. 18.72)
0.0 31.2 44.7 55.6 65.1 73.9 82.3 90.5 98.5 106.6 114.7 123.2 131.9 133.7
0.0 26.5 38.0 47.1 55.2 62.7 69.8 76.8 83.6 90.4 97.4 104.5 111.9 113.5
0.155 0.350 0.502 0.623 0.730 0.829 0.923 1.015 1.105 1.195 1.287 1.381 1.480 1.500
0.000 0.133 0.258 0.376 0.486 0.588 0.680 0.763 0.835 0.896 0.944 0.979 0.998 1.000
1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.39
3. Calculate modified G-factor. GE G∗E GD
= = =
0.448 (Example 17.1) SRF (GE ) = 0.794(0.448) = 0.355 10 (AISC-LRFD 1993)
4. From the alignment chart in Figure 17.4b , we have (KDE )inelastic = 1.75
17.6
Framed Columns—Alternative Methods
17.6.1
LeMessurier Method
Considering that all columns in a story buckle simultaneously and strong columns will brace weak columns (Figure 17.13), a more accurate approach to calculate K-factors for columns in a sidesway frame was developed by LeMessurier [52]. The Ki value for the i-th column in a story can be obtained by the following expression: s P P P + CL P π 2 EIi P (17.76) Ki = PL L2i Pi where Pi is the axial compressive force for member i, subscript i represents the i-th column, and P P is the sum of the axial force of all columns in a story. PL β CL 1999 by CRC Press LLC
c
βEI L2 6 (GA + GB ) + 36 = 2 (GA + GB ) + GA GB + 3 Ko2 = β 2 −1 π =
(17.77) (17.78) (17.79)
FIGURE 17.13: Subassemblage of the LeMessurier method. in which Ko is the effective length factor obtained by the alignment chart for unbraced frames, and PL is only for rigid columns which provide sidesway stiffness.
EXAMPLE 17.6:
Given: A sway frame with columns of unequal height is shown in Figure 17.14a. Determine elastic K-factors for columns by using the LeMessurier method. Member properties are: Member AB BD CD
A in.2 21.5 21.5 7.65
(mm2 ) I in.4 (13,871) 620 (13,871) 620 (4,935) 310
(mm4 × 108 ) L in. (2.58) 240 (2.58) 240 (1.29) 120
Solution
The detailed calculations are listed in Table 17.4. Using Equation 17.76, we obtain: s KAB
= s
1999 by CRC Press LLC
c
P
P+ P
P
CL P
PL
π 2 E(620) 3P + 0.495P = 0.83 0.271E (240)2 (2P ) s P P P + CL P π 2 EICD P = PL L2CD PCD s π 2 E(310) 3P + 0.495P = 1.66 = 0.271E (120)2 (P )
= KCD
π 2 EIAB L2AB PAB
(mm) (6,096) (6,096) (3,048)
FIGURE 17.14: A frame with unequal columns.
17.6.2
Lui Method
A simple and straightforward approach for determining the effective length factors for framed columns without the use of alignment charts and other charts was proposed by Lui [57]. The formulas take into account both the member instability and frame instability effects explicitly. The K-factor for the i-th column in a story was obtained in a simple form: v ! u u π 2 EIi X P 1 11 t P +P Ki = L 5 η H Pi L2i
(17.80)
P P where (P /L) represents the sum of the axial force-to-length ratio of all members in a story, H is the story lateral load producing 11 , 11 is the first-order inter-story deflection, and η is the member 1999 by CRC Press LLC
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TABLE 17.4 Example 17.6—Detailed Calculation by the LeMessurier Method Members
AB
CD
Sum
I in.4 (mm4 × 108 ) L in. (mm) Gtop Gbottom β Kio CL PL P CL P
620 (2.58) 240 (6,096) 1.0 0.0 8.4 1.17 0.165 0.09E 2P 0.33P
310 (1.29) 120 (3,048) 1.0 0.0 8.4 1.17 0.165 0.181E P 0.165P
—
Notes
— — — — — — 0.271E 3P 0.495P
Eq. 18.7 Eq. 18.7 Eq. 18.78 Alignment Chart Eq. 18.79 Eq. 18.77
stiffness index and can be calculated by 3 + 4.8m + 4.2m2 EI η= L3
(17.81)
in which m is the ratio of the smaller to larger end moments of the member; it is taken as positive if the member bends in reverse curvatureP and negative for single curvature. It is important to note that the term H used in Equation 17.80 is not the actual applied lateral load. Rather, it is a small disturbing or fictitious force (taken as a fraction of the story gravity loads) to be applied to each story of the frame. This fictitious force is applied in a direction such that the deformed configuration of the frame will resemble its buckled shape.
EXAMPLE 17.7:
Given: Determine K-factors by using the Lui method for the frame shown in Figure 17.14a. E = 29,000 ksi (200 GPa). Solution
Apply fictitious lateral forces at B and D (Figure 17.14b) and perform a first-order analysis. Detailed calculation is shown in Table 17.5. Using Equation 17.80, we obtain: v ! u u π 2 EIAB X P 1 11 t P +P KAB = L 5 η H PAB L2AB s 1 π 2 (29,000)(620) P + 0.019 = 0.76 = 60 5(56.24) (2P )(240)2 v ! u u π 2 EICD X P 1 11 t P P + KCD = L 5 η H PCD L2CD s 1 π 2 (29,000)(310) P + 0.019 = 1.52 = 60 5(56.24) (P )(120)2
1999 by CRC Press LLC
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TABLE 17.5 Lui Method
17.6.3
Example 17.7—Detailed Calculation by the
Members
AB
CD
Sum
I in.4 (mm4 × 108 ) L in. (mm) H kips (kN) 11 in. (mm) P 11 / H in./kips (mm/kN) Mtop k-in. (kN-m) Mbottom k-in. (kN-m) m η kips/in. (kN/mm) P /L kips/in. (kN/mm)
620 (2.58) 240 (6096) 1.0 (4.448) 0.0286 (0.7264) —
310 (1.29) 120 (3048) 0.5 (2.224) 0.0283 (0.7188) —
—
−38.8 (−4.38) −46.2 (−5.22) 0.84 13.00 (2.28) P /120 P /3048
56.53 (6.39) 81.18 (9.17) 0.69 43.24 (7.57) P /120 P /3048
Notes
— 1.5 (6.672) — 0.019 (0.108) —
Average
— — 56.24 (9.85) P /60 P /1524
Eq. 18.1
Remarks
For comparison, Table 17.6 summarizes K-factors for the frame shown in Figure 17.14a obtained from the alignment chart, the LeMessurier and Lui methods, as well as an eigenvalue analysis. It is seen that errors in alignment chart results are rather significant in this case. Although K-factors predicted by Lui’s and LeMessurier’s formulas are identical in most cases, the simplicity and independence of any chart in the case of Lui’s formulas make it more desirable for design office use [66]. TABLE 17.6 Comparison of K -Factors for the Frame in Figure 17.14a
17.7
Columns
Theoretical
Alignment chart
Lui Eq. 18.80
LeMessurier Eq. 18.76
AB CD
0.70 1.40
1.17 1.17
0.76 1.52
0.83 1.67
Unbraced Frames With Leaning Columns
A column framed with simple connections is often called a leaning column. It has no lateral stiffness or sidesway resistance. A column framed with rigid moment-resisting connections is called a rigid column. It provides the lateral stiffness or sidesway resistance to the frame. When a frame system (Figure 17.15a) includes leaning columns, the effective length factors of rigid columns must be modified. Several approaches to account for the effect of “leaning columns” were reported in the literature [16, 52, 54, 73]. A detailed discussion about the leaning columns for practical applications was presented by Geschwindner [37].
17.7.1
Rigid Columns
1. Yura Method Yura [73] discussed frames with leaning columns and noted the behavior of stronger columns assisting weaker ones in resisting sidesway. He concluded that the alignment 1999 by CRC Press LLC
c
FIGURE 17.15: A frame with leaning columns.
chart gives valid sidesway buckling solutions if the columns are in the elastic range and all columns in a story reach their individual buckling loads simultaneously. For columns that do not satisfy these two conditions, the alignment chart is generally overly conservative. Yura states that (a) The maximum load-carrying capacity of an individual column is limited to the load permitted on that column for braced case K = 1.0. (b) The total gravity loads that produce sidesway are distributed among the columns, which provides lateral stiffness in a story. 2. Lim and McNamara Method Based on the story buckling concept and using the stability functions, Lim and McNamara [54] presented the following formula to account for the leaning column effect. 1999 by CRC Press LLC
c
s
P Q Fo 1+ P P Fn
Kn = Ko
(17.82)
where Kn is the effective length factor accounting for the leaning columns; Ko is the effective length factor determined by P Pthe alignment chart (Figure 17.3b) not accounting for the leaning columns; P and Q are the loads on the restraining columns and on the leaning columns in a story, respectively; and Fo and Fn are the eigenvalue solutions for a frame without and with leaning columns, respectively. For normal column end conditions that fall somewhere between fixed and pinned, Fo /Fn = 1 provides a Kfactor on the conservative side by less than 2% [37]. Using Fo /Fn = 1, Equation 17.82 becomes: s P Q (17.83) Kn = Ko 1 + P P Equation 17.83 gives the same K-factor as the modified Yura approach [37]. 3. LeMessurier and Lui Methods Equation 17.76 developed by LeMessurier [52] and Equation 17.80 proposed by Lui [57] can be used for frames with and without leaning columns. Since the K-factor expressions Equations 17.76 and 17.80 were derived for an entire story of the frame, they are applicable to frames with and without leaning columns. 4. AISC-LRFD Method The current AISC-LRFD [4] commentary adopts the following modified effective length factor, Ki0 , for the i-th rigid column: s Ki0
=
π 2 EIi L2i Pui
P Pu P Pe2
(17.84)
P where Pe2 is the Euler loads of all columns in a story providing lateral stiffness for the frame based on the effective length factor obtained from the alignment chart for an unbraced P frame; Pui is the required axial compressive strength for the i-th rigid column; and Pu is the required axial compressive strength of all columns in a story. When E and L2 are constant for all columns in a story, AISC [4] suggested that: v uP u I P u u i × P I Ki0 = t i Pui 2
(17.85)
Kio
except r Ki0
≥
5 Kio 8
(17.86)
where Kio is the effective length factor of a rigid column based on the alignment chart for unbraced frames.
1999 by CRC Press LLC
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EXAMPLE 17.8:
Given: A frame with a leaning column is shown in Figure 17.16a [59]. Evaluate the K-factor for
FIGURE 17.16: A leaning column frame. column AB using various methods. The bottom of column AB is assumed to be ideally pin-ended 1999 by CRC Press LLC
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for comparison purposes. E = 29,000 ksi (200 GPa). Solution 1. Alignment Chart Method GA
=
GB
=
∞ P Ec Ic /Lc EI /L P = = 2.0 0.5EI /L αk Eg Ig /Lg
From Figure 17.3b, we have KAB = 2.6 2. Lim and McNamara P Method P For this frame, P = Q = P and Ko = 2.6. From Equation 17.83, we have s P √ Q KAB = Ko 1 + P = 2.6 1 + 1 = 3.68 P 3. LeMessurier Method For column AB, GA = ∞ and GB = 2.0; from the alignment chart, Ko = 2.6. According to Equations 17.76 to 17.79 we have, β |GA=∞ X
PL CL
KAB
6(GA + GB ) + 36 6 6 = = |G = 1.5 2(GA + GB ) + GA GB + 3 A=∞ 2 + GB 2+2 βEI EI = (PL )AB = 2 = 1.5 2 L L Ko2 2.62 = β 2 − 1 = (1.5) 2 − 1 = 0.0274 π π s P P s 2 P + CL P π 2 EIAB π EI 2P + 0.0274P P = = PL L2 P 1.5EI /L2 L2AB PAB √ 13.34 = 3.65 = =
4. AISC-LRFD Method Using Equation 17.85 for column AB: v uP u √ u Pu IAB × P I = Kio 2 = 3.68 KAB = t PAB 2 Kio
5. Lui Method (a) Apply a small lateral force, H = 1 kip, as shown in Figure 17.16b. (b) Perform a first-order analysis and find 11 = 0.687 in. (17.45 mm). (c) Calculate η factors from Equation 17.81. Since column CD buckles in a single curvature, m = −1, ηCD = 1999 by CRC Press LLC
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(3 + 4.8m + 4.2m2 )EI (3 − 4.8 + 4.2)EI 2.4EI = = 3 3 L L L3
For column AB, m = 0,
ηAB X
η
(3 + 4.8m + 4.2m2 )EI 3EI = 3 3 L L 2.4EI 3EI 5.4(29,000)(100) + = = L3 L3 (144)3 = 5.245 kips/in. (0.918 kN/mm) =
(d) Calculate the K-factor from Equation 17.80.
KAB
v ! u u π 2 EIAB X P 1 11 t P P + = L 5 η H PAB L2AB s 1 0.687 π 2 (29,000)(100) 2P + = 3.73 = 144 5(5.245) 1 P (144)2
From an eigenvalue analysis, KAB = 3.69 is obtained. It is seen that a direct use of the alignment chart leads to a significant error for this frame, and other approaches give good results. However, the LeMessurier approach requires the use of the alignment chart, and the Lui approach requires a first-order analysis subjected to a fictitious lateral loading.
17.7.2
Leaning Columns
Recognizing that a leaning column is being braced by rigid columns, Lui [57] proposed a model for the leaning column, as shown in Figure 17.15b. Rigid columns provide lateral stability to the whole structure and are represented by a translation spring with a spring stiffness, SK . The K-factor for a leaning column can be obtained as: ( K = larger of
q1
π 2 EI SK L3
(17.87)
For most commonly framed structures, the term (π 2 EI /SK L3 ) normally does not exceed unity and so K = 1 often governs. AISC-LRFD [4] suggests that leaning columns with K = 1 may be used in unbraced frames provided that the lack of lateral stiffness from simple connections to the frame (K = ∞) is included in the design of moment frame columns.
17.7.3
Remarks
Numerical studies by Geschwindner [37] found that the Yura approach gives overly conservative results for some conditions; Lim and McNamara’s approach provides sufficiently accurate results for design, and the LeMessurier approach is the most accurate of the three. The Lim and McNamara approach could be appropriate for preliminary design while the LeMessurier and Lui approaches would be appropriate for final design. 1999 by CRC Press LLC
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17.8
Cross Bracing Systems
Diagonal bracing or X-bracing is commonly used in steel structures to resist horizontal loads. In the current practice, the design of this type of bracing system is based on the assumptions that the compression diagonal has negligible capacity and the tension diagonal resists the total load. The assumption that compression diagonal has a negligible capacity usually results in an overdesign [62, 63]. Picard and Beaulieu [62, 63] reported theoretical and experimental studies on double diagonal cross bracings (Figure 17.6) and found that 1. A general effective length factor equation (Figure 17.17) is given as s K=
0.523 −
0.428 ≥ 0.50 C/T
(17.88)
where C and T represent compression and tension forces obtained from an elastic analysis, respectively.
FIGURE 17.17: Effective length factor of compression diagonal. (From Picard, A. and Beaulieu, D., AISC Eng. J., 24(3), 122, 1987. With permission.)
1999 by CRC Press LLC
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2. When the double diagonals are continuous and attached at their intersection point, the effective length of the compression diagonal is 0.5 times the diagonal length, i.e., K = 0.5, because the C/T ratio is usually smaller than 1.6. EL-Tayem and Goel [27] reported a theoretical and experimental study about the X-bracing system made from single equal-leg angles. They concluded that: 1. Design of an X-bracing system should be based on an exclusive consideration of one half diagonal only. 2. For X-bracing systems made from single equal-leg angles, an effective length of 0.85 times the half diagonal length is reasonable, i.e., K = 0.425.
17.9
Latticed and Built-Up Members
The main difference of behavior between solid-webbed members, latticed members, and built-up members is the effect of shear deformation on their buckling strength. For solid-webbed members, shear deformation has a negligible effect on buckling strength. Whereas for latticed structural members using lacing bars and batten plates, shear deformation has a significant effect on buckling strength. It is a common practice that when a buckling model involves relative deformation produced by shear forces in the connectors, such as lacing bars and batten plates, between individual components, a modified effective length factor, Km , is defined as follows: Km = αv K
(17.89)
in which K is the usual effective length factor of a latticed member acting as a unit obtained from a structural analysis, and αv is the shear factor to account for shear deformation on the buckling strength, or the modified effective slenderness ratio, (KL/r)m should be used in the determination of the compressive strength. Details of the development of the shear factor, αv , can be found in textbooks by Bleich [13] and Timoshenko and Gere [70]. The following section briefly summarizes αv formulas for various latticed members.
17.9.1
Laced Columns
For laced members as shown in Figure 17.18, by considering shear deformation due to the lengthening of diagonal lacing bars in each panel and assuming hinges at joints, the shear factor, αv , has the form: s αv =
1 π 2 EI (KL2 ) Ad Ed sin φ cos2 φ
1+
(17.90)
where Ed is the modulus of elasticity of materials for the lacing bars, Ad is the cross-sectional area of all diagonals in one panel, and φ is the angle between the lacing diagonal and the axis that is perpendicular to the member axis. If the length of the lacing bars is given (Figure 17.18), Equation 17.90 can be rewritten as: s αv =
1+
d3 π 2 EI 2 (KL ) Ad Ed ab2
(17.91)
where a, b, and d are the height of panel, depth of member, and length of diagonal, respectively. 1999 by CRC Press LLC
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FIGURE 17.18: Typical configurations of laced members. The SSRC [36] suggested that a conservative estimate of the influence of 60 or 45◦ lacing, as generally specified in bridge design practice, can be made by modifying the overall effective length factor, K, by multiplying a factor, αv , originally developed by Bleich [13] as follows: For KL r > 40, q (17.92) αv = 1 + 300/ (KL/r)2 For
KL r
≤ 40, αv = 1.1
(17.93)
EXAMPLE 17.9:
Given: A laced column with angles and cover plates is shown in Figure 17.19. Ky = 1.25, L = 30 ft (9144 mm). Determine the modified effective length factor, (Ky )m , by considering the shear deformation effect. Section properties:
1999 by CRC Press LLC
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Iy
=
2259 in.4 (9.4 × 108 mm4 )
E a
Ad = 1.69 in.2 (1090 mm2 ) = Ed = 6 in. (152 mm) b = 11 in. (279 mm)
d
=
12.53 in. (318 mm)
FIGURE 17.19: A laced column. Solution
1. Calculate the shear factor, αv , by Equation 17.91. s αv
=
1+ s
=
1+
π 2 EI d3 (KL)2 Ad Ed ab2 π 2 E(2259)
12.533 = 1.09 (1.25 × 30 × 12)2 1.69E(6)(11)2
2. Calculate (Ky )m by Equation 17.89. Ky m = αv Ky = 1.09(1.25) = 1.36
1999 by CRC Press LLC
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17.9.2
Columns with Battens
The battened column has a greater shear flexibility than either the laced column or the column with perforated cover plates, hence the effect of shear distortion must be taken into account in calculating the effective length of a column [43]. For the battened members shown in Figure 17.20a, assuming that points of inflection in the battens are at the batten midpoints, and that points of inflection in the longitudinal element occur midway between the battens, the shear factor, αv , is obtained as: s π 2 EI a2 ab + (17.94) αv = 1 + 24EIf (KL)2 12Eb Ib where Eb is the modulus of elasticity of materials for the batten plates, Ib is the moment inertia of all the battens in one panel in the buckling plane, and If is the moment inertia of one side of the main components taken about the centroid axis of the flange in the buckling plane.
FIGURE 17.20: Typical configurations of members with battens and with perforated cover plates.
EXAMPLE 17.10:
Given: A battened column is shown in Figure 17.21. Ky = 0.8, L = 30 ft (9144 mm). Determine the modified effective length factor, (Ky )m , by considering the shear deformation effect. Section properties: Iy 1999 by CRC Press LLC
c
=
144 in.4 (6.0 × 107 mm4 )
FIGURE 17.21: A battened column.
E If a
= Eb = 1.98 in.4 (8.24 × 105 mm4 ) = 15 in. (381 mm)
b Ib
= 9 in. (229 mm) = 9 in.4 (3.75 × 106 mm4 )
Solution
1. Calculate the shear factor, αv , by Equation 17.94. 1999 by CRC Press LLC
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s
αv
a2 ab = + 12EIb 24EIf s 152 π 2 E(144) 15(9) + = 1.05 1+ = (0.8 × 30 × 12)2 12E(9) 24E(1.98) π 2 EI 1+ (KL)2
2. Calculate (Ky )m by Equation 17.89. Ky
17.9.3
m
= αv Ky = 1.05(0.8) = 0.84
Laced-Battened Columns
For the laced-battened columns, as shown in Figure 17.20b, considering the shortening of the battens and the lengthening of the diagonal lacing bars in each panel, the shear factor, αv , can be expressed as: s π 2 EI b d3 + (17.95) αv = 1 + aAb Eb (KL)2 Ad Ed ab2 where Eb is the modulus of elasticity of the materials for battens and Ab is the cross-sectional area of all battens in one panel.
17.9.4
Columns with Perforated Cover Plates
For members with perforated cover plates, shown in Figure 17.20c, considering the horizontal cross member as infinitely rigid, the shear factor, αv , has the form: s αv =
1+
π 2 EI (KL)2
9c3 64aEIf
(17.96)
where c is the length of a perforation. It should be pointed out that the usual K-factor based on a solid member analysis is included in Equations 17.90 to 17.96. However, since the latticed members studied previously have pin-ended conditions, the K-factor of the member in the frame was not included in the second terms of the square root of the above equations in their original derivations [13, 70].
EXAMPLE 17.11:
Given: A column with perforated cover plates is shown in Figure 17.22. Ky = 1.3, L = 25 ft (7620 mm). Determine the modified effective length factor, (Ky )m , by considering the shear deformation effect. Section properties: Iy
=
If a
= 35.5 in.4 (1.48 × 106 mm4 ) = 30 in. (762 mm)
c 1999 by CRC Press LLC
c
=
2467 in.4 (1.03 × 108 mm4 )
14 in. (356 mm)
FIGURE 17.22: A column with perforate cover plates.
Solution
1. Calculate the shear factor, αv , by Equation 17.96. s αv
=
1+ s
=
π 2 EI (KL)2
9c3 64aEIf
π 2 E(2467) 1+ (1.3 × 25 × 12)2
9(14)3 64(30)E(35.5)
2. Calculate (Ky )m by Equation 17.89. Ky 1999 by CRC Press LLC
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m
= αv Ky = 1.03(1.3) = 1.34
= 1.03
17.9.5
Built-Up Members with Bolted and Welded Connectors
AISC-LRFD [4] specifies that if the buckling of a built-up member produces shear forces in the connectors between individual component members, the usual slenderness ratio, KL/r, for compression members must be replaced by the modified slenderness ratio, KL r m , in determining the compressive strength. 1. For snug-tight bolted connectors:
KL r
s
m
=
KL r
2 o
+
2 a ri
(17.97)
2. For welded connectors and for fully tightened bolted connectors:
KL r
s
m
=
KL r
2 o
+ 0.82
α2 (1 + α 2 )
a rib
2 (17.98)
KL where KL r o is the slenderness ratio of the built-up member acting as a unit, r m is the modified slenderness ratio of the built-up member, rai is the largest slenderness ratio of the individual components, raib is the slenderness ratio of the individual components relative to its centroidal axis parallel to the axis of buckling, a is the distance between connectors, ri is the minimum radius of gyration of individual components, rib is the radius of gyration of individual components relative to its centroidal axis parallel to the member axis of buckling, α is the separation ratio = h/2rib , and h is the distance between centroids of individual components perpendicular to the member axis of buckling. Equation 17.97 is the same as that used in the current Italian code as well as other European specifications, based on test results [74]. In the equation, the bending effect is considered in the first term in square root, and shear force effect is taken into account in the second term. Equation 17.98 was derived from elastic stability theory and verified by test data [9]. In both cases the end connectors must be welded or slip-critical bolted [9].
EXAMPLE 17.12:
Given: A built-up member with two back-to-back angles is shown in Figure 17.23. Determine the modified slenderness ratio, (KL/r)m , in accordance with AISC-LRFD [4] and Equation 17.98. rib = a = h = (KL/r)o =
0.735 in. (19 mm) 48 in. (1219 mm) 1.603 in. (41 mm) 70
Solution
1. Calculate the separation factor α. α= 1999 by CRC Press LLC
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1.603 h = 1.09 = 2rib 2(0.735)
FIGURE 17.23: A built-up member with back-to-back angles.
2. Calculate the modified slenderness ratio, (KL/r)m , by Equation 17.98.
KL r
s
m
= s = =
17.10
KL r
2 o
+ 0.82
α2 (1 + α 2 )
1.092 (70) + 0.82 (1 + 1.092 ) 82.5 2
a rib
2
48 0.735
2
Tapered Columns
The state-of-the-art design for tapered structural members was provided in the SSRC guide [36]. The charts shown in Figures 17.24 and 17.25 can be used to evaluate the effective length factors for tapered columns restrained by prismatic beams [36]. In these figures, IT and IB are the moment of inertia of the top and bottom beam, respectively; b and L are the length of beam and column, respectively; and γ is the tapering factor as defined by: γ =
d1 − do do
where do and d1 are the section depth of column at the smaller and larger end, respectively. 1999 by CRC Press LLC
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(17.99)
FIGURE 17.24: Effective length factor for tapered columns in braced frames.
17.11
Crane Columns
The columns in mill buildings and warehouses are designed to support overhead crane loads. The cross-section of a crane column may be uniform or stepped (see Figure 17.26). Over the past two decades, a number of simplified procedures have been developed for evaluating the K-factors for crane columns [5, 6, 7, 11, 29, 30, 41, 51, 61]. Those procedures have limitations in terms of column geometry, loading, and boundary conditions. Most importantly, most of these studies ignored the interaction effect between the left and right column of frames and were based on isolated member analyses [59]. Recently, a simple yet reasonably accurate procedure for calculating the K-factors for crane columns with any value of relative shaft length, moment of inertia, loading, and boundary conditions was developed by Lui and Sun [59]. Based on the story stiffness concept and accounting for both member and frame instability effects in the formulation, Lui and Sun [59] proposed the following procedure [see Figure 17.27]: 1. Apply the fictitious lateral loads, αP (α is an arbitrary factor; 0.001 may be used), in such a direction as to create a deflected geometry for the frame that closely approximates its actual buckled configuration. 2. Perform a first-order elastic analysis P on the frame subjected to the fictitious lateral loads (Figure 17.27b). Calculate 11 / H , where 11 is the average lateral P deflection at the intermediate load points (i.e., points B and F) of columns, and H is the sum of all fictitious lateral loads that act at and above the intermediate load points. 3. Calculate η using results obtained from a first-order elastic analysis for lower shafts (i.e., segments AB and F G), according to Equation 17.81. 4. Calculate the K-factor for the lower shafts using Equation 17.80. 5. Calculate the K-factor for upper shafts using the following formula: 1999 by CRC Press LLC
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FIGURE 17.25: Effective length factor for tapered columns in unbraced frames.
FIGURE 17.26: Typical crane columns. (From Lui, E.M. and Sun, M.Q., AISC Eng. J., 32(2), 98, 1995. With permission.)
1999 by CRC Press LLC
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FIGURE 17.27: Crane column model for effective length factor computation. (From Lui, E.M. and Sun, M.Q., AISC Eng. J., 32(2), 98, 1995. With permission.)
KU = KL
LL LU
s
PL + PU PU
IU IL
(17.100)
where P is the applied load and subscripts U and L represent upper and lower shafts, respectively.
EXAMPLE 17.13:
Given: A stepped crane column is shown in Figure 17.28a. The example is the same frame as used by Fraser [30] and Lui and Sun [59]. Determine the effective length factors for all columns using the Lui approach. E = 29,000 ksi (200 GPa). IAB 1999 by CRC Press LLC
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=
IF G = IL = 30,000 in.4 (1.25 × 1010 mm4 )
FIGURE 17.28: A pin-based stepped crane column. (From Lui, E.M. and Sun, M.Q., AISC Eng. J., 32(2), 98, 1995. With permission.)
AAB IBC ABC
= = =
AF G = AL = 75 in.2 (48,387 mm2 ) IEF = ICE = IU = 5,420 in.4 (2.26 × 109 mm4 ) AEF = ACE = AU = 34.14 in.2 (22,026 mm2 )
Solution
1. Apply a set of fictitious lateral forces with α = 0.001 as shown in Figure 17.28b. 1999 by CRC Press LLC
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2. Perform a first-order analysis and find (11 )B = 0.1086 in.(2.76 mm) and (11 )F = 0.1077 in. (2.74 mm) so, (0.1086 + 0.1077)/2 11 P = = 0.198 in./kips (1.131 mm/kN) 0.053 + 0.3 + 0.053 + 0.14 H 3. Calculate η factors from Equation 17.81. Since the bottom of column AB and F G is pin-based, m = 0,
ηAB
= =
X
η
=
(3 + 4.8m + 4.2m2 )EI 3EI = 3 3 L L (3)(29,000)(30,000) = 42.03 kips/in. (7.36 mm/kN) (396)3
ηF G =
42.03 + 42.03 = 84.06 kips/in. (14.72 mm/kN)
4. Calculate the K-factors for columns AB and F G using Equation 17.80. s
KAB
KF G
1 π 2 (29,000)(30,000) 353 + 193 + 0.198 396 5(84.06) (353)(396)2 = 6.55 s 1 π 2 (29,000)(30,000) 353 + 193 + 0.198 = 396 5(84.06) (193)(396)2 = 8.85 =
5. Calculate the K-factors for columns BC and EF using Equation 17.100. s LAB PAB + PBC IBC = KAB LBC PBC IAB s 5420 353 396 = 18.2 = 6.55 156 53 30,000 s LF G PF G + PEF IEF = KF G LEF PEF IF G s 5420 193 396 = 18.2 = 8.85 156 53 30,000
KBC
KEF
The K-factors calculated above are in good agreement with the theoretical values reported by Lui and Sun [59]. 1999 by CRC Press LLC
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17.12
Columns in Gable Frames
For a pin-based gable frame subjected to a uniformly distributed load on the rafter, as shown in Figure 17.29a, Lu [56] presented a graph (Figure 17.29b) to determine the effective length factors of columns. For frames having different member sizes for rafter and columns with (L/ h) of (f/ h)
FIGURE 17.29: Effective length factors for columns in a pin-based gable frame. (From Lu, L.W., AISC Eng. J., 2(2), 6, 1965. With permission.) ratios not covered in Figure 17.29, an approximate method is available for determining K-factors of columns [39]. The method is to find an equivalent portal frame whose span length is equal to 1999 by CRC Press LLC
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twice the rafter length, Lr (see Figure 17.29a). The K-factors can be determined form the alignment c/ h charts using Gtop = IrI/2L and corresponding Gbottom . r
17.13
Summary
This chapter summarizes the state-of-the-art practice of the effective length factors for isolated columns, framed columns, diagonal bracing systems, latticed and built-up members, tapered columns, crane columns, and columns in gable frames. Design implementation with formulas, charts, tables, various modification factors adopted in current codes and specifications, as well as those used in engineering practice are described. Several examples illustrate the steps of practical applications of various methods.
17.14
Defining Terms
Alignment chart: A monograph for determining the effective length factor K for some types of compression members. Braced frame: A frame in which the resistance to lateral load or frame instability is primarily provided by diagonal bracing, shear walls, or equivalent means. Build-up member: A member made of structural metal elements that are welded, bolted, and riveted together. Column: A structural member whose primary function is to carry loads parallel to its longitudinal axis. Crane column: A column that is designed to support overhead crane loads. Effective length factor K: A factor that when multiplied by actual length of the end-restrained column gives the length of an equivalent pin-ended column whose elastic buckling load is the same as that of the end-restrained column. Framed column: A column in a framed structure. Gable frame: A frame with a gabled roof. Latticed member: A member made of two or more rolled-shapes that are connected to one another by means of lacing bars, batten plates, or perforated plates. Leaning column: A column that is connected to a frame with simple connections and does not provide lateral stiffness or sidesway resistance. LRFD (Load and Resistance Factor Design): A method of proportioning structural components (members, connectors, connecting elements, and assemblages) such that no applicable limit state is exceeded when the structure is subjected to all appropriate load combinations. Tapered column: A column that has a continuous reduction in section from top to bottom. Unbraced frame: A frame in which the resistance to lateral loads is provided by the bending stiffness of frame members and their connections.
References [1] American Association of State Highway and Transportation Officials. 1994. LRFD Bridge Design Specifications, 1st ed., AASHTO, Washington, D.C. [2] American Concrete Institute. 1995. Building Code Requirements for Structural Concrete (ACI 318-95) and Commentary (ACI 318R-95). ACI, Farmington Hills, MI. 1999 by CRC Press LLC
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[3] American Institute of Steel Construction. 1989. Allowable Stress Design Specification for Structural Steel Buildings, 9th ed., AISC, Chicago, IL. [4] American Institute of Steel Construction. 1993. Load and Resistance Factor Design Specification for Structural Steel Buildings, 2nd ed., AISC, Chicago, IL. [5] Association of Iron and Steel Engineers. 1991. Guide for the Design and Construction of Mill Buildings, AISE, Technical Report, No. 13, Pittsburgh, PA. [6] Anderson, J.P. and Woodward, J.H. 1972. Calculation of Effective Lengths and Effective Slenderness Ratios of Stepped Columns. AISC Eng. J., 7(4):157-166. [7] Agarwal, K.M. and Stafiej, A.P. 1980. Calculation of Effective Lengths of Stepped Columns. AISC Eng. J., 15(4):96-105. [8] Aristizabal-Ochoa, J.D. 1994. K -Factors for Columns in Any Type of Construction: Nonparadoxical Approach. J. Struct. Eng., 120(4):1272-1290. [9] Aslani, F. and Goel, S.C. 1991. An Analytical Criteria for Buckling Strength of Built-Up Compression Members. AISC Eng. J., 28(4):159-168. [10] Barakat, M. and Chen, W.F. 1991. Design Analysis of Semi-Rigid Frames: Evaluation and Implementation. AISC Eng. J., 28(2):55-64. [11] Bendapudi, K.V. 1994. Practical Approaches in Mill Building Columns Subjected to Heavy Crane Loads. AISC Eng. J., 31(4):125-140. [12] Bjorhovde, R. 1984. Effect of End Restraints on Column Strength—Practical Application, AISC Eng. J., 21(1):1-13. [13] Bleich, F. 1952. Buckling Strength of Metal Structures. McGraw-Hill, New York. [14] Bridge, R.Q. and Fraser, D.J. 1987. Improved G-Factor Method for Evaluating Effective Length of Columns. J. Struct. Eng., 113(6):1341-1356. [15] Chapius, J. and Galambos, T.V. 1982. Restrained Crooked Aluminum Columns, J. Struct. Div., 108(ST12):511-524. [16] Cheong-Siat-Moy, F. 1986. K -Factor Paradox, J. Struct. Eng., 112(8):1647-1760. [17] Chen, W.F. and Lui, E.M. 1991. Stability Design of Steel Frames, CRC Press, Boca Raton, FL. [18] Chu, K.H. and Chow, H.L. 1969. Effective Column Length in Unsymmetrical Frames, Publ. Intl. Assoc. Bridge Struct. Eng., 29(1). [19] Cranston, W.B. 1972. Analysis and Design of Reinforced Concrete Columns. Research Report No. 20, Paper 41.020, Cement and Concrete Association, London. [20] Disque, R.O. 1973. Inelastic K -Factor in Design. AISC Eng. J., 10(2):33-35. [21] Duan, L. and Chen, W.F. 1988. Effective Length Factor for Columns in Braced Frames. J. Struct. Eng., 114(10):2357-2370. [22] Duan, L. and Chen, W.F. 1989. Effective Length Factor for Columns in Unbraced Frames. J. Struct. Eng., 115(1):149-165. [23] Duan, L. and Chen, W.F. 1996. Errata of Paper: Effective Length Factor for Columns in Unbraced Frames. J. Struct. Eng., 122(1):224-225. [24] Duan, L., King, W.S., and Chen, W.F. 1993. K -Factor Equation to Alignment Charts for Column Design. ACI Struct. J., 90(3):242-248. [25] Duan, L. and Lu, Z.G. 1996. A Modified G-Factor for Columns in Semi-Rigid Frames. Research Report, Division of Structures, California Department of Transportation, Sacramento, CA. [26] Dumonteil, P. 1992. Simple Equations for Effective Length Factors. AISC Eng. J., 29(3):111-115. [27] El-Tayem, A.A. and Goel, S.C. 1986. Effective Length Factor for the Design of X-Bracing Systems. AISC Eng. J., 23(4):41-45. [28] European Convention for Constructional Steelwork. 1978. European Recommendations for Steel Construction, ECCS. [29] Fraser, D.J. 1989. Uniform Pin-Based Crane Columns, Effective Length, AISC Eng. J., 26(2):6165.
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[30] Fraser, D.J. 1990. The In-Plane Stability of a Frame Containing Pin-Based Stepped Column. AISC Eng. J., 27(2):49-53. [31] French. 1975. Regles de calcul des constructions en acier CM66, Eyrolles, Paris, France. [32] Furlong, R.W. 1971. Column Slenderness and Charts for Design. ACI Journal, Proceedings, 68(1):9-18. [33] Galambos, T.V. 1960. Influence of Partial Base Fixity on Frame Instability. J. Struct. Div., 86(ST5):85-108. [34] Galambos, T.V. 1964. Lateral Support for Tier Building Frames. AISC Eng. J., 1(1):16-19. [35] Galambos, T.V. 1968. Structural Members and Frames. Prentice-Hall International, London, U.K. [36] Galambos, T.V., Ed. 1988. Structural Stability Research Council, Guide to Stability Design Criteria for Metal Structures, 4th ed., John Wiley & Sons, New York. [37] Geschwindner, L.F. 1995. A Practical Approach to the “Leaning” Column, AISC Eng. J., 32(2):63-72. [38] Gurfinkel, G. and Robinson, A.R. 1965. Buckling of Elasticity Restrained Column. J. Struct. Div., 91(ST6):159-183. [39] Hansell, W.C. 1964. Single-Story Rigid Frames, in Structural Steel Design, Chapt. 20, Ronald Press, New York. [40] Hu, X.Y., Zhou, R.G., King, W.S., Duan, L., and Chen, W.F. 1993. On Effective Length Factor of Framed Columns in ACI Code. ACI Struct. J., 90(2):135-143. [41] Huang, H.C. 1968. Determination of Slenderness Ratios for Design of Heavy Mill Building Stepped Columns. Iron Steel Eng., 45(11):123. [42] Johnson, D.E. 1960. Lateral Stability of Frames by Energy Method. J. Eng. Mech., 95(4):23-41. [43] Johnston, B.G., Ed. 1976. Structural Stability Research Council, Guide to Stability Design Criteria for Metal Structures, 3rd ed., John Wiley & Sons, New York. [44] Jones, S.W., Kirby, P.A., and Nethercot, D.A. 1980. Effect of Semi-Rigid Connections on Steel Column Strength, J. Constr. Steel Res., 1(1):38-46. [45] Jones, S.W., Kirby, P.A., and Nethercot, D.A. 1982. Columns with Semi-Rigid Joints, J. Struct. Div., 108(ST2):361-372 [46] Julian, O.G. and Lawrence, L.S. 1959. Notes on J and L Nomograms for Determination of Effective Lengths. Unpublished report. [47] Kavanagh, T.C. 1962. Effective Length of Framed Column. Trans. ASCE, 127(II):81-101. [48] King, W.S., Duan, L., Zhou, R.G., Hu, Y.X., and Chen, W.F. 1993. K -Factors of Framed Columns Restrained by Tapered Girders in US Codes. Eng. Struct., 15(5):369-378. [49] Kishi, N., Chen, W.F., and Goto, Y. 1995. Effective Length Factor of Columns in Semi-Rigid and Unbraced Frames. Structural Engineering Report CE-STR-95-5, School of Civil Engineering, Purdue University, West Lafayette, IN. [50] Koo, B. 1988. Discussion of Paper “Improved G-Factor Method for Evaluating Effective Length of Columns” by Bridge and Fraser. J. Struct. Eng., 114(12):2828-2830. [51] Lay, M.G. 1973. Effective Length of Crane Columns, Steel Const., 7(2):9-19. [52] LeMessurier, W.J. 1977. A Practical Method of Second Order Analysis. Part 2—Rigid Frames. AISC Eng. J., 14(2):49-67. [53] Liew, J.Y.R., White, D.W., and Chen, W.F. 1991. Beam-Column Design in Steel Frameworks— Insight on Current Methods and Trends. J. Const. Steel. Res., 18:269-308. [54] Lim, L.C. and McNamara, R.J. 1972. Stability of Novel Building System, Structural Design of Tall Steel Buildings, Vol. II-16, Proceedings, ASCE-IABSE International Conference on the Planning and Design of Tall Buildings, Bethlehem, PA, 499-524. [55] Lu, L.W. 1962. A Survey of Literature on the Stability of Frames. Weld. Res. Conc. Bull., New York. [56] Lu, L.W. 1965. Effective Length of Columns in Gable Frame, AISC Eng. J., 2(2):6-7. 1999 by CRC Press LLC
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[57] Lui, E.M. 1992. A Novel Approach for K -Factor Determination. AISC Eng. J., 29(4):150-159. [58] Lui, E.M. and Chen, W.F. 1983. Strength of Columns with Small End Restraints, J. Inst. Struct. Eng., 61B(1):17-26 [59] Lui, E.M. and Sun, M.Q. 1995. Effective Length of Uniform and Stepped Crane Columns, AISC Eng. J., 32(2):98-106. [60] Maquoi, R. and Jaspart, J.P. 1989. Contribution to the Design of Braced Framed with SemiRigid Connections. Proc. 4th International Colloquium, Structural Stability Research Council, 209-220. Lehigh University, Bethlehem, PA. [61] Moore, W.E. II. 1986. A Programmable Solution for Stepped Crane Columns. AISC Eng. J., 21(2):55-58. [62] Picard, A. and Beaulieu, D. 1987. Design of Diagonal Cross Bracings. Part 1: Theoretical Study. AISC Eng. J., 24(3):122-126. [63] Picard, A. and Beaulieu, D. 1988. Design of Diagonal Cross Bracings. Part 2: Experimental Study. AISC Eng. J., 25(4):156-160. [64] Razzaq, Z. 1983. End Restraint Effect of Column Strength. J. Struct. Div., 109(ST2):314-334. [65] Salmon, C.G., Schenker, L., and Johnston, B.G. 1957. Moment-Rotation Characteristics of Column Anchorage. Trans. ASCE, 122:132-154. [66] Shanmugam, N.E. and Chen, W.F. 1995. An Assessment of K Factor Formulas. AISC Eng. J., 32(3):3-11. [67] Sohal, I.S., Yong, Y.K., and Balagura, P.N. 1995. K -Factor in Plastic and SOIA for Design of Steel Frames, Proceeding International Conference on Stability of Structures, ICSS 95, June 7-9, Coimbatore, India, 411-421. [68] Sugimoto, H. and Chen, W.F. 1982. Small End Restraint Effects on Strength of H-Columns, J. Struct. Div., 108(ST3):661-681. [69] Vinnakota, S. 1982. Planar Strength of Restrained Beam Columns. J. Struct. Div., 108(ST11):2349-2516. [70] Timoshenko, S.P. and Gere, J.M. 1961. Theory of Elastic Stability, 2nd ed., McGraw-Hill, New York. [71] Winter, G. et al. 1948. Buckling of Trusses and Rigid Frames, Cornell Univ. Bull. No. 36, Engineering Experimental Station, Cornell University, Ithaca, NY. [72] Wood, R.H. 1974. Effective Lengths of Columns in Multi-Storey Buildings. Struct. Eng., 52(7,8,9):234-244, 295-302, 341-346. [73] Yura, J.A. 1971. The Effective Length of Columns in Unbraced Frames. AISC Eng. J., 8(2):37-42. [74] Zandonini, R. 1985. Stability of Compact Built-Up Struts: Experimental Investigation and Numerical Simulation (in Italian). Construzioni Metalliche, No. 4.
Further Reading [1] Chen, W.F. and Lui, E.M. 1987. Structural Stability: Theory and Implementation, Elsevier, New York. [2] Chen, W.F., Goto, Y. and Liew, J.Y.R. 1996. Stability Design of Semi-Rigid Frames, John Wiley and Sons, New York. [3] Chen, W.F. and Kim, S.E. 1997. LRFD Steel Design Using Advanced Analysis, CRC Press, Boca Raton, FL.
1999 by CRC Press LLC
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Bjorhovde, R. “Stub Girder Floor Systems” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Stub Girder Floor Systems 18.1 Introduction 18.2 Description of the Stub Girder Floor System 18.3 Methods of Analysis and Modeling
General Observations • Preliminary Design Procedure • Choice of Stub Girder Component Sizes • Modeling of the Stub Girder
18.4 Design Criteria For Stub Girders
Reidar Bjorhovde Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, PA
18.1
General Observations • Governing Sections of the Stub Girder • Design Checks for the Bottom Chord • Design Checks for the Concrete Slab • Design Checks for the Shear Transfer Regions • Design of Stubs for Shear and Axial Load • Design of Stud Shear Connectors • Design of Welds between Stub and Bottom Chord • Floor Beam Connections to Slab and Bottom Chord • Connection of Bottom Chord to Supports • Use of Stub Girder for Lateral Load System • Deflection Checks
18.5 Influence of Method of Construction 18.6 Defining Terms References Further Reading
Introduction
The stub girder system was developed in response to a need for new and innovative construction techniques that could be applied to certain parts of all multi-story steel-framed buildings. Originated in the early 1970s, the design concept aimed at providing construction economies through the integration of the electrical and mechanical service ducts into the part of the building volume that is occupied by the floor framing system [11, 12]. It was noted that the overall height of the floor system at times could be large, leading to significant increases in the overall height of the structure, and hence the steel tonnage for the project. At other times the height could be reduced, but only at the expense of having sizeable web penetrations for the ductwork to pass through. This solution was often accompanied by having to reinforce the web openings by stiffeners, increasing the construction cost even further. The composite stub girder floor system subsequently was developed. Making extensive use of relatively simple shop fabrication techniques, basic elements with limited fabrication needs, simple connections between the main floor system elements and the structural columns, and composite action between the concrete floor slab and the steel load-carrying members, a floor system of significant strength, stiffness, and ductility was devised. This led to a reduction in the amount of structural steel that traditionally had been needed for the floor framing. When coupled with the use of continuous, 1999 by CRC Press LLC
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composite transverse floor beams and the shorter erection time that was needed for the stub girder system, this yielded attractive cost savings. Since its introduction, the stub girder floor system has been used for a variety of steel-framed buildings in the U.S., Canada, and Mexico, ranging in height from 2 to 72 stories. Despite this relatively widespread usage, the analysis techniques and design criteria remain unknown to many designers. This chapter will offer examples of practical uses of the system, together with recommendations for suitable design and performance criteria.
18.2
Description of the Stub Girder Floor System
The main element of the system is a special girder, fabricated from standard hot-rolled wide-flange shapes, that serves as the primary framing element of the floor. Hot-rolled wide-flange shapes are also used as transverse floor beams, running in a direction perpendicular to the main girders. The girder and the beams are usually designed for composite action, although the system does not rely on having composite floor beams, and the latter are normally analyzed as continuous beams. As a result, the transverse floor beams normally use a smaller drop-in span within the positive moment region. This results in further economies for the floor beam design, since it takes advantage of continuous beam action. Allowable stress design (ASD) or load and resistance factor design (LRFD) criteria are equally applicable for the design of stub girders, although LRFD is preferable, since it gives lower steel weights and simple connections. The costs that are associated with an LRFD-designed stub girder therefore tend to be lower. Figure 18.1 shows the elevation of a typical stub girder. It is noted that the girder that is shown
FIGURE 18.1: Elevation of a typical stub girder (one half of span is shown).
makes use of four stubs, oriented symmetrically with respect to the midspan of the member. The locations of the transverse floor beams are assumed to be the quarter points of the span, and the supports are simple. In practice many variations of this layout are used, to the extent that the girders may utilize any number of stubs. However, three to five stubs is the most common choice. The locations of the stubs may differ significantly from the symmetrical case, and the exterior ( = end) stubs may have been placed at the very ends of the bottom chord. However, this is not difficult to 1999 by CRC Press LLC
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address in the modeling of the girder, and the essential requirements are that the forces that develop as a result of the choice of girder geometry be accounted for in the design of the girder components and the adjacent structure. These actual forces are used in the design of the various elements, as distinguished from the simplified models that are currently used for many structural components. The choices of elements, etc., are at the discretion of the design team, and depend on the service requirements of the building as seen from the architectural, structural, mechanical, and electrical viewpoints. Unique design considerations must be made by the structural engineer, for example, if it is decided to eliminate the exterior openings and connect the stubs to the columns in addition to the chord and the slab. Figure 18.1 shows the main components of the stub girder, as follows: 1. 2. 3. 4. 5. 6. 7. 8.
Bottom chord Exterior and interior stubs Transverse floor beams Formed steel deck Concrete slab with longitudinal and transverse reinforcement Stud shear connectors Stub stiffeners Beam-to-column connection
The bottom chord should preferably be a hot-rolled wide-flange shape of column-type proportions, most often in the W12 to W14 series of wide-flange shapes. Other chord cross-sections have been considered [19]; for example, T shapes and rectangular tubes have certain advantages as far as welded attachments and fire protection are concerned, respectively. However, these other shapes also have significant drawbacks. The rolled tube, for example, cannot accommodate the shear stresses that develop in certain regions of the bottom chord. Rather than using a T or a tube, therefore, a smaller W shape (in the W10 series, for example) is most likely the better choice under these conditions. The steel grade for the bottom chord, in particular, is important, since several of the governing regions of the girder are located within this member, and tension is the primary stress resultant. It is therefore possible to take advantage of higher strength steels, and 50-ksi-yield stress steel has typically been the choice, although 65-ksi steel would be acceptable as well. The floor beams and the stubs are mostly of the same size W shape, and are normally selected from the W16 and W18 series of shapes. This is directly influenced by the size(s) of the HVAC ducts that are to be used, and input from the mechanical engineer is essential at this stage. Although it is not strictly necessary that the floor beams and the stubs use identical shapes, it avoids a number of problems if such a choice is made. At the very least, these two components of the floor system should have the same height. The concrete slab and the steel deck constitute the top chord of the stub girder. It is made either from lightweight or normal weight concrete, although if the former is available, even at a modest cost premium, it is preferred. The reason is the lower dead load of the floor, especially since the shores that will be used are strongly influenced by the concrete weight. Further, the shores must support several stories before they can be removed. In other words, the stub girders must be designed for shored construction, since the girder requires the slab to complete the system. In addition, the bending rigidity of the girder is substantial, and a major fraction is contributed by the bottom chord. The reduction in slab stiffness that is prompted by the lower value of the modulus of elasticity for the lightweight concrete is therefore not as important as it may be for other types of composite bending members. Concrete strengths of 3000 to 4000 psi are most common, although the choice also depends on the limit state of the stud shear connectors. Apart from certain long-span girders, some local regions in the 1999 by CRC Press LLC
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slab, and the desired mode of behavior of the slab-to-stub connection (which limits the maximum fc0 value that can be used), the strength of the stub girder is not controlled by the concrete. Consequently, there is little that can gained by using high-strength concrete. The steel deck should be of the composite type, and a number of manufacturers produce suitable types. Normal deck heights are 2 and 3 in., but most floors are designed for the 3-in. deck. The deck ribs are run parallel to the longitudinal axis of the girder, since this gives better deck support on the transverse floor beams. It also increases the top chord area, which lends additional stiffness to a member that can span substantial distances. Finally, the parallel orientation provides a continuous rib trough directly above the girder centerline, improving the composite interaction of the slab and the girder. Due to fire protection requirements, the thickness of the concrete cover over the top of the deck ribs is either 4-3/16 in. (normal weight concrete) or 3-1/4 in. (lightweight concrete). This eliminates the need for applying fire protective material to the underside of the steel deck. Stud shear connectors are distributed uniformly along the length of the exterior and interior stubs, as well as on the floor beams. The number of connectors is determined on the basis of the computed shear forces that are developed between the slab and the stubs. This is in contrast to the current design practice for simple composite beams, which is based on the smaller of the ultimate axial loadcarrying capacity of the slab and the steel beam [2, 3]. However, the simplified approach of current specifications is not applicable to members where the cross-section varies significantly along the length (nonprismatic beams). The computed shear force design approach also promotes connector economy, in the sense that a much smaller number of shear connectors is required in the interior shear transfer regions of the girder [5, 7, 21]. The stubs are welded to the top flange of the bottom chord with fillet welds. In the original uses of the system, the design called for all-around welds [11, 12]; subsequent studies demonstrated that the forces that are developed between the stubs and the bottom chord are concentrated toward the end of the stubs [5, 6, 21]. The welds should therefore be located in these regions. The type and locations of the stub stiffeners that are indicated for the exterior stubs in Figure 18.1, as well as the lack of stiffeners for the interior stubs, represent one of the major improvements that were made to the original stub girder designs. Based on extensive research [5, 21], it was found that simple end-plate stiffeners were as efficient as the traditional fitted ones, and in many cases the stiffeners could be eliminated at no loss in strength and stiffness to the overall girder. Figure 18.1 shows that a simple (shear) connection is used to attach the bottom chord of the stub girder to the adjacent structure (column, concrete building core, etc.). This is the most common solution, especially when a duct opening needs to be located at the exterior end of the girder. If the support is an exterior column, the slab will rest on an edge member; if it is an interior column, the slab will be continuous past the column and into the adjacent bay. This may or may not present problems in the form of slab cracking, depending on the reinforcement details that are used for the slab around the column. The stub girder has sometimes been used as part of the lateral load-resisting system of steel-framed buildings [13, 17]. Although this has certain disadvantages insofar as column moments and the concrete slab reinforcement are concerned, the girder does provide significant lateral stiffness and ductility for the frame. As an example, the maintenance facility for Mexicana Airlines at the Mexico City International Airport, a structure utilizing stub girders in this fashion [17], survived the 1985 Mexico City earthquake with no structural damage. Expanding on the details that are shown in Figure 18.1, Figure 18.2 illustrates the cross-section of a typical stub girder, and Figure 18.3 shows a complete girder assembly with lights, ducts, and suspended ceiling. Of particular note are the longitudinal reinforcing bars. They add flexural strength as well as ductility and stiffness to the girder, by helping the slab to extend its service range. The longitudinal rebars are commonly placed in two layers, with the top one just below the heads of the stud shear connectors. The lower longitudinal rebars must be raised above the deck proper, 1999 by CRC Press LLC
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FIGURE 18.2: Cross-sections of a typical stub girder (refer to Figure 18.1 for section location).
FIGURE 18.3: Elevation of a typical stub girder, complete with ductwork, lights, and suspended ceiling (duct sizes, etc., vary from system to system).
using high chairs or other means. This assures that the bars are adequately confined. The transverse rebars are important for adding shear strength to the slab, and they also help in the shear transfer from the connectors to the slab. The transverse bars also increase the overall ductility of the stub girder, and placing the bars in a herring bone pattern leads to a small improvement in the effective width of the slab. The common choices for stub girder floor systems have been 36- or 50-ksi-yield stress steel, with a preference for the latter, because of the smaller bottom chord size that can be used. Due to its function in the girder, there is no reason why steels such as ASTM A913 (65 ksi) cannot be used for the bottom chord. However, all detail materials (stiffeners, connection angles, etc.) are made from 36-ksi steel. Welding is usually done with 70-grade low hydrogen electrodes, using either the SMAW, 1999 by CRC Press LLC
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FCAW, or GMAW process, and the stud shear connectors are welded in the normal fashion. All of the work is done in the fabricating shop, except for the shear connectors, which are applied in the field, where they are welded directly through the steel deck. The completed stub girders are then shipped to the construction site.
18.3
Methods of Analysis and Modeling
18.3.1
General Observations
In general, any number of methods of analysis may be used to determine the bending moments, shear forces, and axial forces throughout the components of the stub girder. However, it is essential to bear in mind that the modeling of the girder, or, in other words, how the actual girder is transformed into an idealized structural system, should reflect the relative stiffness of the elements. This means that it is important to establish realistic trial sizes of the components, through an appropriate preliminary design procedure. The subsequent modeling will then lead to stress resultants that are close to the magnitudes that can be expected in actual stub girders. Based on this approach, the design that follows is likely to require relatively few changes, and those that are needed are often so small that they have no practical impact on the overall stiffness distribution and final member forces. The preliminary design procedure is therefore a very important step in the overall design. However, it will be shown that by using an LRFD approach, the process is simple, efficient, and accurate.
18.3.2
Preliminary Design Procedure
Using the LRFD approach for the preliminary design, it is not necessary to make any assumptions as regards the stress distribution over the depth of the girder, other than to adhere to the strength model that was developed for normal composite beams [3, 15]. The stress distribution will vary anyway along the span because of the openings. The strength model of Hansell et al. [15] assumes that when the ultimate moment is reached, all or a portion of the slab is failing in compression, with a uniformly distributed stress of 0.85fc0 . The steel shape is simultaneously yielding in tension. Equilibrium is therefore maintained, and the internal stress resultants are determined using first principles. Tests have demonstrated excellent agreement with theoretical analyses that utilize this approach [5, 7, 15, 21]. The LRFD procedure uses load and resistance factors in accordance with the American Institute of Steel Construction (AISC) LRFD specification [3]. The applicable resistance factor is given by the AISC LRFD specification, Section D1, for the case of gross cross-section yielding. This is because the preliminary design is primarily needed to find the bottom chord size, and this component is primarily loaded in tension [5, 7, 10, 21]. The load factors of the LRFD specification are those of the American Society of Civil Engineers (ASCE) load standard [4], for the combination of dead plus live load. The load computations follow the choice of the layout of the floor framing plan, whereby girder and floor beam spans are determined. This gives the tributary areas that are needed to calculate the dead and live loads. The load intensities are governed by local building code requirements or by the ASCE recommendations, in the absence of a local code. Reduced live loads should be used wherever possible. This is especially advantageous for stub girder floor systems, since spans and tributary areas tend to be large. The ASCE load standard [4] makes use of a live load reduction factor, RF , that is significantly simpler to use, and also less conservative than that of earlier codes. The standard places some restrictions on the value of RF , to the effect that the reduced live load cannot be less than 50% of the nominal value for structural members that 1999 by CRC Press LLC
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support only one floor. Similarly, it cannot be less than 40% of the nominal live load if two or more floors are involved. Proceeding with the preliminary design, the stub girder and its floor beam locations determine the magnitudes of the concentrated loads that are to be applied at each of the latter locations. The following illustrative example demonstrates the steps of the solution.
FIGURE 18.4: Stub girder layout used for preliminary design example.
EXAMPLE 18.1:
Given: Figure 18.4 shows the layout of the stub girder for which the preliminary sizes are needed. Other computations have already given the sizes of the floor beam, the slab, and the steel deck. The span of the girder is 40 ft, the distance between adjacent girders is 30 ft, and the floor beams are located at the quarter points. The steel grade remains to be chosen (36- and 50-ksi-yield stress steel are the most common); the concrete is lightweight, with wc = 120 pcf and a compressive strength of fc0 = 4000 psi. Solution Loads: Estimated dead load = 74 psf Nominal live load = 50 psf Live load reduction factor: p RF = 0.25 + 15/ [2 × (30 × 30)] = 0.60 Reduced live load: RLL = 0.60 × 50 = 30 psf Load factors (for D + L combination): For dead load: 1.2 For live load: 1.6 1999 by CRC Press LLC
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Factored distributed loads: Dead Load, DL = 74 × 1.2 = 88.8 psf Live Load, LL = 30 × 1.6 = 48.0 psf Total = 136.8 psf Concentrated factored load at each floor beam location: Due to the locations of the floor beams and the spacing of the stub girders, the magnitude of each load, P , is: P = 136.8 × 30 × 10 = 41.0 kips Maximum factored midspan moment: The girder is symmetric about midspan, and the maximum moment therefore occurs at this location: Mmax = 1.5 × P × 20 − P × 10 = 820 k-ft Estimated interior moment arm for full stub girder cross-section at midspan (refer to Figure 18.2 for typical details): The interior moment arm (i.e., the distance between the compressive stress resultant in the concrete slab and the tensile stress resultant in the bottom chord) is set equal to the distance between the slab centroid and the bottom chord (wide-flange shape) centroid. This is simplified and conservative. In the example, the distance is estimated as Interior moment arm: d = 27.5 in. This is based on having a 14 series W shape for the bottom chord, W16 floor beams and stubs, a 3-in.-high steel deck, and 3-1/4 in. of lightweight concrete over the top of the steel deck ribs (this allows the deck to be used without having sprayed-on fire protective material on the underside). These are common sizes of the components of a stub girder floor system. In general, the interior moment arm varies between 24.5 and 29.5 in., depending on the heights of the bottom chord, floor beams/stubs, steel deck, and concrete slab. Slab and bottom chord axial forces, F (these are the compressive and tensile stress resultants): F = Mmax /d = (820 × 12)/27.5 = 357.9 kips Required cross-sectional area of bottom chord, As : The required cross-sectional area of the bottom chord can now be found. Since the chord is loaded in tension, the φ value is 0.9. It is also important to note that in the vierendeel analysis that is commonly used in the final evaluation of the stub girder, the member forces will be somewhat larger than those determined through the simplified preliminary procedure. It is therefore recommended that an allowance of some magnitude be given for the vierendeel action. This is done most easily by increasing the area, As , by a certain percentage. Based on experience [7, 10], an increase of one-third is suitable, and such has been done in the computations that follow. On the basis of the data that have been developed, the required area of the bottom chord is: (Mmax /d) 4 4 F × = × As = φ × Fy 3 0.9 × Fy 3 1999 by CRC Press LLC
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which gives As values for 36-ksi and 50-ksi steel of As
=
As
=
357.9 4 × = 14.73 in.2 (Fy = 36 ksi) 0.9 × 36 3 4 357.9 × = 10.60 in.2 (Fy = 50 ksi) 0.9 × 50 3
Conclusions: If 36-ksi steel is chosen for the bottom chord of the stub girder, the wide-flange shapes W12x50 and W14x53 will be suitable. If 50-ksi steel is the choice, the sections may be W12x40 or W14x38. Obviously the final decision is up to the structural engineer. However, in view of the fact that the W12 series shapes will save approximately 2 in. in net floor system height, per story of the building, this would mean significant savings if the overall structure is 10 to 15 stories or more. The differences in stub girder strength and stiffness are not likely to play a role [7, 10, 14].
18.3.3
Choice of Stub Girder Component Sizes
Some examples have been given in the preceding for the choices of chord and floor beam sizes, deck height, and slab configuration. These were made primarily on the basis of acceptable geometries, deck size, and fire protection requirements, to mention some examples. However, construction economy is critical, and the following guidelines will assist the user. The data that are given are based on actual construction projects. Economical span lengths for the stub girder range from 30 to 50 ft, although the preferable spans are 35 to 45 ft; 50-ft span girders are erectable, but these are close to the limit where the dead load becomes excessive, which has the effect of making the slab govern the overall design. This is usually not an economical solution. Spans shorter than 30 ft are known to have been used successfully; however, this depends on the load level and the type of structure, to mention the key considerations. Depending on the type and configuration of steel deck that has been selected, the floor beam spacing should generally be maintained between 8 and 12 ft, although larger values have been used. The decisive factor is the ability of the deck to span the distance between the floor beams. The performance of the stub girder is not particularly sensitive to the stub lengths that are used, as long as these are kept within reasonable limits. In this context it is important to observe that it is usually the exterior stub that controls the behavior of the stub girder. As a practical guideline, the exterior stubs are normally 5 to 7 ft long; the interior stubs are considerably shorter, normally around 3 ft, but components up to 5 ft long are known to have been used. When the stub lengths are chosen, it is necessary to bear in mind the actual purpose of the stubs and how they carry the loads on the stub girder. That is, the stubs are loaded primarily in shear, which explains why the interior stubs can be kept so much shorter than the exterior ones. The shear connectors that are welded to the top flange of the stub, the stub web stiffeners, and the welds between the bottom flange of the stub and the top flange of the bottom chord are crucial to the function of the stub girder system. For example, the first application of stub girders utilized fitted stiffeners at the ends and sometimes at midlength of all of the stubs. Subsequent research demonstrated that the midlength stiffener did not perform any useful function, and that only the exterior stubs needed stiffeners in order to provide the requisite web stability and shear capacity [5, 21]. Regardless of the span of the girder, it was found that the interior stubs could be left unstiffened, even when they were made as short as 3 ft [7, 14]. Similar savings were realized for the welds and the shear connectors. In particular, in lieu of allaround fillet welds for the connection between the stub and the bottom chord, the studies showed 1999 by CRC Press LLC
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that a significantly smaller amount of welding was needed, and often only in the vicinity of the stub ends. However, specific weld details must be based on appropriate analyses of the stub, considering overturning, weld capacity at the tension end of the stub, and adequate ability to transfer shear from the slab to the bottom chord.
18.3.4
Modeling of the Stub Girder
The original work of Colaco [11, 12] utilized a vierendeel modeling scheme for the stub girder to arrive at a set of stress resultants, which in turn were used to size the various components. Elastic finite element analyses were performed for some of the girders that had been tested, mostly to examine local stress distributions and the correlation between test and theory. However, the finite element solution is not a practical design tool. Other studies have examined approaches such as nonprismatic beam analysis [6, 21] and variations of the finite element method [16]. The nonprismatic beam solution is relatively simple to apply. On the other hand, it is not as accurate as the vierendeel approach, since it tends to overlook some important local effects and overstates the service load deflections [5, 21]. On the whole, therefore, the vierendeel modeling of the stub girder has been found to give the most accurate and consistent results, and the correlation with test results is good [5, 6, 11, 14, 21]. Finally, it offers the best physical similarity with actual girders; many designers have found this to be an important advantage. There are no “simple” methods of analysis that can be used to find the bending moments, shear forces, and axial forces in vierendeel girders. Once the preliminary sizing has been accomplished, a computer solution is required for the girder. In general, all that is required for the vierendeel evaluation is a two-dimensional plane frame program for elastic structural analysis. This gives moments, shears, and axial forces, as well as deflections, joint rotations, and other displacement characteristics. The stress resultants are used to size the girder and its elements and connections; the displacements reflect the serviceability of the stub girder. Once the stress resultants are known, the detailed design of the stub girder can proceed. A final run-through of the girder model should then be done, using the components that were chosen, to ascertain that the performance and strength are sufficient in all respects. Under normal circumstances no alterations are necessary at this stage. As an illustration of the vierendeel modeling of a stub girder, the girder itself is shown in Figure 18.5a and the vierendeel model in Figure 18.5b. The girder is the same as the one used for the preliminary design example. It has four stubs and is symmetrical about midspan; therefore, only half is illustrated. The boundary conditions are shown in Figure 18.5b. The bottom chord of the model is assigned a moment of inertia equal to the major axis I value, Ix , of the wide-flange shape that was chosen in the preliminary design. However, some analysts believe that since the stub is welded to the bottom chord, a portion of its flexural stiffness should be added to that of the moment of inertia of the wide-flange shape [5, 7, 14, 21] This approach is identical to treating the bottom chord W shape as if it has a cover plate on its top flange. The area of this cover plate is the same as the area of the bottom flange of the stub. This should be done only in the areas where the stubs are placed. In the regions of the interior and exterior stubs it is therefore realistic to increase the moment of inertia of the bottom chord by the parallel-axis value of Af × df2 , where Af designates the area of the bottom flange of the stub and df is the distance between the centroids of the flange plate and the W shape. The contribution to the overall stub girder stiffness is generally small. The bending stiffness of the top vierendeel chord equals that of the effective width portion of the slab. This should include the contributions of the steel deck as well as the reinforcing steel bars that are located within this width. In particular, the influence of the deck is important. The effective width is determined from the criteria in the AISC LRFD specification, Section I3.1 [3]. It is noted 1999 by CRC Press LLC
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FIGURE 18.5: An actual stub girder and its vierendeel model (due to symmetry, only one half of the span is shown). that these were originally developed on the basis of analyses and tests of prismatic composite beams. The approach has been found to give conservative results [5, 21], but should continue to be used until more accurate criteria are available. In the computations for the slab, the cross-section is conveniently subdivided into simple geometrical shapes. The individual areas and moments of inertia are determined on the basis of the usual transformation from concrete to steel, using the modular ratio n = E/Ec , where E is the modulus of elasticity of the steel and Ec is that of concrete. The latter must reflect the density of the concrete that is used, and can be computed from [1]: p (18.1) Ec = 33 × wc1.5 × fc0 The shear connectors used for the stub are required to develop 100% interaction, since the design is based on the computed shear forces, rather than the axial capacity of the steel beam or the concrete slab, as is used for prismatic beams in the AISC Specifications [2, 3]. However, it is neither common nor proper to add the moment of inertia contribution of the top flange of the stub to that of the slab, contrary to what is done for the bottom chord. The reason for this is that dissimilar materials are joined, and some local concrete cracking and/or crushing can be expected to take place around the shear connectors. The discretization of the stubs into vertical vierendeel girder components is relatively straightforward. Considering the web of the stub and any stiffeners, if applicable (for exterior stubs, most commonly, since interior stubs usually can be left unstiffened), the moment of inertia about an axis that is perpendicular to the plane of the web is calculated. As an example, Figure 18.6 shows the stub and stiffener configuration for a typical case. The stub is a 5-ft long W16x26 with 5-1/2x1/2-in. end-plate stiffeners. The computations give: Moment of inertia about the Z − Z axis: h i IZZ = 0.25 × (60)3 /12 + 2 × 5.5 × 0.5 × (30)2 = 1999 by CRC Press LLC
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9450 in.4
FIGURE 18.6: Horizontal cross-section of stub with stiffeners. Depending on the number of vierendeel truss members that will represent the stub in the model, the bending stiffness of each is taken as a fraction of the value of IZZ . For the girder shown in Figure 18.5, where the stub is discretized as three vertical members, the magnitude of Ivert is found as: Moment of inertia of vertical member: Ivert = IZZ /(no. of verticals ) = 9450/3 = 3150 in.4 The cross-sectional area of the stub, including the stiffeners, is similarly divided between the verticals: Area of vertical member: Avert
= [Aweb + 2 × Ast ] /(no. of verticals) = [0.25 × (60 − 2 × 0.5) + 2 × 5.5 × 0.5] /3 = 6.75 in.2
Several studies have aimed at finding the optimum number of vertical members to use for each stub. However, the strength and stiffness of the stub girder are only insignificantly affected by this choice, and a number between 3 and 7 is usually chosen. As a rule of thumb, it is advisable to have one vertical per foot length of stub, but this should serve only as a guideline. The verticals are placed at uniform intervals along the length of the stub, usually with the outside members close to the stub ends. Figure 18.5 illustrates the approach. As for end conditions, these vertical members are assumed to be rigidly connected to the top and bottom chords of the vierendeel girder. One vertical member is placed at each of the locations of the floor beams. This member is assumed to be pinned to the top and bottom chords, as shown in Figure 18.5, and its stiffness is conservatively set equal to the moment of inertia of a plate with a thickness equal to that of the web of the floor beam and a length equal to the beam depth. In the example, tw = 0.25 in.; the beam depth is 15.69 in. This gives a moment of inertia of i h 15.69 × 0.253 /12 = 0.02 in.4 and the cross-sectional area is (15.69 × 0.25) = 3.92 in.2 The vierendeel model shown in Figure 18.5b indicates that the portion of the slab that spans across the opening between the exterior end of the exterior stub and the support for the slab (a column, or a corbel of the core of the structural frame) has been neglected. This is a realistic simplification, considering the relatively low rigidity of the slab in negative bending. 1999 by CRC Press LLC
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Figure 18.5b also shows the support conditions that are used as input data for the computer analysis. In the example, the symmetrical layout of the girder and its loads make it necessary to analyze only one-half of the span. This cannot be done if there is any kind of asymmetry, and the entire girder must then be analyzed. For the girder that is shown, it is known that only vertical displacements can take place at midspan; horizontal displacements and end rotations are prevented at this location. At the far ends of the bottom chord only horizontal displacements are permitted, and end rotations are free to occur. The reactions that are found are used to size the support elements, including the bottom chord connections and the column. The structural analysis results are shown in Figure 18.7, in terms of the overall bending moment, shear force, and axial force distributions of the vierendeel model given in Figure 18.5b. Figure 18.7d repeats the layout details of the stub girder, to help identify the locations of the key stress resultant magnitudes with the corresponding regions of the girder.
FIGURE 18.7: Distributions of bending moments, shear forces, and axial forces in a stub girder (see Figure 18.5) (dead load = 74 psf; nominal live load = 50 psf).
The design of the stub girder and its various components can now be done. This must also include deflection checks, even though research has demonstrated that the overall design will never be gov1999 by CRC Press LLC
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erned by deflection criteria [7, 14]. However, since the girder has to be built in the shored condition, the girder is often fabricated with a camber, approximately equal to the dead load deflection [7, 10].
18.4
Design Criteria For Stub Girders
18.4.1
General Observations
In general, the design of the stub girder and its components must consider overall member strength criteria as well as local checks. For most of these, the AISC Specifications [2, 3] give requirements that address the needs. Further, although LRFD and ASD are equally applicable in the design of the girder, it is recommended that LRFD be used exclusively. The more rational approach of this specification makes it the method of choice. In several important areas there are no standardized rules that can be used in the design of the stub girder, and the designer must rely on rational engineering judgment to arrive at satisfactory solutions. This applies to the parts of the girder that have to be designed on the basis of computed forces, such as shear connectors, stiffeners, stub-to-chord welds, and slab reinforcement. The modeling and evaluation of the capacity of the central portion of the concrete slab are also subject to interpretation. However, the design recommendations that are given in the following are based on a wide variety of practical and successful applications. It is again emphasized that the design throughout is based on the stress resultants that have been determined in the vierendeel or other analysis, rather than on idealized code criteria. However, the capacities of materials and fasteners, as well as the requirements for the stability and strength of tension and compression members, adhere strictly to the AISC Specifications. Any interpretations that have been made are always to the conservative side.
18.4.2
Governing Sections of the Stub Girder
Figures 18.5 and 18.7 show certain circled numbers at various locations throughout the span of the stub girder. These reflect the sections of the girder that are the most important, for one reason or another, and are the ones that must be examined to determine the required member size, etc. These are the governing sections of the stub girder and are itemized as follows: 1. Points 1, 2, and 3 indicate the critical sections for the bottom chord. 2. Points 4, 5, and 6 indicate the critical sections for the concrete slab. 3. Point 7, which is a region rather than a specific point, indicates the critical shear transfer region between the slab and the exterior stub. The design checks that must be made for each of these areas are discussed in the following.
18.4.3
Design Checks for the Bottom Chord
The size of the bottom chord is almost always governed by the stress resultants at midspan, or point 3 in Figures 18.5 and 18.7. This is also why the preliminary design procedure focused almost entirely on determining the required chord cross-section at this location. As the stress resultant distributions in Figure 18.7 show, the bottom chord is subjected to combined positive bending moment and tensile force at point 3, and the design check must consider the beam-tension member behavior in this area. The design requirements are given in Section H1.1, Eqs. (H1-1a) and (H1-1b), of the AISC LRFD Specification [3]. Combined bending and tension must also be evaluated at point 2, the exterior end of the interior stub. The local bending moment in the chord is generally larger here than at midspan, but the axial 1999 by CRC Press LLC
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force is smaller. Only a computation can confirm whether point 2 will govern in lieu of point 3. Further, although the location at the interior end of the exterior stub (point 2a) is rarely critical, the combination of negative moment and tensile force should be evaluated. At point 1 of the bottom chord, which is located at the exterior end of the exterior stub, the axial force is equal to zero. At this location the bottom chord must therefore be checked for pure bending, as well as shear. The preceding applies only to a girder with simple end supports. When it is part of the lateral load-resisting system, axial forces will exist in all parts of the chord. These must be resisted by the adjacent structural members.
18.4.4
Design Checks for the Concrete Slab
The top chord carries varying amounts of bending moment and axial force, as illustrated in Figure 18.7, but the most important areas are indicated as points 4 to 6. The axial forces are always compressive in the concrete slab; the bending moments are positive at points 5 and 6, but negative at point 4. As a result, this location is normally the one that governs the performance of the slab, not the least because the reinforcement in the positive moment region includes the substantial cross-sectional area of the steel deck. The full effective width of the slab must be analyzed for combined bending and axial force at all of points 4 through 6. Either the composite beam-column criteria of the AISC LRFD specification [3] or the criteria of the reinforced concrete structures code of the American Concrete Institute (ACI) [1] may be used for this purpose.
18.4.5
Design Checks for the Shear Transfer Regions
Region 7 is the shear transfer region between the concrete slab and the exterior stub, and the combined shear and longitudinal compressive capacity of the slab in this area must be determined. The shear transfer region between the slab and the interior stub always has a smaller shear force. Region 7 is critical, and several studies have shown that the slab in this area will fail in a combination of concrete crushing and shear [5, 6, 7, 21]. The shear failure zone usually extends from corner to corner of the steel deck, over the top of the shear connectors, as illustrated in Figure 18.8. This also
FIGURE 18.8: Shear and compression failure regions in the slab of the stub girder. 1999 by CRC Press LLC
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emphasizes why the placement of the longitudinal reinforcing steel bars in the central flute of the steel deck is important, as well as the location of the transverse bars: both groups should be placed just below the level of the top of the shear connectors (see Figure 18.2). The welded wire mesh reinforcement that is used as a matter of course, mostly to control shrinkage cracking in the slab, also assists in improving the strength and ductility of the slab in this region.
18.4.6
Design of Stubs for Shear and Axial Load
The shear and axial force distributions indicate the governing stress resultants for the stub members. It is important to note that since the vierendeel members are idealized from the real (i.e., continuous) stubs, bending is not a governing condition. Given the sizes and locations of the individual vertical members that make up the stubs, the design checks are easily made for axial load and shear. For example, referring to Figure 18.7, it is seen that the shear and axial forces in the exterior and interior stubs, and the axial forces in the verticals that represent the floor beams, are the following: Exterior stub verticals: Shear forces: Axial forces:
103 kips −18 kips
63 kips 0.4 kips
99 kips 3 kips
Shear forces: Axial forces:
38 kips −5 kips
19 kips 0.8 kips
20 kips 4 kips
Interior stub verticals:
Floor beam verticals: Exterior: Axial force = −39 kips Interior: Axial force = −12 kips Shear forces are zero in these members. The areas and moments of inertia of the verticals are known from the modeling of the stub girder. Figure 18.7 also shows the shear and axial forces in the bottom and top chords, but the design for these elements has been addressed earlier in this chapter. The design checks that are made for the stub verticals will also indicate whether there is a need for stiffeners for the stubs, since the evaluations for axial load capacity should always first be made on the assumption that there are no stiffeners. However, experience has shown that the exterior stubs always must be stiffened; the interior stubs, on the other hand, will almost always be satisfactory without stiffeners, although exceptions can occur. The axial forces that are shown for the stub verticals in the preceding are small, but typical, and it is clear that in all probability only the exterior end of the exterior stub really requires a stiffener. This was examined in one of the stub girder research studies, where it was found that a single stiffener would suffice, although the resulting lack of structural symmetry gave rise to a tensile failure in the unstiffened area of the stub [21]. Although this occurred at a very late stage in the test, the type of failure represents an undesirable mode of behavior, and the use of single stiffeners therefore was discarded. Further, by reason of ease of fabrication and erection, stiffeners should always be provided at both stub ends. It is essential to bear in mind that if stiffeners are required, the purpose of such elements is to add to the area and moment of inertia of the web, to resist the axial load that is applied. There is no need to provide bearing stiffeners, since the load is not transmitted in this fashion. The most economical solution is to make use of end-plate stiffeners of the kind that is shown in Figure 18.1; extensive research evaluations showed that this was the most efficient and economical choice [5, 6, 21]. 1999 by CRC Press LLC
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The vertical stub members are designed as columns, using the criteria of Section E1 of the AISC Specification [3]. For a conservative solution, an effective length factor of 1.0 may be used. However, it is more realistic to utilize a K value of 0.8 for the verticals of the stubs, recognizing the end restraint that is provided by the connections between the chords and the stubs. The K-factor for the floor beam verticals must be 1.0, due to the pinned ends that are assumed in the modeling of these components, as well as the flexibility of the floor beam itself in the direction of potential buckling of the vertical member.
18.4.7
Design of Stud Shear Connectors
The shear forces that must be transferred between the slab and the stubs are given by the vierendeel girder shear force diagram. These are the factored shear force values which are to be resisted by the connectors. The example shown in Figure 18.7 indicates the individual shear forces for the stub verticals, as listed in the preceding section. However, in the design of the overall shear connection, the total shear force that is to be transmitted to the stub is used, and the stud connectors are then distributed uniformly along the stub. The design strength of each connector is determined in accordance with Section I5.3 of the LRFD Specification [3], including any deck profile reduction factor (Section I3.5). Analyzing the girder whose data are given in Figure 18.7, the following is known: Exterior stub: Total shear force = Ves = 103 + 63 + 99 = 265 kips Interior stub: Total shear force = Vis = 38 + 19 + 20 = 77 kips The nominal strength, Qn , of the stud shear connectors is given by Eq. (I5-1) in Section I5.3 of the LRFD Specification, thus: p (18.2) Qn = 0.5 × Asc fc0 × Ec ≤ Asc × Fu where Asc is the cross-sectional area of the stud shear connector, fc0 and Ec are the compressive strength and modulus of elasticity of the concrete, and Fu is the specified minimum tensile strength of the stud shear connector steel, or 60 ksi (ASTM A108). In the equation for Qn , the left-hand side reflects the ultimate limit state of shear yield failure of the connector; the right-hand side gives the ultimate limit state of tension fracture of the stud. Although shear almost always governs and is the desirable mode of behavior, a check has to be made to ensure that tension fracture will not take place. This as achieved by the appropriate value of Ec , setting Fu = 60 ksi, and solving for fc0 from Equation 18.2. The requirement that must be satisfied in order for the stud shear limit state to govern is given by Equation 18.3: fc0 ≤
57,000 wc
(18.3)
This gives the limiting values for concrete strength as related to the density; data are given in Table 18.1. For concrete with wc = 120 pcf and fc0 = 4,000 psi, as used in the design example, Ec = 2,629,000 psi. Using 3/4-in. diameter studs, the nominal shear capacity is: h ip h i Qn = 0.5 π(0.75)2 /4 (4 × 2,629) ≤ π(0.75)2 /4 60 which gives Qn = 22.7 kips < 26.5 kips 1999 by CRC Press LLC
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TABLE 18.1 Concrete Strength Limitations for Ductile Shear Connector Failure Concrete density, wc (pcf)
Maximum concrete strength, fc0 (psi)
145 (= N W ) 120 110 100 90
4000 4800 5200 5700 6400
Note: N W = normal weight.
The LRFD Specification [3] does not give a resistance factor for shear connectors, on the premise that the φ value of 0.85 for the overall design of the composite member incorporates the stud strength variability. This is not satisfactory for composite members such as stub girders and composite trusses. However, a study was carried out to determine the resistance factors for the two ultimate limit states for stud shear connectors [20]. Briefly, on the basis of extensive analyses of test data from a variety of sources, and using the Qn equation as the nominal strength expression, the values of the resistance factors that apply to the shear yield and tension fracture limit states, respectively, are: Stud shear connector resistance factors: Limit state of shear yielding: φconn = 0.90 Limit state of tension fracture: φconn = 0.75 The required number of shear connectors can now be found as follows, using the total stub shear forces, Ves and Vis , computed earlier in this section: Exterior stub: nes
= Ves /(0.9 × Qn ) = Ves /(φconn Qn ) = 265/(0.9 × 22.7) = 13.0
i.e., use nes = 14-3/4-in. diameter stud shear connectors, placed in pairs and distributed uniformly along the length of the top flange of each of the exterior stubs. Interior stub: nis
= Vis /(0.9 × Qn ) = Vis /(φconn Qn ) = 77/(0.9 × 22.7) = 3.8
i.e., use nis = 4-3/4-in. diameter stud shear connectors, placed singly and distributed uniformly along the length of the top flange of each of the interior stubs. Considering the shear forces for the stub girder of Figures 18.5 and 18.7, the number of connectors for the exterior stub is approximately three times that for the interior one, as expected. Depending on span, loading, etc., there are instances when it will be difficult to fit the required number of studs on the exterior stub, since typical usage entails a double row, spaced as closely as permitted (four diameters in any direction [Section I5.6, AISC LRFD Specification [3]]). Several avenues may be followed to remedy such a problem; the easiest one is most likely to use a higher strength concrete, as long as the limit state requirements for Qn and Table 18.1 are satisfied. This entails only minor reanalysis of the girder. 1999 by CRC Press LLC
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18.4.8
Design of Welds between Stub and Bottom Chord
The welds that are needed to fasten the stubs to the top flange of the bottom chord are primarily governed by the shear forces that are transferred between these components of the stub girder. The shear force distribution gives these stress resultants, which are equal to those that must be transferred between the slab and the stubs. Thus, the factored forces, Ves and Vis , that were developed in Section 18.4.7 are used to size the welds. Axial loads also act between the stubs and the chord; these may be compressive or tensile. In Figure 18.7 it is seen that the only axial force of note occurs in the exterior vertical of the exterior stub (load = 18 kips); the other loads are very small compressive or tensile forces. Unless a significant tensile force is found in the analysis, it will be a safe simplification to ignore the presence of the axial forces insofar as the weld design is concerned. The primary shear forces that have to be taken by the welds are developed in the outer regions of the stubs, although it is noted that in the case of Figure 18.5, the central vertical element in both stubs carries forces of some magnitude (63 and 19 kips, respectively). However, this distribution is a result of the modeling of the stubs; analyses of girders where many more verticals were used have confirmed that the major part of the shear is transferred at the ends [7, 10, 21]. The reason is that the stub is a full shear panel, where the internal moment is developed through stress resultants that act at points toward the ends, in a form of bending action. Tests have also verified this characteristic of the girder behavior [6, 21]. Finally, concentrating the welds at the stub ends will have significant economic impact [5, 7, 21]. In view of these observations, the most effective placement of the welds between the stubs and the bottom chord is to concentrate them across the ends of the stubs and along a short distance of both sides of the stub flanges. For ease of fabrication and structural symmetry, the same amount of welding should be placed at both ends, although the forces are always smaller at the interior ends of the stubs. Such U-shaped welds were used for a number of the full-size girders that were tested [5, 6, 21], with only highly localized yielding occurring in the welds. A typical detail is shown in Figure 18.9; this reflects what is recommended for use in practice. Prior to the research that led to the change of the welded joint design, the stubs were welded with all-around fillet welds for the exterior as well as the interior elements. The improved, U-shaped detail provided for weld metal savings of approximately 75% for interior stubs and around 50% for exterior stubs. For the sample stub girder, W16x26 shapes are used for the stubs. The total forces to be taken by the welds are: Exterior stub: Ves = 265 kips Interior stub: Vis = 77 kips Using E70XX electrodes and 5/16-in. fillet welds (the fillet weld size must be smaller than the thickness of the stub flange, which is 3/8 in. for the W16x26), the total weld length for each stub is Lw , given by (refer to Figure 18.9): Lw = 2(bf s + 2`) since U-shaped welds of length (bf s +2`) are placed at each stub end. The total weld lengths required for the stub girder in question are therefore: Exterior stub: (Lw )es
1999 by CRC Press LLC
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=
Ves /(0.707aφw Fw )
=
265/ [0.707(5/16) × 0.75(0.6 × 70)] = 38.1 in.
FIGURE 18.9: Placement of U-shaped fillet weld for attachment at each end of stub to bottom chord.
Interior stub: (Lw )is
= Vis /(0.707aφw Fw ) = 77/ [0.707(5/16) × 0.75(0.6 × 70)] = 11.1 in.
In the above expressions, a = 5/16 in. = fillet weld size, φw = 0.75, and Fw = 0.6FEXX = 0.6 × 70 = 42 ksi for E70XX electrodes (Table J2.3, AISC LRFD Specification [3]). The total U-weld lengths at each stub end are therefore: Exterior stub: LU es = 19.1 in. Interior stub: LU is = 5.6 in. With a flange width for the W16x26 of 5.50 in., the above lengths can be simplified as: LU es = 5.50 + 7.0 + 7.0 where `es is chosen as 7.0 in. For the interior stub: LU is = 5.50 + 2.0 + 2.0 where `is is chosen as 2.0 in. The details chosen are a matter of judgment. In the example, the interior stub for all practical purposes requires no weld other than the one across the flange, although at least a minimum weld return of 1/2 in. should be used. 1999 by CRC Press LLC
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18.4.9
Floor Beam Connections to Slab and Bottom Chord
In the vierendeel model, the floor beam is represented as a pinned-end compression member. It is designed using a K-factor of 1.0, and the floor beam web by itself is almost always sufficient to take the axial load. However, the floor beam must be checked for web crippling and web buckling under shoring conditions. No shear is transferred from the beam to the slab or the bottom chord. In theory, therefore, any attachment device between the floor beam and the other components should not be needed. However, due to construction stability requirements, as well as the fact that the floor beam usually is designed for composite action normal to the girder, fasteners are needed. In practice, these are not actually designed; rather, one or two stud shear connectors are placed on the top flange of the beam, and two high-strength bolts attach the lower flange to the bottom chord.
18.4.10
Connection of Bottom Chord to Supports
In the traditional use of stub girders, the girder is supported as a simple beam, and the bottom chord end connections need to be able to transfer vertical reactions to the supports. The latter structural elements may be columns, or the girder may rest on corbels or other types of supports that are part of the concrete core of the building. For both of these cases the reactions that are to be carried to the adjacent structure are given by the analysis, and the response needs for the supports are clear. Any shear-type beam connections may be used to connect the bottom chord to a column or a corbel or similar bracket. It is important to ascertain that the chord web shear capacity is sufficient, including block shear (Section J5 of the AISC LRFD Specification [3]). Some designers prefer to use slotted holes for the connections, and to delay the final tightening of the bolts until after the shoring has been removed. This is done on the premise that the procedure will leave the slab essentially stress free from the construction loads, leading to less cracking in the slab during service. Other designers specify additional slab reinforcement to take care of any cracking problem. Experience has shown that both methods are suitable. The slab may be supported on an edge beam or similar element at the exterior side of the floor system. There is no force transfer ability required of this support. In the interior of the building the slab will be continuously cast across other girders and around columns; this will almost always lead to some cracking, both in the vicinity of the columns as well as along beams and girders. With suitable placement of floor slab joints, this can be minimized, and appropriate transverse reinforcement for the slab will reduce, if not eliminate, the longitudinal cracks. Data on the effects of various types of cracks in composite floor systems are scarce. Current opinion appears to be that the strength may not be influenced very much. In any case, the mechanics of the short- and long-term service response of composite beams is not well understood. Recent studies have developed models for the cracking mechanism and the crack propagation [18]; the correlation with a wide variety of laboratory tests is good. However, a comprehensive study of concrete cracking and its implications for structural service and strength needs to be undertaken.
18.4.11
Use of Stub Girder for Lateral Load System
The stub girder was originally conceived only as being part of the vertical load-carrying system of structural frames, and the use of simple connections, as discussed in Section 18.4.9, came from this development. However, because a deep, long-span member can be very effective as a part of the lateral load-resisting system for a structure, several attempts have been made to incorporate the stub girder into moment frames and similar systems. The projects of Colaco in Houston [13] and Martinez-Romero [17] in Mexico City were successful, although the designers noted that the cost premium could be substantial. For the Colaco structure, his applications reduced drift, as expected, but gave much more complex 1999 by CRC Press LLC
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beam-to-column connections and reinforcement details in the slab around the columns. Thus, the exterior stubs were moved to the far ends of the girders, and moment connections were designed for the full depth. For the Mexico City building, the added ductility was a prime factor in the survival of the structure during the 1985 earthquake. The advantages of using the stub girders in moment frames are obvious. Some of the disadvantages have been outlined; in addition, it must be recognized that the lack of room for perimeter HVAC ducts may be undesirable. This can only be addressed by the mechanical engineering consultant. As a general rule, a designer who wishes to use stub girders as part of the lateral load-resisting system should examine all structural effects, but also incorporate nonstructural considerations such as are prompted by HVAC and electronic communication needs.
18.4.12
Deflection Checks
The service load deflections of the stub girder are needed for several purposes. First, the overall dead load deflection is used to assess the camber requirements. Due to the long spans of typical stub girders, as well as the flexibility of the framing members and the connections during construction, it is important to end up with a floor system that is as level as possible by the time the structure is ready to be occupied. Thus, the girders must be built in the shored condition, and the camber should be approximately equal to the dead load deflection. Second, it is essential to bear in mind that each girder will be shored against a similar member at the level below the current construction floor. This member, in turn, is similarly shored, albeit against a girder whose stiffness is greater, due to the additional curing time of the concrete slab. This has a cumulative effect for the structure as a whole, and the dead load deflection computations must take this response into account. In other words, the support for the shores is a flexible one, and deflections therefore will occur in the girder as a result of floor system movements of the structure at levels in addition to the one under consideration. Although this is not unique to the stub girder system, the span lengths and the interaction with the frame accentuate the influence on the girder design. Depending on the structural system, it is also likely that the flexibility of the columns and the connections will add to the vertical displacements of the stub girders. The deflection calculations should incorporate these effects, preferably by utilizing realistic modified Ec values and determining displacements as they occur in the frame. Thus, the curing process for the concrete might be considered, since the strength development as a function of time is directly related to the value of Ec [1]. This is a subject that is open for study, although similar criteria have been incorporated in studies of the strength and behavior of composite frames [8, 9]. However, detailed evaluations of the influence of time-dependent stiffness still need to be made for a wide variety of floor systems and frames. The cumulative deflection effects can be significant for the construction of the building, and consequently also must enter into the contractor’s planning. This subject is addressed briefly in Section 18.5. Third, the live load deflections must be determined to assess the serviceability of the floor system under normal operating conditions. However, several studies have demonstrated that such displacements will be significantly smaller than the L/360 requirement that is normally associated with live load deflections [6, 7, 10, 14, 21]. It is therefore rarely possible to design a girder that meets the strength and the deflection criteria simultaneously [14]. In other words, strength governs the overall design. Finally, although they rarely play a role in the overall response of the stub girder, the deflections and end rotations of the slab across the openings of the girder should also be checked. This is primarily done to assess the potential for local cracking, especially at the stub ends and at the floor beams. However, proper placement of the longitudinal girder reinforcement is usually sufficient to prevent problems of this kind, since the deformations tend to be small. 1999 by CRC Press LLC
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18.5
Influence of Method of Construction
A number of construction-related considerations have already been addressed in various sections of this chapter. The most important ones relate to the fact that the stub girders must be built in the shored condition. The placement and removal of the shores may have a significant impact on the performance of the member and the structure as a whole. In particular, too early shore removal may lead to excessive deflections in the girders at levels above the one where the shores were located. This is a direct result of the low stiffness of “green” concrete. It can also lead to “ponding” of the concrete slab, producing larger dead loads than accounted for in the original design. Finally, larger girder deflections can be translated into an “inward pulling” effect on the columns of the frame. However, this is clearly a function of the framing system. On the other hand, the use of high early strength cement and similar products can reduce this effect significantly. Further, since the concrete usually is able to reach about 75% of the 28-day strength after 7 to 10 days, the problem is less severe than originally thought [5, 7, 10]. In any case, it is important for the structural engineer to interact with the general contractor, in order that the influence of the method of construction on the girders as well as the frame can be quantified, however simplistic the analysis procedure may be. Due to the larger loads that can be expected for the shores, the latter must either be designed as structural members or at least be evaluated by the structural engineer. The size of the shores is also influenced by the number of floors that are to have these supports left in place. As a general rule, when stub girders are used for multi-story frames, the shores should be left in place for at least three floor levels. Some designers prefer a larger number; however, any choices of this kind should be based on computations for sizes and effects. Naturally, the more floors that are specified, the larger the shores will have to be.
18.6
Defining Terms
Composite: Steel and concrete acting in concert. Formed steel deck: A thin sheet of steel shaped into peaks and valleys called corrugations. Green concrete: concrete that has just been placed. HVAC: Heating, ventilating, and air conditioning. Lightweight: Refers to concrete with unit weights between 90 and 120 pcf. Normal weight: Refers to concrete with unit weights of 145 lb per cubic foot (pcf). Prismatic beam: A beam with a constant size cross-section over the full length. Rebar: An abbreviated name for reinforcing steel bars. Serviceability: The ability of a structure to function properly under normal operating condictions. Shoring: Temporary support. Vierendeel girder: A girder with top and bottom chords attached to each other through fully welded connections to vertical (generally) members.
References [1] American Concrete Institute. 1995. Building Code Requirements for Reinforced Concrete, ACI Standard No. 318-95, ACI, Detroit, MI. [2] American Institute of Steel Construction. 1989. Specification for the Allowable Stress Design, Fabrication, and Erection of Structural Steel for Buildings, 9th ed., AISC, Chicago, IL. 1999 by CRC Press LLC
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[3] American Institute of Steel Construction. 1993. Specification for the Load and Resistance Factor Design, Fabrication, and Erection of Structural Steel for Buildings, 2nd ed., AISC, Chicago, IL. [4] American Society of Civil Engineers. 1995. Minimum Design Loads for Buildings and Other Structures, ASCE/ANSI Standard No. 7-95, ASCE, New York. [5] Bjorhovde, R., and Zimmerman, T.J. 1980. Some Aspects of Stub Girder Design, AISC Eng. J., 17(3), Third Quarter, September (pp. 54-69). [6] Bjorhovde, R. 1981. Full-Scale Test of a Stub Girder, Report submitted to Dominion Bridge Company, Calgary, Alberta, Canada. Department of Civil Engineering, University of Alberta, Edmonton, Alberta, Canada, June. [7] Bjorhovde, R. 1985. Behavior and Strength of Stub Girder Floor Systems, in Composite and Mixed Construction, ASCE Special Publication, ASCE, New York. [8] Bjorhovde, R. 1987. Design Considerations for Composite Frames, Proceedings 2nd International and 5th Mexican National Symposium on Steel Structures, IMCA and SMIE, Morelia, Michoacan, Mexico, November 23-24. [9] Bjorhovde, R. 1994. Concepts and Issues in Composite Frame Design, Steel Structures, Journal of the Singapore Society for Steel Structures, 5(1), December (pp. 3-14). [10] Chien, E.Y.L. and Ritchie, J.K. 1984. Design and Construction of Composite Floor Systems, Canadian Institute of Steel Construction (CISC), Willowdale (Toronto), Ontario, Canada. [11] Colaco, J.P. 1972. A Stub Girder System for High-Rise Buildings, AISC Eng. J., 9(2), Second Quarter, July (pp. 89-95). [12] Colaco, J. P. 1974. Partial Tube Concept for Mid-Rise Structures, AISC Eng. J., 11(4), Fourth Quarter, December (pp. 81-85). [13] Colaco, J.P. and Banavalkar, P.V. 1979. Recent Uses of the Stub Girder System, Proceedings 1979 National Engineering Conference, American Institute of Steel Construction, Chicago, IL, May. [14] Griffis, T.C. 1983. Stiffness Criteria for Stub Girder Floor Systems, M.S. thesis, University of Arizona, Tucson, AZ. [15] Hansell, W.C., Galambos, T.V., Ravindra, M.K., and Viest, I.M. 1978. Composite Beam Criteria in LRFD, J. Structural Div., ASCE, 104(ST9), September (pp. 1409-1426). [16] Hrabok, M.M. and Hosain, M.U. 1978. Analysis of Stub Girders Using Sub-Structuring, Intl. J. Computers and Structures, 8(5), 615-619. [17] Martinez-Romero, E. 1983. Continuous Stub Girder Structural System for Floor Decks, Technical report, EMRSA, Mexico City, Mexico, February. [18] Morcos, S.S. and Bjorhovde, R. 1995. Fracture Modeling of Concrete and Steel, J. Structural Eng., ASCE, 121(7), 1125-1133. [19] Wong, A.F. 1979. Conventional and Unconventional Composite Floor Systems, M.Eng. thesis, University of Alberta, Edmonton, Alberta, Canada. [20] Zeitoun, L.A. 1984. Development of Resistance Factors for Stud Shear Connectors, M.S. thesis, University of Arizona, Tucson, AZ. [21] Zimmerman, T.J. and Bjorhovde, R. 1981. Analysis and Design of Stub Girders, Structural Engineering Report No. 90, University of Alberta, Edmonton, Alberta, Canada, March.
Further Reading The references that accompany this chapter are all-encompassing for the literature on stub girders. Primary references that should be studied in addition to this chapter are [5, 7, 10, 11], and [13].
1999 by CRC Press LLC
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Elgaaly, M. “Plate and Box Girders” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Plate and Box Girders 19.1 Introduction 19.2 Stability of the Compression Flange Vertical Buckling • Lateral Buckling Compression Flange of a Box Girder
Mohamed Elgaaly Department of Civil & Architectural Engineering, Drexel University, Philadelphia, PA
19.1
•
Torsional Buckling
•
19.3 Web Buckling Due to In-Plane Bending 19.4 Nominal Moment Strength 19.5 Web Longitudinal Stiffeners for Bending Design 19.6 Ultimate Shear Capacity of the Web 19.7 Web Stiffeners for Shear Design 19.8 Flexure-Shear Interaction 19.9 Steel Plate Shear Walls 19.10In-Plane Compressive Edge Loading 19.11Eccentric Edge Loading 19.12Load-Bearing Stiffeners 19.13Web Openings 19.14Girders with Corrugated Webs 19.15Defining Terms References
Introduction
Plate and box girders are used mostly in bridges and industrial buildings, where large loads and/or long spans are frequently encountered. The high torsional strength of box girders makes them ideal for girders curved in plan. Recently, thin steel plate shear walls have been effectively used in buildings. Such walls behave as vertical plate girders with the building columns as flanges and the floor beams as intermediate stiffeners. Although traditionally simply supported plate and box girders are built up to 150 ft span, several three-span continuous girder bridges have been built in the U.S. with center spans exceeding 400 ft. In its simplest form a plate girder is made of two flange plates welded to a web plate to form an I section, and a box girder has two flanges and two webs for a single-cell box and more than two webs in multi-cell box girders (Figure 19.1). The designer has the freedom in proportioning the cross-section of the girder to achieve the most economical design and taking advantage of available high-strength steels. The larger dimensions of plate and box girders result in the use of slender webs and flanges, making buckling problems more relevant in design. Buckling of plates that are adequately supported along their boundaries is not synonymous with failure, and these plates exhibit post-buckling strength that can be several times their buckling strength, depending on the plate slenderness. Although plate 1999 by CRC Press LLC
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FIGURE 19.1: Plate and box girders.
buckling has not been the basis for design since the early 1960s, buckling strength is often required to calculate the post-buckling strength. The trend toward limit state format codes placed the emphasis on the development of new design approaches based on the ultimate strength of plate and box girders and their components. The post-buckling strength of plates subjected to shear is due to the diagonal tension field action. The post-buckling strength of plates subjected to uniaxial compression is due to the change in the stress distribution after buckling, higher near the supported edges. An effective width with a uniform stress, equal to the yield stress of the plate material, is used to calculate the post-buckling strength [40]. The flange in a box girder and the web in plate and box girders are often reinforced with stiffeners to allow for the use of thin plates. The designer has to find a combination of plate thickness and stiffener spacing that will optimize the weight and reduce the fabrication cost. The stiffeners in most cases are designed to divide the plate panel into subpanels, which are assumed to be supported along the stiffener lines. Recently, the use of corrugated webs resulted in employing thin webs without the need for stiffeners, thus reducing the fabrication cost and also improving the fatigue life of the girders. The web of a girder and load-bearing diaphragms can be subjected to in-plane compressive patch loading. The ultimate capacity under this loading condition is controlled by web crippling, which can occur prior to or after local yielding. The presence of openings in plates subjected to in-plane loads is unavoidable in some cases, and the presence of openings affects the stability and ultimate strength of plates.
19.2
Stability of the Compression Flange
The compression flange of a plate girder subjected to bending usually fails in lateral buckling, local torsional buckling, or yielding; if the web is slender the compression flange can fail by vertical buckling into the web (Figure 19.2). 1999 by CRC Press LLC
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FIGURE 19.2: Compression flange modes of failure.
19.2.1
Vertical Buckling
The following limiting value for the web slenderness ratio to preclude this mode of failure [4] can be used, q p Aw /Af (19.1) h/tw ≤ 0.68E/ Fyf Fyf + Fr where h and tw are the web height and thickness, respectively; Aw is the area of the web; Af is the area of the flange; E is Young’s modulus of elasticity; Fyf is the yield stress of the flange material; and Fr is the residual tension that must be overcome to achieve uniform yielding in compression. This limiting value may be too conservative since vertical buckling of the compression flange into the web occurs only after general yielding of the flange. This limiting value, however, can be helpful to avoid fatigue cracking under repeated loading due to out-of-plane flexing, and it also facilitates fabrication. The American Institute of Steel Construction (AISC) specification [32] uses Equation 19.1 when the spacing between the vertical stiffeners, a, is more than 1.5 times the web depth, h (a/ h > 1.5). In such a case the specification recommends that q (19.2) h/tw ≤ 14,000/ Fyf (Fyf + 16.5) where a minimum value of Aw /Af = 0.5 was assumed and the residual tension was taken to be 16.5 ksi. Furthermore, when a/ h is less than or equal to 1.5, higher web slenderness is permitted, namely p (19.3) h/tw ≤ 2000/ Fyf
19.2.2
Lateral Buckling
When a flange is not adequately supported in the lateral direction, elastic lateral buckling can occur. The compression flange, together with an effective area of the web equal to Aw /6, can be treated as a column and the buckling stress can be calculated from the Euler equation [2]: Fcr = π 2 E/(λ)2
(19.4)
where λ is the slenderness ratio, which is equal to Lb /rT ; Lb is the length of the unbraced segment of the beam; and rT is the radius of gyration of the compression flange plus one-third of the compression portion of the web. The AISC specification adopted Equation 19.4, rounding π 2 E to 286,000 and p assuming that elastic buckling will occur when the slenderness ratio, λ, is greater than λr (= 756/ Fyf ). Furthermore, Equation 19.4 is based on uniform compression; in most cases the bending is not uniform within the length of the unbraced segment of the beam. To account for nonuniform bending, 1999 by CRC Press LLC
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Equation 19.4 should be multiplied by a coefficient, Cb [25], where Cb = 12.5Mmax /(2.5Mmax + 3MA + 4MB + 3MC )
(19.5)
where Mmax = absolute value of maximum moment in the unbraced beam segment = absolute value of moment at quarter point of the unbraced beam segment MA = absolute value of moment at centerline of the unbraced beam segment MB = absolute value of moment at three-quarter point of thepunbraced beam segment MC When the slenderness ratio, λ, is less than or equal to λp (= 300/ Fyf ), the flange will yield before it buckles, and Fcr = Fyf . When the flange slenderness ratio, λ, is greater than λp and smaller than or equal to λr , inelastic buckling will occur and a straight line equation must be adopted between yielding (λ ≤ λp ) and elastic buckling (λ > λr ) to calculate the inelastic buckling stress, namely (19.6) Fcr = Cb Fyf 1 − 0.5(λ − λp )/(λr − λp ) ≤ Fyf
19.2.3
Torsional Buckling
If the outstanding width-to-thickness ratio of the flange is high, torsional buckling may occur. If one neglects any restraint provided by the web to the flange rotation, then the flange can be treated as a long plate, which is simply supported (hinged) at one edge and free at the other, subjected to uniaxial compression in the longitudinal direction. The elastic buckling stress under these conditions can be calculated from (19.7) Fcr = kc π 2 E/12(1 − µ2 )λ2 where kc is a buckling coefficient equal to 0.425 for a long plate simply supported and free at its longitudinal edges; λ is equal to bf /2tf ; bf and tf are the flange width and thickness, respectively; and E and µ are Young’s modulus of elasticity and the Poisson ratio, respectively. 2 2 The AISC √ specification adopted Equation 19.7, rounding π E/12(1−µ ) to 26,200 and assuming kc = 4 h/tw , where 0.35 ≤ kc ≥ 0.763. Furthermore, to allow for nonuniform bending, the buckling stress has to be multiplied by Cb , given by Equation p 19.5. Elastic torsional buckling of the compression flange will occur if λ is greater than λr (= 230/ Fyf /kc ). When λ is less than or equal p to λp (= 65/ Fyf ), the flange will yield before it buckles, and Fcr = Fyf . When λp < λ ≤ λr , inelastic buckling will occur and Equation 19.6 shall be used.
19.2.4
Compression Flange of a Box Girder
Lateral-torsional buckling does not govern the design of the compression flange in a box girder. Unstiffened flanges and flanges stiffened with longitudinal stiffeners can be treated as long plates supported along their longitudinal edges and subjected to uniaxial compression. In the AASHTO (American Association of State Highway and Transportation Officials ) specification [1], the nominal flexural stress,p Fn , for the compression flange is calculated as follows: If w/t ≤ 0.57 kE/Fyf , then the flange will yield before it buckles, and Fn = Fyf p If w/t > 1.23 kE/Fyf , then the flange will elastically buckle, and
(19.8)
Fn = kπ 2 E/12(1 − µ2 )(w/t)2 or Fn = 26,200 k(t/w)2 1999 by CRC Press LLC
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(19.9)
p p If 0.57 kE/Fyf < w/t ≤ 1.23 kE/Fyf , then the flange buckles inelastically, and Fn = 0.592Fyf [1 + 0.687 sin(cπ/2)]
(19.10)
In Equations 19.8 to 19.10, = the the longitudinal stiffeners, or the flange width for unstiffened flanges spacing between p = 1.23 − (w/t) Fyf /kE /0.66 1/3 ≤ 4.0, for n = 1 k = 8Is /wt 3 1/3 3 4 ≤ 4.0, for n = 2, 3, 4, or 5 k = 14.3Is /wt n n = number of equally spaced longitudinal stiffeners = the moment of inertia of the longitudinal stiffener about an axis parallel to the flange and Is taken at the base of the stiffener w c
The nominal stress, Fn , shall be reduced for hybrid girders to account for the nonlinear variation of stresses caused by yielding of the lower strength steel in the web of a hybrid girder. Furthermore, another reduction is made for slender webs to account for the nonlinear variation of stresses caused by local bend buckling of the web. The reduction factors for hybrid girders and slender webs will be given in Section 19.3. The longitudinal stiffeners shall be equally spaced across the compression flange width and shall satisfy the following requirements [1]. The projecting width, bs , of the stiffener shall satisfy: p (19.11) bs ≤ 0.48ts E/Fyc where = thickness of the stiffener ts = specified minimum yield strength of the compression flange Fyc The moment of inertia, Is , of each stiffener about an axis parallel to the flange and taken at the base of the stiffener shall satisfy: (19.12) Is ≥ 9wt 3 where 9 = = n = w =
0.125k 3 for n = 1 0.07k 3 n4 for n = 2, 3, 4, or 5 number of equally spaced longitudinal compression flange stiffeners larger of the width of compression flange between longitudinal stiffeners or the distance from a web to the nearest longitudinal stiffener t = compression flange thickness k = buckling coefficient as defined in connection with Equations 19.8 to 19.10 The presence of the in-plane compression in the flange magnifies the deflection and stresses in the flange from local bending due to traffic loading. The amplification factor, 1/(1 − σa /σcr ), can be used to increase the deflections and stresses due to local bending; where σa and σcr are the in-plane compressive and buckling stresses, respectively.
19.3
Web Buckling Due to In-Plane Bending
Buckling of the web due to in-plane bending does not exhaust its capacity; however, the distribution of the compressive bending stress changes in the post-buckling range and the web becomes less efficient. Only part of the compression portion of the web can be assumed effective after buckling. A reduction in the girder moment capacity to account for the web bend buckling can be used, and the following reduction factor [4] has been suggested: p (19.13) R = 1 − 0.0005(Aw /Af )(h/t − 5.7 E/Fyw ) 1999 by CRC Press LLC
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p It must be noted that when h/t = 5.7 E/Fyw , the web will yield before it buckles and there is no reduction in the moment capacity. This can be determined by equating the bend buckling stress to the web yield stress, i.e., h i (19.14) kπ 2 E/ 12(1 − µ2 )(h/t)2 = Fyw where k is the web bend buckling coefficient, which is equal to 23.9 if the flange simply supports the web and 39.6 if one assumes that the flange provides full fixity; the 5.7 in Equation 19.13 is based on a k value of 36. The AISC specification replaces the reduction factor given in Equation 19.13 by p (19.15) RP G = 1 − [ar /(1,200 + 300ar )] (h/t − 970/ Fcr ) √ where ar is equal to Aw /Af and 970 is equal to 5.7 29000; it must be noted that the yield stress in Equation 19.13 was replaced by the flange critical buckling stress, which can be equal to or less than the yield stress as discussed earlier. It must also be noted that in homogeneous girders the yield stresses of the web and flange materials are equal; in hybrid girders another reduction factor, Re , [39] shall be used: h i (19.16) Re = 12 + ar (3m − m3 ) /(12 + 2ar ) where ar is equal to the ratio of the web area to the compression flange area (≤ 10) and m is the ratio of the web yield stress to the flange yield or buckling stress.
19.4
Nominal Moment Strength
The nominal moment strength can be calculated as follows. Based on tension flange yielding: Mn = Sxt Re Fyt
(19.17a)
Mn = Sxc RP G Re Fcr
(19.17b)
or Based on compression flange buckling:
where Sxc and Sxt are the section moduli referred to the compression and tension flanges, respectively; Fyt is the tension flange yield stress; Fcr is the compression flange buckling stress calculated according to Section 19.2; RP G is the reduction factor calculated using Equation 19.15; and Re is a reduction factor to be used in the case of hybrid girders and can be calculated using Equation 19.16.
19.5
Web Longitudinal Stiffeners for Bending Design
Longitudinal stiffeners can increase the bending strength of plate girders. This increase is due to the control of the web lateral deflection, which increases its flexural stress capacity. The presence of the stiffener also improves the bending resistance of the flange due to a greater web restraint. If one longitudinal stiffener is used, its optimum location is 0.20 times the web depth from the compression flange. In this case the web plate elastic bend buckling stress increases more than five times that without the stiffener. Tests [8] showed that an adequately proportioned longitudinal stiffener at 0.2h from the compression flange eliminates bend buckling in girders with web slenderness, h/t, as large as 450. Girders with larger slenderness will require two or more longitudinal stiffeners to eliminate 1999 by CRC Press LLC
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web bend buckling. It must be noted that the increase in the bending strength of a longitudinally stiffened thin-web girder is usually small because the web contribution to the bending strength is small. However, longitudinal stiffeners can be important in a girder subjected to repeated loads because they reduce or eliminate the out-of-plane bending of the web, which increases resistance to fatigue cracking at the web-to-flange juncture and allows more slender webs to be used [42]. The AISC specification does not address longitudinal stiffeners; on the other hand, the AASHTO specification states that longitudinal stiffeners should consist of either a plate welded longitudinally to one side of the web or a bolted angle, and shall be located at a distance of 0.4 Dc from the inner surface of the compression flange, where Dc is the depth of the web in compression at the section with the maximum compressive flexural stress. Continuous longitudinal stiffeners placed on the opposite side of the web from the transverse intermediate stiffeners, as shown in Figure 19.3, are preferred. If longitudinal and transverse stiffeners must be placed on the same side of the web, it is preferable
FIGURE 19.3: Longitudinal stiffener for flexure.
that the longitudinal stiffener not be interrupted for the transverse stiffener. Where the transverse stiffeners are interrupted, the interruptions must be carefully detailed with respect to fatigue. To prevent local buckling, the projecting width, bs of the stiffener shall satisfy the requirements of Equation 19.11. The section properties of the stiffener shall be based on an effective section consisting of the stiffener and a centrally located strip of the web not exceeding 18 times the web thickness. The moment of inertia of the longitudinal stiffener and the effective web strip about the edge in contact with the web, Is , and the corresponding radius of gyration, rs , shall satisfy the following requirements: i h (19.18) Is ≥ htw3 2.4(a/ h)2 − 0.13 and
p rs ≥ 0.234a Fyc /E
(19.19)
where a = spacing between transverse stiffeners
19.6
Ultimate Shear Capacity of the Web
As stated earlier, in most design codes buckling is not used as a basis for design. Minimum slenderness ratios, however, are specified to control out-of-plane deflection of the web. These ratios are derived to give a small factor of safety against buckling, which is conservative and in some cases extravagant. 1999 by CRC Press LLC
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Before the web reaches its theoretical buckling load the shear is taken by beam action and the shear stress can be resolved into diagonal tension and compression. After buckling, the diagonal compression ceases to increase and any additional loads will be carried by the diagonal tension. In very thin webs with stiff boundaries, the plate buckling load is very small and can be ignored and the shear is carried by a complete diagonal tension field action [41]. In welded plate and box girders the web is not very slender and the flanges are not very stiff; in such a case the shear is carried by beam action as well as incomplete tension field action. Based on test results, the analytical model shown in Figure 19.4 can be used to calculate the ultimate shear capacity of the web of a welded plate girder [5]. The flanges are assumed to be too flexible to
FIGURE 19.4: Tension field model by Basler.
support the vertical component from the tension field. The inclination and width of the tension field were defined by the angle 2, which is chosen to maximize the shear strength. The ultimate shear capacity of the web, Vu , can be calculated from Vu = τcr + 0.5σyw (1 − τcr /τyw ) sin 2d Aw (19.20) where = critical buckling stress in shear τcr = yield stress in shear τyw σyw = web yield stress 2d = angle of panel diagonal with flange = area of the web Aw In Equation 19.20, if τcr ≥ 0.8τyw , the buckling will be inelastic and p τcr = τcri = 0.8τcr τyw
(19.21)
It was shown later [23] that Equation 19.20 gives the shear strength for a complete tension field instead of the limited band shown in Figure 19.4. The results obtained from the formula, however, were in good agreement with the test results, and the formula was adopted in the AISC specification. Many variations of this incomplete tension field model have been developed; are view can be found in the SSRC Guide to Stability Design Criteria for Metal Structures [22]. The model shown in Figure 19.5 [36, 38] gives better results and has been adopted in codes in Europe. In the model shown in Figure 19.5, near failure the tensile membrane stress, together with the buckling stress, causes yielding, and failure occurs when hinges form in the flanges to produce a combined mechanism that includes the yield zone ABCD. The vertical component of the tension field is added to the shear at buckling and combined with the frame action shear to calculate the ultimate shear strength. The 1999 by CRC Press LLC
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FIGURE 19.5: Tension field model by Rockey et al.
ultimate shear strength is determined by adding the shear at buckling, the vertical component of the tension field, and the frame action shear, and is given by Vu = τcr Aw + σt Aw [(2c/ h) + cot 2 − cot 2d ] sin2 2 + 4Mp /c
(19.22)
where q 2 + (2.25 sin2 22 − 3)τ 2 σt = −1.5τcr sin 22 + σyw cr √ c = (2/ sin 2) Mp/(σt tw ) 0 ≤ c ≤ a = plastic moment capacity of the flange with an effective depth of the web, be , given by Mp = 30tw [1 − 2(τcr /τyw )] be where (τcr /τyw ) ≤ 0.5; reduction in Mp due to the effect of the flange axial compression shall be considered and when τcr > 0.8τyw , τcr = τcri = τyw [1 − 0.16(τyw /τcr )] The maximum value of Vu must be found by trial; 2 is the only independent variable in Equation 19.22, and the optimum is not difficult to determine by trial since it is between 2d /2 and 45 degrees, and Vu is not sensitive to small changes from the optimum 2. Recently [2, 33], it has been argued that the post-buckling strength arises not due to a diagonal tension field action, but by redistribution of shear stresses and local yielding in shear along the boundaries. A case in between is to model the web panel as a diagonal tension strip anchored by corner zones carrying shear stresses and act as gussets connecting the diagonal tension strip to the vertical stiffeners which are in compression [9]. On the basis of test results, it can be concluded that unstiffened webs possess a considerable reserve of post-buckling strength [16, 24]. The incomplete diagonal tension field approach, however, is only reasonably accurate up to a maximum aspect ratio (stiffeners spacing: web depth) equal to 6. Research is required to develop an appropriate method of predicting the post-buckling strength of unstiffened girders. In the AISC specification, the shear capacity of a plate girder web can be calculated, using the model shown in Figure p 19.4, as follows: For h/tw ≤ 187 kv /Fyw , the web yields before buckling, and Vn = 0.6Aw Fyw 1999 by CRC Press LLC
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(19.23)
p For h/tw > 187 kv /Fyw , the web will buckle and a tension field will develop, and q 2 Vn = 0.6Aw Fyw Cv + (1 − Cv )/1.15 1 + (a/ h)
(19.24)
where kv = buckling coefficient = 5 + 5/(a/ h)2 = 5, if (a/ h) > 3 or [260/(h/tw )]2 Cv = ratiopof the web buckling stresspto the shear yield stress of pthe web = 187 kv /Fyw /(h/tw ), for 187 kv /Fyw ≤ h/tw ≤ 234 kv /Fyw p = 44,000 kv /[(h/tw )2 Fyw ], for h/tw > 234 kv /Fyw p It must be noted that, in the above, the web buckling is elastic when h/tw > 234 kv /Fyw . The AISC specification does not permit the consideration of tension field action in end panels, hybrid and web-tapered plate girders, and when a/ h exceeds 3.0 or [260/(h/tw)]2 . This is contrary to the fact that a tension field can develop in all these cases; however, little or no research was conducted. Furthermore, tension field can be considered for end panels if the end stiffener is designed for this purpose. When neglecting the tension field action, the nominal shear capacity can be calculated from Vn = 0.6Aw Fyw Cv
(19.25)
Care must be exercised in applying the tension field models developed primarily for welded plate girders to the webs of a box girder. The thin flange of a box girder can provide very little or no resistance against movements in the plane of the web. If the web of a box girder is transversely stiffened and if the model shown in Figure 19.4 is used, it may overpredict the web strength. Hence, it is advisable to use the model shown in Figure 19.5, assuming the plastic moment capacity of the flange to be negligible.
19.7
Web Stiffeners for Shear Design
Transverse stiffeners must be stiff enough to prevent out-of-plane displacement along the panel boundaries in computing shear buckling of plate girder webs. To provide the out-of-plane support an equation, developed for an infinitely long web with simply supported edges and equally spaced stiffeners, to calculate the required moment of inertia of the stiffeners, Is , namely for a ≤ h, Is = 2.5htw3 [(h/a) − 0.7(a/ h)]
(19.26)
The AASHTO formula for load-factor design is Is = J atw3
(19.27)
where J = [2.5(h/a)2 − 2] ≥ 0.5 Equation 19.27 is the same as Equation 19.26 except that the coefficient of (a/ h) in the second term between brackets is 0.8 instead of 0.7. Equation 19.27 was adopted by the AISC specification as well. The moment of inertia of the transverse stiffener shall be taken about the edge in contact with the web for single-sided stiffeners and about the mid-thickness of the web for double-sided stiffeners. To prevent local buckling of transverse stiffeners, the width, bs , of each projecting stiffener element shall satisfy the requirements of Equation 19.11 using the yield stress of the stiffener material rather than that of the flange, as in Equation 19.11. Furthermore, bs shall also satisfy the following requirements: (19.28) 16.0ts ≥ bs ≤ 0.25bf 1999 by CRC Press LLC
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where bf = the full width of the flange Transverse stiffeners shall consist of plates or angles welded or bolted to either one or both sides of the web. Stiffeners that are not used as connection plates shall be a tight fit at the compression flange, but need not be in bearing with the tension flange. The distance between the end of the webto-stiffener weld and the near edge of the web-to-flange fillet weld shall not be less than 4tw or more than 6tw . Stiffeners used as connecting plates for diaphragms or cross-frames shall be connected by welding or bolting to both flanges. In girders with longitudinal stiffeners the transverse stiffener must also support the longitudinal stiffener as it forces a horizontal node in the bend buckling configuration of the web. In such a case it is recommended that the transverse stiffener section modulus, ST , be equal to SL (h/a), where SL is the section modulus of the longitudinal stiffener and h and a are the web depth and the spacing between the transverse stiffeners, respectively. In the AASHTO specification, the moment of inertia of transverse stiffeners used in conjunction with longitudinal stiffeners shall also satisfy It ≥ (bt /bl )(h/3a)Il
(19.29)
where bt and bl = projecting width of transverse and longitudinal stiffeners, respectively, and It and Il = moment of inertia of transverse and longitudinal stiffeners, respectively. Transverse stiffeners in girders that rely on a tension field must also be designed for their role in the development of the diagonal tension. In this situation they are compression members, and so must be checked for local buckling. Furthermore, they must have cross-sectional area adequate for the axial force that develops. The axial force, Fs , can be calculated based on the analytical model [5] shown in Figure 19.4, and is given by (19.30) Fs = 0.5Fyw atw 1 − τcr /τyw (1 − cos 2d ) The AISC and AASHTO specifications assume that a width of the web equal to 18tw acts with the stiffener and give the following formula for the cross-sectional area, As , of the stiffeners: h i (19.31) As ≥ 0.15Bhtw (1 − Cv )Vu /0.9Vn − 18tw2 (Fyw /Fys ) where the new notations are 0.9Vu = shear due to factored loads B = 1.0 for double-sided stiffeners = 1.8 for single-sided angle stiffeners = 2.4 for single-sided plate stiffeners If longitudinally stiffened girders are used, h in Equation 19.31 shall be taken as the depth of the web, since the tension field will occur between the flanges and the transverse stiffeners. The optimum location of a longitudinal stiffener that is used to increase resistance to shear buckling is at the web mid-depth. In this case the two subpanels buckle simultaneously and the increase in the critical stress is substantial. To obtain the tension field shear resistance one can assume that only one tension field is developed between the flanges and transverse stiffeners even if longitudinal stiffeners are used.
19.8
Flexure-Shear Interaction
The shear capacity of a girder is independent of bending as long as the applied moment is less than the moment that can be taken by the flanges alone, Mf = σyf Af h; any larger moment must be resisted in part by the web, which reduces the shear capacity of the girder. When the girder is subjected to 1999 by CRC Press LLC
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pure bending with no shear, the maximum moment capacity is equal to the plastic moment capacity, Mp , due to yielding of the girder’s entire cross-section. In view of the tension field action and based on test results [3], a simple conservative interaction equation is given by (V /Vu )2 + (M − Mf )/(Mp − Mf ) = 1
(19.32)
where M and V are the applied moment and shear, respectively. In the AISC specification, plate girders with webs designed for tension field action and when the ultimate shear, Vu , is between 0.54 and 0.9Vn , and the ultimate moment, Mu , is between 0.675 and 0.9Mn , where Vn and Mn are the nominal shear and moment capacities in absence of one another; the following interaction equation must be satisfied, Mu /0.9Mn + 0.625(Vu /0.9Vn ) ≤ 1.375
19.9
(19.33)
Steel Plate Shear Walls
Although the post-buckling behavior of plates under monotonic loads has been under investigation for more than half a century, post-buckling strength of plates under cyclic loading has not been investigated until recently [7]. The results of this investigation indicate that plates can be subjected to few reversed cycles of loading in the post-buckling domain, without damage. In steel plate shear walls, the boundary members are stiff and the plate is relatively thin; in such cases a complete tension field can be developed. The plate can be modeled as a series of tension bars inclined at an angle, φ [27]. The angle of inclination, φ, is a function of the panel length and height, the plate thickness, the cross-sectional areas of the surrounding beams and columns, and the moment of inertia of the columns. It can be determined by applying the principle of least work and is given by h i (19.34) tan4 α = [(2/tw L) + (1/Ac )] / (2/tw L) + (2h/Ab L) + (h4 /180Ic L2 ) where α = angle of inclination of tension field with the vertical axis L = panel length h = panel height tw = wall thickness Ab = cross-sectional area of beam Ac = cross-sectional area of column = moment of inertia of column about axis perpendicular to the plane of the wall Ic Although this model can predict the ultimate capacity to a reasonable degree of accuracy it cannot depict the load-deflection characteristics to the same degree of accuracy. Based on test results and finite element analysis [17, 18], the stresses in the inclined tensile plate strips are not uniform but are higher near the supporting boundaries than the center of the plate, and yielding of these strips starts near their ends and propagates toward the midlength. The following method can be used to calculate the ultimate capacity and determine the load-deflection characteristics of a thin-steel-plate shear wall. The plate in the shear wall is replaced by a series of truss elements in the diagonal tension direction, as shown in Figure 19.6. A minimum of four truss members shall be used to replace the plate panel in order to depict the panel behavior to a reasonable degree of accuracy; however, six members are recommended. The stress-strain relationship for the truss elements shall be assumed to be bilinearly elastic perfectly plastic, as shown in Figure 19.7, where E is Young’s modulus of elasticity and σy is the tensile yield stress of the plate material. In Figure 19.7 the first slope represents the elastic 1999 by CRC Press LLC
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FIGURE 19.6: Steel plate shear wall analytical model.
FIGURE 19.7: Stress-strain relationship for truss element. response and the second represents the reduced stiffness caused by partial yielding; σy1 and E2 can be determined using a semi-empirical approach for welded as well as bolted shear walls, and can be calculated using the following equations:
where
σy1 = (0.423 + 0.816be /L)σy
(19.35)
h i0.5 be = 14.6π 2 E/12(1 − ν 2 )τy t
(19.36)
L = length of the strip t = thickness of the plate and E2 = (σy2 − σy1 )(3 − σy1 )/σy2 (2 + α) E 1999 by CRC Press LLC
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(19.37)
where α is the ratio between the plastic strain, εp , and the strain at the initiation of yielding, εy . This ratio is in the range of 5 to 20, depending on the stiffness of the columns relative to the thickness of the plate; a value of 10 can be used in design. In the derivation of Equations 19.35 and 19.36 the inclination angle of the equivalent truss elements was assumed to be 45 degrees. The angle of inclination is usually in the range of 38 to 43 degrees and the effect of this assumption on the overall behavior of the wall is negligible. In order to define the load displacement relationship of the truss elements in a bolted shear wall, the parameters Py1 , K1 , Py , and K2 shown in Figure 19.8 need to be calculated. For a bolted plate,
FIGURE 19.8: Load-displacement relationship for truss element.
the initial yielding can occur when the plate at the bolted connection starts slipping or when it locally yields near the boundaries as in the welded plate. The load due to slippage is controlled by the friction coefficient between the connected surfaces and the normal force applied by the bolts. In case the bolts were pre-tensioned to 70% of their ultimate tensile strength, slip will occur at a load equal to Py1 = n(0.7µFub Ab )
(19.38)
where n is the number of bolts at one end of the truss element; Fub and Ab are the ultimate tensile strength and cross-section area of the bolt, respectively; and µ is the friction coefficient between the connected surfaces. The load that causes initial yielding at the ends of the strip is the same as for the welded plate, and can be obtained from Equation 19.35 by multiplying the stress, σy1 , by the strip cross-sectional area, Ap . The usable load is the smaller of the loads that causes slippage or initial yielding at the ends of the strip. The initial stiffness of the equivalent truss element can be calculated as follows: K1 = EAp /L
(19.39)
The ultimate load is controlled by the total yielding of the plate strip, tearing at the bolt holes, or shearing of the bolts. The smallest failure load is the controlling ultimate capacity of the truss element. The ultimate load due to the strip yielding along its entire length, Py0 = σy Ap The bolted shear wall should be designed such that plate yielding controls. 1999 by CRC Press LLC
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(19.40)
Tearing at the bolt holes will occur when the edge distance is small or the bolt spacing is large, which should be avoided in design because it is a brittle failure. The tearing load, denoted as Py00 , can be calculated using the following formula: Py00 = 1.4nσu (Le − D/2)t
(19.41)
where n is the number of bolts at the end of the truss element, σu is the ultimate tensile strength of the plate material, Le is the distance from the edge of the plate to the centroid of the bolt hole, D is the diameter of the bolt, and t is the thickness of the plate. Note that in the above formula the ultimate strength of the plate material in shear was assumed to be 0.7, its ultimate strength in tension. The shear failure of the bolts can occur if the shear strength of the bolts is small or the spacing between them is large. In such case the ultimate capacity of the truss element, which is denoted as Py000 , can be calculated from one of the following formulas: Py000
=
0.45nFub Ab
(19.42a)
Py000
=
0.60nFub Ab
(19.42b)
where n is the number of bolts and Fub and Ab are the ultimate strength and the gross cross-section area of the bolts, respectively. Equation 19.42a is used if the shear plane is within the threaded part of the bolt and Equation 19.42b is used if the shear plane is not within the threaded part of the bolt. The ultimate load of the truss element is the smallest value of the plate total yielding capacity, the tearing capacity, and the bolt shearing capacity, i.e., Py = min(Py0 , Py00 , Py000 )
(19.43)
As stated earlier, the plate yielding shall control and the designer must ensure that Py0 is the smallest of the three values. As can be seen in Figure 19.8, in order to define the stiffness, K2 , the displacement of the truss element when the load reaches the ultimate capacity, Py , needs to be determined. This ultimate displacement includes the stretching of the element as well as the slippage and the bearing deformation of the plate and the bolts at the connections. The plate strip represented by the truss element is stretched under load; the elongation includes both elastic and plastic deformations. As discussed earlier, due to the nonuniform strain distribution along the length of the strip, the plastic deformation will occur mostly near the ends. If one assumes that the strain distribution along the length of the strip is a second-degree parabola, and using a plastic deformation factor a, the elongation of the truss element due to the plate elastic and plastic deformations can be calculated using the following equation: 1def = (σy /3E)(2 + α)L
(19.44)
The elongation of the truss element due to slippage can be approximated by two times the hole clearance, taking into consideration the slippage at both ends of the element [26]. The local deformation at the bolt holes includes the effects of shearing, bending, and bearing deformations of the fastener as well as local deformation of the connected plates, and can be taken as 0.2 times the bolt diameter [21]. Having defined the ultimate elongation of the truss element, its reduced stiffness after slippage and/or initial yielding can be obtained using the following equation: K2 = (Py − Py1 )/ 0.125 + 0.2D + σy (α − 1)L/(3E)
1999 by CRC Press LLC
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(19.45)
19.10
In-Plane Compressive Edge Loading
Webs of plate and box girders and load-bearing diaphragms in box girders can be subjected to local in-plane compressive loads. Vertical (transverse) stiffeners can be provided at the location of the load to prevent web crippling; however, this is not always possible, such as in the case of a moving load, and it involves higher cost. Failure of the web under this loading is always due to crippling [10], as shown in Figure 19.9; in thin webs crippling occurs before yielding of the web and in stocky webs after yielding. The formula in the AISC specification predicts the crippling load, Pcr , to a reasonable
FIGURE 19.9: Deformed shape under in-plane edge loading (only half of the beam is shown).
degree of accuracy [37]; this formula is h 1.5 i 0.5 Fyw tf /tw Pcr = 135tw2 1 + 3(N/d) tw /tf
(19.46)
The formula given by Equation 19.46 is applicable if the load is applied at a distance not less than half the member’s depth from its end; if the load is at a distance less than half the member’s depth, the following formulae shall be used [14]: h 1.5 i 0.5 (19.47a) Fyw tf /tw For N/d ≤ 0.2, Pcr = 68tw2 1 + 3(N/d) tw /tf h i 0.5 For N/d > 0.2, Pcr = 68tw2 1 + (4N/d − 0.2)(tw /tf )1.5 Fyw tf /tw (19.47b)
In addition to web crippling, the AISC specification requires a web yielding check; furthermore, when the relative lateral movement between the loaded compression flange and the tension flange is not restrained at the point of load application, sidesway buckling must be checked. 1999 by CRC Press LLC
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19.11
Eccentric Edge Loading
Eccentricities in loading with respect to the plane of the web are unavoidable, and it was found that there is a reduction in the web capacity due to the presence of an eccentricity [13, 14]; for example, in one case, an eccentricity of 0.5 in. reduced the web ultimate capacity to about half its capacity under in-plane load. Furthermore, it was found that the effect of the load eccentricity in reducing the ultimate capacity decreases as the ratio of the flange-to-web thickness increases. A deformed beam subjected to eccentric load near failure is shown in Figure 19.10. Web strength reduction factors for various eccentricities as a function of the flange width and for various flange-to-web thickness ratios are given in Figure 19.11.
FIGURE 19.10: Deformed shape under eccentric edge loading.
The failure mechanism in the case of eccentric loading is different from that for in-plane loading. The flange twisting moment acting at the web-flange intersection can cause failure due to bending rather than crippling of the web, if the eccentricity is large enough. In most cases, however, the failure mode is due to a combination of web bending and crippling. Failure mechanisms were developed, and formulas to calculate the ultimate capacity of the web under eccentric edge loading were derived [15]. Currently, the AISC specification does not address the effect of the eccentricity on the reduction of the web crippling load. Eccentricities can also arise due to moments applied to the top flange in addition to vertical loads. An example would be a beam resting on the top flange of another beam and the two flanges are welded together. Rotation of the supported beam will impose a twisting moment in the flange of the supporting beam and bending of its web, which will reduce its crippling load. 1999 by CRC Press LLC
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FIGURE 19.11: Eccentricity reduction factor.
19.12
Load-Bearing Stiffeners
Webs of girders are often strengthened with transverse stiffeners at points of concentrated loads and over intermediate and end supports. The AISC specification requires that these stiffeners be double sided, extend at least one-half the beam depth, and either bear on or be welded to the loaded flange. The specification, further, requires that they be designed as axially loaded members with an effective length equals to 0.75 times the web depth; and a strip of the web, with a width equal to 25 times its thickness for intermediate stiffeners and 12 times the thickness for end stiffeners, shall be considered in calculating the geometric properties of the stiffener. The failure, in cases where the stiffener depth is less than 75% of the depth of the web, can be due to crippling of the web below the stiffener [14, 15], as shown in Figure 19.12. The failure, otherwise,
FIGURE 19.12: Web crippling below stiffener. 1999 by CRC Press LLC
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is due to global buckling of the stiffener, provided that the thickness of the stiffener is adequate to prevent local buckling. The optimum depth of the stiffener is 0.75 times the web depth. The AISC specification does not account for factors such as the stiffener depth and load eccentricity. In box girders intermediate diaphragms are provided to limit cross-sectional deformation and loadbearing diaphragms are used at the supports to transfer loads to the bridge bearings. Diaphragm design is treated in the BS 5400: Part 3 (1983) [6] and discussed in Chapter 7 of the SSRC guide [22].
19.13
Web Openings
Openings are frequently encountered in the webs of plate and box girders. Research on the buckling and ultimate strength of plates with rectangular and circular openings subjected to in-plane loads has been performed by many investigators. The research has included reinforced and unreinforced openings. A theoretical method of predicting the ultimate capacity of slender webs containing circular and rectangular holes, and subjected to shear, has been developed [34, 35]. The solution is obtained by considering the equilibrium of two tension bands, one above and the other below the opening. These bands have been chosen to conform to the failure pattern observed in tested plate girders with holes. Experimental results showed that the method gives satisfactory and safe predictions. The calculated values were found to be between 5 and 30% below the test results. Solutions for transversely stiffened webs subjected to shear and bending with centrally located holes are available [28, 30] and are applicable for webs with depth-to-thickness ratios of 120 to 360, panel aspect ratio between 0.7 and 1.5, hole depth greater than 1/10th of the web depth, and for circular, elongated circular, and rectangular holes.
19.14
Girders with Corrugated Webs
Corrugated webs can be used in an effort to decrease the weight of steel girders and reduce its fabrication cost. Studies have been conducted in Europe and Japan and girders with corrugated webs have been used in these countries [12]. The results of the studies indicate that the fatigue strength of girders with corrugated webs can be 50% higher compared to girders with flat stiffened webs. In addition to the improved fatigue life, the weight of girders with corrugated webs can be as much as 30 to 60% less than the weight of girders with flat webs and have the same capacity. Due to the weight savings, larger clear spans can be achieved. Beams and girders with corrugated webs are economical to use and can improve the aesthetics of the structure. Beams manufactured and used in Germany for buildings have a web thickness that varies between 2 and 5 mm, and the corresponding web height-to-thickness ratio is in the range of 150 to 260. The corrugated webs of two bridges built in France were 8 mm thick and the web height-to-thickness ratio was in the range of 220 to 375. Failure in shear is usually due to buckling of the web and the failure in bending is due to yielding of the compression flange and its vertical buckling into the corrugated web, which buckles [19, 20]. The shear buckling mode is global for dense corrugation and local for course corrugation, as shown in Figure 19.13. The load-carrying capacity of the specimens drops after buckling, with some residual load-carrying capacity after failure. In the local buckling mode, the corrugated web acts as a series of flat-plate subpanels that mutually support each other along their vertical (longer) edges and are supported by the flanges at their horizontal (shorter) edges. These flat-plate subpanels are subjected to shear, and the elastic buckling stress is given by h i τcre = ks π 2 E/12(1 − µ2 )(w/t)2
1999 by CRC Press LLC
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(19.48)
FIGURE 19.13: Local and global buckling.
where ks = buckling coefficient, which is a function of the panel aspect ratio, h/w, and the boundary support conditions h = the web depth t = the web thickness w = the flat-plate subpanel width — the horizontal or the inclined, whichever is bigger E = Young’s modulus of elasticity µ = the Poisson ratio The buckling coefficient, ks, is given by ks = 5.34 + 2.31(w/ h) − 3.44(w/ h)2 + 8.39(w/ h)3 , for the longer edges simply supported and the shorter edges clamped ks = 8.98 + 5.6(w/ h)2 , in the case where all edges are clamped An average local buckling stress, τav (= 0.5[τssf + τf x ]), is recommended, and in the case of τcre ≥ 0.8τy , inelastic buckling will occur and the inelastic buckling stress, τcri , can be calculated by τcri = (0.8 ∗ τcre ∗ τy )0.5 , where τcri ≤ τy . As stated earlier, the mode of failure is local and/or global buckling; when global buckling controls, the buckling stress can be calculated for the entire corrugated web panel, using orthotropic-plate buckling theory. The global elastic buckling stress, τcre , can be calculated from h 0.75 i /th2 τcre = ks (Dx )0.25 Dy where Dx Dy Iy ks
= = = =
(19.49)
(q/s)Et 3 /12 EIy /q 2bt (hr /2)2 + {t (hr )3 /6 sin 2} Buckling coefficient, equal to 31.6 for simply supported boundaries and 59.2 for clamped boundaries t = corrugated plate thickness b, hr , q, s, and 2 are as shown in Figure 19.14. 1999 by CRC Press LLC
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In the aforementioned, when τcre ≥ 0.8τy , inelastic buckling will occur and the inelastic buckling stress, τcri , can be calculated by τcri = (0.8 ∗ τcre ∗ τy )0.5 , where τ cri ≤ τy . For design, it is recommended that the local and global buckling values be calculated and the smaller value controls. As stated earlier, the failure in bending is due to compression flange yielding and vertical buckling into the web, as shown in Figure 19.15. The failure is sudden, with no appreciable residual strength. The web offers negligible contribution to the moment carrying capacity of the beam, and for design,
FIGURE 19.14: Dimensions of corrugation profile.
FIGURE 19.15: Bending failure of a beam with corrugate web.
the ultimate moment capacity can be calculated based on the flange yielding, ignoring any contribution from the web. The stresses in the web are equal to zero except near the flanges. This is because the corrugated web has no stiffness perpendicular to the direction of the corrugation, except for a 1999 by CRC Press LLC
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very small distance which is adjacent to and restrained by the flanges, and the stresses are appreciable only within the horizontal folds of the corrugation. It must be noted that the common practice is to fillet weld the web to the flanges from one side only; under static loading this welding detail was found to be adequate and there is no need to weld from both sides. Finally, the bracing requirements of the compression flange in beams and girders with corrugated webs are less severe compared to conventional beams and girders with flat webs. Lateral-torsional buckling of beams and girders with corrugated webs has been investigated [20].
19.15
Defining Terms
AASHTO: American Association of State Highway and Transportation Officials. AISC: American Institute of Steel Construction. Buckling load: The load at which a compressed element or member assumes a deflected position. Effective width: Reduced flat width of a plate element due to buckling, the reduced width is termed the effective width. Corrugated web: A web made of corrugated steel plates, where the corrugations are parallel to the depth of the girder. Factored load: The nominal load multiplied by a load factor to account for unavoidable deviations of the actual load from the nominal load. Limit state: A condition at which a structure or component becomes unsafe (strength limit state) or no longer useful for its intended function (serviceability limit state). LRFD (Load and Resistance Factor Design): A method of proportioning structural components such that no applicable limit state is exceeded when the structure is subjected to all appropriate load combinations. Shear wall: A wall in a building to carry lateral loads from wind and earthquakes. Stress: Force per unit area. Web crippling: Local buckling of the web plate under local loads. Web slenderness ratio: The depth-to-thickness ratio of the web.
References [1] American Association of State Highway and Transportation Officials. 1994. AASHTO LRFD Bridge Design Specifications, Washington, D.C. [2] Ajam, W. and Marsh, C. 1991. Simple Model for Shear Capacity of Webs, ASCE Struct. J., 117(2). [3] Basler, K. 1961. Strength of Plate Girders Under Combined Bending and Shear, ASCE J. Struct. Div., October, vol. 87. [4] Basler K. and Th¨urlimann, B. 1963. Strength of Plate Girders in Bending, Trans. ASCE, 128. [5] Basler, K. 1963. Strength of Plate Girders in Shear, Trans. ASCE, Vol. 128, Part II, 683. [6] British Standards Institution. 1983. BS 5400: Part 3, Code of Practice for Design of Steel Bridges, BSI, London. [7] Caccese, V., Elgaaly, M., and Chen, R. 1993. Experimental Study of Thin Steel-Plate Shear Walls Under Cyclic Load, ASCE J. Struct. Eng., February. [8] Cooper, P.B. 1967. Strength of Longitudinally Stiffened Plate Girders, ASCE J. Struct. Div., 93(ST2), 419-452. 1999 by CRC Press LLC
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[9] Dubas, P. and Gehrin, E. 1986. Behavior and Design of Steel Plated Structures, ECCS Publ. No. 44, TWG 8.3, 110-112. [10] Elgaaly, M. 1983. Web Design Under Compressive Edge Loads, AISC Eng. J., Fourth Quarter. [11] Elgaaly, M. and Nunan W. 1989. Behavior of Rolled Sections Webs Under Eccentric Edge Compressive Loads, ASCE J. Struct. Eng., 115(7). [12] Elgaaly, M. and Dagher, H. 1990. Beams and Girders with Corrugated Webs, Proceedings of the SSRC Annual Technical Session, St. Louis, MO. [13] Elgaaly, M. and Salkar, R. 1990. Behavior of Webs Under Eccentric Compressive Edge Loads, Proceedings of IUTAM Symposium, Prague, Czechoslovakia. [14] Elgaaly, M. and Salkar, R. 1991. Web Crippling Under Edge Loading, Proceedings of the AISC National Steel Construction Conference, Washington, D.C. [15] Elgaaly, M., Salkar, R., and Eash, M. 1992. Unstiffened and Stiffened Webs Under Compressive Edge Loads, Proceedings of the SSRC Annual Technical Session, Pittsburgh, PA. [16] Evans, H.R. and Mokhtari, A.R. 1992. Plate Girders with Unstiffened or Profiled Web Plates, J. Singapore Struct. Steel Soc., 3(1), December. [17] Elgaaly, M., Caccese, V., and Du, C. 1993. Postbuckling Behavior of Steel-Plate Shear Walls Under Cyclic Loads, ASCE J. Struct. Eng., 119(2). [18] Elgaaly, M., Liu, Y., Caccese, V., Du, C., Chen, R., and Martin, D. 1994. Non-Linear Behavior of Steel Plate Shear Walls, Computational Structural Engineering for Practice, edited by Papadrakakis and Topping, Civil-Comp Press, Edinburgh, UK. [19] Elgaaly, M., Hamilton, R., and Seshadri, A. 1996. Shear Strength of Beams with Corrugated Webs, J. Struct. Eng., 122(4). [20] Elgaaly, M., Seshadri, A., and Hamilton, R. 1996. Bending Strength of Beams with Corrugated Webs, J. Struct. Eng., 123(6). [21] Fisher, J.W. 1965. Behavior of Fasteners and Plates with Holes, J. Struct. Eng., ASCE, 91(6). [22] Galambos, T.V., Ed. 1988. Guide to Stability Design Criteria for Metal Structures, 4th ed., Wiley Interscience, New York. [23] Gaylord, E.H. 1963. Discussion of K. Basler Strength of Plate Girders in Shear, Trans. ASCE, 128, Part II, 712. [24] Hoglund, T. 1971. Behavior and Load Carrying Capacity of Thin Plate I-Girders, Division of Building Statics and Structural Engineering, Royal Institute of Technology, Bulletin No. 93, Stockholm. [25] Kirby, P.A. and Nethercat, D.A. 1979. Design for Structural Stability, John Wiley & Sons, New York. [26] Kulak, G.L., Fisher, J.W., and Struik, J.H.A. 1987. Guide to Design Criteria for Bolted and Riveted Joints, 2nd ed., Wiley Interscience, New York. [27] Kulak, G.L. 1985. Behavior of Steel Plate Shear Walls, Proc. of the AISC Int. Eng. Symp. on Struct. Steel, American Institute of Steel Construction (AISC), Chicago, IL. [28] Lee, M.M.K., Kamtekar, A.G., and Little, G.H. 1989. An Experimental Study of Perforated Steel Web Plates, Struct. Engineer, 67(2/24). [29] Lee, M.M.K. 1990. Numerical Study of Plate Girder Webs with Holes, Proc. Inst. Civ. Eng., Part 2. [30] Lee, M.M.K. 1991. A Theoretical Model for Collapse of Plate Girders with Perforated Webs, Struct. Eng., 68(4). [31] Lindner, J. 1990. Lateral-Torsional Buckling of Beams with Trapezoidally Corrugated Webs, Proceedings of the 4th International Colloquium on Stability of Steel Structures, Budapest, Hungary. [32] American Institute of Steel Construction. 1993. Load and Resistance Factor Design Specification for Structural Steel Buildings, AISC, Chicago. [33] Marsh, C. 1985. Photoelastic Study of Postbuckled Shear Webs, Canadian J. Civ. Eng., 12(2). 1999 by CRC Press LLC
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[34] Narayanan, R. and Der Avanessian, N.G.V. 1983. Strength of Webs Containing Cut-Outs, IABSE Proceedings P-64/83. [35] Narayanan, R. and Der Avanessian, N.G.V. 1983. Equilibrium Solution for Predicting the Strength of Webs with Rectangular Holes, Proc. ICE, Part 2. [36] Porter, D.M., Rockey, K.C., and Evans, H.R. 1975. The Collapse Behavior of Plate Girders Loaded in Shear, Struct. Eng., 53(8), 313-325. [37] Roberts, T.M. 1981. Slender Plate Girders Subjected to Edge Loading, Proc. Inst. Civil Eng., Part 2, 71. [38] Rockey, K.C. and Skaloud, M. 1972. The Ultimate Load Behavior of Plate Girders Loaded in Shear, Struct. Eng., 50(1). [39] Schilling, C.G. and Frost, R.W. 1964. Behavior of Hybrid Beams Subjected to Static Loads, ASCE, J. Struct. Div., 90(ST3), 55-88. [40] Von Karman, T., Sechler, E.F., and Donnell, L.H. 1932. The Strength of Thin Plates in Compression, Trans. ASME, 54(2). [41] Wagner, H. 1931. Flat Sheet Metal Girder with Very Thin Metal Web, NACA Tech. Memo. Nos. 604, 605, 606. [42] Yen, B.T. and Mueller, J.A. 1966. Fatigue Tests of Large-Size Welded Plate Girders, Weld. Res. Counc. Bull., No. 118, November.
1999 by CRC Press LLC
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Durkee, J. “Steel Bridge Construction” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Steel Bridge Construction 20.1 Introduction 20.2 Construction Engineering in Relation to Design Engineering 20.3 Construction Engineering Can Be Critical 20.4 Premises and Objectives of Construction Engineering 20.5 Fabrication and Erection Information Shown on\break Design Plans 20.6 Erection Feasibility 20.7 Illustrations of Challenges in Construction\break Engineering 20.8 Obstacles to Effective Construction Engineering 20.9 Examples of Inadequate Construction Engineering Allowances and Effort 20.10Considerations Governing Construction Engineering Practices 20.11Two General Approaches to Fabrication and\break Erection of Bridge Steelwork 20.12Example of Arch Bridge Construction 20.13Which Construction Procedure Is To Be Preferred? 20.14Example of Suspension Bridge Cable Construction 20.15Example of Cable-Stayed Bridge Construction 20.16Field Checking at Critical Erection Stages 20.17Determination of Erection Strength Adequacy 20.18Philosophy of the Erection Rating Factor 20.19Minimum Erection Rating Factors 20.20Deficiencies of Typical Construction Procedure\break Drawings and Instructions 20.21Shop and Field Liaison by Construction Engineers 20.22Construction Practices and Specifications— The Future 20.23Concluding Comments Jackson Durkee Consulting Structural Engineer, Bethlehem, 20.24Further Illustrations References PA
20.1
Introduction
This chapter addresses some of the principles and practices applicable to the construction of mediumand long-span steel bridges — structures of such size and complexity that construction engineering becomes an important or even the governing factor in the successful fabrication and erection of the superstructure steelwork. 1999 by CRC Press LLC
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We begin with an explanation of the fundamental nature of construction engineering, then go on to explain some of the challenges and obstacles involved. Two general approaches to the fabrication and erection of bridge steelwork are described, with examples from experience with arch bridges, suspension bridges, and cable-stayed bridges. The problem of erection-strength adequacy of trusswork under erection is considered, and a method of appraisal offered that is believed to be superior to the standard working-stress procedure. Typical problems in respect to construction procedure drawings, specifications, and practices are reviewed, and methods for improvement suggested. Finally, we take a view ahead, to the future prospects for effective construction engineering in the U.S. This chapter also contains a large number of illustrations showing a variety of erection methods for several types of steel bridges.
20.2
Construction Engineering in Relation to Design Engineering
With respect to bridge steelwork the differences between construction engineering and design engineering should be kept firmly in mind. Design engineering is of course a concept and process well known to structural engineers; it involves preparing a set of plans and specifications — known as the contract documents — that define the structure in its completed configuration, referred to as the geometric outline. Thus, the design drawings describe to the contractor the steel bridge superstructure that the owner wants to see in place when the project is completed. A considerable design engineering effort is required to prepare a good set of contract documents. Construction engineering, however, is not so well known. It involves governing and guiding the fabrication and erection operations needed to produce the structural steel members to the proper cambered or “no-load” shape, and get them safely and efficiently “up in the air” in place in the structure, such that the completed structure under the deadload conditions and at normal temperature will meet the geometric and stress requirements stipulated on the design drawings. Four key considerations may be noted: (1) design engineering is widely practiced and reasonably well understood, and is the subject of a steady stream of technical papers; (2) construction engineering is practiced on only a limited basis, is not as well understood, and is hardly ever discussed; (3) for medium- and long-span bridges, the construction engineering aspects are likely to be no less important than design engineering aspects; and (4) adequately staffed and experienced construction engineering offices are a rarity.
20.3
Construction Engineering Can Be Critical
The construction phase of the total life of a major steel bridge will probably be much more hazardous than the service-use phase. Experience shows that a large bridge is more likely to suffer failure during erection than after completion. Many decades ago, steel bridge design engineering had progressed to the stage where the chance of structural failure under service loadings became altogether remote. However, the erection phase for a large bridge is inherently less secure, primarily because of the prospect of inadequacies in construction engineering and its implementation at the job site. Indeed, the hazards associated with the erection of large steel bridges will be readily apparent from a review of the illustrations in this chapter. For significant steel bridges the key to construction integrity lies in the proper planning and engineering of steelwork fabrication and erection. Conversely, failure to attend properly to construction engineering constitutes an invitation to disaster. In fact, this thesis is so compelling that whenever a steel bridge failure occurs during construction (see for example Figure 20.1), it is reasonable to assume 1999 by CRC Press LLC
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that the construction engineering investigation was either inadequate, not properly implemented, or both.
FIGURE 20.1: Failure of a steel girder bridge during erection, 1995. Steel bridge failures such as this one invite suspicion that the construction engineering aspects were not properly attended to.
20.4
Premises and Objectives of Construction Engineering
Obviously, when the structure is in its completed configuration it is ready for the service loadings. However, during the erection sequences the various components of major steel bridges are subject to stresses that may be quite different from those provided for by the designer. For example, during construction there may be a derrick moving and working on the partially erected structure, and the structure may be cantilevered out some distance causing tension-designed members to be in compression and vice versa. Thus, the steelwork contractor needs to engineer the bridge members through their various construction loadings, and strengthen and stabilize them as may be necessary. Further, the contractor may need to provide temporary members to support and stabilize the structure as it passes through its successive erection configurations. In addition to strength problems there are also geometric considerations. The steelwork contractor must engineer the construction sequences step by step to ensure that the structure will fit properly together as erection progresses, and that the final or closing members can be moved into position and connected. Finally, of course, the steelwork contractor must carry out the engineering studies needed to ensure that the geometry and stressing of the completed structure under normal temperature will be in accordance with the requirements of the design plans and specifications.
1999 by CRC Press LLC
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20.5
Fabrication and Erection Information Shown on Design Plans
Regrettably, the level of engineering effort required to accomplish safe and efficient fabrication and erection of steelwork superstructures is not widely understood or appreciated in bridge design offices, nor indeed by a good many steelwork contractors. It is only infrequently that we find a proper level of capability and effort in the engineering of construction. The design drawings for an important bridge will sometimes display an erection scheme, even though most designers are not experienced in the practice of erection engineering and usually expend only a minimum or even superficial effort on erection studies. The scheme portrayed may not be practical, or may not be suitable in respect to the bidder or contractor’s equipment and experience. Accordingly, the bidder or contractor may be making a serious mistake if he relies on an erection scheme portrayed on the design plans. As an example of misplaced erection effort on the part of the designer, there have been cases where the design plans show cantilever erection by deck travelers, with the permanent members strengthened correspondingly to accommodate the erection loadings; but the successful bidder elected to use waterborne erection derricks with long booms, thereby obviating the necessity for most or all of the erection strengthening provided on the design plans. Further, even in those cases where the contractor would decide to erect by cantilevering as anticipated on the plans, there is hardly any way for the design engineer to know what will be the weight and dimensions of the contractor’s erection travelers.
20.6
Erection Feasibility
Of course, the bridge designer does have a certain responsibility to his client and to the public in respect to the erection of the bridge steelwork. This responsibility includes (1) making certain, during the design stage, that there is a feasible and economical method to erect the steelwork; (2) setting forth in the contract documents any necessary erection guidelines and restrictions; and (3) reviewing the contractor’s erection scheme, including any strengthening that may be needed, to verify its suitability. It may be noted that this latter review does not relieve the contractor from responsibility for the adequacy and safety of the field operations. Bridge annals include a number of cases where the designing engineer failed to consider erection feasibility. In one notable instance the design plans showed the 1200-ft (366-m) main span for a long crossing over a wide river as an esthetically pleasing steel tied-arch. However, erection of such a span in the middle of the river was impractical; one bidder found that the tonnage of falsework required was about the same as the weight of the permanent steelwork. Following opening of the bids, the owner found the prices quoted to be well beyond the resources available, and the tied-arch main span was discarded in favor of a through-cantilever structure, for which erection falsework needs were minimal and practical. It may be noted that designing engineers can stand clear of serious mistakes such as this one, by the simple expedient of conferring with prospective bidders during the preliminary design stage of a major bridge.
20.7
Illustrations of Challenges in Construction Engineering
Space does not permit comprehensive coverage of the numerous and difficult technical challenges that can confront the construction engineer in the course of the erection of various types of major steel bridges. However, some conception of the kinds of steelwork erection problems, the methods available to resolve them, and hazards involved can be conveyed by views of bridges in various stages of erection; refer to the illustrations in the text. 1999 by CRC Press LLC
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20.8
Obstacles to Effective Construction Engineering
There is an unfortunate tendency among designing engineers to view construction engineering as relatively unimportant. This view may be augmented by the fact that few designers have had any significant experience in the engineering of construction. Further, managers in the construction industry must look critically at costs, and they can readily develop the attitude that their engineers are doing unnecessary theoretical studies and calculations, detached from the practical world. (And indeed, this may sometimes be the case.) Such management apprehension can constitute a serious obstacle to staff engineers who see the need to have enough money in the bridge tender to cover a proper construction engineering effort for the project. There is the tendency for steelwork construction company management to cut back the construction engineering allowance, partly because of this apprehension and partly because of the concern that other tenderers will not be allotting adequate money for construction engineering. This effort is often thought of by company management as “a necessary evil” at best — something they would prefer not to be bothered with or burdened with. Accordingly, construction engineering tends to be a difficult area of endeavor. The way for staff engineers to gain the confidence of management is obvious — they need to conduct their investigations to a level of technical proficiency that will command management respect and support, and they must keep management informed as to what they are doing and why it is necessary. As for management’s concern that other bridge tenderers will not be putting into their packages much money for construction engineering, this concern is no doubt usually justified, and it is difficult to see how responsible steelwork contractors can cope with this problem.
20.9
Examples of Inadequate Construction Engineering Allowances and Effort
Even with the best of intentions, the bidder’s allocation of money to construction engineering can be inadequate. A case in point involved a very heavy, long-span cantilever truss bridge crossing a major river. The bridge superstructure carried a contract price of some $30 million, including an allowance of $150,000, or about one-half of 1%, for construction engineering of the permanent steelwork (i.e., not including such matters as design of erection equipment). As fabrication and erection progressed, many unanticipated technical problems came forward, including brittle-fracture aspects of certain grades of the high-strength structural steel, and aerodynamic instability of H-shaped vertical and diagonal truss members. In the end the contractor’s construction engineering effort mounted to about $1.3 million, almost nine times the estimated cost. Another significant example — this one in the domain of buildings — involved a design-andconstruct project for airplane maintenance hangars at a prominent airport. There were two large and complicated buildings, each 100 × 150 m (328 × 492 ft) in plan and 37 m (121 ft) high with a 10-m (33-ft) deep space-frame roof. Each building contained about 2300 tons of structural steelwork. The design-and-construct steelwork contractor had submitted a bid of about $30 million, and included therein was the magnificent sum of $5000 for construction engineering, under the expectation that this work could be done on an incidental basis by the project engineer in his “spare time”. As the steelwork contract went forward it quickly became obvious that the construction engineering effort had been grossly underestimated. The contractor proceeded of staff-up appropriately and carried out in-depth studies, leading to a detailed erection procedure manual of some 270 pages showing such matters as erection equipment and its positioning and clearances; falsework requirements; lifting tackle and jacking facilities; stress, stability, and geometric studies for gravity and wind loads; step-by-step instructions for raising, entering, and connecting steelwork components; closing and swinging the roof structure and portal frame; and welding guidelines and procedures. This 1999 by CRC Press LLC
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erection procedure manual turned out to be a key factor in the success of the fieldwork. The cost of this construction engineering effort amounted to ten times the estimate, but still came to a mere one-fifth of 1% of the total contract cost. In yet another example a major steelwork general contractor was induced to sublet the erection of a long-span cantilever truss bridge to a reputable erection contractor, whose quoted price for the work was less than the general contractor’s estimated cost. During the erection cycle the general contractor’s engineers made some visits to the job site to observe progress, and were surprised and disconcerted to observe how little erection engineering and planning had been accomplished. For example, the erector had made no provision for installing jacks in the bottom-chord jacking points for closure of the main span; it was left up to the field forces to provide the jack bearing components inside the bottom-chord joints and to find the required jacks in the local market. When the job-built installations were tested it was discovered that they would not lift the cantilevered weight, and the job had to be shut down while the field engineer scouted around to find larger-capacity jacks. Further, certain compression members did not appear to be properly braced to carry the erection loadings; the erector had not engineered those members, but just assumed they were adequate. It became obvious that the erector had not appraised the bridge members for erection adequacy and had done little or no planning and engineering of the critical evolutions to be carried out in the field. Many further examples of inadequate attention to construction engineering could be presented. Experience shows that the amounts of money and time allocated by steelwork contractors for the engineering of construction are frequently far less than desirable or necessary. Clearly, effort spent on construction engineering is worthwhile; it is obviously more efficient and cheaper, and certainly much safer, to plan and engineer steelwork construction in the office in advance of the work, rather than to leave these important matters for the field forces to work out. Just a few bad moves on site, with the corresponding waste of labor and equipment hours, will quickly use up sums of money much greater than those required for a proper construction engineering effort — not to mention the costs of any job accidents that might occur. The obvious question is “Why is construction engineering not properly attended to?” Do not contractors learn, after a bad experience or two, that it is both necessary and cost effective to do a thorough job of planning and engineering the construction of important bridge projects? Experience and observation would seem to indicate that some steelwork contractors learn this lesson, while many do not. There is always pressure to reduce bid prices to the absolute minimum, and to add even a modest sum for construction engineering must inevitably reduce the chance of being the low bidder.
20.10
Considerations Governing Construction Engineering Practices
There are no textbooks or manuals that define how to accomplish a proper job of construction engineering. In bridge construction (and no doubt in building construction as well), the engineering of construction tends to be a matter of each firm’s experience, expertise, policies and practices. Usually there is more than one way to build the structure, depending on the contractor’s ingenuity and engineering skill, his risk appraisal and inclination to assume risk, the experience of his fabrication and erection work forces, his available equipment, and his personal preferences. Experience shows that each project is different; and although there will be similarities from one bridge of a given type to another, the construction engineering must be accomplished on an individual project basis. Many aspects of the project at hand will turn out to be different from those of previous similar jobs, and also there may be new engineering considerations and requirements for a given project that did not come forward on previous similar work. During the estimating and bidding phase of the project the prudent, experienced bridge steelwork contractor will “start from scratch” and perform his own fabrication and erection studies, irrespective 1999 by CRC Press LLC
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of any erection schemes and information that may be shown on the design plans. These studies can involve a considerable expenditure of both time and money, and thereby place that contractor at a disadvantage in respect to those bidders who are willing to rely on hasty, superficial studies, or — where the design engineer has shown an erection scheme — to simply assume that it has been engineered correctly and proceed to use it. The responsible contractor, on the other hand, will appraise the feasible construction methods and evaluate their costs and risks, and then make his selection. After the contract has been executed the contractor will set forth how he intends to fabricate and erect, in detailed plans that could involve a large number of calculation sheets and drawings along with construction procedure documents. It is appropriate for the design engineer on behalf of his client to review the contractor’s plans carefully, perform a check of construction considerations, and raise appropriate questions. Where the contractor does not agree with the designer’s comments the two parties get together for review and discussion, and in the end they concur on essential factors such as fabrication and erection procedures and sequences, the weight and positioning of erection equipment, the design of falsework and other temporary components, erection stressing and strengthening of the permanent steelwork, erection stability and bracing of critical components, any erection check measurements that may be needed, and span closing and swinging operations. The designing engineer’s approval is needed for certain fabrication plans, such as the cambering of individual members; however, in most cases the designer should stand clear of actual approval of the contractor’s construction plans since he is not in a position to accept construction responsibility, and too many things can happen during the field evolutions over which the designer has no control. It should be emphasized that even though the designing engineer has usually had no significant experience in steelwork construction, the contractor should welcome his comments and evaluate them carefully and respectfully. In major bridge projects many matters can get out of control or can be improved upon, and the contractor should take advantage of every opportunity to improve his prospects and performance. The experienced contractor will make sure that he works constructively with the designing engineer, standing well clear of antagonistic or confrontational posturing.
20.11
Two General Approaches to Fabrication and Erection of Bridge Steelwork
As has been stated previously, the objective in steel bridge construction is to fabricate and erect the structure so that it will have the geometry and stressing designated on the design plans, under full dead load at normal temperature. This geometry is known as the geometric outline. In the case of steel bridges there have been, over the decades, two general procedures for achieving this objective: 1. The “field adjustment” procedure — Carry out a continuing program of field surveys and measurements, and perform certain adjustments of selected steelwork components in the field as erection progresses, in an attempt to discover fabrication and erection deficiencies and compensate for them. 2. The “shop control” procedure — Place total reliance on first-order surveying of span baselines and pier elevations, and on accurate steelwork fabrication and erection augmented by meticulous construction engineering; and proceed with erection without any field adjustments, on the basis that the resulting bridge deadload geometry and stressing will be as good as can possibly be achieved. Bridge designers have a strong tendency to overestimate the capability of field forces to accomplish accurate measurements and effective adjustments of the partially erected structure, and at the same time they tend to underestimate the positive effects of precise steel bridgework fabrication and 1999 by CRC Press LLC
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erection. As a result, we continue to find contract drawings for major steel bridges that call for field evolutions such as the following: 1. Continuous trusses and girders — At the designated stages, measure or “weigh” the reactions on each pier, compare them with calculated theoretical values, and add or remove bearing-shoe shims to bring measured values into agreement with calculated values. 2. Arch bridges — With the arch ribs erected to midspan and only the short, closing “crown sections” not yet in place, measure thrust and moment at the crown, compare them with calculated theoretical values, and then adjust the shape of the closing sections to correct for errors in span-length measurements and in bearing-surface angles at skewback supports, along with accumulated fabrication and erection errors. 3. Suspension bridges — Following erection of the first cable wire or strand across the spans from anchorage to anchorage, survey its sag in each span and adjust these sags to comport with calculated theoretical values. 4. Arch bridges and suspension bridges — Carry out a deck-profile survey along each side of the bridge under the steel-load-only condition, compare survey results with the theoretical profile, and shim the suspender sockets so as to render the bridge floorbeams level in the completed structure. 5. Cable-stayed bridges — At each deck-steelwork erection stage, adjust tensions in the newly erected cable stays so as to bring the surveyed deck profile and measured stay tensions into agreement with calculated theoretical data. There are two prime obstacles to the success of “field adjustment” procedures of whatever type: (1) field determination of the actual geometric and stress conditions of the partially erected structure and its components will not necessarily be definitive, and (2) calculation of the corresponding “proper” or “target” theoretical geometric and stress conditions will most likely prove to be less than authoritative.
20.12
Example of Arch Bridge Construction
In the case of the arch bridge closing sections referred to heretofore, experience on the construction of two major fixed-arch bridges crossing the Niagara River gorge from the U.S. to Canada—the Rainbow and the Lewiston-Queenston arch bridges (see Figures 20.2 through 20.5)—has demonstrated the difficulty, and indeed the futility, of attempts to make field-measured geometric and stress conditions agree with calculated theoretical values. The broad intent for both structures was to make such adjustments in the shape of the arch-rib closing sections at the crown (which were nominally about 1 ft [0.3 m] long) as would bring the arch-rib actual crown moments and thrusts into agreement with the calculated theoretical values, thereby correcting for errors in span-length measurements, errors in bearing-surface angles at the skewback supports, and errors in fabrication and erection of the arch-rib sections. Following extensive theoretical investigations and on-site measurements the steelwork contractor found, in the case of each Niagara arch bridge, that there were large percentage differences between the field-measured and the calculated theoretical values of arch-rib thrust, moment, and line-of-thrust position, and that the measurements could not be interpreted so as to indicate what corrections to the theoretical closing crown sections, if any, should be made. Accordingly, the contractor concluded that the best solution in each case was to abandon any attempts at correction and simply install the theoretical-shape closing crown sections. In each case, the contractor’s recommendation was accepted by the designing engineer. Points to be noted in respect to these field-closure evolutions for the two long-span arch bridges 1999 by CRC Press LLC
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FIGURE 20.2: Erection of arch ribs, Rainbow Bridge, Niagara Falls, New York, 1941. Bridge span is 950 ft (290 m), with rise of 150 ft (46 m); box ribs are 3 × 12 ft (0.91 × 3.66 m). Tiebacks were attached starting at the end of the third tier and jumped forward as erection progressed (see Figure 20.3). Much permanent steelwork was used in tieback bents. Derricks on approaches load steelwork on material cars that travel up arch ribs. Travelers are shown erecting last full-length arch-rib sections, leaving only the short, closing crown sections to be erected. Canada is at right, the U.S. at left. (Courtesy of Bethlehem Steel Corporation.)
are that accurate jack-load closure measurements at the crown are difficult to obtain under field conditions; and calculation of corresponding theoretical crown thrusts and moments are likely to be questionable because of uncertainties in the dead loading, in the weights of erection equipment, and in the steelwork temperature. Therefore, attempts to adjust the shape of the closing crown sections so as to bring the actual stress condition of the arch ribs closer to the theoretical condition are not likely to be either practical or successful. It was concluded that for long, flexible arch ribs, the best construction philosophy and practice is (1) to achieve overall geometric control of the structure by performing all field survey work and steelwork fabrication and erection operations to a meticulous degree of accuracy, and then (2) to rely on that overall geometric control to produce a finished structure having the desired stressing and geometry. For the Rainbow arch bridge, these practical construction considerations were set forth definitively by the contractor in [2]. The contractor’s experience for the Lewiston-Queenston arch bridge was similar to that on Rainbow, and was reported — although in considerably less detail — in [10].
20.13
Which Construction Procedure Is To Be Preferred?
The contractor’s experience on the construction of the two long-span fixed-arch bridges is set forth at length since it illustrates a key construction theorem that is broadly applicable to the fabrication 1999 by CRC Press LLC
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1999 by CRC Press LLC
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FIGURE 20.3: Rainbow Bridge, Niagara Falls, New York, showing successive arch tieback positions. Arch-rib erection geometry and stressing were controlled by means of measured tieback tensions in combination with surveyed arch-rib elevations.
FIGURE 20.4: Lewiston-Queenston arch bridge, near Niagara Falls, New York, 1962. The world’s longest fixed-arch span, at 1000 ft (305 m); rise is 159 ft (48 m). Box arch-rib sections are typically about 3 × 13-1/2 ft (0.9 × 4.1 m) in cross-section and about 44-1/2 ft (13.6 m) long. Job was estimated using erection tiebacks (same as shown in Figure 20.3), but subsequent studies showed the long, sloping falsework bents to be more economical (even if less secure looking). Much permanent steelwork was used in the falsework bents. Derricks on approaches load steelwork onto material cars that travel up arch ribs. The 115-ton-capacity travelers are shown erecting the last full-length arch-rib sections, leaving only the short, closing crown sections to be erected. Canada is at left, the U.S. at right. (Courtesy of Bethlehem Steel Corporation.) and erection of steel bridges of all types. This theorem holds that the contractor’s best procedure for achieving, in the completed structure, the deadload geometry and stressing stipulated on the design plans, is generally as follows: 1. Determine deadload stress data for the structure, at its geometric outline and under normal temperature, based on accurately calculated weights for all components. 2. Determine the cambered (i.e., “no-load”) dimensions of each component. This involves determining the change of shape of each component from the deadload geometry, as its deadload stressing is removed and its temperature is changed from normal to the “shop-tape” temperature. 3. Fabricate, with all due precision, each structural component to its proper no-load dimensions — except for certain flexible components such as wire rope and strand members, which may require special treatment. 4. Accomplish shop assembly of members and “reaming assembled” of holes in joints, as needed. 5. Carry out comprehensive engineering studies of the structure under erection at each key erection stage, determining corresponding stress and geometric data, and prepare a step-by-step erection procedure plan, incorporating any check measurements that may be necessary or desirable. 6. During the erection program, bring all members and joints to the designated alignment prior to bolting or welding. 1999 by CRC Press LLC
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1999 by CRC Press LLC
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FIGURE 20.5: Lewiston-Queenston arch bridge, near Niagara Falls, New York. Crawler cranes erect steelwork for spans 1 and 6 and erect material derricks thereon. These derricks erect traveler derricks, which move forward and erect supporting falsework and spans 2, 5, and 4. Traveler derricks erect arch-rib sections 1 and 2 and supporting falsework at each skewback, then set up creeper derricks, which erect arches to midspan.
7. Enter and connect the final or closing structural components, following the closing procedure plan, without attempting any field measurements thereof or adjustments thereto. In summary, the key to construction success is to accomplish the field surveys of critical baselines and support elevations with all due precision, perform construction engineering studies comprehensively and shop fabrication accurately, and then carry the erection evolutions through in the field without any second guessing and ill-advised attempts at measurement and adjustment. It may be noted that no special treatment is accorded to statically indeterminate members; they are fabricated and erected under the same governing considerations applicable to statically determinate members, as set forth above. It may be noted further that this general steel bridge construction philosophy does not rule out check measurements altogether, as erection goes forward; under certain special conditions, measurements of stressing and/or geometry at critical erection stages may be necessary or desirable in order to confirm structural integrity. However, before the erector calls for any such measurements he should make certain that they will prove to be practical and meaningful.
20.14
Example of Suspension Bridge Cable Construction
In order to illustrate the “shop control” construction philosophy further, its application to the main cables of the first Wm. Preston Lane, Jr., Memorial Bridge, crossing the Chesapeake Bay in Maryland, completed in 1952 (Figure 20.6), will be described. Suspension bridge cables constitute one of the most difficult bridge erection challenges. Up until “first Chesapeake” the cables of major suspension bridges had been adjusted to the correct position in each span by means of a sag survey of the first-erected cable wires or strands, using surveying instruments and target rods. However, on first Chesapeake, with its 1600-ft (488-m) main span, 661-ft (201-m) side spans, and 450-ft (l37-m) back spans, the steelwork contractor recommended abandoning the standard cable-sag survey and adopting the “setting-to-mark” procedure for positioning the guide strands — a significant new concept in suspension bridge cable construction.
FIGURE 20.6: Suspension spans of first Chesapeake Bay Bridge, Maryland, 1952. Deck steelwork is under erection and is about 50% complete. A typical four-panel through-truss deck section, weighing about 100 tons, is being picked in west side span, and also in east side span in distance. Main span is 1600 ft (488 m) and side spans are 661 ft (201 m); towers are 324 ft (99 m) high. Cables are 14 in. (356 mm) in diameter and are made up of 61 helical bridge strands each (see Figure 20.8).
The steelwork contractor’s rationale for “setting to marks” was spelled out in a letter to the designing engineer (see Figure 20.7). (The complete letter is reproduced because it spells out significant 1999 by CRC Press LLC
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construction philosophies.) This innovation was accepted by the designing engineer. It should be noted that the contractor’s major argument was that setting to marks would lead to a more accurate cable placement than would the sag survey. The minor arguments, alluded to in the letter, were the resulting savings in preparatory office engineering work and in the field engineering effort, and most likely in construction time as well. Each cable consisted of 61 standard helical-type bridge strands, as shown in Figure 20.8. To implement the setting-to-mark procedure each of three lower-layer “guide strands” of each cable (i.e., strands 1, 2, and 3) was accurately measured in the manufacturing shop under the simulated full-deadload tension, and circumferential marks were placed at the four center-of-saddle positions of each strand. Then, in the field, the guide strands (each about 3955 ft [1205 m] long) were erected and positioned according to the following procedure: 1. Place the three guide strands for each cable “on the mark” at each of the four saddles and set normal shims at each of the two anchorages. 2. Under conditions of uniform temperature and no wind, measure the sag differences among the three guide strands of each cable, at the center of each of the five spans. 3. Calculate the “center-of-gravity” position for each guide-strand group in each span. 4. Adjust the sag of each strand to bring it to the center-of-gravity position in each span. This position was considered to represent the correct theoretical guide-strand sag in each span. The maximum “spread” from the highest to the lowest strand at the span center, prior to adjustment, was found to be 1-3/4 in. (44 mm) in the main span, 3-1/2 in. (89 mm) in the side spans, and 3-3/4 in. (95 mm) in the back spans. Further, the maximum change of perpendicular sag needed to bring the guide strands to the center-of-gravity position in each span was found to be 15/16 in. (24 mm) for the main span, 2-1/16 in. (52 mm) for the side spans, and 2-1/16 in. (52 mm) for the back spans. These small adjustments testify to the accuracy of strand fabrication and to the validity of the setting-to-mark strand adjustment procedure, which was declared to be a success by all parties concerned. It seems doubtful that such accuracy in cable positioning could have been achieved using the standard sag-survey procedure. With the first-layer strands in proper position in each cable, the strands in the second and subsequent layers were positioned to hang correctly in relation to the first layer, as is customary and proper for suspension bridge cable construction. This example provides good illustration that the construction engineering philosophy referred to as the shop-control procedure can be applied advantageously not only to typical rigid-type steel structures, such as continuous trusses and arches, but also to flexible-type structures, such as suspension bridges. There is, however, an important caveat: the steelwork contractor must be a firm of suitable caliber and experience.
20.15
Example of Cable-Stayed Bridge Construction
In the case cable-stayed bridges, the first of which were built in the 1950s, it appears that the governing construction engineering philosophy calls for field measurement and adjustment as the means for control of stay-cable and deck-structure geometry and stressing. For example, we have seen specifications calling for the completed bridge to meet the following geometric and stress requirements: 1. The deck elevation at midspan shall be within 12 in. (305 mm) of theoretical. 2. The deck profile at each cable attachment point shall be within 2 in. (50 mm) of a parabola passing through the actual (i.e., field-measured) midspan point. 1999 by CRC Press LLC
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1999 by CRC Press LLC
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FIGURE 20.7: Setting cable guide strands to marks.
FIGURE 20.7: (Continued) Setting cable guide strands to marks.
1999 by CRC Press LLC
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FIGURE 20.8: Main cable of first Chesapeake Bay suspension bridge, Maryland. Each cable consists of 61 helical-type bridge strands, 55 of 1-11/16 in. (43 mm) and 6 of 29/32 in. (23 mm) diameter. Strands 1, 2, and 3 were designated “guide strands” and were set to mark at each saddle and to normal shims at anchorages. 3. Cable-stay tensions shall be within 5% of the “corrected theoretical” values. Such specification requirements introduce a number of problems of interpretation, field measurement, calculation, and field correction procedure, such as the following: 1. Interpretation: • The specifications are silent with respect to transverse elevation differentials. Therefore, two deck-profile control parabolas are presumably needed, one for each side of the bridge. 2. Field measurement of actual deck profile: • The temperature will be neither constant nor uniform throughout the structure during the survey work. • The survey procedure itself will introduce some inherent error. 3. Field measurement of cable-stay tensions: • Hydraulic jacks, if used, are not likely to be accurate within 2%, perhaps even 5%; further, the exact point of “lift off ” will be uncertain. • Other procedures for measuring cable tension, such as vibration or strain gaging, do not appear to define tensions within about 5%. • All cable tensions cannot be measured simultaneously; an extended period will be needed, during which conditions will vary and introduce additional errors. 1999 by CRC Press LLC
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4. Calculation of “actual” bridge profile and cable tensions: • Field-measured data must be transformed by calculation into “corrected actual” bridge profiles and cable tensions, at normal temperature and without erection loads. • Actual dead weights of structural components can differ by perhaps 2% from nominal weights, while temporary erection loads probably cannot be known within about 5%. • The actual temperature of structural components will be uncertain and not uniform. • The mathematical model itself will introduce additional error. 5. “Target condition” of bridge: • The “target condition” to be achieved by field adjustment will differ from the geometric condition, because of the absence of the deck wearing surface and other such components; it must therefore be calculated, introducing additional error. 6. Determining field corrections to be carried out by erector, to transform “corrected actual” bridge into “target condition” bridge: • The bridge structure is highly redundant, and changing any one cable tension will send geometric and cable-tension changes throughout the structure. Thus, an iterative correction procedure will be needed. It seems likely that the total effect of all these practical factors could easily be sufficient to render ineffective the contractor’s attempts to fine tune the geometry and stressing of the as-erected structure in order to bring it into agreement with the calculated bridge target condition. Further, there can be no assurance that the specifications requirements for the deck-profile geometry and cable-stay tensions are even compatible; it seems likely that either the deck geometry or the cable tensions may be achieved, but not both. Specifications clauses of the type cited seem clearly to constitute unwarranted and unnecessary field-adjustment requirements. Such clauses are typically set forth by bridge designers who have great confidence in computer-generated calculations, but do not have a sufficient background in and understanding of the practical factors associated with steel bridge construction. Experience has shown that field procedures for major bridges developed unilaterally by design engineers should be reviewed carefully to determine whether they are practical and desirable and will in fact achieve the desired objectives. In view of all these considerations, the question comes forward as to what design and construction principles should be followed to ensure that the deadload geometry and stressing of steel cable-stayed bridges will fall within acceptable limits. Consistent with the general construction-engineering procedures recommended for other types of bridges, we should abandon reliance on field measurements followed by adjustments of geometry and stressing, and instead place prime reliance on proper geometric control of bridge components during fabrication, followed by accurate erection evolutions as the work goes forward in the field. Accordingly, the proper construction procedure for cable-stayed steel bridges can be summarized as follows: 1. Determine the actual bridge baseline lengths and pier-top elevations to a high degree of accuracy. 2. Fabricate the bridge towers, cables, and girders to a high degree of geometric precision. 3. Determine, in the fabricating shop, the final residual errors in critical fabricated dimensions, including cable-stay lengths after socketing, and positions of socket bearing surfaces or pinholes. 1999 by CRC Press LLC
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4. Determine “corrected theoretical” shims for each individual cable stay. 5. During erection, bring all tower and girder structural joints into shop-fabricated alignment, with fair holes, etc. 6. At the appropriate erection stages, install “corrected theoretical” shims for each cable stay. 7. With the structure in the all-steel-erected condition (or other appropriate designated condition), check it over carefully to determine whether any significant geometric or other discrepancies are in evidence. If there are none, declare conditions acceptable and continue with erection. This construction engineering philosophy can be summarized by stating that if the steelwork fabrication and erection are properly engineered and carried out, the geometry and stressing of the completed structure will fall within acceptable limits; whereas, if the fabrication and erection are not properly done, corrective measurements and adjustments attempted in the field are not likely to improve the structure, or even to prove satisfactory. Accordingly, in constructing steel cable-stayed bridges we should place full reliance on accurate shop fabrication and on controlled field erection, just as is done on other types of steel bridges, rather than attempting to make measurements and adjustments in the field to compensate for inadequate fabrication and erection.
20.16
Field Checking at Critical Erection Stages
As has been stated previously, the best governing procedure for steel bridge construction is generally the shop control procedure, wherein full reliance is placed on accurate fabrication of the bridge components as the basis for the integrity of the completed structure. However, this philosophy does not rule out the desirability of certain checks in the field as erection goes forward, with the objective of providing assurance that the work is on target and no significant errors have been introduced. It would be impossible to catalog those cases during steel bridge construction where a field check might be desirable; such cases will generally suggest themselves as the construction engineering studies progress. We will only comment that these field-check cases, and the procedures to be used, should be looked at carefully, and even skeptically, to make certain that the measurements will be both desirable and practical, producing meaningful information that can be used to augment job integrity.
20.17
Determination of Erection Strength Adequacy
Quite commonly, bridge member forces during the erection stages will be altogether different from those that will prevail in the completed structure. At each critical erection stage the bridge members must be reviewed for strength and stability, to ensure structural integrity as the work goes forward. Such a construction engineering review is typically the responsibility of the steelwork erector, who carries out thorough erection studies of the structure and calls for strengthening or stabilizing of members as needed. The erector submits the studies and recommendations to the designing engineer for review and comment, but normally the full responsibility for steelwork structural integrity during erection rests with the erector. In the U.S., bridgework design specifications commonly require that stresses in steel structures under erection shall not exceed certain multiples of design allowable stresses. Although this type of erection stress limitation is probably safe for most steel structures under ordinary conditions, it is not necessarily adequate for the control of the erection stressing of large monumental-type bridges. The key point to be understood here is that fundamentally, there is no logical fixed relationship between design allowable stresses, which are based upon somewhat uncertain long-term service 1999 by CRC Press LLC
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FIGURE 20.9: Cable-stayed orthotropic-steel-deck bridge over Mississippi River at Luling, La., 1982; view looking northeast. The main span is 1222 ft (372 m); the A-frame towers are 350 ft (107 m) high. A barge-mounted ringer derrick erected the main steelwork, using a 340-ft (104-m) boom with a 120-ft (37-m) jib to erect tower components weighing up to 183 tons, and using a shorter boom for deck components. Cable stays at the ends of projecting cross girders are permanent; others are temporary erection stays. Girder section 16-west of north portion of bridge, erected a few days previously, is projecting at left; companion girder section 16-east is on barge ready for erection (see Figure 20.10).
loading requirements along with some degree of assumed structural deterioration, and stresses that are safe and economical during the bridge erection stages, where loads and their locations are normally well defined and the structural material is in new condition. Clearly, the basic premises of the two situations are significantly different, and “factored design stresses” must therefore be considered unreliable as a basis for evaluating erection safety. There is yet a further problem with factored design stresses. Large truss-type bridges in various erection stages may undergo deflections and distortions that are substantial compared with those occurring under service conditions, thereby introducing apprehension regarding the effect of the secondary bending stresses that result from joint rigidity. Recognizing these basic considerations, the engineering department of a major U.S. steelwork contractor went forward in the early 1970’s to develop a logical philosophy for erection strength appraisal of large structural steel frameworks, with particular reference to long-span bridges, and implemented this philosophy with a stress analysis procedure. The effort was successful and the results were reported in a paper published by the American Society of Civil Engineers in 1977 [5]. This stress analysis procedure, designated the erection rating factor (ERF) procedure, is founded directly upon basic structural principles, rather than on bridge-member design specifications, which are essentially irrelevant to the problem of erection stressing. 1999 by CRC Press LLC
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FIGURE 20.10: Luling Bridge deck steelwork erection, 1982; view looking northeast (refer to Figure 20.9) The twin box girders are 14 ft (4.3 m) deep; the deck plate is 7/16 in. (11 mm) thick. Girder section 16-east is being raised into position (lower right) and will be secured by large-pin hinge bars prior to fairing-up of joint holes and permanent bolting. Temporary erection stays are jumped forward as girder erection progresses.
It may be noted that a significant inducement toward development of the ERF procedure was the failure of the first Quebec cantilever bridge in 1907 (see Figures 20.11 and 20.12). It was quite obvious that evaluation of the structural safety of the Quebec bridge at advanced cantilever erection stages, such as that portrayed in Figure 20.11, by means of the factored design stress procedure, would inspire no confidence and would not be justifiable. The erection rating factor (ERF) procedure can be summarized as follows: 1. Assume either (a) pin-ended members (no secondary bending), (b) plane-frame action (rigid truss joints, secondary bending in one plane), or (c) space-frame action (bracingmember joints also rigid, secondary bending in two planes), as engineering judgment dictates. 2. Determine, for each designated erection stage, the member primary forces (axial) and secondary forces (bending) attributable to gravity loads and wind loads. 3. Compute the member stresses induced by the combined erection axial forces and bending moments. 4. Compute the ERF for each member at three or five locations: at the middle of the member; at each joint, inside the gusset plates (usually at the first row of bolts); and, where upset member plates or gusset plates are used, at the stepped-down cross-section outside each joint. 5. Determine the minimum computed ERF for each member and compare it with the stipulated minimum value. 1999 by CRC Press LLC
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FIGURE 20.11: First Quebec railway cantilever bridge, 23 August 1907. Cantilever erection of south main span, 6 days before collapse. The tower traveler erected the anchor span (on falsework) and then the cantilever arm; then erected the top-chord traveler, which is shown erecting suspended span at end of cantilever arm. The main span of 1800 ft (549 m) was the world’s longest of any type. The sidespan bottom chords second from pier (arrow) failed in compression because latticing connecting chord corner angles was deficient under secondary bending conditions.
6. Where the computed minimum ERF equals or exceeds the stipulated minimum value, the member is considered satisfactory. Where it is less, the member may be inadequate; the critical part of it is reevaluated in greater detail and the ERF recalculated for further comparison with the stipulated minimum. (Initially calculated values can often be increased significantly.) 7. When the computed minimum ERF remains less than the stipulated minimum, the member must be strengthened as required. Note that member forces attributable to wind are treated the same as those attributable to gravity loads. The old concept of “increased allowable stresses” for wind is not considered to be valid for erection conditions and is not used in the ERF procedure. Maximum acceptable `/r and b/t values are included in the criteria. ERFs for members subjected to secondary bending moments are calculated using interaction equations.
20.18
Philosophy of the Erection Rating Factor
In order that the structural integrity or reliability of a steel framework can be maintained throughout the erection program, the minimum probable (or “minimum characteristic”) strength value of each member must necessarily be no less than the maximum probable (or “maximum characteristic”) force value, under the most adverse erection condition. In other words, the following relationship is 1999 by CRC Press LLC
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FIGURE 20.12: Wreckage of south anchor span of first Quebec railway cantilever bridge, 1907. View looking north from south shore a few days after collapse of 29 August 1907, the worst disaster in the history of bridge construction. About 20,000 tons of steelwork fell into the St. Lawrence River, and 75 workmen lost their lives.
required: S − 1S ≥ F + 1F where S 1S F 1F
= = = =
(20.1)
computed or nominal strength value for the member maximum probable member strength underrun from the computed or nominal value computed or nominal force value for the member maximum probable member force overrun from the computed or nominal value
Equation 20.1 states that in the event the actual strength of the structural member is less than the nominal strength, S, by an amount 1S, while at same time the actual force in the member is greater than the nominal force, F , by an amount 1F , the member strength will still be no less than the member force, and so the member will not fail during erection. This equation provides a direct appraisal of erection realities, in contrast to the allowable-stress approach based on factored design stresses. Proceeding now to rearrange the terms in Equation 20.1, we find that 1F 1S ≥F 1+ ; S 1− S F 1999 by CRC Press LLC
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1 + 1F S F ≥ F 1 − 1S S
(20.2)
The ERF is now defined as ERF ≡
S F
(20.3)
that is, the nominal strength value, S, of the member divided by its nominal force value, F . Thus, for erection structural integrity or reliability to be maintained, it is necessary that ERF ≥
20.19
1+ 1−
1F F 1S S
(20.4)
Minimum Erection Rating Factors
In view of possible errors in (1) the assumed weight of permanent structural components, (2) the assumed weight and positioning of erection equipment, and (3) the mathematical models assumed for purposes of erection structural analysis, it is reasonable to assume that the actual member force for a given erection condition may exceed the computed force value by as much as 10%; that is, it is reasonable to take 1F /F as equal to 0.10. For tension members, uncertainties in (1) the area of the cross-section, (2) the strength of the material, and (3) the member workmanship, indicate that the actual member strength may be up to 15% less than the computed value; that is, 1S/S can reasonably be taken as equal to 0.15. The additional uncertainties associated with compression member strength suggest that 1S/S be taken as 0.25 for those members. Placing these values into Equation 20.4, we obtain the following minimum ERFs: Tension members: ERFt,min = (1 + 0.10)/(1 − 0.15) = 1.294, say 1.30 Compression members: ERFc,min = (1 + 0.10)/(1 − 0.25) = 1.467, say 1.45 The proper interpretation of these expressions is that if, for a given tension (compression) member, the ERF is calculated as 1.30 (1.45) or more, the member can be declared safe for the particular erection condition. Note that higher, or lower, values of erection rating factors may be selected if conditions warrant. The minimum ERFs determined as indicated are based on experience and judgment, guided by analysis and test results. They do not reflect any specific probabilities of failure and thus are not based on the concept of an acceptable risk of failure, which might be considered the key to a totally rational approach to structural safety. This possible shortcoming in the ERF procedure might be at least partially overcome by evaluating the parameters 1F /F and 1S/S on a statistical basis; however, this would involve a considerable effort, and it might not even produce significant results. It is important to recognize that the ERF procedure for determining erection strength adequacy is based directly on fundamental strength and stability criteria, rather than being only indirectly related to such criteria through the medium of a design specification. Thus, the procedure gives uniform results for the erection rating of framed structural members irrespective of the specification that was used to design the members. Obviously, the end use of the completed structure is irrelevant to its strength adequacy during the erection configurations, and therefore the design specification should not be brought into the picture as the basis for erection appraisal. Experience with application of the ERF procedure to long-span truss bridges has shown that it places the erection engineer in much better contact with the physical significance of the analysis than can be obtained by using the factored design stress procedure. Further, the ERF procedure takes account of secondary stresses, which have generally been neglected in erection stress analysis. 1999 by CRC Press LLC
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Although the ERF procedure was prepared for application to truss bridge members, the simple governing structural principle set forth by Equation 20.1 could readily be applied to bridge components of any type.
20.20
Deficiencies of Typical Construction Procedure Drawings and Instructions
At this stage of the review it is appropriate to bring forward a key problem in the realm of bridge construction engineering: the strong tendency for construction procedure drawings to be insufficiently clear, and for step-by-step instructions to be either lacking or less than definitive. As a result of these deficiencies it is not uncommon to find the contractor’s shop and field evolutions to be going along under something less than suitable control. Shop and field operations personnel who are in a position to speak frankly to construction engineers will sometimes let them know that procedure drawings and instructions often need to be clarified and upgraded. This is a pervasive problem, and it results from two prime causes: (1) the fabrication and erection engineers responsible for drawings and instructions do not have adequate on-the-job experience, and (2) they are not sufficiently skilled in the art of setting forth on the documents, clearly and concisely, exactly what is to be done by the operations forces—and, sometimes of equal importance, what is not to be done. This matter of clear and concise construction procedure drawings and instructions may appear to be a pedestrian matter, but it is decidedly not. It is a key issue of utmost importance to the success of steel bridge construction.
20.21
Shop and Field Liaison by Construction Engineers
In addition to the need for well-prepared construction procedure drawings and instructions, it is essential for the staff engineers carrying out construction engineering to set up good working relations with the shop and field production forces, and to visit the work sites and establish effective communication with the personnel responsible for accomplishing what is shown on the documents (see Figure 20.13). Construction engineers should review each projected operation in detail with the work forces, and upgrade the procedure drawings and instructions as necessary, as the work goes forward. Further, engineers should be present at the work sites during critical stages of fabrication and erection. As a component of these site visits, the engineers should organize special meetings of key production personnel to go over critical operations in detail—complete with slides and blackboard as needed— thereby providing the work forces with opportunities to ask questions and discuss procedures and potential problems, and providing engineers the opportunity to determine how well the work forces understand the operations to be carried out. This matter of liaison between the office and the work sites—like the preceding issue of clear construction procedure documents—may appear to be somewhat prosaic; again, however, it is a matter of paramount importance. Failure to attend to these two key issues constitutes a serious problem in steel bridge construction, and opens the door to high costs and delays, and even to erection accidents.
20.22
Construction Practices and Specifications—The Future
The many existing differences of opinion and procedures in respect to proper governance of steelwork fabrication and erection for major steel bridges raises the question: How do proper bridge 1999 by CRC Press LLC
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FIGURE 20.13: Visiting the work site. It is of first-order importance for bridge construction engineers to visit the site regularly and confer with the job superintendent and his foremen regarding practical considerations. Construction engineers have much to learn from the work forces in shop and field, and vice versa. (Courtesy of Bethlehem Steel Corporation.)
construction guidelines come into existence and find their way into practice and into bridge specifications? Looking back over the period roughly from 1900 to 1975, we find that the major steelwork construction companies in the U.S. developed and maintained competent engineering departments that planned and engineered large bridges (and smaller ones as well) through the fabrication and erection processes with a high degree of proficiency. Traditionally, the steelwork contractor’s engineers worked in cooperation with design-office engineers to develop the full range of bridgework technical factors, including construction procedures and practices. However, times have changed during the last two decades; since 1970s major steel bridge contractors have all but disappeared in the U.S., and further, very few bridge design offices have on their staffs engineers experienced in fabrication and erection engineering. As a result, construction-engineering often receives less attention and effort than it needs and deserves, and this is not a good omen for the future of the design and construction of large bridges in the U.S. Bridge construction engineering is not a subject that is or can be taught in the classroom; it must be learned on the job with major steelwork contractors. The best route for an aspiring young construction engineer is to spend significant amounts of time in the fabricating shop and at the job site, interspersed with time doing construction-engineering technical work in the office. It has been pointed out previously that although construction engineering and design engineering are related, they constitute different practices and require diverse backgrounds and experience. Design engineering can essentially be learned in the design office; construction engineering, however, cannot—it requires a background of experience at work sites. Such experience, it may be noted, is valuable also for design engineers; however, it is not as necessary for them as it is for construction engineers. The training of future steelwork construction engineers in the U.S. will be handicapped by the demise of the “Big Two” steelwork contractors in the 1970s. Regrettably, it appears that surviving steelwork contractors in the U.S. generally do not have the resources for supporting strong engineering 1999 by CRC Press LLC
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departments, and so there is some question as to where the next generation of steel bridge construction engineers in the U.S. will be coming from.
20.23
Concluding Comments
In closing this review of steel bridge construction it is appropriate to quote from the work of an illustrious British engineer, teacher, and author, the late Sir Alfred Pugsley [14]: A further crop of [bridge] accidents arose last century from overloading by traffic of various kinds, but as we have seen, engineers today concentrate much of their effort to ensure that a margin of strength is provided against this eventuality. But there is one type of collapse that occurs almost as frequently today as it has over the centuries: collapse at a late stage of erection. The erection of a bridge has always presented its special perils and, in spite of everincreasing care over the centuries, few great bridges have been built without loss of life. Quite apart from the vagaries of human error, with nearly all bridges there comes a critical time near completion when the success of the bridge hinges on some special operation. Among such are . . . the fitting of a last section . . . in a steel arch, the insertion of the closing central [members] in a cantilever bridge, and the lifting of the roadway deck [structure] into position on a suspension bridge. And there have been major accidents in many such cases. It may be wondered why, if such critical circumstances are well known to arise, adequate care is not taken to prevent an accident. Special care is of course taken, but there are often reasons why there may still be “a slip betwixt cup and lip”. Such operations commonly involve unusually close cooperation between constructors and designers, and between every grade of staff, from the laborers to the designers and directors concerned; and this may put a strain on the design skill, on detailed inspection, and on practical leadership that is enough to exhaust even a Brunel. In such circumstances it does well to . . . recall [the] dictum . . . that “it is essential not to have faith in human nature. Such faith is a recent heresy and a very disastrous one.” One must rely heavily on the lessons of past experience in the profession. Some of this experience is embodied in professional papers describing erection processes, often (and particularly to young engineers) superficially uninteresting. Some is crystallized in organizational habits, such as the appointment of resident engineers from both the contracting and [design] sides. And some in precautions I have myself endeavored to list . . . . It is an easy matter to list such precautions and warnings, but quite another for the senior engineers responsible for the completion of a bridge to stand their ground in real life. This is an area of our subject that depends in a very real sense on the personal qualities of bridge engineers . . . . At bottom, the safety of our bridges depends heavily upon the integrity of our engineers, particularly the leading ones.
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20.24
Further Illustrations of Bridges Under Construction, Showing Erection Methods
FIGURE 20.14: Royal Albert Bridge across River Tamar, Saltash, England, 1857. The two 455-ft (139-m) main spans, each weighing 1060 tons, were constructed on shore, floated out on pairs of barges, and hoisted about 100 ft (30 m) to their final position using hydraulic jacks. Pier masonry was built up after each 3-ft (1-m) lift.
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FIGURE 20.15: Eads Bridge across the Mississippi River, St. Louis, Mo., 1873. The first important metal arch bridge in the U.S., supported by four planes of hingeless trussed arches having chromesteel tubular chords. Spans are 502-520-502 ft (153-158-153 m). During erection, arch ribs were tied back by cables passing over temporary towers built on the piers. Arch ribs were packed in ice to effect closure.
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FIGURE 20.16: Glasgow (Missouri) railway truss bridge, 1879. Erection on full supporting falsework was commonplace in the 19th century. The world’s first all-steel bridge, with five 315-ft (96-m) through-truss simple spans, crossing the Missouri River.
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FIGURE 20.17: Niagara River railway cantilever truss bridge, near Niagara Falls, New York 1883. Massive wood erection traveler constructed side span on falsework, then cantilevered half of main span to midspan. Erection of other half of bridge was similar. First modern-type cantilever bridge, with 470-ft (143-m) clear main span having a 120-ft (37-m) center suspended span.
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FIGURE 20.18: Construction of monumental Forth Bridge, Scotland, 1888. Numerous small movable booms were used, along with erection travelers for cantilevering the two 1710-ft (521-m) main spans. The main compression members are tubes 12 ft (3.65 m) in diameter; many other members are also tubular. Total steelwork weight is 51,000 tons. Records are not clear regarding such essentials as cambering and field fitting of individual members in this heavily redundant railway bridge. The Forth is arguably the world’s greatest steel structure.
FIGURE 20.19: Pecos River railway viaduct, Texas, 1892. Erection by massive steam-powered wood traveler having many sets of falls and very long reach. Cantilever-truss main span has 185-ft (56-m) clear opening. 1999 by CRC Press LLC
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FIGURE 20.20: Raising of suspended span, Carquinez Strait Bridge, California, 1927. The 433-ft (132-m) suspended span, weighing 650 tons, was raised into position in 35 min., driven by four counterweight boxes having a total weight of 740 tons.
FIGURE 20.21: First Cooper River cantilever bridge, Charleston, S.C., 1929. Erection travelers constructed 450-ft (137-m) side spans on falsework, then went on to erect 1050-ft (320-m) main span (including 437.5-ft [133-m] suspended span) by cantilevering to midspan.
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FIGURE 20.22: Erecting south tower of Golden Gate Bridge, San Francisco, 1935. A creeper traveler with two 90-ft (27-m) booms erects a tier of tower cells for each leg, then is jumped to the top of that tier and proceeds to erect the next tier. The tower legs are 90 ft (27 m) center-to-center and 690 ft (210 m) high. When the traveler completed the north tower (in background) it erected a Chicago boom on the west tower leg, which dismantled the creeper, erected tower-top bracing, and erected two small derricks (one shown) to service cable erection. Each tower contains 22,200 tons of steelwork.
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FIGURE 20.23: Balanced-cantilever erection, Governor O.K. Allen railway/highway cantilever bridge, Baton Rouge, La., 1939. First use of long balanced-cantilever erection in the U.S. On each pier 650 ft (198 m) of steelwork, about 4000 tons, was balanced on the 40-ft (12-m) base formed by a sloping falsework bent. The compression load at the top of the falsework bent was measured at frequent intervals and adjusted by positioning a counterweight car running at bottom-chord level. The main spans are 848-650-848 ft (258-198-258 m); 650 ft span shown. (Courtesy of Bethlehem Steel Corporation.)
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FIGURE 20.24: Tower erection, second Tacoma Narrows Bridge, Washington, 1949. This bridge replaced first Tacoma Narrows bridge, which blew down in a 40-mph (18-m/sec) wind in 1940. The tower legs are 60 ft (18 m) on centers and 462 ft (141 m) high. The creeper traveler is shown erecting the west tower, in background. On the east tower, the creeper erected a Chicago boom at the top of the south leg; this boom dismantled the creeper, then erected the tower-top bracing and a stiffleg derrick, which proceeded to dismantle the Chicago boom. The tower manhoist can be seen at the second-from-topmost landing platform. Riveting cages are approaching the top of the tower. Note tower-base erection kneebraces, required to ensure tower stability in free-standing condition (see Figure 20.27).
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FIGURE 20.25: Aerial spinning of parallel-wire main cables, second Tacoma Narrows suspension bridge, Washington, 1949. Each main cable consists of 8702 parallel galvanized high-strength wires of 0.196-in. (4.98-mm) diameter, laid up as 19 strands of mostly 460 wires each. Following compaction the cable became a solid round mass of wires with a diameter of 20-1/4 in. (514 mm).
Figure 20.25a Tramway starts across from east anchorage carrying two wire loops. Three 460-wire strands have been spun, with two more under construction. Tramway spinning wheels pull wire loops across the three spans from east anchorage to west anchorage. Suspended footbridges provide access to cables. Spinning goes on 24 hours per day.
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Figure 20.25b Tramway arrives at west anchorage. Wire loops shown in Figure 20.25a are removed from spinning wheels and placed around strand shoes at west anchorage. This tramway then returns empty to east anchorage, while tramway for other “leg” of endless hauling rope brings two wire loops across for second strand that is under construction for this cable.
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Figure 20.26a
Erection of individual wire loops.
Figure 20.26b
Adjustment of individual wire loops.
FIGURE 20.26: Cable-spinning procedure for constructing suspension bridge parallel-wire main cables, showing details of aerial spinning method for forming individual 5-mm wires into strands containing 400 to 500 wires. Each wire loop is erected as shown in Figure 20.26a (refer to Figure 20.25), then adjusted to the correct sag as shown in Figure 20.26b. Each completed strand is banded with tape, then adjusted to the correct sag in each span. With all strands in place, they are compacted to form a solid round homogeneous mass of wires. The aerial spinning method was developed by John Roebling in the mid-19th century.
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FIGURE 20.27: Erection of suspended deck steelwork, second Tacoma Narrows Bridge, Washington, 1950. The Chicago boom on the tower raises deck steelwork components to deck level, where they are transported to deck travelers by material cars. Each truss double panel is connected at top-chord level to previously erected trusses, and left open at bottom-chord level to permit temporary upward deck curvature, which results from the partial loading condition of the main suspension cables. The main span (at right) is 2800 ft (853 m), and side spans are 1100 ft (335 m). The stiffening trusses are 33 ft (10 m) deep and 60 ft (18 m) on centers. Tower-base kneebraces (see Figure 20.24) show clearly here.
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FIGURE 20.28: Moving deck traveler forward, second Tacoma Narrows Bridge, Washington, 1950. The traveler pulling-falls leadline passes around the sheave beams at the forward end of the stringers, and is attached to the front of the material car (at left). The material car is pulled back toward the tower, advancing the traveler two panels to its new position at the end of the deck steelwork. Arrows show successive positions of material car. (a) Traveler at star of move, (b) traveler advanced one panel, and (c) traveler at end of move.
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FIGURE 20.29: Erecting closing girder sections of Passaic River Bridge, New Jersey Turnpike, 1951. Huge double-boom travelers, each weighing 270 tons, erect closing plate girders of the 375-ft (114-m) main span. The closing girders are 14 ft (4.3 m) deep and 115 ft (35 m) long and weigh 146 tons each. Sidewise entry was required (as shown) because of long projecting splice material. Longitudinal motion was provided at one pier, where girders were jacked to effect closure. Closing girders were laterally stable without floor steel fill-in, such that derrick falls could be released immediately. (Courtesy of Bethlehem Steel Corporation.)
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FIGURE 20.30: Floating-in erection of a truss span, first Chesapeake Bay Bridge, Maryland, 1951. Erected 300-ft (91-m) deck-truss spans form erection dock, providing a work platform for two derrick travelers. A permanent deck-truss span serves as a falsework truss supported on barges and is shown carrying the 470-ft (143-m) anchor arm of the through-cantilever truss. This span is floated to its permanent position, then landed onto its piers by ballasting the barges. (a) Float leaves erection dock, and (b) float arrives at permanent position. (Courtesy of Bethlehem Steel Corporation.)
FIGURE 20.31: Floating-in erection of a truss span, first Chesapeake Bay Bridge, Maryland, 1952. A 480-ft (146-m) truss span, weighing 850 tons, supported on falsework consisting of a permanent decktruss span along with temporary members, is being floated-in for landing onto its piers. Suspension bridge cables are under construction in background. (Courtesy of Bethlehem Steel Corporation.)
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FIGURE 20.32: Erection of a truss span by hoisting, first Chesapeake Bay Bridge, Maryland, 1952. A 360-ft (110-m) truss span is floated into position on barges and picked clear using four sets of lifting falls. Suspension bridge deck is under construction at right. (Courtesy of Bethlehem Steel Corporation.)
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FIGURE 20.33: Erection of suspension bridge deck structure, first Chesapeake Bay Bridge, Maryland, 1952. A typical four-panel through-truss deck section, weighing 99 tons, has been picked from the barge and is being raised into position using four sets of lifting falls attached to main suspension cables. The closing deck section is on the barge, ready to go up next. (Courtesy of Bethlehem Steel Corporation.)
FIGURE 20.34: Greater New Orleans cantilever bridge, Louisiana, 1957. Tall double-boom deck travelers started at ends of main bridge and erected anchor spans on falsework, then the 1575-ft (480-m) main span by cantilevering to midspan. (Courtesy of Bethlehem Steel Corporation.)
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FIGURE 20.35: Tower erection, second Delaware Memorial Bridge, Wilmington, Del., 1966. The tower erection traveler has reached the topmost erecting position and swings into place the 23-ton closing top-strut section. The tower legs were jacked apart about 2 in. (50 mm) to provide entering clearance. The traveler jumping beams are in the topmost working position, above the cable saddles. The tower steelwork is about 418 ft (127 m) high. Cable anchorage pier is under construction at right. First Delaware Memorial Bridge (1951) is at left. The main span of both bridges is 2150 ft (655 m). (Courtesy of Bethlehem Steel Corporation.)
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FIGURE 20.36: Erecting orthotropic-plate decking panel, Poplar Street Bridge, St. Louis, Mo., 1967. A five-span, 2165-ft (660-m) continuous box-girder bridge, main span 600 ft (183 m). Projecting box ribs are 5-1/2 × 17 ft (1.7 × 5.2 m) in cross-section, and decking section is 27 × 50 ft (8.2 × 15.2 m). Decking sections were field welded, while all other connections were field bolted. Box girders are cantilevered to falsework bents using overhead “positioning travelers” (triangular structure just visible above deck at left) for intermediate support. (Courtesy of Bethlehem Steel Corporation.)
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FIGURE 20.37: Erection of parallel-wire-strand (PWS) cables, the Newport Bridge suspension spans, Narragansett Bay, R.I., 1968. Bridge engineering history was made at Newport with the development and application of shop-fabricated parallel-wire socketed strands for suspension bridge cables. Each Newport cable was formed of seventy-six 61-wire PWS, about 4512 ft (1375 m) long. Individual wires are 0.202 in. (5.13 mm) in diameter and are zinc coated. Parallel-wire cables can be constructed of PWS faster and at lower cost than by traditional air spinning of individual wires (see Figures 20.25 and 20.26). (Courtesy of Bethlehem Steel Corporation.)
Figure 20.37a Aerial tramway tows PWS from west anchorage up side span, then on across other spans to east anchorage. Strands are about 1-3/4 in. (44 mm) in diameter.
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Figure 20.37b Cable formers maintain strand alignment in cables prior to compaction. Each finished cable is about 15-1/4 in. (387 mm) in diameter. (Courtesy of Bethlehem Steel Corporation.)
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FIGURE 20.38: Pipe-type anchorage for parallel-wire-strand (PWS) cables, the Newport Bridge suspension spans, Narragansett Bay, R.I., 1967. Pipe anchorages shown will be embedded in anchorage concrete. The socketed end of each PWS is pulled down its pipe from the upper end, then seated and shim-adjusted against the heavy bearing plate at the lower end. The pipe-type anchorage is much simpler and less costly than the standard anchor-bar type used with aerial-spun parallel-wire cables (see Figure 20.25b). (Courtesy of Bethlehem Steel Corporation.)
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FIGURE 20.39: Manufacturing facility for production of shop-fabricated parallel-wire strands (PWS). Prior to 1966, parallel-wire suspension bridge cables had to be constructed wire-by-wire in the field using the aerial spinning procedure developed by John Roebling in the mid-19th century (refer to Figures 20.25 and 20.26). In the early 1960s a major U.S. steelwork contractor originated and developed a procedure for manufacturing and reeling parallel-wire strands, as shown in these patent drawings. A PWS can contain up to 127 wires (see Figures 20.45 and 20.46). (a) Plan view of PWS facility. Turntables 11 contain “left-hand” coils of wire and turntables 13 contain “right-hand” coils, such that wire cast is balanced in the formed strand. Fairleads 23 and 25 guide the wires into half-layplates 27 and 29, followed by full layplates 31 and 32 whose guide holes delineate the hexagonal shape of final strand 41. (b) Elevation view of PWS facility. Hexagonal die 33 contains six spring-actuated rollers that form the wires into regular-hexagon shape; and similar roller dies 47, 49, 50, and 51 maintain the wires in this shape as PWS 41 is pulled along by hexagonal dynamic clamp 53. The PWS is bound manually with plastic tape at about 3-ft (1-m) intervals as it passes along between roller dies. The PWS passes across roller table 163, then across traverse carriage 168, which is operated by traverse mechanism 161 to direct the PWS properly onto reel 159. Finally, the reeled PWS is moved off-line for socketing. Note that wire measuring wheels (201) can be installed and used for control of strand length.
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FIGURE 20.40: Suspended deck steelwork erection, the Newport Bridge suspension spans, Narragansett Bay, R.I., 1968. The closing mainspan deck section is being raised into position by two cable travelers, each made up of a pair of 36-in. (0.91-m) wide-flange rolled beams that ride the cables on wooden wheels. The closing section is 40-1/2 ft (12 m) long at top-chord level, 66 ft (20 m) wide and 16 ft (5 m) deep, and weighs about 140 tons. (Courtesy of Bethlehem Steel Corporation.)
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FIGURE 20.41: Erection of Kansas City Southern Railway box-girder bridge, near Redland, Okla., by “launching”, 1970. This nine-span continuous box-girder bridge is 2110 ft (643 m) long, with a main span of 330 ft (101 m). Box cross-section is 11 × 14.9 ft (3.35 × 4.54 m). The girders were launched in two “trains”, one from the north end and one from the south end. A “launching nose” was used to carry the leading end of each girder train up onto the skidway supports as the train was pushed out onto successive piers. Closure was accomplished at center of main span. (Courtesy of Bethlehem Steel Corporation.)
Figure 20.41a Leading end of north girder train moves across 250-ft (76-m) span 4, approaching pier 5. Main span is to right of pier 5.
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Figure 20.41b
Launching nose rides up onto pier 5 skidway units, removing girder-train leading-end sag.
Figure 20.41c
Leading end of north girder train is now supported on pier 5.
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Figure 20.42a compression.
Typical assumed erection loading of box-girder web panels in combined moment, shear, and transverse
Figure 20.42b
Launch of north girder train from pier 4 to pier 5.
Figure 20.42c to pier 5.
Negative-moment envelopes occurring simultaneously with reaction, for launch of north girder train
FIGURE 20.42: Erection strengthening to withstand launching, Kansas City Southern Railway boxgirder bridge, near Redland, Okla. (see Figure 20.41).
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FIGURE 20.43: Erection of west arch span of twin-arch Hernando de Soto Bridge, Memphis, Tenn., 1972. The two 900-ft (274-m) continuous-truss tied-arch spans were erected by a high-tower derrick boat incorporating a pair of barges. West-arch steelwork (shown) was cantilevered to midspan over two pile-supported falsework bents. Projecting east-arch steelwork (at right) was then cantilevered to midspan (without falsework) and closed with falsework-supported other half-arch. (Courtesy of Bethlehem Steel Corporation.)
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FIGURE 20.44: Closure of east side span, Commodore John Barry cantilever truss bridge, Chester, Pa., 1973. A high-tower derrick boat (in background) started erection of trusses at both main piers, supported on falsework; then erected top-chord travelers for main and side spans. The sidespan traveler carried steelwork erection to closure, as shown, and the falsework bent was then removed. The mainspan traveler then cantilevered the steelwork (without falsework) to midspan, concurrently with erection by the west-half mainspan traveler, and the trusses were closed at midspan. Commodore Barry has a 1644-ft (501-m) main span, the longest cantilever span in the U.S., and 822-ft (251-m) side spans. (Courtesy of Bethlehem Steel Corporation.)
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FIGURE 20.45: Reel of parallel-wire strand (PWS), Akashi Kaikyo suspension bridge, Kobe, Japan, 1994. Each socketed PWS is made up of 127 0.206-in. (5.23-mm) wires, is 13,360 ft (4073 m) long, and weighs 96 tons. Plastic-tape bindings secure the strand wires at 1-m intervals. Sockets can be seen on right side of reel. These PWS are the longest and heaviest ever manufactured. (Courtesy of Nippon Steel—Kobe Steel.)
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FIGURE 20.46: Parallel-wire-strand main cable, Akashi Kaikyo suspension bridge, Kobe, Japan, 1994. The main span is 6529 ft (1990 m), by far the world’s longest. The PWS at right is being towed across the spans, supported on rollers. The completed cable is made up of 290 PWS, making a total of 36,830 wires, and has a diameter of 44.2 in. (1122 mm) following compaction—the largest bridge cables built to date. Each 127-wire PWS is about 2-3/8 in. (60 mm) in diameter. (Courtesy of Nippon Steel—Kobe Steel.)
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FIGURE 20.47: Artist’s rendering of proposed Messina Strait suspension bridge, Italy. The Messina Strait crossing has been under discussion since about 1850, under investigation since about 1950, and under active design since about 1980. The enormous bridge shown would connect Sicily to mainland Italy with a single span of 10,827 ft (3300 m). Towers are 1250 ft (380 m) high. The bridge construction problems for such a span will be tremendously challenging. (Courtesy of Stretto di Messina, S.p.A.)
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References [1] Conditions of Contract and Forms of Tender, Agreement and Bond for Use in Connection with Works of Civil Engineering Construction, 6th ed. (commonly known as “ICE Conditions of Contract”), Inst. Civil Engrs. (U.K.), 1991. [2] Copp, J.I., de Vries, K., Jameson, W.H., and Jones, J. 1945. Fabrication and Erection Controls, Rainbow Arch Bridge Over Niagara Gorge — A Symposium, Transactions ASCE, vol. 110. [3] Durkee, E.L, 1945. Erection of Steel Superstructure, Rainbow Arch Bridge Over Niagara Gorge — A Symposium, Transactions ASCE, vol. 110. [4] Durkee, J.L. 1972. Railway Box-Girder Bridge Erected by Launching, J. Struct. Div., ASCE, July. [5] Durkee, J.L. and Thomaides, S.S. 1977. Erection Strength Adequacy of Long Truss Cantilevers, J. Struct. Div., ASCE, January. [6] Durkee, J.L. 1977. Needed: U.S. Standard Conditions for Contracting, J. Struct. Div., ASCE, June. [7] Durkee, J.L. 1982. Bridge Structural Innovation: A Firsthand Report, J. Prof. Act., ASCE, July. [8] Durkee, J.L. 1966. Advancements in Suspension Bridge Cable Construction, Proceedings, International Symposium on Suspension Bridges, Laboratorio Nacional de Engenharia Civil, Lisbon. [9] Enquiry into the Basis of Design and Methods of Erection of Steel Box Girder Bridges. Final Report of Committee, 4 vols. (commonly known as “The Merrison Report”), HMSO (London), 1973/4. [10] Feidler, L.L., Jr. 1962. Erection of the Lewiston-Queenston Bridge, Civil Engrg., ASCE, November. [11] Freudenthal, A.M., Ed. 1972. The Engineering Climatology of Structural Accidents, Proceedings of the International Conference on Structural Safety and Reliability, Pergamon Press, Elmsford, N.Y. [12] Holgate, H., Kerry, J.G.G., and Galbraith, J. 1908. Royal Commission Quebec Bridge Inquiry Report, Sessional Paper No. 154, vols. I and II, S.E. Dawson, Ottawa, Canada. [13] Petroski, H. 1993. Predicting Disaster, American Scientist, vol. 81, March. [14] Pugsley, A. 1968. The Safety of Bridges, The Structural Engineer, U.K., July. [15] Ratay, R.T., Ed. 1996. Handbook of Temporary Structures in Construction, 2nd ed., McGrawHill, New York. [16] Schneider, C.C. 1905. The Evolution of the Practice of American Bridge Building, Transactions ASCE, vol.54. [17] Sibly, P.G. and Walker, A.C. 1977. Structural Accidents and Their Causes, Proc. Inst. Civil Engrs. (U.K.), vol. 62(1), May. [18] Smith, D.W. 1976. Bridge Failures, Proc. Inst. Civil Engrs. (U.K.), vol. 60(1), August.
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Blodgett, O.W. and Miller, D.K. “Basic Principles of Shock Loading” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Basic Principles of Shock Loading
O.W. Blodgett and D.K. Miller The Lincoln Electric Company, Cleveland, OH
21.1
21.1 Introduction 21.2 Requirements for Optimum Design 21.3 Absorbing Kinetic Energy 21.4 Material Properties for Optimum Design 21.5 Section Properties for Optimum Design 21.6 Detailing and Workmanship for Shock Loading 21.7 An Example of Shock Loading 21.8 Conclusions 21.9 Defining Terms References Further Reading
Introduction
Shock loading presents an interesting set of problems to the design engineer. In the engineering community, design for static loading traditionally has been the most commonly used design procedure. Designing for shock loading, however, requires a change of thinking in several areas. The objectives of this discussion are to introduce the basic principles of shock loading and to consider their effect upon the integrity of structures. To understand shock loading, one must first establish the various loading scheme definitions. There are four loading modes that are a function of the strain rate and the number of loading cycles experienced by the member. They are the following: • • • •
static loading fatigue loading shock loading shock/fatigue combination loading
Static loading occurs when a force is slowly applied to a member. This is a slow or constant loading process that is equivalent to loading the member over a period of 1 min. Designing for static loading traditionally has been the approach used for a wide variety of components, such as buildings, water towers, dams, and smoke stacks. Fatigue loading occurs when the member experiences alternating, repeated, or fluctuating stresses. Fatigue loading generally is classified by the number of loading cycles on the member and the stress level. Low cycle fatigue usually is associated with stress levels above the yield point, and fracture initiates in less than 1000 cycles. High cycle fatigue involves 10,000 cycles or more and overall stress levels that are below the yield point, although stress raisers from notch-like geometries and/or the residual stresses of welding result in localized plastic deformation. High cycle fatigue failures often occur with overall maximum stresses below the yield strength of the material. 1999 by CRC Press LLC
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In shock loading, an impact-type force is applied over a short instant of time. The yield and ultimate tensile strengths of a given member can be higher for loads with accelerated strain rates when compared to static loading (see Figure 21.1) [4]. The total elongation, however, remains constant at strain rates above approximately 10−4 in./in./s.
FIGURE 21.1: Influence of the strain rate on tensile properties. (From Manjoine, M.J., J. Applied Mech., 66, A211, 1944. With permission.)
Shock/fatigue combination loading is equivalent to a shock load applied many times. It exhibits the problems of both fatigue loading and shock loading. Some examples of this loading mode are found in jack hammers, ore crushers, pile drivers, foundry “shake-outs”, and landing gear on aircraft.
21.2
Requirements for Optimum Design
For an optimum design, shock loading requires that the entire volume of the member be stressed to the maximum (i.e., the yield point). This refers to both the entire length of the member and the entire cross-section of that member. In practice, however, most members do not have a geometry that allows for the member to be maximally loaded for its entire length, or stressed to the maximum across the entire cross-section.
21.3
Absorbing Kinetic Energy
In Figure 21.2, the member on the left acts as a cantilever beam. Under shock loading, the beam is deflected, which allows that force to be absorbed over a greater distance. The result is a decrease in 1999 by CRC Press LLC
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the impact intensity applied to the beam. The member on the right, however, is hit straight on as a column, and is restricted to little deflection. Therefore, as the column absorbs the same amount of energy, the resulting force is extremely high.
FIGURE 21.2: Absorbing kinetic energy. (From The Lincoln Electric Co. With permission.)
In shock loading, the energy of the applied force is ultimately absorbed, or transferred, to the structural component designed to resist the force. For example, a vehicle striking a column illustrates this transfer of energy, as shown in Figure 21.3. Prior to impact, the vehicle has a certain value of kinetic energy that is equal to 1/2mV 2 . The members that absorb the kinetic energy applied by the vehicle are illustrated in Figure 21.4, where the spring represents the absorbed potential energy, Ep , in “a”. Figure 21.4b illustrates a vertical stop much like Figure 21.2a. The entire kinetic energy must be absorbed by the receiving member (less any insignificant conversion byproducts—heat, noise, etc.) and momentarily stored as elastic potential energy. This elastic potential energy is stored in the member due to its stressed or deflected condition. In Figure 21.5, the maximum potential energy (Ep ) that can be elastically absorbed by a simply supported beam is derived. The maximum potential energy that may be stored in the loaded (deflected) beam when stressed to yield is Ep = where Ep = σY = I = L = E = C =
σY2 I L 6EC 2
(21.1)
maximum elastic capacity (max. absorbed potential energy) yield strength of material moment of inertia of section length of member modulus of elasticity of material distance to outer fiber from neutral axis of section
Figure 21.6 shows various equations that can be used to determine the amount of energy stored in a particular beam, depending upon the specific end conditions and point of load application. 1999 by CRC Press LLC
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FIGURE 21.3: Kinetic energy prior to impact. (From The Lincoln Electric Co. With permission.)
FIGURE 21.4: Kinetic energy absorbed by the recovery member. (From The Lincoln Electric Co. With permission.)
FIGURE 21.5: Maximum potential energy that can be absorbed by a simply supported beam. (From The Lincoln Electric Co. With permission.)
Figure 21.7 addresses the shock loading potential in the case of a simply supported beam with a concentrated force applied in the center. To analyze this problem, two issues must be considered. First, the property of the material is σY2 /E, and second, the property of the cross-section is I /C 2 . The objective for optimum performance in shock loading is to ensure maximal energy absorption capability. Both contributing elements, material properties and section properties, will be examined later in this chapter. 1999 by CRC Press LLC
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FIGURE 21.6: Energy storage capability for various end conditions and loading. (From The Lincoln Electric Co. With permission.) 1999 by CRC Press LLC
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FIGURE 21.7: An example of shock loading for a simply supported beam. (From The Lincoln Electric Co. With permission.)
Plotted in Figure 21.8 is the applied strain on the horizontal axis and the resisting stress on the vertical axis. If this member is strained up to the yield point stress, the area under that triangle represents the maximum energy that can be absorbed elastically (Ep ). When the load is removed, it will return to its original position. The remaining area, which is not cross-hatched, represents the plastic (or inelastic) energy if the member is strained. A tremendous amount of plastic energy can be absorbed, but of course, by that time the member is deformed (permanently set) and may be of no further use.
FIGURE 21.8: Stress-strain curve. (From The Lincoln Electric Co. With permission.)
The stress-strain diagram can help illustrate the total elastic potential capability of a material. The area of the triangular region is σY εY /2. Since E = σY /εY and σY = EεY , then this area is equal to σY2 /2E. This gives the elastic energy potential per unit volume of material. As shown in Figure 21.7, the property of the material was σY2 /E (as defined by Equation 21.1), which is directly proportional to the triangular area in Figure 21.8. The stress-strain curve for two members (one smooth and one notched) is illustrated in Figure 21.9. At the top of the illustration, the stress-strain curve is shown. Extending from the stress-strain curve is the length of the member. If the member is smooth, as on the lower left, it can be stressed up to the yield point, resulting in the highest strain that can be attained and the maximum elastic energy per unit volume of material. Applying this along the length of the member, the solid prism represents the total energy absorbed by that member. On the right side of the illustration, note that the notch shown gives a stress raiser. If the stress raiser is assigned an arbitrary value of three-to-one, the area under the notch will display stresses three times greater than the average stress. Therefore, to keep the stress in the notched region below the yield stress, the applied stress must be reduced to 1/3 of the original amount. This will reduce the 1999 by CRC Press LLC
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FIGURE 21.9: Smooth vs. notched. (From The Lincoln Electric Co. With permission.)
strain to 1/3, which corresponds to only 1/9 of the energy that will be absorbed. In other words, 1/9 of the elastic energy can be absorbed if the stress at the notch is kept to the yield stress. Notches and cracks in members subject to shock loading can become very dangerous, especially if the stress flow is normal to the notch. Therefore, structural members in shock loading require good workmanship and detailing.
21.4
Material Properties for Optimum Design
As shown previously, in Figure 21.8, the elastic energy absorbed during a shock load is equal to the triangular-shaped area under the stress-strain curve up to the yield point. This energy is equal to σY2 /2E, which is directly a function of the material. In order to maximize the amount of energy that can be absorbed (Ep ), with regard to material properties, one must maximize this triangular-shaped area under the stress-strain curve. In order to compare the energy absorption capability of various materials in shock loading, a list of materials and their corresponding mechanical properties is compiled in Table 21.1. For each TABLE 21.1
Material
Yield strength, σY (ksi)
Ultimate tensile strength, σu (ksi)
Modulus of elasticity, E (psi)
Percent elongation (%)
Absorbed elastic energy, σY2 /2E (psi)
Mild steel Medium carbon steel High carbon steel A514 Steel Gray cast iron Malleable cast iron 5056-H18 Aluminum alloy
35 45 75 100 6 20 59
60 85 120 115–135 20 50 63
30 × 106 30 × 106 30 × 106 30 × 106 15 × 106 23 × 106 10 × 106
35 25 8 18 5 10 10
20.4 33.7 94.0 166.7 1.2 8.7 174.1
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Mechanical Properties of Common Design Materials
material listed in column 1, typical yield and ultimate tensile strengths are tabulated in columns 2 and 3. The modulus of elasticity is recorded in column 4, and column 5 lists the typical percent elongation. Column 6 details the calculated value of σY2 /2E for each material. Higher strength steel will provide a maximum value of σY2 /2E, since the modulus of elasticity is approximately constant for all steels. However, if aluminum is an option, 5xxx series alloys will provide moderate-to-high strengths with high values of σY2 /2E. In addition, the 5xxx series alloys usually have the highest welded strengths among aluminum alloys with good corrosion resistance [3]. Some materials, such as cast iron, should not be used for shock loading. Cast iron has a very low value of σY2 /2E and will absorb only a small amount of elastic energy. It is ironic that some engineers believe that cast iron is a better material for shock loading than steel or aluminum alloys; this is not the case. When choosing a material for shock loading applications, the designer should also remember to consider other characteristics of the material such as machineability, weldability, corrosion resistance, etc. For example, very high strength steels are appropriate for shock loading applications; however, they are not very weldable.
21.5
Section Properties for Optimum Design
It was indicated earlier that most engineers are well versed in the principles of steady loading. To demonstrate the differences between steady and shock loading, an example is shown in Figure 21.10. On the left, a steady load is applied to a simply supported beam, and the maximum force that the member can carry for a given stress is F = 4σ I /LC. From this equation, it is important to realize that if the length of the beam (L) is doubled, then the beam would be able to carry only half of the load at the same stress.
FIGURE 21.10: Steady load vs. shock load. (From The Lincoln Electric Co. With permission.)
With the shock loading equation, Ep = σY2 I L/6EC 2 , notice that the length is in the numerator. Therefore, under shock loading, doubling the span of the beam will also double the amount of energy that it can absorb for the same stress level. This is just the opposite of what happens under static loading. Under static loading conditions with a fixed load, doubling the length of the beam also doubles the stress level, but with shock loading, doubling the length of the beam reduces the stress level to only 70% of what it was before. In another example, consider the case of a variable-depth girder shown in the lower portion of Figure 21.11. On the top left there is a prismatic beam (a beam of the same cross-section throughout) under steady load. If, for instance, this is a crane beam in the shop, and there is concern about the weight of the beam, the beam depth can be reduced at the ends. This reduces the dead weight, but leaves the strength capacity of the beam unchanged under a static loading condition. The right-hand side of Figure 21.11 illustrates what happens under shock loading. The equations show that by 1999 by CRC Press LLC
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reducing the beam depth, thus increasing the amount of bending stresses away from the center of the beam to a maximum, Ep is actually doubled.
FIGURE 21.11: Steady load vs. shock load for a variable-depth beam. (From The Lincoln Electric Co. With permission.)
In Table 21.2, the properties of two beams are analyzed. Beam A, a 12-in. deep, 65-lb beam, was first suggested. It has a moment of inertia of 533.4 in.4 . The steady load section modulus will equal S = I /C, which results in a value of 88 in.3 . Taking a more conventional (i.e., static loading) approach, beam B represents an attempt to create a better beam for shock loading. In this case, the beam size is increased to a depth of 24 in. and a weight of 76 lb. Its moment of inertia is now 2000 in.4 , or four times higher than that of beam A. However, in terms of shock loading alone, beam B has virtually no advantage over beam A because the value of I /C 2 has not changed. From Equation 21.1, I /C 2 is directly proportional to the amount of potential energy that can be absorbed, therefore, since I /C 2 has not changed, no additional energy can be absorbed by beam B as compared to beam A. TABLE 21.2
Properties of Beam A and Beam B
Section property Moment of inertia, I C Steady load strength, I S= C Shock load strength, I C2
Beam A 12 in. WF 65 lb
Beam B 24 in. WF 76 lb
533.4 in.4 6.06 in.
2096.4 in.4 11.96 in.
533.4 = 88.2 in.3 6.06
2096.4 = 175 in.3 11.96
533.4 = 14.5 in.2 (16.067)2
2096.4 = 14.6 in.2 (11.96)2
To determine the relative efficiency or economy of the configuration, it is possible to divide I /C 2 by the cross-sectional area (A). Since the material costs are directly related to the material weight (weight is equal to the cross-sectional area multiplied by the density and the length), I /C 2 A is a good measure of the configuration’s efficiency. Figure 21.12 shows that, for the same flange dimension and web thickness, changing the overall depth of section has little effect upon the shock load strength, I /C 2 A. For this reason, when a beam is made deeper, it also becomes more rigid. When a shock load strikes it, the resultant force will be extremely high due to the small deflection. This example 1999 by CRC Press LLC
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demonstrates the importance of discarding the preconceived ideas that come from experience with static loads, and embracing a new way of thinking that is appropriate for shock loading situations.
FIGURE 21.12: Effect of depth on shock loading capacity. (From The Lincoln Electric Co. With permission.)
FIGURE 21.13: Effect of web thickness on shock loading capacity. (From The Lincoln Electric Co. With permission.)
The effect of the beam web thickness is examined in Figure 21.13. For the same overall depth of section and flange size, decreasing the web thickness increases the shock load strength, I /C 2 A. This 1999 by CRC Press LLC
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example makes it clear that decreasing the web thickness actually increases the efficiency of shock load absorption for the same cross-sectional area. Figure 21.14, on the other hand, demonstrates what happens when the flange width is considered a variable. For the same overall depth of section, increasing the flange width increases the shock load strength, I /C 2 A. This puts the steel on the outer fibers, which is an improvement for shock loading. Two different section geometries are considered in Figure 21.15. The box-shaped section has a much higher moment of inertia than the H-shaped section; however, the distance to the outer fiber in both cases is equal. By implementing the box-shaped design, there is an increased capability to absorb a shock load over the H section. In the examples previously presented, it is important to realize that improvements were made by changing the section, not the material.
FIGURE 21.14: Effect of flange width on shock loading capacity. (From The Lincoln Electric Co. With permission.)
FIGURE 21.15: Effect of section geometry on shock loading capacity. (From The Lincoln Electric Co. With permission.)
1999 by CRC Press LLC
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FIGURE 21.16: An example of shock loading. (From The Lincoln Electric Co. With permission.)
21.6
Detailing and Workmanship for Shock Loading
The fabrication of structures subjected to shock loading is generally no different from that required for slowly loaded structures, except that good workmanship is even more important. Wherever shock loading is a factor, the smallest fabrication problem can have negative, sometimes disastrous, consequences. For welded structures, postweld nondestructive inspection of these assemblies is recommended to verify the integrity of the welds.
21.7
An Example of Shock Loading
Figure 21.16 illustrates a real-world example of shock loading. The drawing shows ore falling into a primary gyratory crusher. When the ore comes down the chute, it strikes a bumper plate rather than falling with great impact directly into the crusher. This transforms some of the kinetic energy into the structure that supports the bumper. If the bumper plate is considered as a design problem, the falling ore has a normal velocity component to the surface of the bumper plate. Prior to impact, the normal velocity of the ore could equal 213 in./s. When the ore strikes the bumper plate, this normal velocity becomes zero. If the force caused by the moving ore is found in terms of an incremental time (1t), the “impulse” (force multiplied by incremental time) is equal to the change in momentum of the falling ore. After determining the force applied to the bumper plate, the plate size can be calculated. Equations 21.2 through 21.7 below outline the calculation procedure for this problem. Mass of ore: m=
AV1 (1t)δ W = g g
(21.2)
Impulse: I mp = F (1t) 1999 by CRC Press LLC
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(21.3)
Change in momentum: 1M = mVn − mV0
(21.4)
Impulse equals change in momentum: I mp
=
F (1t)
=
F
=
1M
AV1 (1t)δ Vn − mV0 1M = mVn − mV0 = g AV1 δVn g
(21.5) (21.6) (21.7)
where m = mass of ore W = weight of ore t = time ore falls A = cross-sectional area of ore stream δ = density of ore g = acceleration of gravity V1 = ore velocity prior to impact V0 = ore velocity after impact (assume V0 = 0) Vn = normal velocity of ore at impact F = force of impact This force, F , can then be used to size the bumper plate and the supporting structure of the crusher.
21.8
Conclusions
In structural design, historically, the first tool an engineer utilizes is one applicable to static loading. However, in shock loading applications, new techniques are required to prevent failure and to utilize energy absorption. For example, optimum shock loading performance occurs when a member’s complete volume is stressed to a maximum. When sizing a structural member, remember to consider both the material properties and the section properties. The amount of absorbed kinetic energy is directly related to both of these properties. Detailing and workmanship are also crucial because flaws and cracks act as stress raisers that can reduce the member’s ability to withstand shock loads.
21.9
Defining Terms
Elastic capacity: The maximum amount of potential energy that can be stored in a member. Fatigue loading: Alternating, repeated, or fluctuating stresses that a member experiences over its operating life. Impulse: The product of the force on a body and the time during which that force acts; equal to the change in momentum. Kinetic energy: Energy associated with motion. Moment of inertia: A measure of the resistance of a body to angular acceleration about a given axis that is equal to the sum of the products of each element of mass in the body and the square of the element’s distance from the axis. Momentum: A property of a moving body that the body has by virtue of its mass and motion and that is equal to the product of the body’s mass and velocity. 1999 by CRC Press LLC
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Potential energy: Energy that a piece of matter has because of its position. Shock/fatigue combination loading: The application of a shock load repeated many times. Shock loading: The application of an extremely high force over a very short duration of time. Static loading: Constant loading of a member.
References [1] Blodgett, O.W. 1996. Design of Welded Structures, The James F. Lincoln Arc Welding Foundation. [2] Blodgett, O.W. 1976. Design of Weldments, The James F. Lincoln Arc Welding Foundation. [3] Kissell, J. Randolph. 1996. “Aluminum and Its Alloys”, Marks’ Standard Handbook for Mechanical Engineers. 10th ed., pp. 6.53–6.60, McGraw-Hill, New York. [4] Manjoine, M.J. 1944. “Influence of Rate of Strain and Temperature on Yield Stresses of Mild Steel”, J. Appl. Mech., 66, A211–A218. [5] Shigley, J.E. and Mischke, C.R. 1989. Mechanical Engineering Design, 5th ed., McGraw-Hill, New York. [6] Stout, R.D. and Doty, W.D. 1971. Weldability of Steels, 2nd ed., Welding Research Council.
Further Reading [1] Fatigue Crack Propagation: A Symposium Presented at the Sixty-Ninth Annual Meeting. 1967. ASTM Special Technical Publication No. 415. [2] Harris, C.M. and Crede, C.E. 1976. Shock and Vibration Handbook, 2nd ed., McGraw-Hill, New York. [3] Potter, J.M. and McHenry, H.I. 1990. Fatigue and Fracture Testing of Weldments, ASTM STP 1058. [4] Sandor, B.I. 1972. Fundamentals of Cyclic Stress and Strain, The University of Wisconsin Press.
1999 by CRC Press LLC
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Blodgett, O.W. and Miller, D.K. “Welded Connections” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Welded Connections
O.W. Blodgett and D. K. Miller The Lincoln Electric Company, Cleveland, OH
22.1
22.1 Introduction 22.2 Joint and Weld Terminology 22.3 Determining Weld Size 22.4 Principles of Design 22.5 Welded Joint Details 22.6 Design Examples of Specific Components 22.7 Understanding Ductile Behavior 22.8 Materials 22.9 Connection Details 22.10Achieving Ductile Behavior in Seismic Sections 22.11Workmanship Requirements 22.12Inspection 22.13Post-Northridge Assessment 22.14Defining Terms References Further Reading
Introduction
Arc welding has become a popular, widely used method for making steel structures more economical. Although not a new process, welding is still often misunderstood. Perhaps some of the confusion results from the complexity of the technology. To effectively and economically design a building that is to be welded, the engineer should have a knowledge of metallurgy, fatigue, fracture control, weld design, welding processes, welding procedure variables, nondestructive testing, and welding economics. Fortunately, excellent references are readily available, and industry codes specify the minimum standards that are required to be met. Finally, the industry is relatively mature. Although new developments are made every year, the fundamentals of welding are well understood, and many experienced engineers may be consulted for assistance. Welding is the only joining method that creates a truly one-piece member. All the components of a welded steel structure act in unison, efficiently and effectively transferring loads from one piece to another. Only a minimum amount of material is required when welding is used for joining. Alternative joining methods, such as bolting, are generally more expensive and require the use of lapped plates and angles, increasing the number of pieces required for construction. With welded construction, various materials with different tensile strengths may be mixed, and otherwise unattainable shapes can be achieved. Along with these advantages, however, comes one significant drawback: any problems experienced in one element of a member may be transferred to another. For example, a crack that exists in the flange of a beam may propagate through welds into a column flange. This means 1999 by CRC Press LLC
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that, particularly in a dynamically loaded structure that is to be joined by welding, all details must be carefully controlled. Interrupted, non-continuous backing bars, tack welds, and even seemingly minor arc strikes have resulted in cracks propagating through primary members. In order to best utilize the unique capabilities of welding, it is imperative to consider the entire design–fabrication–erection sequence. A properly designed welded connection not only transfers stresses safely, but also is economical to fabricate. Successful integration of design, welding processes, metallurgical considerations, inspection criteria, and in-service inspection depends upon mutual trust and free communication between the engineer and the fabricator.
22.2
Joint and Weld Terminology
A welded connection consists of two or more pieces of base metal joined by weld metal. Engineers determine joint type and generally specify weld type and the required throat dimension. Fabricators select the joint details to be used.
22.2.1
Joint Types
When pieces of steel are brought together to form a joint, they will assume one of the five configurations presented in Figure 22.1. Of the five, butt, tee, corner, and lap joints are common in construction. Coverplates on rolled beams, and angles to gusset plates would be examples of lap joints. Edge joints are more common for sheet metal applications. Joint types are merely descriptions of the relative positioning of the materials; the joint type does not imply a specific type of weld.
FIGURE 22.1: Joint types. (Courtesy of The Lincoln Electric Company. With permission.)
22.2.2
Weld Types
Welds may be placed into three major categories: groove welds, fillet welds, and plug or slot welds (see Figure 22.2). For groove welds, there are two subcategories: complete joint penetration (CJP) groove welds and partial joint penetration (PJP) groove welds (see Figure 22.3). Plug welds are commonly used to weld decking to structural supports. Groove and fillet welds are of prime interest for major structural connections. In Figure 22.4, terminology associated with groove welds and fillet welds is illustrated. Of great interest to the designer is the dimension noted as the “throat.” The throat is theoretically the weakest plane in the weld. This generally governs the strength of the welded connection. 1999 by CRC Press LLC
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FIGURE 22.2: Major weld types. (Courtesy of The Lincoln Electric Company. With permission.)
FIGURE 22.3: Types of groove welds. (Courtesy of The Lincoln Electric Company. With permission.)
22.2.3
Fillet Welds
Fillet welds have a triangular cross-section and are applied to the surface of the materials they join. Fillet welds by themselves do not fully fuse the cross-sectional areas of parts they join, although it is still possible to develop full-strength connections with fillet welds. The size of a fillet weld is usually determined by measuring the leg size, even though the weld is designed by determining the required throat size. For equal-legged, flat-faced fillet welds applied to plates that are oriented 90◦ apart, the throat dimension is found by multiplying the leg size by 0.707 (i.e., sine 45◦ ).
22.2.4
Complete Joint Penetration (CJP) Groove Welds
By definition, CJP groove welds have a throat dimension equal to the thickness of the plate they join (see Figure 22.3). For prequalified welding procedure specifications, the American Welding Society (AWS) D1.1-96 [9] Structural Welding Code requires backing (see Weld Backing) if a CJP weld is made from one side, and back gouging if a CJP weld is made from both sides. This ensures complete fusion throughout the thickness of the material being joined. Otherwise, procedure qualification testing is required to prove that the full throat is developed. A special exception to this is applied to tubular connections whose CJP groove welds may be made from one side without backing.
1999 by CRC Press LLC
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FIGURE 22.4: Weld terminology. (Courtesy of The Lincoln Electric Company. With permission.)
22.2.5
Partial Joint Penetration (PJP) Groove Welds
A PJP groove weld is one that, by definition, has a throat dimension less than the thickness of the materials it joins (see Figure 22.3). An “effective throat” is associated with a PJP groove weld (see Figure 22.5). This term is used to delineate the difference between the depth of groove preparation
FIGURE 22.5: PJP groove welds: “E” vs. “S”. (Courtesy of The Lincoln Electric Company. With permission.) 1999 by CRC Press LLC
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and the probable depth of fusion that will be achieved. When submerged arc welding (which has inherently deep penetration) is used, and the weld groove included angle is 60◦ , the D1.1-96 [9] code allows the designer to rely on the full depth of joint preparation to be used for delivering the required throat dimension. When other processes with less penetration are used, such as shielded metal arc welding, and when the groove angle is restricted to 45◦ , it is doubtful that fusion to the root of the joint will be obtained. Because of this, the D1.1-96 code assumes that 1/8 in. of the PJP joint may not be fused. Therefore, the effective throat is assumed to be 1/8 in. less than the depth of preparation. This means that for a given included angle, the depth of joint preparation must be increased to offset the loss of penetration. The effective throat on a PJP groove weld is abbreviated utilizing a capital “E”. The required depth of groove preparation is designated by a capital “S”. Since the engineer does not normally know which welding process a fabricator will select, it is necessary for the engineer to specify only the dimension for E. The fabricator then selects the welding process, determines the position of welding, and thus specifies the appropriate S dimension, which will be shown on the shop drawings. In most cases, both the S and E dimensions will be contained on the welding symbols of shop drawings, the effective throat dimension showing up in parentheses.
22.2.6
Double-Sided Welds
Welds may be single or double. Double welds are made from both sides of the member (see Figure 22.6). Double-sided welds may require less weld metal to complete the joint. This, of course, has advantages with respect to cost and is of particular importance when joining thick members. However, double-sided joints necessitate access to both sides. If the double joint necessitates overhead welding, the economies of less weld metal may be lost because overhead welding deposition rates are inherently slower. For joints that can be repositioned, this is of little consequence. There are also distortion considerations, where the double-sided joints have some advantages in balancing weld shrinkage strains.
FIGURE 22.6: Single- vs. double-sided joints. (Courtesy of The Lincoln Electric Company. With permission.)
22.2.7
Groove Weld Preparations
Within the groove weld category, there are several types of preparations (see Figure 22.7). If the joint contains no preparation, it is known as a square groove. Except for thin sections, the square groove is rarely used. The bevel groove is characterized by one plate cut at a 90◦ angle and a second plate with 1999 by CRC Press LLC
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FIGURE 22.7: Groove weld preparation. (Courtesy of The Lincoln Electric Company. With permission.)
a bevel cut. A vee groove is similar to a bevel, except both plates are bevel cut. A J-groove resembles a bevel, except the root has a radius, as opposed to a straight cut. A U-groove is similar to two J-grooves put together. For butt joints, vee and U-groove details are typically used when welding in the flat position since it is easier to achieve uniform fusion when welds are placed upon the inclined surfaces of these details versus the vertical edge of one side of the bevel or J-groove counterparts. Properly made, any CJP groove preparation will yield a connection equal in strength to the connected material. The factors that separate the advantages of each type of preparation are largely fabrication related. Preparation costs of the various grooves differ. The flat surfaces of vee and bevel groove weld preparations are generally more economical to produce than the U and J counterparts, although less weld metal is usually required in the later examples. For a given plate thickness, the volume of weld metal required for the different types of grooves will vary, directly affecting fabrication costs. As the volume of weld metal cools, it generates residual stresses in the connection that have a direct effect on the extent of distortion and the probability of cracking or lamellar tearing. Reducing weld volume is generally advantageous in limiting these problems. The decision as to which groove type will be used is usually left to the fabricator who, based on knowledge, experience, and available equipment, selects the type of groove that will generate the required quality at a reasonable cost. In fact, design engineers should not specify the type of groove detail to be used, but rather determine whether a weld should be a CJP or a PJP.
22.2.8
Interaction of Joint Type and Weld Type
Not every weld type can be applied to every type of joint. For example, butt joints can be joined only with groove welds. A fillet weld cannot be applied to a butt joint. Tee joints may be joined with fillet welds or groove welds. Similarly, corner joints may be joined with either groove welds or fillet welds. Lap joints would typically be joined with fillet welds or plug/slot welds. Table 22.1 illustrates possible combinations.
1999 by CRC Press LLC
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TABLE 22.1
Weld Type/Joint Type Interaction
Courtesy of Lincoln Electric Company. With permission.
22.3
Determining Weld Size
22.3.1
Strength of Welded Connections
A welded connection can be designed and fabricated to have a strength that matches or exceeds that of the steel it joins. This is known as a full-strength connection and can be considered 100% efficient; that is, it has strength equivalent to that of the base metal it joins. Welded connections can be designed so that if loaded to destruction, failure would occur in the base material. Poor weld quality, however, may adversely affect weld strength. A connection that duplicates the base metal capacity is not always necessary and when unwarranted, its specification unnecessarily increases fabrication costs. In the absence of design information, it is possible to specify welds that have strengths equivalent to the base metal capacity. Assuming the base metal thickness has been properly selected, a weld that duplicates the strength of the base metal will be adequate as well. This, however, is a very costly approach. Economical connections cannot be designed on this basis. Unfortunately, the overuse of the CJP detail and the requirement of “matching filler metal” (i.e., weld metal of a strength that is equal to that of the base metal) serves as evidence that this is often the case.
22.3.2
Variables Affecting Welded Connection Strength
The strength of a welded connection is dependent on the weld metal strength and the area of weld that resists the load. Weld metal strength is a measure of the capacity of the deposited weld metal itself, measured in units such as ksi (kips per square inch). The connection strength reflects the combination of weld metal strength and cross-sectional area, and would be expressed as a unit of force, such as kips. If the product of area times the weld metal strength exceeds the loads applied, the weld should not fail in static service. For cyclic dynamic service, fatigue must be considered as well. The area of weld metal that resists fracture is the product of the theoretical throat multiplied by the length. The theoretical weld throat is defined as the minimum distance from the root of the weld to its theoretical face. For a CJP groove weld, the theoretical throat is assumed to be equal to the 1999 by CRC Press LLC
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thickness of the plate it joins. Theoretical throat dimensions of several types of welds are shown in Figure 22.8.
FIGURE 22.8: Theoretical throats. (Courtesy of The Lincoln Electric Company. With permission.)
For fillet welds or partial joint penetration groove welds, using filler metal with strength levels equal to or less than the base metal, the theoretical failure plane is through the weld throat. When the same weld is made using filler metal with a strength level greater than that of the base metal, the failure plane may shift into the fusion boundary or heat-affected zone. Most designers will calculate the load capacity of the base metal, as well as the capacity of the weld throat. The fusion zone and its capacity is not generally checked, as this is unnecessary when matching or undermatching weld metal is used. When overmatching weld metal is specifically selected, and the required weld size is deliberately reduced to take advantage of the overmatched weld metal, the designer must check the capacity of the fusion zone (controlled by the base metal) to ensure adequate capacity in the connection. Complete joint penetration groove welds that utilize weld metal with strength levels exactly equal to the base metal will theoretically fail in either the weld or the base metal. Even with matching weld metal, the weld metal is generally slightly higher in strength than the base metal, so the theoretical failure plane for transversely loaded connections is assumed to be in the base metal.
22.3.3
Determining Throat Size for Tension or Shear Loads
Connection strength is governed by three variables: weld metal strength, weld length, and weld throat. The weld length is often fixed, due to the geometry of the parts being joined, leaving one variable to be determined, namely, the throat dimension. For tension or shear loads, the required capacity the weld must deliver is simply the force divided by the length of the weld. The result, in units of force per length (such as kips per inch) can be divided by the weld metal strength, in units of force per area (such as kips per square inch). The final result would be the required throat, in inches. Weld metal allowables that incorporate factors of safety can be used instead of the actual weld metal capacity. This directly generates the required throat size. To determine the weld size, it is necessary to consider what type of weld is to be used. Assume the preceding calculation determined the need for a 1-in. throat size. If a single fillet weld is to be used, a throat of 1 in. would necessitate a leg size of 1.4 in., shown in Figure 22.9. For double-sided fillets, 1999 by CRC Press LLC
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FIGURE 22.9: Weld combinations with equal throat dimensions. (Courtesy of The Lincoln Electric Company. With permission.)
two 0.7-in. leg size fillets could be used. If a single PJP groove weld is used, the effective throat would have to be 1 in. The actual depth of preparation of the production joint would be 1 in. or greater, depending on the welding procedure and included angle used. A double PJP groove weld would require two effective throats of 0.5 in. each. A final option would be a combination of partial joint penetration groove welds and external fillet welds. As shown in Figure 22.9, a 60◦ included angle was utilized for the PJP groove weld and an unequal leg fillet weld was applied externally. This acts to shift the effective throat from the normal 45◦ angle location to a 30◦ throat. If the plates being joined are 1 in. thick, a CJP groove weld is the only type of groove weld that will effectively transfer the stress, since the throat on a CJP weld is equal to the plate thickness. PJP groove welds would be incapable of developing adequate throat dimensions for this application, although the use of a combination PJP-fillet weld would be a possibility.
22.3.4
Determining Throat Size for Compressive Loads
When joints are subject only to compression, the unwelded portion of the joint may be milled-tobear, reducing the required weld throat. Typical of these types of connections are column splices where PJP groove welds frequently are used for static structures.
22.3.5
Determining Throat Size for Bending or Torsional Loads
When a weld, or group of welds, is subject to bending or torsional loads, the weld(s) will not be uniformly loaded. In order to determine the stress on the weld(s), a weld size must be assumed and the resulting stress distribution calculated. An iterative approach may be used to optimize the weld size. 1999 by CRC Press LLC
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A simpler approach is to treat the weld as a line with no throat. Standard design formulas may be used to determine bending, vertical shear, torsion, etc. These formulas normally result in unit stresses. When applied to welds treated as a line, the formulas result in a force on the welds, measured in pounds per linear inch, from which the capacity of the weld metal, or applicable allowable values, may be used to determine the required throat size. The following is a simple method used to determine the correct amount of welding required to provide adequate strength for either a bending or a torsional load. In this method, the weld is treated as a line, having no area but having a definite length and cross-section. This method offers the following advantages: 1. It is not necessary to consider throat areas. 2. Properties of the weld are easily found from a table without knowledge of weld leg size. 3. Forces are considered per unit length of weld, rather than converted to stresses. This facilitates dealing with combined-stress problems. 4. Actual values of welds are given as force per unit length of weld instead of unit stress on throat of weld. Visualize the welded connection as a line (or lines), following the same outline as the connection but having no cross-sectional area. In Figure 22.10, the desired area of the welded connection, Aw ,
FIGURE 22.10: Treating the weld as a line for a twisting or bending load: Aw = length of weld (in.), Zw = section modulus of weld (in.2 ), Jw = polar moment of inertia of weld (in.3 ). (Courtesy of The Lincoln Electric Company. With permission.)
can be presented by just the length of the weld. The stress on the weld cannot be determined unless the weld size is assumed; but by following the proposed procedure, which treats the weld as a line, the solution is more direct, is much simpler, and becomes basically one of determining the force on the weld(s).
1999 by CRC Press LLC
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22.3.6
Treating the Weld as a Line to Find Weld Size
By inserting this property of the welded connection into the standard design formula used for a particular type of load (Table 22.2), the unit force on the weld is found in terms of pounds per linear inch of weld. TABLE 22.2
Standard Design Formulas Used for Determining Force on Weld
Courtesy of The Lincoln Electric Company. With permission.
Normally, use of these standard design formulas results in a unit stress, in pounds per square inch, but with the weld treated as a line, these formulas result in a unit force on the weld, in units of pounds per linear inch. For problems involving bending or twisting loads, Table 22.3 is used. It contains the section modulus, Sw , and polar moment of inertia, Jw , of 13 typical welded connections with the weld treated as a line. For any given connection, two dimensions are needed: width, b, and depth, d. Section modulus, Sw , is used for welds subjected to bending; polar moment of inertia, Jw , for welds 1999 by CRC Press LLC
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subjected to twisting. Section modulus, Sw , in Table 22.3 is shown for symmetric and asymmetric connections. For asymmetric connections, Sw values listed differentiate between top and bottom, and the forces derived therefrom are specific to location, depending on the value of Sw used. When more than one load is applied to a welded connection, they are combined vectorially, but must occur at the same location on the welded joint.
22.3.7
Use Allowable Strength of Weld to Find Weld Size
Weld size is obtained by dividing the resulting unit force on the weld by the allowable strength of the particular type of weld used, obtained from Table 22.4 or 22.5. For a joint that has only a transverse load applied to the weld (either fillet or butt weld), the allowable transverse load may be used from the applicable table. If part of the load is applied parallel (even if there are transverse loads in addition), the allowable parallel load must be used.
22.3.8
Applying the System to Any Welded Connection
1. Find the position on the welded connection where the combination of forces will be maximum. There may be more than one that must be considered. 2. Find the value of each of the forces on the welded connection at this point. Use Table 22.2 for the standard design formula to find the force on the weld. Use Table 22.3 to find the property of the weld treated as a line. 3. Combine (vectorially) all the forces on the weld at this point. 4. Determine the required weld size by dividing this value (step 3) by the allowable force in Table 22.4 or 22.5.
22.3.9
Sample Calculations Using This System
The example in Figure 22.11 illustrates the application of this procedure.
22.3.10
Weld Size for Longitudinal Welds
Longitudinal welds include the web-to-flange welds on I-shaped girders and the welds on the corners of box girders. These welds primarily transmit horizontal shear forces resulting from the change in moment along the member. To determine the force between the members being joined, the following equation may be used: V ay f = In where f = force on weld per unit length V = total shear on section at a given position along the beam a = area of flange connected by the weld y = distance from the neutral axis of the whole section to the center of gravity of the flange I = moment of inertia of the whole section n = number of welds joining the flange to webs per joint The resulting force per unit length is then divided by the allowable stress in the weld metal and the weld throat is attained. This particular procedure is emphasized because the resultant value for the weld throat is nearly always less than the minimum allowable weld size. The minimum size then becomes the controlling factor. 1999 by CRC Press LLC
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TABLE 22.3
Properties of Welded Connection; Treating Weld as a Line
Courtesy of The Lincoln Electric Company. With permission.
1999 by CRC Press LLC
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TABLE 22.4
Stress Allowables for Weld Metal
Type of weld
Complete joint penetration groove welds
Partial joint penetration groove welds
Stress in weld
Allowable connection stress
Required filler metal strength level
Tension normal to the effective area
Same as base metal
Matching filler metal shall be used
Compression normal to the effective area
Same as base metal
Filler metal with a strength level equal to or one classification (10 ksi [69 MPa]) less than matching filler metal may be used
Tension or compression parallel to the axis of the weld
Same as base metal
Shear on the effective areas
0.30 × nominal tensile strength of filler metal, except shear stress on base metal shall not exceed 0.40 × yield strength of base metal
Compression normal to effective area
Joint not designed to bear
0.50 × nominal tensile strength of filler metal, except stress on base metal shall not exceed 0.60 × yield strength of base metal
Joint designed to bear
Same as base metal
Filler metal with a strength level equal to or less than matching filler metal may be used
Tension or compression parallel to the axis of the welda
Same as base metal
Filler metal with strength level equal to or less than matching filler metal may be used
Shear parallel to axis of weld
0.30 × nominal tensile strength of filler metal, except shear stress on base metal shall not exceed 0.40 × yield strength of base metal
Tension normal to effective area
0.30 × nominal tensile strength of filler metal, except tensile stress on base metal shall not exceed 0.60 × yield strength of base metal
Shear on effective area
0.30 × nominal tensile strength of filler metal
Fillet weld
Tension or compression parallel to axis of welda
Same as base metal
Filler metal with a strength level equal to or less than matching filler metal may be used
Plug and slot welds
Shear parallel to faying surfaces (on effective area)
0.30 × nominal tensile strength of filler metal, except shear stress on base metal shall not exceed 0.40 × yield strength of base metal
Filler metal with a strength level equal to or less than matching filler metal may be used
a Fillet weld and partial joint penetration groove welds joining the component elements of built-up members, such as flange-to-web
connections, may be designed without regard to the tensile or compressive stress in these elements parallel to the axis of the welds. From American Welding Society. Structural Welding Code: Steel: ANSI/AWS D1.1-96. Miami, Florida, 1996. With permission.
1999 by CRC Press LLC
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TABLE 22.5
AISC Fatigue Allowables
From American Institute of Steel Construction, Chicago, IL, 1996.
1999 by CRC Press LLC
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TABLE 22.5
AISC Fatigue Allowables (continued)
From American Institute of Steel Construction, Chicago, IL, 1996.
1999 by CRC Press LLC
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FIGURE 22.11: Sample problem using steps outlined in this approach to determine weld size. (Courtesy of The Lincoln Electric Company. With permission.)
22.3.11
Minimum Weld Size
Many codes specify minimum weld sizes that are a function of plate thickness. These are not designrelated requirements, but rather reflect the inherent interaction of heat input and weld size.
22.3.12
Heat Input and Weld Size
Heat input and weld bead size (or cross-sectional area) are directly related. Heat input is typically calculated with the following equation: H = where H = heat input (kJ/in.) 1999 by CRC Press LLC
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60EI 1000S
E = arc volts I = amperage S = travel speed (in./min) In order to create a larger weld in one pass, two approaches may be used: higher amperages (I) or slower travel speeds (S) must be employed. Notice that either procedure modification results in a higher heat input. Welding codes have specified minimum acceptable weld sizes with the primary purpose of dictating minimum heat input levels. For example, almost independent of the welding process used, a 1/4-in. fillet weld will require a heat input of approximately 20–30 kJ/in. By prescribing a minimum fillet weld size, these specifications have, in essence, specified a minimum heat input. Understanding that the minimum fillet weld size is related to heat input, we must also note that there is an inherent interaction of preheat and heat input. The prescribed minimum fillet weld sizes assume the required preheats are also applied. If a situation arises where it is impossible to construct the minimum fillet weld size, it may be appropriate to increase the required preheat to compensate for the reduced energy of welding. The minimum fillet weld size need never exceed the thickness of the thinner part. It is important to recognize the implications of this requirement. In some extreme circumstances, the connection might involve a very thin plate being joined to an extremely thick plate. The code requirements would dictate that the weld need not exceed the size of the thinner part. However, under these circumstances, additional preheat based upon the thicker material may be justified.
22.3.13
Required Weld vs. Minimum Weld Sizes
When welds are properly sized based upon the forces they are required to transfer, the appropriate weld size frequently is found to be surprisingly small. Even on bridge plate girders that may be 18 to 20 ft deep, with flange thicknesses exceeding 2 in., the required fillet weld size to transmit the horizontal shear forces may be in the range of a 3/32-in. continuous fillet. Intuition indicates that something would be wrong when trying to apply this small weld to join a flange that may be 2 in. thick to a web that is 3/4 in. thick. This is not to indicate a fault with the method used to determine weld size, but rather reveals the small shear forces involved. However, when attempts are made to fabricate this plate girder with these small weld sizes, extremely high travel speeds or very low currents would be required. This naturally would result in an extremely low heat input value. The cooling rates that would be experienced by the weld metal and the base material, specifically the heat-affected zone, would be exceedingly high. A brittle microstructure could be formed. To avoid this condition, the minimum weld size would dictate that a larger weld is required. This is frequently the case for longitudinal welds that resist shear. Any further increase in specified weld size is unnecessary and directly increases fabrication costs.
22.3.14
Single-Pass Minimum Sized Welds
Controlling the heat input by specifying the minimum fillet weld size necessitates that this minimum fillet weld be made in a single pass. If multiple passes are used to construct the minimum sized fillet weld, the intent of the requirement is circumvented. In the past, some recommendations included minimum fillet weld sizes of 3/8 in. and larger. A single-pass 3/8-in. fillet weld can be made only in the flat or vertical position. In the horizontal position, multiple passes are required, and the spirit of the requirement is invalidated. For this reason, the largest minimum fillet weld in Table 5.8 of the AWS D1.1-96 code [9] is 5/16-in. However, even this weld may necessitate multiple passes, depending on the particular welding process used. For example, a quality 5/16-in. fillet weld cannot be made in a single pass with the shielded metal arc welding process utilizing l/8-in.-diameter electrodes, except perhaps in the vertical plane. It may not be possible to make the required minimum sized fillet weld in a single weld pass under 1999 by CRC Press LLC
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all conditions. For example, it is impossible to make a 5/16-in. fillet weld in a single pass in the overhead position. Under these conditions, it is important to remember the principles that underlie the code requirements. For the preceding example, the overhead fillet weld would necessitate three weld passes. Each weld pass would be made with approximately one-third of the heat input normally associated with the 5/16-in. fillet weld. In order to ensure satisfactory results, it would be desirable to utilize additional preheat to offset the naturally resulting lower heat input that would result from each of these weld passes.
22.3.15
Minimum Sized Groove Welds
When CJP groove welds are made, there is no need to specify the minimum weld size, because the weld size will be the thickness of the base material being joined. This is not the case, however, for PJP, groove welds, so the various codes typically specify minimum PJP groove weld sizes as well. When making CJP groove welds, it is a good practice to make certain that the individual passes applied to the groove meet or exceed the minimum weld size for PJP groove welds.
22.4
Principles of Design
Many welding-related problems have at their root a violation of basic design principles. For dynamically loaded structures, attention to detail is particularly critical. This applies equally to high-cycle fatigue loading, short duration abrupt-impact loading, and seismic loading. The following constitutes a review of basic welding engineering principles that apply to all construction.
22.4.1
Transfer of Forces
Not all welds are evenly loaded. This applies to weld groups that are subject to bending as well as those subject to variable loads along their length. The situation is less obvious when steels of different geometries are joined by welding. A rule of thumb is to assume the transfer of force takes place from one member, through the weld, to the member that lies parallel to the force that is applied. Some examples are illustrated in Figure 22.12. For most simple static loading applications, redistribution of stress throughout the member accommodates the variable loading levels. For dynamically loaded members, however, this is an issue that must be carefully addressed in the design. The addition of stiffeners or continuity plates to column webs helps to unify the distribution of stress across the groove weld.
22.4.2
Minimize Weld Volumes
A good principle of welded design is to always use the smallest amount of weld metal possible for a given application. This not only has sound economic implications, but it reduces the level of residual stress in the connection due to the welding process. All heat-expanded metal will shrink as it cools, inducing residual stresses in the connection. These tendencies can be minimized by reducing the volume of weld metal. Details that will minimize weld volumes for groove welds generally involve minimum root openings, minimum included angles, and the use of double-sided joints.
22.4.3
Recognize Steel Properties
Steel is not a perfectly isotropic material. The best mechanical properties usually are obtained in the same orientation in which the steel was originally rolled, called the X axis. Perpendicular to the X axis is the width of the steel, or the Y axis. Through the thickness, or the Z axis, the steel will exhibit the 1999 by CRC Press LLC
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FIGURE 22.12: Examples of transfer of force. (a) The leg welded under the beam has direct force transfer when oriented parallel to, and directly under, the beam web. (b) The same leg rotated 90◦ will result in an uneven distribution of stress along the weld length, unless stiffeners are added. The stiffeners could be triangular in shape, since the purpose is to provide a path for force transfer into the weld. (c) For hollow box sections, a lug attached perpendicular to the beam’s longitudinal axis results in an unevenly loaded weld until an internal diaphragm is added. (d) Wrapping the lug around the outside of the box section permits it to be directly welded to the section that is parallel to the load, i.e., the vertical sides. (e) Side plates are added to this lug in order to provide a path for force transfer to the vertical sides of the box section. (Courtesy of The Lincoln Electric Company. With permission.)
1999 by CRC Press LLC
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least amount of ductility, lowest strength, and lowest toughness properties. It is always desirable, if possible, to allow the residual stresses of welding to elongate the steel in the X direction. Of particular concern are large welds placed on either side of the thickness of the steel where the weld shrinkage stress will act in the Z axis. This can result in lamellar tearing during fabrication, or under extreme loading conditions, can result in subsurface fracture.
22.4.4
Provide Ample Access for Welding
It is essential that the design provide adequate access for both welder and welding equipment, as well as good visibility for the welder. As a general rule, if the welder cannot see the joint, neither can the inspector; weld quality will naturally suffer. It is important that adequate access be provided for the proper placement of the welding electrode with respect to the joint. This is a function of the welding process. Gas-shielded processes, for example, must have ample access for insertion of the shielding gas nozzle into the weld joint. Overall access to the joint is a function of the configuration of the surrounding material. The prequalified groove weld details listed in AWS D1.1-96 [9] take these issues into consideration.
22.4.5
No Secondary Members in Welded Design
A fundamental premise of welding design is that there are no secondary members. Anything that is joined by welding can, and will, transfer stress between joined materials. For instance, segmented pieces of steel used for weld backing can result in a stress concentration at the interface of the backing. Attachments that are simply tack welded in place may become major load-carrying members, resulting in the initiation of fracture and propagation throughout the structure. These details must be considered in the design phase of every project, and also controlled during fabrication and erection.
22.4.6
Residual Stresses in Welding
As heat-expanded weld metal and the surrounding base metal cool to room temperature, they shrink volumetrically. Under most conditions, this contraction is restrained or restricted by the surrounding material, which is relatively rigid and resists the shrinkage. This causes the weld to induce a residual stress pattern, where the weld metal is in residual tension and the surrounding base metal is in residual compression. The residual stress pattern is three dimensional since the metal shrinks volumetrically. The residual stress distribution becomes more complex when multiple-pass welding is performed. The final weld pass is always in residual tension, but subsequent passes will induce compression in previous weld beads that were formerly in tension. For relatively flexible assemblages, these residual stresses induce distortion. As assemblages become more rigid, the same residual stresses can cause weld cracking, typically occurring shortly after fabrication. If distortion does not occur, or when cracking does not occur, the residual stresses do not relieve themselves, but are “locked in”. Residual stresses are considered to be at the yield point of the material involved. Because any area that is subject to residual tensile stress is surrounded by a region of residual compressive stress, there is no loss in overall capacity of as-welded structures. However, this reduces the fatigue life for low-stress-range, high-cycle applications. Small welded assemblies can be thermally stress relieved by heating the steel to 1150◦ F, holding it for a predetermined length of time (typically 1 h/in. of thickness), and allowing it to return to room temperature. Residual stresses can be reduced by this method, but they are never totally eliminated. This approach is not practical for large assemblies, and care must be exercised to ensure that the components being stress relieved have adequate support when at the elevated temperature, where the yield strength and the modulus of elasticity are greatly reduced, as opposed to room temperature properties. For most structural applications, residual stresses cause no particular problem to the 1999 by CRC Press LLC
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performance of the system, and due to the complexity of stress relief activities, welded structures commonly are used in the as-welded condition. When loads are applied to as-welded structures, there is some redistribution or gradual decrease in the residual stress patterns. Usually called “shake down”, the thermal expansion and contraction experienced by a typical structure as it goes through a climatic season, as well as initial service loads applied to the building, result in a gradual reduction in the residual stresses from welding. These residual stresses should be considered in any structural application. On a macro level, they will affect the erector’s overall sequence of assembling of a building. On a micro level, they will dictate the most appropriate weld bead sequencing in a particular groove-welded joint. For welding applications involving repair, control of residual stresses is particularly important, since the degree of restraint associated with weld repair conditions is inevitably very high. Under these conditions, as well as applications involving heavy, highly restrained, very thick steel for new construction, the experience of a competent welding engineer can be helpful in avoiding the creation of unnecessarily high residual stresses.
22.4.7
Triaxial Stresses and Ductility
The commonly reported values for ductility of steel generally are obtained from uniaxial tensile coupons. The same degree of ductility cannot be achieved under biaxial or triaxial loading conditions. This is particularly significant since residual stresses are always present in any as-welded structure. A more detailed discussion on this subject is found in Section 22.7.
22.4.8
Flat Position Welding
Whenever possible, weld details should be oriented so that the welding can be performed in the flat position, taking advantage of gravity, which helps hold the molten weld metal in place. Flat position welds are made with a lower requirement for operator skill, and at the higher deposition rates that correspond to economical fabrication. This is not to say, however, that overhead welding should be avoided at all costs. An overhead weld may be advantageous if it allows for double-sided welding, with a corresponding reduction in the weld volume. High-quality welds can be made in the vertical plane, and, with the welding consumables available today, can be made at an economical rate.
22.5
Welded Joint Details
22.5.1
Selection of Fillet vs. PJP Groove Welds
For applications where either fillet welds or PJP groove welds are acceptable, the selection is usually based on cost. A variety of factors must be considered in order to determine the most economical weld type. For welds with equal throat dimensions, the PJP configuration requires one-half the volume of weld metal required by the fillet weld. Alternatively, for equal weld metal volumes, the PJP option is approximately 40% stronger than the fillet weld. Additional factors must be considered, however. For PJP welds, the bevel surface must be prepared prior to welding, increasing joint preparation cost. Typically achieved by flame cutting, this additional operation requires fuel gas, oxygen, and, most costly of all, labor. In general, fillet welds are the easiest welds to produce. Access into the more narrow included angles of groove welds usually requires more careful control of welding parameters, commonly resulting in slower welding speeds. The root pass of a PJP groove weld, made into a joint with no root opening, necessitates sufficient included groove angles to avoid centerline cracking tendencies due to poor cross-sectional bead shape. Slag removal may be difficult in root passes as well. These problems do 1999 by CRC Press LLC
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not exist in fillet welds when applied to 90◦ intersections of T joint members. Such issues can be of concern for skewed T joints, particularly when the acute angle side is less than 60◦ . Typical shop practices have generated a general rule of thumb suggesting that fillet welds are the most cost-effective details for connections requiring throats of 1/2 in. or less, which equates to a leg size of 3/4 in. PJP groove welds are generally the best choice for throat sizes of 3/4 in. or greater. This would roughly equate to a 1-in. fillet weld. In general, fillet welds should not exceed 1-in., nor should PJP groove welds be specified for throat dimensions less than 1/2 in. Between these boundaries, specific shop practices will determine the most economical approach.
22.5.2
Weld Backing
When there is a gap between two members to be joined, it is difficult to bridge the space with weld metal. On the other hand, when two members are tightly abutted to each other, it is difficult to obtain complete fusion. To overcome these problems, weld backing is added behind the members to act as a support for the weld metal (Figure 22.13). Weld backing fits into one of two categories: fusible-permanent steel backing or removable backing.
FIGURE 22.13: Weld backing. (Courtesy of The Lincoln Electric Company. With permission.)
22.5.3
Fusible Backing
Fusible steel backing, commonly known as backing bars, becomes part of the final structure when left in place, so steel that would meet quality requirements for primary members should be used for backing. In general, however, notch toughness properties are not specified for backing. The backing must be continuous for the length of the joint. If multiple pieces of steel backing are to be used in a single joint, they must be joined with CJP groove welds before being applied to the joint they are to back. Welds joining segments of backing bars should be inspected with radiography or ultrasonography to ensure soundness. Interrupted backing bars have been the source of fracture, as well as fatigue crack initiation, and are unacceptable. For building construction, steel backing is frequently used to compensate for dimensional variations that inevitably occur under field conditions. To maintain plumb columns, there will be slight variations in the dimensions between the columns in a bay. Since the beams are cut to length before 1999 by CRC Press LLC
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the exact dimensions are known, an oversized gap will often result between the beam and the column. Steel backing is inserted underneath this gap, and weld metal is used to bridge this space. It is important to remember, however, that the steel backing becomes part of the final structure if it is left in place.
22.5.4
Removable Backing
Removable backing includes fiberglass tapes, ceramic tiles, and fluxes attached to flexible tape. Removable backing generally is applied when the joint is to be welded with an open arc process such as flux core or shielded metal arc welding. Such backing is applied to the joint with some type of adhesive before the joint is welded. Upon completion of welding, the temporary backing is removed. Removable backing may be less costly for the fabricator than using the alternatives of doublesided joints or fusible backing. A major obstacle in the use of many of these types of backing is the adhesive that holds the material in place. This is particularly a concern when preheat is required. In some situations, mechanical means have been used to assist in holding the backing in place. When supports are attached by tack welds, care should be exercised to ensure that appropriate techniques are employed.
22.5.5
Copper Backing
Another type of removable backing would be a copper chill bar placed under the joint. Because of the high thermal conductivity of copper, the large difference in melting points of copper versus steel, and physical and chemical differences between the metals, molten weld metal can be supported by copper and the two materials rarely fuse together. This makes copper an attractive material to use for weld backing. However, this practice is discouraged or prohibited by many codes, because of the possibility of the arc impinging itself on the copper and drawing some of the melted copper into the weld metal. Copper promotes centerline cracking. This would, of course, be unacceptable. As a practical matter, fabricators avoid this practice simply because the copper backing is extremely expensive, and is rapidly ruined when the arc melts a portion of the copper. Copper backing can be used successfully under controlled conditions, which generally involve mechanized welding and joints that do not utilize root openings. In some situations, the fabricator will mill a groove in a copper chill bar, and fill the groove with clean, dry submerged arc flux. The flux then acts as the backing, and ensures the arc does not melt any of the copper. This is an efficient method and does not have the same ramifications as welding directly against copper. To ensure tight fit of the copper to the back of the joint, pneumatic, mechanical, or hydraulic pressure may be applied to achieve close alignment. Any temporary welds made to attach the backing system to the structural member must employ appropriate welding techniques.
22.5.6
Weld Tabs
Weld tabs, commonly known as starting and run off tabs, are added to the ends of joints in order to facilitate quality welding for the full length of the joint. The start and finish ends of weld beads are known to be more defect prone than the continuous weld between these points. Under starting conditions, the weld pool must be established, adequate shielding developed, and thermal equilibrium established. At the termination of a weld, the crater experiences rapid cooling with the extinguishing arc. Shielding is reduced. Cracks and porosity are more likely to occur in craters than at other points of the weld. Starts and stops can be placed on these extension tabs and subsequently removed upon the completion of the weld (see Figure 22.14). It is preferable to attach the weld tabs by tack welding within the joint (in Figure 22.14, notice the 1999 by CRC Press LLC
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FIGURE 22.14: Examples of weld tabs. (Courtesy of The Lincoln Electric Company. With permission.)
tack welds in the third example). Preheat requirements must be met when attaching weld tabs, unless the production weld is made with the submerged arc welding process, which will remelt these zones. It is important for weld tabs to have the same geometry as the weld joint to ensure the full throat or plate thickness dimension is maintained at the ends of the weld joint. When a weld tab containing weld metal of questionable quality is left in place, a fracture can initiate in these regions and propagate along the length of the weld. Weld tabs are removed for bridge fabrications, and since 1989, weld tab removal has been required by American Institute of Steel Construction (AISC) specifications when “jumbo” sections or heavy built-up sections are joined in tension applications by CJP groove welds.
22.5.7
Weld Access Holes
Weld access holes are provided in the web of beam sections to be joined to columns. The access hole in the upper flange connection permits the application of weld backing. The lower weld access hole permits access for the welder to make the bottom flange groove weld. AISC and AWS prescribe minimum weld access hole sizes for these connections ([9], para. 5.17, Figure 5.2). It must be emphasized that these minimum dimensions can be increased for specific requirements necessitated by the weld process, overall geometry, etc. However, the designer must be certain that the resultant section loss is acceptable. In order to provide ample access for electrode placement, visibility of the joint, and effective cleaning of the weld bead, it is imperative to provide adequate access. In addition to offering access for welding operations, properly sized weld access holes provide an important secondary function: they prevent the interaction of the residual stress fields generated by the vertical weld associated with the web connection and the horizontal weld between the beam flange and column face. The weld access hole acts as a physical barrier to preclude the interaction of these residual stress fields, which 1999 by CRC Press LLC
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can result in cracking. It is best for the weld access hole to terminate in an area of residual compressive stress [21]. More ductile behavior can be obtained under these conditions. Weld access holes must be properly made. Nicks, gouges, and other geometric discontinuities can act as stress raisers, increasing local stress levels and acting as points of fracture initiation. AISC requires that weld access holes be ground to a bright finish on applications where tension splices are applied to heavy sections. Although not mandated by the codes, these requirements for tension members may be needed for successful fabrication of compression members when connection details typically associated with tension members are applied to compression members (e.g., CJP groove welds) [22].
22.5.8
Lamellar Tearing
Lamellar tearing is a welding-related type of cracking that occurs in the base metal. It is caused by the shrinkage strains of welding acting perpendicular to planes of weakness in the steel. These planes are the result of inclusions in the base metal that have been flattened into very thin plates that are roughly parallel to the surface of the steel. When stressed perpendicular to the direction of rolling, the metallurgical bonds across these plates can separate. Since the various plates are not on the same plane, a fracture may jump between the plates, resulting in a stair-stepped pattern of fractures, illustrated in Figure 22.15. This type of fracture generally occurs near the time of fabrication, and can be confused with underbead cracking.
FIGURE 22.15: Lamellar tearing. (Courtesy of The Lincoln Electric Company. With permission.)
Several approaches can be taken to overcome lamellar tearing. The first variable is the steel itself. Lower levels of inclusions within the steel will help mitigate this tendency. This generally means lower sulfur levels, although the characteristics of the sulfide inclusion are also important. Manganese sulfide is relatively soft, and when the steel is rolled at hot working temperatures of 1600–2000◦ F, the sulfide inclusions flatten significantly. If steel is first treated to reduce the sulfur, and then calcium treated, for example, the resultant sulfide is harder than the surrounding steel, and during the rolling process, is more likely to remain spherical. This type of material will have much less of a tendency toward lamellar tearing. Current developments in steel-making practice have helped to minimize lamellar tearing tendencies. With continuously cast steel, the degree of rolling after casting is diminished. The reduction in the amount of rolling has directly affected the degree to which these laminations are flattened, and has correspondingly reduced lamellar tearing tendencies. The second variable involves the weld joint design. For a specific joint detail, it may be possible to alternate the weld joint to minimize lamellar tearing tendencies. For example, on corner joints it 1999 by CRC Press LLC
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is preferred to bevel the member in which lamellar tearing would be expected, that is, the plate that will be strained in the through-thickness direction. This is illustrated in Figure 22.16.
FIGURE 22.16: Lamellar tearing. (Courtesy of The Lincoln Electric Company. With permission.)
A reduction in the volume of weld metal used will help to reduce the stress that is imposed in the through-thickness direction. For example, a single bevel groove weld with a 3/8-in. root opening and 30◦ included angle will require approximately 22% less weld metal for a 1-1/2-in.-thick plate, compared to a 1/4-in. root opening and a 45◦ joint. The corresponding reduction in shrinkage stresses may be sufficient to eliminate lamellar tearing. In extreme cases, it may be necessary to resort to special measures to minimize lamellar tearing, which may involve peening. This technique involves the mechanical deformation of the weld surface, which results in compressive residual stresses that minimize the magnitude of the residual tensile stresses that naturally occur after welding. In order for peening to be effective, it is generally performed when the weld metal is warm (above 300◦ F), and must cause plastic deformation of the weld surface. Peening is restricted from being applied to root passes (because the partially completed weld joint could easily crack), as well as final weld layers, because the peening can inhibit appropriate visual weld inspection and embrittle the weld metal, which will not be reheated ([9], para. 5.27). Another specialized technique that can be used to overcome lamellar tearing tendencies is the “buttering layer” technique. With this approach, the surface of the steel where there might be a risk of lamellar tearing is milled to produce a slight cavity in which the butter layer can be applied. Individual weld beads are placed into this cavity. Since the weld beads are not constrained by being attached to a second surface, they solidify and cool, and thereby shrink, with a minimum level of applied stress to the material on which they are placed. After the butter layer is in place, it is possible to weld upon that surface with much less concern about lamellar tearing. This concept is illustrated in Figure 22.17. Lamellar tearing tendencies are aggravated by the presence of hydrogen. When such tendencies are encountered, it is important to review the low hydrogen practice, examining the electrode selec1999 by CRC Press LLC
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tion, care of electrodes, application of preheat, and interpass temperature. Additional preheat can minimize lamellar tearing tendencies.
FIGURE 22.17: “Buttered” surface. (Courtesy of The Lincoln Electric Company. With permission.)
22.6
Design Examples of Specific Components
To demonstrate the design principles of welded connections, five examples are presented. The objective of each example is to determine either the weld leg size or the weld length. These are representative of several beam-to-column design concepts. For further details and examples consult [20].
22.6.1
Flexible Seat Angles
determine maximum unit horizontal force on weld (Fn ) M= 1999 by CRC Press LLC
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2 R ef = Lv P 2 3
also P =
1 2 Fn Lv 2 3
from this Fn =
2.25Rer L2v
unit vertical force on weld Fv =
R 2Lv
resultant unit force on weld (at top) s q R 2 R q 2 2.25Rer 2 2 2 + = Lv + 20.25ef2 fr = fn + fv = 2Lv L2v 2L2v leg size of fillet weld W =
22.6.2
q R L2v + 20.25ef2 2L2v (.707)(.30EXX)
Stiffened Seat Brackets
In this particular connection, the shear reaction is taken as bearing through the lower flange of the beam. There is no welding directly on the web. For this reason it cannot be assumed that the web can be stressed up to its yield in bending throughout its full depth. Since full plastic moment cannot be assumed, the bending stress allowable is held to σ = .60σy , or 22 ksi. AISC Sect. 1.5.1.4.1. Check the bending stress in the beam: σ =
1100 in.-kip M = = 20.1 ksi < .60σy or 22 ksi OK S 54.7 in.
Bending force in the connection plate: F = 1999 by CRC Press LLC
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1100 in.-kip M = = 78.0 kip d 14.12 in.
Area of the top connection plate: Ap =
78.0 kip F = = 3.54 in.2 σ 22 ksi
or use a 5 × 3 in. plate which gives a value of Ap = 3.75 in.2 > 3.54 in.2 OK If a 3/8 in. fillet weld is used to connect top plate to upper beam flange: fw = (.707)(3/8 in.)(21 ksi) = 5.56 kips for linear inch of weld. Length of fillet weld: L=
78.0 kip F = 14.1 in. = fw 5.56 kip/in.
or use 5 in. across the end of the plate, and 5 in. along each side, a total length of 15 in. > 14.1 in. OK
22.6.3
Web Framing Angles
Twisting J Cv Jw J Cn = Jw
(horizontal) Fn = (vertical) Fv1 Shear (vertical) Fv2 = Resultant force Fr =
q Fn2 − (Fv1 + Fv2 )2
Leg size of fillet weld W = 1999 by CRC Press LLC
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R 2(2b − Lv )
Fr .707(.30EXX)
22.6.4
Top Plate Connections
The welding of the flanges and nearly full depth of the web would allow the beam to develop its full plastic moment. This will allow the “compact” beam to have a 10% higher bending allowable, or σ = .66σy . This also allows the end of the beam, and its welded connection, to be designed for 90% of end moment due to gravity loading. AISC Sect. 1.5.1.4.1. Check the bending stress in the beam: σ =
.9(1100 in.-kip) .9M = 23.6 ksi < .66σy or 24 ksi OK = S 41.9 in.3
Bending force in the top connecting plate: F =
.9(1100 in.-kip) .9M = = 71.5 kip d 13.86 in.
Area of top connection plate: Ap =
71.5 kip F = = 2.98 in.2 σ 24 ksi
Or use a 5-1/2 in. by 5/8-in. plate, Ap = 3.44 in.2 > 2.98 in.2 OK If a 3/8-in. fillet weld is used to connect the top plate to the upper beam flange: fw = (.707)(3/8 in.)(21 ksi) = 5.56 kip per linear inch of weld Length of fillet weld L=
71.5 kip F = 12.9 in. = fw 5.56 kip/in.
or use 5-1/2 in. across the end of the plate end 4 in. along each side, a total length of 13-1/2 in. The lower flange of the beam is butt welded directly to the flange of the column. Since the web angle carries the shear reaction, no further work is required on this lower portion of the connection. The seat angle simply serves to provide temporary support for the beam during erection and a backing for the flange groove butt weld. 1999 by CRC Press LLC
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22.6.5
Directly Connected Beam-to-Column Connections
Design a fully welded beam-to-column connection for a W14x30 beam a W8x31 column to transfer an end moment of M = 1000 in.-kips, and a vertical shear of V = 20 kips. This example will be considered with several variations. Use A36 steel and E70 filler metal.
The welding of the flanges and full depth of the web would allow the beam to develop its full plastic moment. This will allow the “compact” beam to have a 10% higher bending moment, or σ = .66σy . This also allows the end of the beam, and its welded connection, to be designed for 90% of the end moment due to gravity loading. AISC Sect. 1.5.1.4.1. 6.733 65 65 b = = 8.79 AISC allowable √ = √ = 10.83 OK 2tf 2(.383) σy 36 13.86 − 2(.383) d 64D σx σ = = 48.5 AISC allowable = 1 − 3.74 actual t .290 t σy σy = 106.7 OK
actual
hence this beam has a “compact” section. σ =
.9(1100 in.-kip) .9M = 23.63 ksi < .66σy or 24 ksi OK = S 41.9 in.3
The weld on the web must be able to stress the web in bending to yield (σy ) throughout its depth (see the bending stress distribution above). Unit force this weld: fw =
20 kip V = = .764 kip per linear inch 2L 2[13.86 − (2 × .383)]
Leg size of fillet weld: w=
.764 kip/in. = .05 in. .707(21 ksi)
However, this is welded to a .433-in.-thick flange of the column, so the minimum fillet weld size for this would be 3/16 in. 1999 by CRC Press LLC
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22.7
Understanding Ductile Behavior
“Ductility” can mean different things to different people. Materials such as cast iron are generally considered “brittle”, while steel is called “ductile”. A physical metallurgist may talk in terms of cleavage and ductile dimpling to define material behavior on a microscopic level. This is of little benefit to the structural engineer, who is more concerned with global deformation than microscopic behavior. Global deformation would include buckling, plastic hinge formation, stretching of members, and other inelastic behaviors that are visually observable. To achieve ductile behavior, the structural engineer, will select ductile materials for construction. To assume, however, that global deformation will occur simply because a ductile material has been selected can lead to unexpected brittle fracture of even ductile materials. It is essential, therefore, that global behavior be separated from microscopic behavior. A material that fails by low-energy cleavage fracture cannot be made to function in a globally ductile manner, although it is possible for a structural element to fail with little or no deformation, and yet the fracture surface would exhibit the characteristics of ductile dimpling. Microscopic ductility and global ductility are separate issues, and the structural engineer must understand what conditions lead to global ductile behavior. This is particularly important where welding is applied, since welding introduces residual stresses and geometric influences that can affect the achievement of ductile behavior. For global ductility to be possible, the following conditions must be achieved: 1. There must be a shear stress component (τ ) that results from the applied load. 2. The shear stress must be of sufficient magnitude so as to exceed the critical shear stress of the material. 3. The shear stress must result in an inelastic shear strain that acts in a direction to relieve the particular stress that is applied. 4. There must be a sufficient length of unrestrained material to permit a reduction in the cross-sectional area (i.e., to allow for “necking” to occur). These conditions are met in the specimens typically used to measure ductility of steels. As illustrated in Figure 22.18, the preceding four principles will be applied to the uniaxial tensile specimen. In Figure 22.18a, the specimen has been stretched to a point so that the resultant stresses, σ1 , are below the yield point, σy . The stress, σ1 , has caused a shear stress, τ1−2 , that acts on a 45◦ plane to the applied stress. Rather than focusing on σ1 being less than σy , it is better to realize the resultant shear stress, τ1−2 , is less than the critical shear stress, τcr . A Mohr’s Circle diagram assists in visualizing this behavior. Once τcr is exceeded, slippage along shear planes can occur, resulting in elongation. In Figure 22.18a, a shear stress has resulted from the applied stress (i.e., condition 1 from above was achieved), but the shear stress is not sufficient to exceed the critical value (i.e., condition 2 has not been achieved). All behavior under these conditions would be elastic, and although brittle fracture would not occur, neither would ductile behavior be achieved. In Figure 22.18b, the load, F, has been increased so that the resultant shear, τ1−2 , exceeds τcr , resulting in slip on the plane oriented at 45◦ . This slip results in elongation, or stretching of the member. Global ductility is seen. This behavior occurs because condition 2 has been achieved. Figure 22.18c illustrates the continued application of force, resulting in slip occurring on multiple planes, eventually resulting in a reduction in cross-section, or necking. This is possible because all four conditions have been achieved. A further increase in load, illustrated in Figure 22.18d, causes the critical tensile strength, σt , to be exceeded. The sample eventually breaks, and the final fracture surface exhibits little deformation. This occurs because, due to the localized deformations occurring in the necking region, the stresses in the other two principle directions are no longer zero. This is triaxial stress, and as illustrated in the 1999 by CRC Press LLC
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FIGURE 22.18: Shear stresses in simple, uniaxial tensile specimens. (Courtesy of The Lincoln Electric Company. With permission.)
Mohr’s Circle, there is a resultant decrease in the shear stress. No longer is condition 1 maintained, and brittle fracture (in a global sense) occurs across the necked region. These sample principles can be applied to various connection details. Consider, for example, the taper required for tension members that have thickness or width transitions. As seen in Figure 22.19, the sharp 90◦ transition results in a biaxial stress state near the transition. While ductile behavior could occur in the area where uniaxial stress exists, the second stress will reduce the shear stress, reducing its ductility capacity. The tapered transition allows for essentially uniaxial stresses to be maintained through the transition range, encouraging shear stresses capable of producing ductile behavior.
22.7.1
Two Residual Stresses Isolated
Figure 22.20 illustrates that two important residual stresses exist in the weld access hole’s termination zone. This butt joint in the flange has a residual stress, σ3 , longitudinal to the length of the flange, as well as a stress transverse to the flange, σ1 . The longitudinal stress is tensile along the center line of the flange where the weld access hole terminates. It can be compared to tightening a steel cable lengthwise in the center in tension, with compression spread out on both sides. The transverse stress, σ1 , is positive (tensile) in the weld zone, as well as in an adjacent portion of the plate going through zero, and then compression. Beyond the adjacent plate, it becomes zero and then negative (compression). This transverse stress, σ1 , is also similar to tightening a steel cable. 1999 by CRC Press LLC
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FIGURE 22.19: Stress state in transition connections. (Courtesy of The Lincoln Electric Company. With permission.)
FIGURE 22.20: Resultant residual stress of welding. (Courtesy of The Lincoln Electric Company. With permission.) 1999 by CRC Press LLC
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22.7.2
Residual Stresses Applied
These residual stresses may be applied to a weld detail having a narrow weld access hole, as shown in Figure 22.21. This hole terminates at a point where σ1 and σ3 are in tension. Since the web at the edge of the weld access hole offers some restraint against movement in the through-thickness direction of the flange plate, stress in the σ2 direction may have an appreciable value. All of the circles will be small. Neither τ2−3 nor τ1−3 will probably ever reach the critical shear stress value, and plastic strain or ductility will not occur, as the lower portion of Figure 22.21 illustrates.
FIGURE 22.21: Mohr’s circle of stress for element 2. (Courtesy of The Lincoln Electric Company. With permission.)
If the weld access hole can be cut with circular ends, sometimes called a pear-shaped opening, the stress, σ2 , in the through thickness of the flange plate will be greatly reduced, probably to zero in this critical section, as shown in Figure 22.22. This will produce a very large circle with σ3 and the resulting shear stress, τ2−3 , will be very high — high enough to exceed the critical value well before σ3 reaches its critical value for failure. This would result in a more ductile behavior. If the weld access hole can be made wider, so that it terminates in a zone where the transverse residual stress, σ1 , is compressive (see Figure 22.23), then a more favorable stress condition will result in greater ductility in the σ3 direction. In this case, shear stress, τ1−3 , will be high as shown on Mohr’s Circle of stress, and the critical shear value will be reached at a much lower tensile stress or load value. This will produce more ductility in the σ3 direction, greatly reducing the chance of a transverse crack in the flange at the termination of the weld access hole. If a pear-shaped wide weld access hole is used, and the through-thickness stress, σ2 , becomes zero, it simply increases the shear stress τ2−3 and would seem to improve ductility (see Figure 22.24). However, looking at the resulting stress-strain curve of the flange plate at the termination of the weld access hole, it appears that rounding the ends of the wide access hole in this case does not appreciably increase the ductility. This is probably because the wide weld access hole already has excellent ductility. 1999 by CRC Press LLC
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FIGURE 22.22: Mohr’s circle of stress for element 3. (Courtesy of The Lincoln Electric Company. With permission.)
FIGURE 22.23: Mohr’s circle of stress for element 4. (Courtesy of The Lincoln Electric Company. With permission.)
Figure 22.25 shows stress-strain curves of the four different weld access hole details just discussed. The principles outlined herein can be applied to other details, evaluating the potential of biaxial or triaxial stresses and their effect on shear stress development. Consideration of these principles can assist in avoiding brittle fracture by encouraging ductile behavior. Special Considerations for Welded Structures Subject to Seismic Loading
1999 by CRC Press LLC
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FIGURE 22.24: Mohr’s circle of stress for element 5. (Courtesy of The Lincoln Electric Company. With permission.)
FIGURE 22.25: Stress-strain curves of four different weld access hole details. (Courtesy of The Lincoln Electric Company. With permission.)
22.7.3
Unique Aspects of Seismically Loaded Structures
Demands on Structural Systems
During an earthquake, even structures specifically designed for seismic resistance are subject to extreme demands. Any structure designed with a response modification factor, Rw , greater than unity will be loaded beyond the yield stress of the material. This is far more demanding than other anticipated types of loading. Due to the inherent ductility of steel, stress concentrations within a 1999 by CRC Press LLC
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steel structure are gradually distributed by plastic deformation. If the steel has a moderate degree of notch toughness, this redistribution eliminates localized areas of high stress, whether due to design, material, or fabrication irregularities. For statically loaded structures, the redistribution of stresses is relatively inconsequential. For cyclically loaded structures, repetition of this redistribution can lead to fatigue failure. In seismic loading, however, it is expected that portions of the structure will be loaded well beyond the elastic limit, resulting in plastic deformation. Localized areas of high stress will not simply be spread out over a larger region by plastic deformation. The resultant design, details, materials, fabrication, and erection must be carefully controlled in order to resist these extremely demanding loading conditions. Demand for Ductility
Seismic designs have relied on ductility to protect structures during earthquakes. Unfortunately, much confusion exists regarding the measured property of ductility in steel, and ductility can be experienced in steel configured in various ways. It is essential that a fundamental understanding of ductility be achieved in order to ensure ductile behavior in the steel in general, and particularly in the welded connections. Requirements for Efficient Welded Structures
Five elements are present in any efficient welded structure: • • • • •
Good overall design Good materials Good details Good workmanship Good inspection
Each element is important, and emphasis on one will not overcome deficiencies in others. Both the Northridge earthquake in 1994 and the Kobe earthquake in 1995 showed that deficiencies in one or more of the preceding areas may have contributed to the degredation in performance of Steel Moment-Resisting Frames (SMRFs).
22.8
Materials
22.8.1
Base Metal
Base metal properties are particularly important in structures subject to seismic loading. Unlike most static designs, seismically resistant structures depend on acceptable material behavior beyond the elastic limit. The basic premise of seismic design is to absorb seismic energies through yielding of the material. For static design, additional yield strength capacity in the steel may be desirable, but for applications where yielding is the desired method for achieving energy absorption, higher than expected yield strengths may have a dramatic negative effect. This is especially important as it relates to connections, both bolted and welded. Figure 22.26 illustrates five material zones that occur near the groove weld in a beam-to-column connection. If it is assumed that the web is incapable of transferring any moment, it is essential that the plastic section modulus of the flanges (Zf ) times the tensile strength be greater than the entire plastic section property (Z) times the yield strength in the beam. All five material properties must be considered in order for the connection to behave satisfactorily. Note that this was the standard connection detail used for special moment-resisting frame (SMRF) systems prior to the Northridge earthquake. 1999 by CRC Press LLC
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FIGURE 22.26: Five material zones that occur near the groove weld in a beam-to-column connection. (Courtesy of The Lincoln Electric Company. With permission.)
Current American Society for Testing and Materials (ASTM) specifications do not place an upper limit on the yield strength for most structural steels, but specify only a minimum acceptable value. For instance, for ASTM A36 steel, the minimum acceptable yield strength is 36 ksi. This precludes a steel that has a yield strength of 35.5 ksi as being acceptable, but does nothing to prohibit the delivery of a 60-ksi steel. The tensile strength range is specified as 58–80 ksi. Although A36 is commonly specified for beams, columns are typically specified to be of ASTM A572 grade 50. With a 50-ksi minimum yield strength and a minimum tensile strength of 65 ksi, many designers were left with the false impression that the yield strength of the beam could naturally be less than that of the column. Due to the specification requirements, it is possible to produce steel that meets the requirements of both A36 and A572 grade 50. This material has been commercially promoted as “dual-certified” material. However, no matter what the material is called, it is critical for the connection illustrated in Figure 22.26 to have controls on material properties that are more rigorous than the current ASTM standards impose. Much of the focus of post-Northridge research has related to the beam yield-to-tensile ratio, commonly denoted as Fy /Fu . This is often compared to the ratio of Zf /Z , with the desired relationship being Fy Zf > Z Fu This suggests that not only is Fy (yield strength) important, but the ratio is important as well. For rolled W shapes, Zf /Z ranges from 0.6 to 0.9. Based on ASTM minimum specified properties, Fy /Fu is as follows: A36 0.62 A572Gr50 0.77 However, when actual properties of the steel are used, this ratio may increase. In the case of one building damaged in Northridge, mill test reports indicated the ratio to be 0.83. ASTM steel specifications need further controls to limit the upper value of acceptable yield strengths for materials as well as the ratio of Fy /Fu . A new ASTM specification has been proposed to address these issues, although its approval will probably not be achieved before 1997. In Figure 22.26, five zones have been identified in the area of the connection, with the sixth material property being located in the beam. Thus far, only two have been discussed: the beam yield strength and the beam ultimate strength. These are designated with the subscript X to indicate that these are 1999 by CRC Press LLC
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the properties in the orientation of the longitudinal axis of the beam. When the beam is produced, the longitudinal direction is considered the “direction of rolling”. In general, steel exhibits its best mechanical properties in this orientation. When this axis is designated as the X axis, the width of the beam would be known as the Y direction and the Z axis is through the thickness of the flanges. For beam properties, the X axis is the one of interest. The properties of interest with respect to the column are oriented in the column Z axis, which will exhibit the least desirable mechanical properties. Current ASTM specifications do not require measurement of properties in this orientation. While there are ASTM standards for the measurement of through-thickness properties (ASTM 770), these are not normally applied for structural applications. It is this through-thickness strength, however, that is important to the performance of the connection. Notch toughness is defined as the ability of a material to resist propagation of a preexisting cracklike flaw while under tensile stress. Pre-Northridge specifications did not include notch toughness requirements for either base materials or weld metals. When high loads are applied, and when notch-like details or imperfections exist, notch toughness is the material property that resists crack propagation from that discontinuity. Rolled shapes routinely produced today, specifically for lighter weight shapes in the group 1, 2, and 3 categories, generally are able to deliver a minimum notch toughness of 15 ft.-lb at 40◦ F. This is probably adequate toughness, although additional research should be performed in this area. For heavy columns made of group 4 and 5 shapes, this level of notch toughness may not be routinely achieved in standard production. After Northridge, many engineers began to specify the supplemental requirements for notch toughness that are invoked by AISC specifications for welded tension splices in jumbo sections (group 4 and 5 rolled shapes). This requirement for 20 ft.-lb at 70◦ F is obtained from a Charpy V-notch specimen extracted from the web/flange interface, an area expected to have the lowest toughness in the cross-section of the shape. Since columns are not designed as tension members under most conditions, this requirement would not automatically be applied for column applications. However, as an interim specification, it seems reasonable to ensure minimum levels of notch toughness for heavy columns also.
22.8.2
Weld Metal Properties
Significant properties of weld metal are yield strength, tensile strength, toughness, and elongation. These properties usually may be obtained from data on the particular filler metal that will be employed to make the connection. The American Welding Society (AWS) filler metal classification system defines the minimum acceptable properties for the weld metal when deposited under very specific conditions. Most “70” series electrodes (e.g., E7018, E70T-1, E70T-6) have a minimum specified yield strength of 58 ksi and a minimum tensile strength of 70 ksi. As in the specifications for steel, there are no upper limits on the yield strength. However, in welded design, it is generally assumed that the weld metal properties will exceed those of the base metal, and any yielding that would occur in the connection should be concentrated in the base metal, not in the weld metal, since the base metal is assumed to be more homogeneous and more likely to be free of discontinuities than the weld. Most commercially available filler metals have a “70” classification, exceeding the minimum specified strength properties of the commonly used A36 and A572 grade 50. These weld metal properties are obtained under very specific testing conditions prescribed by the AWS A5 Filler Metal Specifications. Weld metal properties are a function of many variables, including preheat and interpass temperatures, welding parameters, base metal composition, and joint design. Deviations in these conditions from those obtained for the test welds may result in differences in mechanical properties. Most of these changes will result in an increase in yield and tensile strength, along with a corresponding decrease in elongation and, in general, a decrease in toughness. When weld metal properties exceed those of the base metal, and when the connection is loaded into the 1999 by CRC Press LLC
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inelastic range, plastic deformations would be expected to occur in the base metal, not in the weld metal itself. The increase in the strength of the weld metal compensates for the loss in ductility. The general trend to strength levels higher than those obtained under the testing conditions is of little consequence in actual fabrication. There are conditions that may result in lower levels of strength, and the Northridge earthquake experience revealed that this may be more commonplace and more significant than originally thought. The interpass temperature is the temperature of the steel when the arc is initiated for subsequent welding. There are two aspects to the interpass temperature: the minimum level, which should always be the minimum preheat temperature, and the maximum level, beyond which welding should not be performed. Because of the relatively short length of beam-to-column flange welds, an operator may continue welding at a pace that will allow the temperature of the steel at the connection to increase to unacceptably high levels. After one or two weld passes, this temperature may approach the 1000◦ F range. In such a case, the strength of the weld deposit will be rapidly decreased. Weld metal toughness is an area of particular interest in the post-Northridge specifications. Previous specifications did not include any requirement for minimum notch toughness levels in the weld deposits, allowing for the use of filler metals that have no minimum specified requirements. For connections that are subject to inelastic loading, it now appears that minimum levels of notch toughness must be specified. The actual limits on notch toughness have not been experimentally determined. With the AWS filler metal classifications in effect in 1996, electrodes are classified as either having no minimum specified notch toughness or having notch toughness values of 20 ft.-lb at a temperature of 0◦ F or lower. As an interim specification, 20 ft.-lb at 0◦ F or lower has been recommended. It should be noted that the more demanding notch toughness requirements impose several undesirable consequences upon fabrication, including increased cost of materials, lower deposition rates, less operator appeal, and greater difficulty in obtaining sound weld deposits. Therefore, ultra-conservative requirements imposed “just to be safe” may actually be unacceptable. Research will be conducted to determine precise toughness requirements. Until then, based upon practical issues of availability, 20 ft.-lb at 0◦ F is a reasonable specification.
22.8.3
Heat-Affected Zones
As illustrated in Figure 22.26, the base metal heat-affected zones (HAZs) represent material that may affect connection performance as well. The HAZ is that base metal that has been thermally changed due to the energy introduced into it by the welding process. In the small region immediately adjacent to the weld, the base metal has experienced a different thermal history than the rest of the base material. For most hot-rolled steels, the area of concern is a HAZ that is cooled too rapidly, resulting in a hardened HAZ. For quenched-and-tempered steels, the HAZ may be cooled too slowly, resulting in a softening of the area. In columns, the HAZ of interest is the Z direction area immediately adjacent to the groove weld. For the beam, these are oriented in the X direction. Excessively high heat input can negatively affect HAZ properties by causing softening in these areas. Excessively low heat input can result in hardening of the HAZs. Weld metal properties may be negatively affected by extremely high heat input welding procedures, causing a decrease in both the yield strength and tensile strength, as well as the notch toughness of the weld deposit. Excessively low heat input may result in high-strength weld metal and also decrease the notch toughness of the weld deposit. Optimum mechanical properties are generally achieved if the heat input is maintained in the 30–80 kJ/in. range. Post-Northridge evaluation of fractured connections has revealed that excessively high heat input welding procedures were often used, confirmed by the presence of very large weld beads that sometimes exceeded the maximum limits prescribed by the D1.1-96 code. These may have had some corollary effects on weld metal and HAZ properties.
1999 by CRC Press LLC
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22.9
Connection Details
Since there are no secondary members in welded construction, any material connected by a weld participates in the structural system — positively or negatively. Unexpected load paths can be developed by the unintentional metallurgical path resulting from the one-component system created by welding. This phenomenon is particularly significant in detailing.
22.9.1
Weld Backing
Pre-Northridge specifications typically allowed steel backing to be left in place. Most of the fractures experienced in Northridge initiated immediately above the naturally occurring unfused region, between the backing and the column face. When this area experienced tensile loading due to lateral displacements, this region would result in a stress concentration and a notch that served as a crack initiation point (see Figure 22.27).
FIGURE 22.27: Weld backing and fracture initiation. (Courtesy of The Lincoln Electric Company. With permission.)
After Northridge, many specifications began to call for the removal of steel backing from the bottom beam-flange-to-column connection. This activity not only eliminates the notch-like condition, it permits gouging the weld root to sound metal, and allows for the depositing of a reinforcing fillet weld that provides a more gradual transition in the 90◦ interface between the beam and the column. Not all backing is required to be removed. For welds subject to horizontal shear (such as corner joints in box columns), backing can be left in place. In butt joints, the degree of stress amplification that occurs due to backing left in place is much less severe than what occurs in T joints. Backing removal is expensive and, particularly when done in the overhead position, requires considerable welder skill. Some recommendations have not required removal of top beam-flange-to-column 1999 by CRC Press LLC
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connection backing because the removal operation (gouge, clean, inspect, and reweld) must be performed through the weld access hole. This difficult operation may do more harm than good. There is increased interest in ceramic backing. Welding procedures that employ ceramic backing must be qualified by test for work done in accordance with D1.1-96, para. 5.10. Welders must be trained in the proper use of these materials. Ceramic is nonconductive, requiring that the welder establish a “bridge” between the two steel members to be welded in order to maintain the electrical arc between the two members. While this can be accomplished fairly readily with small root opening dimensions (such as 1/4 in.), it becomes increasingly difficult with larger root openings (such as 1/2 in.). Wide, thin root passes on ceramic-backed joints may crack due to high shrinkage stresses imposed on small weld throats. One benefit of the activity of fusible backing removal is that it permits the weld joint to be back gouged to sound material. The root of the weld joint is always the most problem-prone region. The center of the length of the bottom beam-flange-to-column weld is difficult to make, since the welder must work through the weld access hole. This is also one of the most difficult areas to inspect with confidence. In a typical beam-to-column connection, the bottom beam-flange-to-column weld must be interrupted midlength due to interference with the web. This area is particularly sensitive to workmanship problems, and is also a difficult region to inspect with ultrasonic testing. The backgouging operations provide the opportunity for visual verification that sound weld metal has been obtained, particularly in the center of the joint length. This is similar to the D1.1-96 code requirement for back gouging of double-sided joints.
22.9.2
Weld Tabs
Weld tabs are auxiliary pieces of metal on which the welding arc may be started or stopped. For statically loaded structures, these are usually left in place. For seismic construction, weld tabs should be removed from critical connections that are subject to inelastic loading, because metal of questionable integrity may be deposited in the region of these weld tabs. Weld tab removal is probably most important on beam-to-column connections where the column flange width is greater than the beam flange width. It is reasonable to expect that stress flow would take place through the left-in-place weld tab. However, for butt splices where the same width of material is joined, weld tabs extending beyond the width of the joint would not be expected to carry significant stress, making weld tab removal less critical. It is unlikely that tab removal from continuity plate welds would be justified. For beam-to-column connections where columns are box shapes, the natural stress distribution causes the ends of the groove weld between the beam and column to be loaded to the greatest level, the same region as would contain the weld tab. Just the opposite condition exists when columns are composed of I-shaped members. The center of the weld is loaded most severely, causing the areas in which the weld tabs would be located to have the lowest stress level. For welds subject to high levels of stress, however, weld tabs should be removed.
22.9.3
Welds and Bolts Sharing Loads
Welding provides a continuous metallurgical path that relies upon the internal metallurgical structure of the fused metal to provide continuity and strength. Rivets and bolts rely on friction, shear of the fastening element, or bearing of the joint material to provide for transfer of loads between members. When mechanical fasteners such as bolts are combined with welds, caution must be exercised in assigning load-carrying capacity to each joining method. Traditionally, it was thought that welds used in conjunction with bolts should be designed to carry the full load, assuming that the mechanical fasteners have no load-carrying capacity until the weld fails. The development of high-strength fasteners, however, created the assumption that loads can 1999 by CRC Press LLC
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be shared equally between welds and fasteners. This has led to connection details that employ both joining systems. Specifically, the welded flange, bolted web detail used for many beam-to-column connections in SMRFs assumes that the bolted web is able to share loads equally with the welded flanges. Although most analyses suggest that vertical loads are transferred through the shear tab connection (bolted) and moments are transferred through the flanges (welded), the web does have some moment capacity. Depending on the particular rolled shape involved, the moment capacity of the web can be significant. Testing of specimens with the welded web detail, as compared to the bolted web detail, generally has yielded improved performance results. This has called into question the adequacy of the assumption of high-strength bolts sharing loads with welds when subject to inelastic loading. Post-Northridge research provides further evidence that the previously accepted assumptions may have been inadequate. Previous design rules regarding the capacity of bolted connections should be reexamined. This may necessitate additional fasteners, or larger sizes of shear tabs (both in thickness and in width). Stipulations regarding the addition of supplemental fillet welds on shear tabs, currently a function of the ratio of Zf /Z, are probably also inadequate and will require revision. Pending further research, the conservative approach is to utilize welded web details. This does not preclude the use of a bolted shear tab for erection purposes, but would rely on welds as a singular element connecting the web to the column.
22.9.4
Weld Access Holes
The performance of a connection during seismic loading can be limited by poorly made, or improperly sized, weld access holes. In the beam-to-column connection illustrated in Figure 22.28, a welded web connection has been assumed. As the flange groove weld shrinks volumetrically, a residual stress field will develop perpendicular to the longitudinal axis of the weld, as illustrated in direction X in the figure. Concurrently, as the groove weld shrinks longitudinally, a residual stress pattern is established along the length of the weld, designated as direction Y. When the web weld is made, the longitudinal shrinkage of this weld results in a stress pattern in the Z direction. These three residual stress patterns meet at the intersection of the web and flange of the beam with the face of the column. When steel is loaded in all three orthogonal directions simultaneously, even the most ductile steel cannot exhibit ductility. At the intersection of these three welds, cracking tendencies would be significant. A generous weld access hole, however, will physically interrupt the interaction of the Z axis stress field and the biaxial (X and Y) stress field, thereby increasing the resistance to cracking during fabrication. The quality of weld access holes may affect both resistance to fabrication-related cracking and resistance to cracking that may result from seismic events. Access holes usually are cut into the steel by a thermal cutting process, either oxy-fuel or plasma arc. Both processes rely on heating the steel to a high temperature and removing the heated material by pressurized gases. In the case of oxy-fuel cutting, oxidation of the steel is a key ingredient in this process. In either process, the steel on either side of the cut (called the “kerf ”) has been heated to an elevated temperature and rapidly cooled. In the case of oxy-fuel cutting, the surface may be enriched with carbon. For plasma cut surfaces, metallic compounds of oxygen and nitrogen may be present on this surface. The resultant surface may be hard and crack sensitive, depending on the combinations of the cutting procedure, base metal chemistry, and thickness of the materials involved. Under some conditions, the surface may contain small cracks, which can be the points of stress amplification that cause further cracking during fabrication or during seismic events. Nicks or gouges may be introduced during the cutting process, particularly when the cutting torch is manually propelled during the formation of the access hole. These nicks may act as stress amplification points, increasing the possibility of cracking. To decrease the likelihood of notches and/or microcracks on thermally cut surfaces, AISC has specific provisions for making access holes in heavy group 4 and 5 rolled shapes. These provisions include the need for a preheat before cutting, requirements 1999 by CRC Press LLC
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FIGURE 22.28: A generous weld access hole in this beam-to-column connection provides resistance to cracking. (Courtesy of The Lincoln Electric Company. With permission.)
for grinding of these surfaces, and inspection of these surfaces with magnetic particle (MT) or dye penetrant (PT) inspection. Whether these provisions should be required for all connections that may be subject to seismic energies is unknown at this time. However, for connection details that impose high levels of stress on the connection, and specifically those that demand inelastic performance, it is apparent that every detail in the access hole region is a critical variable. In the Northridge earthquake, some cracking initiated from weld access holes.
22.10
Achieving Ductile Behavior in Seismic Sections
22.10.1
System Options
Several systems may be employed to achieve seismic resistance, including eccentrically braced frames (EBFs), concentrically braced frames (CBFs), SMRFs, and base isolation. Of the four mentioned, only base isolation is expected to reduce demand on the structure. The other three systems assume that at some point within the structure, plastic deformations will occur in members, thus absorbing seismic energy. In a CBF, the brace member is expected to be subject to inelastic deformations. The welded connections at the termination of a brace are subject to significant tension or compression loads, although rotation demands in the connections are fairly low. Designing these connections requires the engineer to develop the capacity of the brace member in compression and tension. Recent experiences with CBF systems have reaffirmed the importance of the brace dimensions (b/t ratio), and the importance of good details in the connection itself. Problems appear to be associated with misplaced welds, undersized welds, missing welds, or welds of insufficient throat due to construction 1999 by CRC Press LLC
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methods. In order to place the brace into the building frame, a gusset plate is usually welded into the corners of the frame. The brace is slit along its longitudinal axis and rotated into place. To maintain adequate dimensions for field assembly, the slot in the tube must be oversized, compared to the gusset, resulting in natural gaps between the tube and the gusset plate. When this dimension increases, as illustrated in Figure 22.29, it is important to consider the effect of the root opening on the strength of the fillet weld. For gaps exceeding 1/16 in., the D1.1-96 code requires that the weld leg size be increased by the amount of the gap, ensuring a constant actual throat dimension is maintained.
FIGURE 22.29: Effect of root openings (gaps) on fillet weld throat dimensions. (Courtesy of The Lincoln Electric Company. With permission.)
EBFs and SMRFs are significantly different structural systems, but some welding design principles apply equally to both systems. It is possible to design an EBF so that the “link” consists simply of a rolled steel member. In Figure 22.30, these examples are illustrated by the links designated as c1. In other EBF systems, however, the connection itself can be part of the link, as illustrated by c2. When this design method is used, the welded connections become critical since the expected loading on the connection is in the inelastic region. Much of the discussion under SMRF may be applied to these situations.
FIGURE 22.30: Examples of EBF systems. (From American Institute of Steel Construction. Seismic Provisions for Steel Buildings. 1992.) 1999 by CRC Press LLC
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The SMRF system is commonly applied to low-rise structures. Advantages of this type of system include desirable architectural elements that leave the structure free of interrupting diagonal members. Extremely high demands for inelastic behavior in the connections are inherent to this system. When subject to lateral displacements, the structure assumes a shape as shown in Figure 22.31. Note that the highest moments shown in Figure 22.31 are applied at the connection. Figure 22.31 shows a plot of the section properties. Section properties are at their lowest value at the column face, because of the weld access holes that permit the deposition of the CJP beam-flange-to-column-flange welds. These section properties may be further reduced by deleting the beam web from the calculation of section properties. This is a reasonable assumption when the beam-web-to-column-shear tab is connected by the means of high-strength bolts. Greater capacity is achieved when the beam web is directly welded to the column flange with a CJP weld. The section properties at the end of the beam are least, precisely an area where the moment levels are the greatest, leading to the highest level of stresses. A plot of stress distribution is shown in Figure 22.31. The weld is therefore in the area of highest stress, making it critical to the performance of the connection. Details in either EBF or SMRF structures that place this type of demand on the weld require careful scrutiny.
22.10.2
Ductile Hinges in Connections
The SMRF concept is based on the premise that plastic hinges will form in the beams, absorbing seismically induced energies by inelastically stretching and deforming the steel. The connection is not expected to break. Following the Northridge earthquake, however, there was little or no evidence of hinge formation. Instead, the connections or portions of the connection experienced brittle fracture. Most of the ductility data is obtained from smooth, slowly loaded, uniaxially loaded tensile specimens that are free to neck down. If a notch is placed in the specimen, perpendicular to the applied load, the specimen will be unable to exhibit its normal ductility, usually measured as elongation. The presence of notch-like conditions in the Northridge connections reduced the ductile behavior. In 1994, initial research on SMRF connections attempted to eliminate the issues of notch-like conditions in the test specimens by removing weld backing and weld tabs and controlling weld soundness. Even with these changes, brittle fractures occurred when the standard details were tested. The testing program then evaluated several modified details with short cover plates, with better success. The beam-to-column connection will be examined with respect to the previously outlined conditions required for ductility (see Section 22.7). Figure 22.32 shows two regions in question. Point A is at the weld joining the beam flange to the face of the column flange. Here there is restraint against strain (movement) across the width of the beam flange (ε1 ) as well as through the thickness of the beam flange (ε2 ). Point B is along the length of the beam flange away from the connecting weld. There is no restraint across the width of the flange or through its thickness. The following equations can be found in most texts concerning strength of materials:
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ε3
=
ε2
=
ε1
=
1 (σ3 − µσ2 − µσ1 ) E 1 (−µσ3 + σ2 − µσ1 ) E 1 (−µσ3 − µσ2 + σ1 ) E
(22.1a) (22.1b) (22.1c)
FIGURE 22.31: Analysis of SMRF behavior. (Courtesy of The Lincoln Electric Company. With permission.)
1999 by CRC Press LLC
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FIGURE 22.32: Regions to be analyzed relative to potential for ductile behavior. (Courtesy of The Lincoln Electric Company. With permission.)
It can be shown that: σ1
=
σ2
=
σ3
=
E [µε3 + µε2 + (1 − µ)ε1 ] (1 + µ)(1 − 2µ) E [µε3 + (1 − µ)ε2 + µε1 ] (1 + µ)(1 − 2µ) E [(1 − µ)ε3 + µε2 + µε1 ] (1 + µ)(1 − 2µ)
(22.2a) (22.2b) (22.2c)
The unit cube in Figure 22.33 is an element of the beam flange from point B in Figure 22.32. The applied force due to the moment is σ3 . Assuming strain in direction 3 to be + 0.001 in./in., and Poisson’s ratio of µ = 0.3 for steel, ε2 and ε3 can be found to be equal to −0.0003 in./in. Using these strains, from Equations 22.2a to 22.2c, it is found that σ1 σ2 σ3
= = =
0 ksi 0 ksi 30 ksi
These stresses are plotted as a dotted circle on Figure 22.34. These values are then extrapolated to the point where fracture would occur, that is, where the net tensile strength is 70 ksi. The larger solid line circle is for a stress of 70 ksi or ultimate tensile stress. The resulting maximum shear stresses, τ1−3 and τ2−3 , are the radii of these two circles, or 35 ksi. The ratio of shear to tensile stress for steel is 0.5. Figure 22.35 plots this as line B. At a yield point of 55 ksi, the critical shear value is half of this, or 27.5 ksi. When this critical shear stress is reached, plastic straining takes place and ductile behavior will result up to the ultimate tensile strength, here 70 ksi. Figure 22.38 shows a predicated stress-strain curve indicating ample ductility. Figure 22.36 shows an element from point A of Figure 22.32 at the junction of the beam and column flange. Whether weld metal or the material in the column or beam is considered makes little 1999 by CRC Press LLC
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FIGURE 22.33: Unit cube showing applied stress from Figure 22.32. (Courtesy of The Lincoln Electric Company. With permission.)
FIGURE 22.34: A plot of the tensile stress and shear stress from Figure 22.32. (Courtesy of The Lincoln Electric Company. With permission.) difference. This region is highly restrained. Suppose it is assumed: ε3 ε2 ε1
= = =
+ 0.001 in./in. (as before) 0 (since it is highly restrained 0 with little strain possible)
Then, from the given equations, the following stresses are found: σ1 σ2 σ3
= = =
17.31 ksi 17.31 ksi 40.38 ksi
The stresses are plotted as a dotted circle in Figure 22.37. 1999 by CRC Press LLC
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FIGURE 22.35: The ratio of shear to tensile stress. (Courtesy of The Lincoln Electric Company. With permission.)
FIGURE 22.36: The highly restrained region at the junction of the beam and column flange shown in Figure 22.32. (Courtesy of The Lincoln Electric Company. With permission.) If these stresses are increased to the ultimate tensile strength, it is found that σ1 σ2 σ3
= = =
30.0 ksi 30.0 ksi 70.0 ksi
The solid line circle in Figure 22.37 is a plot of stresses for this condition. The maximum shear stresses are τ1−3 = τ2−3 = 20 ksi. Since these are less than the critical shear stress (27.5 ksi), no plastic movement, or ductility, would be expected. In this case, the ratio of shear to tensile stress is 0.286. In Figure 22.35, this condition is plotted as line A. It never exceeds the value of the critical shear stress (27.5 ksi); therefore, there will be no plastic strain or movement, and it will behave as a brittle material. Figure 22.38 shows a predicated stress-strain curve going upward as a straight line (A) (elastic) until the ultimate tensile stress is 1999 by CRC Press LLC
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FIGURE 22.37: A plot of the tensile stress and shear stress from Figure 22.35. (Courtesy of The Lincoln Electric Company. With permission.)
FIGURE 22.38: Stress-strain curve. (Courtesy of The Lincoln Electric Company. With permission.)
reached in a brittle manner. It would therefore be expected that, at the column face or in the weld where high restraint exists, little ductility would result. This is where brittle fractures have occurred, both in the laboratory and in actual Northridge structures. In the SMRF system, the greatest moment (due to lateral forces) will occur at the column face. This moment must be resisted by the beam’s section properties, which because of weld access holes are lowest at the column face. Thus, the highest stresses occur at this point, the point where analysis shows ductility to be impossible. 1999 by CRC Press LLC
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In Figure 22.32, material at point B was expected to behave as shown in Figure 22.34a, and as line B in Figure 22.35, and curve B in Figure 22.38; that is, with ample ductility. Plastic hinges must be forced to occur in this region. Several post-Northridge designs have employed details that encourage use of this potential ductility. The coverplated design illustrated in Figure 22.39 accomplishes two important things: first, the stress
FIGURE 22.39: Coverplate detail takes advantage of the region where ductility is possible. (Courtesy of The Lincoln Electric Company. With permission.)
level at point A is reduced as a result of the increased cross-section at the weld. This region, incapable of ductility, must be kept below the critical tensile stress and the increase in area accomplishes this goal. Second, and most significant, the most highly stressed region is now at point B, the region of the beam that is capable of exhibiting ductility. The real success of this connection will depend upon getting the adjacent beam to plastically bend before this critical section cracks. The way in which a designer selects structural details under particular load conditions greatly influences whether the condition provides enough shear stress component so that the critical shear value may be exceeded first, producing sufficient plastic movement before the critical normal stress value is exceeded. This will result in a ductile detail and minimize the chances of cracking.
22.11
Workmanship Requirements
In welded construction, the performance of the structural system often depends on the ability of skilled welders to deposit sound weld metal. As the level of loading increases, dependence on high-quality fabrication increases. For severely loaded connections, good workmanship is a key contributor to acceptable performance. Design and fabrication specifications such as the AISC Manual of Steel Construction and the AWS D1.1 Structural Welding Code: Steel [9] communicate minimum acceptable practices. It is impossible for any code to cover every situation that will ever be contemplated. It is the responsibility of the engineer to specify any additional requirements that supersede minimum acceptable standards. The D1.1-96 code does not specifically address seismic issues, but does establish a minimum level of quality that must be achieved in seismic applications. Additional requirements are probably 1999 by CRC Press LLC
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warranted. These would include requirements for nondestructive testing, notch tough weld deposits, and additional requirements for in-process verification inspection.
22.11.1
Purpose of the Welding Procedure Specification
The welding procedure specification (WPS) is somewhat analogous to a cook’s recipe. It outlines the steps required to make a good-quality weld under specific conditions. It is the primary method to ensure the use of welding variables essential to weld quality. In addition, it permits inspectors and supervisors to verify that the actual welding is performed in conformance with the constraints of the WPS. Examples of WPSs are shown in Figures 22.40 and 22.41. WPSs typically are submitted to the inspector for review prior to the start of welding. For critical projects, the services of welding engineers may be needed. WPSs are intended to be communication tools for maintenance of weld quality. All parties involved with the fabrication sequence must have access to these documents to ensure conformance to their requirements.
22.11.2
Effect of Welding Variables
Specific welding variables that determine the quality of the deposited weld metal are a function of the particular welding process being used, but the general trends outlined below are applicable to all welding processes. Amperage is a measure of the amount of current flowing through the electrode and the work. An increase in amperage generally means higher deposition rates, deeper penetration, and more melting of base metal. The role of amperage is best understood in the context of heat input and current density, which are described below. Arc voltage is directly related to arc length. As the voltage increases, the arc length increases. Excessively high voltages may result in weld metal porosity, while extremely low voltages will produce poor weld bead shapes. In an electrical circuit, the voltage is not constant, but is composed of a series of voltage drops. Therefore, it is important to monitor voltage near the arc. Travel speed is the rate at which the electrode is moved relative to the joint. Travel speed, which has an inverse effect on the size of weld beads, is a key variable used in determining heat input. Polarity is a definition of the direction of current flow. Positive (or reverse) polarity is achieved when the electrode lead is connected to the positive terminal of the direct current power supply. The work lead would be connected to the negative terminal. Negative (or straight) polarity occurs when the electrode is connected to the negative terminal. For most welding processes, the required electrode polarity is a function of the design of the electrode. For submerged arc welding, either polarity could be utilized. Current density is determined by dividing the welding amperage by the cross-sectional area of the electrode. The current density is therefore proportional to I /d 2 . As the current density increases, both deposition rates and penetration increase. Preheat and interpass temperatures are used to control cracking tendencies, typically in the base material. Excessively high preheat and interpass temperatures will reduce the yield and tensile strength of the weld metal as well as the toughness. When base metals receive little or no preheat, the resultant rapid cooling can promote cracking as well as excessively high yield and tensile properties in the weld metal, and a corresponding reduction in toughness and elongation. The WPS defines and controls all of the preceding variables. Conformance to the WPS is particularly important in the case of seismically loaded structures, because of the high demand placed on welded connections under these situations.
1999 by CRC Press LLC
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FIGURE 22.40: Example of Welding Procedure Specification (WPS). (From American Welding Society. Structural Welding Code: Steel: ANSI/AWS D1.1-96. Miami, Florida, 1996. With permission.)
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FIGURE 22.41: Example of Welding Procedure Specification (WPS). (From American Welding Society. Structural Welding Code: Steel: ANSI/AWS D1.1-96. Miami, Florida, 1996. With permission.)
1999 by CRC Press LLC
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22.11.3
Fit-Up
The orientation of the various pieces prior to welding is known as “fit-up”. The AWS D1.1-96 code [9] contains specific tolerances that are applied to the as-fit dimensions of a joint prior to welding. There must be ample access to the root of the joint to ensure good, uniform fusion between the members being joined. Excessively small root openings or included angles in groove welds do not permit uniform fusion. Excessively large root openings or included angles result in the need for greater volumes of weld metal, with corresponding increases in shrinkage stresses, which in turn increases distortion and cracking tendencies. The D1.1-96 tolerances for fit-up are generally measured in 1/16-in. increments.
22.11.4
Field vs. Shop Welding
Many believe that the highest quality welding is obtained under shop welding conditions. The greatest differences between field and shop welding are related to control. For shop fabrication, the work force is generally more stable. Supervision practices are well understood and communication is generally more efficient. Under field welding conditions, control of a project seems to be more difficult. While there are environmental challenges to field conditions, including temperature, wind, and moisture, these seem to pose fewer problems than do the management issues. For field welding, the gasless welding processes such as self-shielded flux cored welding and shielded metal arc welding usually are preferred. Gas metal arc, gas tungsten arc, and gas-shielded flux cored arc welding are all limited due to their sensitivity to wind-related gas disturbances. It is imperative that field welding conditions receive an appropriate increase in the monitoring and control area to ensure consistent quality. D1.1-96 imposes the same requirements on field welding as on shop welding. This includes qualification of welders, the use of welding procedures, and the resultant quality requirements.
22.12
Inspection
The AWS D1.1-96 code requires that all welds be inspected, specifically by means of visual inspection. In addition, at the engineer’s discretion and as identified in contract documents, nondestructive testing may be required for finished weldments. This enables the engineer with a knowledge of the complexity of the project to specify additional inspection methodologies commensurate with the degree of confidence required for a particular project. In the case of seismically loaded structures, and connections subject to high stress levels, the need for inspection increases.
22.12.1
In-Process Visual Inspection
D1.1-96 mandates the use of in-process visual inspection. Before welding, the inspector reviews welder qualification records, welding procedure specifications, and the contract documents to confirm that applicable requirements are met. Before welding is performed, the inspector verifies fit-up and joint cleanliness, examines the welding equipment to ensure it is in proper working order, verifies that the materials involved meet the various requirements, and confirms that the required levels of preheat have been properly applied. During welding, the inspector confirms that the WPS is being carried out and that the intermediate weld passes meet the various requirements. After welding is finished, final bead shapes and welding integrity can be visually confirmed. Effective visual inspection is a critical component for ensuring consistent weld quality.
1999 by CRC Press LLC
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22.12.2
Nondestructive Testing
Four major nondestructive testing methods may be used to verify weld integrity after welding operations are completed. Each should be used in conjunction with effective visual inspection. No process is 100% capable of detecting all discontinuities in a weld. Dye penetrant (PT) inspection involves the application of a liquid that is drawn into a surfacebreaking discontinuity, such as a crack or porosity, by capillary action. When the excess residual dye is removed from the surface, a developer is applied that will absorb the penetrant contained within the discontinuity. The result is a stain in the developer that shows that a discontinuity is present. PT testing is limited to surface-breaking discontinuities. It cannot read subsurface discontinuities, but it is highly effective in accenting very small discontinuities. Magnetic particle (MT) inspection utilizes the change in magnetic flux that occurs when a magnetic field is present in the vicinity of a discontinuity. The change will show up as a different pattern when magnetic dust-like particles are applied to the surface of the part. The process is highly effective in locating discontinuities that are on the surface or slightly subsurface. The magnetic field can be created in the material in one of two ways: the current is directly passed through the material or the magnetic field is induced through a coil on a yoke. Since the process is most sensitive to discontinuities that lie perpendicular to the magnetic flux path, it is necessary to energize the part in two directions in order to fully inspect the component. Radiographic (RT) inspection uses X-rays or gamma rays that are passed through the weld to expose a photographic film on the opposite side of the joint. High-voltage generators produce X-rays, while gamma rays are created by atomic disintegration of radioisotopes. Whenever radiographic inspection is employed, workers must be protected from exposure to excessive radiation. RT relies on the ability of the material to pass some of the radiation through, while absorbing part of this energy within the material. Different materials have different absorption rates. As the different levels of radiation are passed through the material, portions of the film are exposed to a greater or lesser degree. When this film is developed, the resulting radiograph will bear the image of the cross-section of the part. The radiograph is actually a negative. The darkest regions are those that were most exposed when the material being inspected absorbed the least amount of radiation. Porosity will show up as small dark round circles. Slag is generally dark and will look similar to porosity, but will have irregular shapes. Cracks appear as dark lines. Excessive reinforcement will result in a light region. A radiographic test is most effective for detecting volumetric discontinuities such as slag or porosity. When cracks are oriented perpendicular to the direction of a radiographic source, they may be missed with the RT method. Therefore, RT inspection is most appropriate for butt joints and is generally not appropriate for inspection of corner or T joints. Radiographic testing has the advantage of generating a permanent record for future reference. In ultrasonic (UT) inspection, solid discontinuity-free materials will transmit high-frequency sound waves throughout the part in an uninterrupted manner. A receiver “hears” the sound reflected off of the back surface of the part being inspected. If there is a discontinuity between the transmitter and the back of the part, an intermediate signal will be sent to the receiver indicating its presence. The pulses are read on a CRT screen. The magnitude of the signal received from the discontinuity indicates its size. UT is most sensitive to planar discontinuities, i.e., cracks. UT effectiveness is dependent on the operator’s skill, so UT technician training and certification is essential. With currently available technology, UT is capable of reading a variety of discontinuities that would be acceptable for many applications. Acceptance criteria must be clearly communicated to the inspection technicians so unnecessary repairs are avoided.
1999 by CRC Press LLC
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22.12.3
Applications for Nondestructive Testing Methods
Visual inspection is the most comprehensive method available to verify conformance with the wide variety of issues that can affect weld quality and should be thoroughly applied on every welding project. To augment visual inspection, nondestructive testing can be specified to verify the integrity of the deposited weld metal. The selection of the inspection method should reflect the probable discontinuities that would be encountered, and the consequences of undetected discontinuities. Consideration must be given to the conditions under which the inspection would be performed, such as field vs. shop conditions. The nature of the joint detail (butt, T, corner, etc.) and the weld type (CJP, PJP, fillet weld) will determine the choice of the inspection process in many situations. MT inspection is usually preferred over PT inspection because of its relative simplicity. Cleanup is easy, and the process is sensitive. PT is normally reserved for applications where the material is nonmagnetic, and MT would not be applicable. While MT is suitable for detection of surface or slightly subsurface discontinuities only, it is in these areas that many welding defects are located. It is very effective in crack detection, and can be utilized to ensure complete crack removal before subsequent welding is performed on damaged structures. UT inspection has become the primary nondestructive testing method used for most building applications. It can be utilized to inspect butt, T, and corner joints, is relatively portable, and is free from the radiation concerns associated with RT inspection. UT is especially sensitive to the identification of cracks, the most significant defect in a structural system. Although it may not detect spherical or cylindrical voids such as porosity, nondetection of these types of discontinuities has fewer consequences.
22.13
Post-Northridge Assessment
Prior to the Northridge earthquake, the SMRF with the “pre-Northridge” beam-to-column detail was unchallenged regarding its ability to perform as expected. This confidence existed in spite of a fairly significant failure rate when these connections had been tested in previous research. The pre-Northridge detail consisted of the following: • CJP groove welds of the beam flanges to the column face, with weld backing and weld tabs left in place. • No specific requirement for minimum notch toughness properties in the weld deposit. • A bolted web connection with or without supplemental fillet welds of the shear tab to the beam web. • Standard ASTM A36 steel for the beam and ASTM 572 grade 50 for the column (i.e., no specific limits on yield strength or the Fy /Fu ratio). As a result of the Northridge earthquake, and research performed immediately afterward, confidence in this detail has been severely shaken. Whether any variation of this detail will be suitable for use in the future is currently unknown. More research must be done, but one can speculate that, with the possible exception of small-sized members, some modification of this detail will be required. Although testing of this configuration had a fairly high failure rate in pre-Northridge tests, many successful results were obtained. Further research will determine which variables are the most significant in predicting performance success. Some changes in materials and design practice also should be considered. In recent years, recycling of steel has become a more predominant method of manufacture. This is not only environmentally responsible, it is economical. However, in recycling, residual alloys can accumulate in the scrap charge, inadvertently increasing steel strength levels. In the past 20 years, the average yield strength of ASTM A36 steel has increased approximately 15%. Testing done with lower yield strength steel would be expected to exhibit different behavior than 1999 by CRC Press LLC
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test specimens made of today’s higher strength steels (in spite of the same ASTM designation). For practical reasons, laboratory specimens tend to be small in size. Success in small-sized specimens was extrapolated to apply to very large connection assemblies in actual structures. The design philosophy that led to fewer SMRFs throughout a structure required that each of the remaining frames be larger in size. This corresponded to heavier and deeper beams, and much heavier columns, with an increase in the size of the weld between the two rolled members. The effect of size on restraint and triaxial stresses was not researched, resulting in some new discoveries about the behavior of the large-sized assemblages during the Northridge earthquake. The engineering community generally agrees that the pre-Northridge connection (as defined above) is no longer adequate and some modification will be required. Any deviation from the definition above will constitute a modification for the purposes of this discussion.
22.13.1
Minor Modifications to the SMRF Connection
With the benefit of hindsight, several aspects of the pre-Northridge connection detail appear to be obviously deficient. Weld backing left in place in a connection subject to both positive and negative moments where the root of the flange weld can be put into tension creates high-stress concentrations that may result in cracking. Failure to specify minimum toughness levels for weld metal for heavily loaded connections is another deficiency. The superior performance of the welded web vs. the bolted web in past testing draws into question the assumption of load sharing between welds and bolts. Now it seems that tighter control of the strength properties of the beam steel and the relationship to the column is essential. Some preliminary tests suggest that tightly controlling all of these variables may result in acceptable performance. However, the authors know of no test of unmodified beam-to-column connections where the connection zone has remained crack free when acceptable rotation limits were achieved. For smaller sized members, this approach may be technically possible, although the degree of control necessary on both the material properties and the welding operations may make it impractical.
22.13.2
Coverplated Designs
This concept uses short coverplates that are added to the top and bottom flanges of the beam. Fillet welds transfer the coverplate forces to the beam flanges. The bottom flange coverplate is shop welded to the column flange, and the bottom beam flange is field welded to the column flange and to the coverplate. Both the top flange and the top flange coverplate are field welded to the column flange with a common weld. The web connection may be welded or high-strength bolted. These connections have been tested to a limited extent, with generally favorable results. Following Northridge, the coverplate approach received significant attention because it offered early promise of a viable solution. Other methods may prove to be superior as time passes. While the coverplate solution treats the beam in the same way as other approaches (i.e., it moves the plastic hinge into a region where ductility can be demonstrated), it concentrates all the loading to the column into a relatively short distance. Other alternatives may treat the column in a more gentle manner.
22.13.3
Flange Rib Connections
This concept utilizes one or two vertical ribs attached between the beam flanges and column face. The intent of the rib plates is to reduce the demand on the weld at the column flange and to shift the plastic hinge from the column face. In limited testing, flange rib connections have demonstrated acceptable levels of plastic rotation provided that the girder flange welding is done correctly. Vertical ribs appear to function very similarly to the coverplated designs, but offer the additional advantage of spreading the load over a greater portion of the column. The single-rib designs appear to 1999 by CRC Press LLC
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be better than the twin-rib approaches because the stiffening device is in alignment with the column web (for I-shaped columns) and facilitates easy access to either side of the device for welding. It is doubtful that the rib design would be appropriate for box column applications.
22.13.4
Top and Bottom Haunch Connections
In this configuration, haunches are placed on both the top and bottom flanges. In two tests of the top and bottom haunch connection, it has exhibited extremely ductile behavior, achieving plastic rotations as great as 0.07 rad. Tests of single, haunched beam–column connections have not been as conclusive; further tests are planned. Although they are costly, haunches appear to be the most straightforward approach to obtaining the desired behavior out of the connection. The treatment to the column is particularly desirable, greatly increasing the portion of the column participating in the transfer of moment.
22.13.5
Reduced Beam Section Connections
In this configuration, the cross-section of the beam is deliberately reduced within a segment to produce a plastic hinge within the beam span, away from the column face. A variant of this approach produces the so-called “dog bone” profile. Reduced section details offer the prospect of a low-cost connection and increased performance out of detailing that is very similar to the pre-Northridge connection. Control of material properties of the beam will still be a major variable if this detail is used. Lateral bracing will probably be required in the area of the reduced section to prevent buckling, particularly at the bottom flange when loaded in compression.
22.13.6
Partially Restrained Connections
Some have suggested that partially restrained (PR) connection details will offer a performance advantage over the SMRF. The relative merits of a PR frame vs. a rigid frame are beyond the scope of this work. However, many engineers immediately think of bolted PR connections when it is possible to utilize welded connections for PR performance as well. Illustrated in Figure 22.42 is a detail that can be employed utilizing the PR concept. Detailing rules must be developed, and tests done, before these details are employed. They are supplied to offer welded alternatives to bolted PR connections.
22.14
Defining Terms
As-welded: The condition of weld metal, weld joints, and weldments after welding, but prior to any subsequent thermal, mechanical, or chemical treatments. Autogenous weld: A fusion weld made without the addition of filler metal. Back gouging: The removal of weld metal and base metal from the other side of a partially welded joint to facilitate complete fusion and complete joint penetration upon subsequent welding from that side. Backing: A material or device placed against the back side of the joint, or at both sides of a weld in electroslag and electrogas welding, to support and retain molten weld metal. The material may be partially fused or remain unfused during welding and may be either metal or nonmetal. Base metal: The material to be welded, brazed, soldered, or cut. 1999 by CRC Press LLC
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FIGURE 22.42: A partially restrained welded connection detail. (Courtesy of The Lincoln Electric Company. With permission.)
Deposition rate: The weight of material deposited in a unit of time. Effective throat: The minimum distance minus any convexity between the weld root and the face of a fillet weld. Filler metal: The metal to be added in making a welded, brazed, or soldered joint. Heat affected zone (HAZ): That portion of the base metal that has not been melted, but whose mechanical properties or microstructure have been altered by the heat of welding, brazing, soldering, or cutting. Nugget: The weld metal joining the workpieces in spot, roll spot, seam, or projection welds. Postheating: The application of heat to an assembly after welding, brazing, soldering, thermal spraying, or thermal cutting. Preheating: The application of heat to the base metal immediately before welding, brazing, soldering, thermal spraying, or cutting. Residual stress: Stress present in a member that is free of external forces or thermal gradients. Theoretical weld throat: The distance from the beginning of the joint root perpendicular to the hypotenuse of the largest right triangle that can be inscribed within the cross-section of a fillet weld. This dimension is based on the assumption that the root opening is equal to zero. Weldability: The capacity of material to be welded under the imposed fabrication conditions into a specific, suitable designed structure and to perform satisfactorily in the intended service. Weldment: An assembly whose component parts are joined by welding. Weld metal: That portion of a weld that has been melted during welding. Weld pool: The localized volume of molten metal in a weld prior to its solidification as weld metal.
1999 by CRC Press LLC
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References [1] Alexander, W.G. 1991. Designing Longitudinal Welds for Bridge Members, Eng. J., 28(1), 29-36. [2] American Association of State Highway and Transportation Officials and American Welding Society. 1995. Bridge Welding Code: ANSI/AASHTO/AWS D1.5-95. Miami, FL. [3] American Institute of Steel Construction. 1994. Manual of Steel Construction: Load & Resistance Factor Design, 2nd ed., Chicago, IL. [4] American Society for Metals. 1983. Metals Handbook, Ninth Edition, Volume 6: Welding, Brazing, and Soldering, Metals Park, OH. [5] American Welding Society. 1988. Guide for the Visual Inspection of Welds: ANSI/AWS B1.1188, Miami, FL. [6] American Welding Society. 1991. Standard Symbols for Welding, Brazing and Nondestructive Examination: ANSI/AWS A2.4-93, Miami, FL. [7] American Welding Society. 1989. Standard Welding Terms and Definitions: ANSI/AWS A3.089, Miami, FL. [8] American Welding Society. 1989. Structural Welding Code: Sheet Steel: ANSI/AWS D1.3-89, Miami, FL. [9] American Welding Society. 1996. Structural Welding Code: Steel: ANSI/AWS D1.1-96, Miami, FL. [10] American Welding Society. 1995. Structural Welding Committee Position Statement on Northridge Earthquake Welding Issues, Miami, FL. [11] American Welding Society. 1987. Welding Handbook, Eighth Edition, Volume 1: Welding Technology, L.P. Conner, Ed., Miami, FL. [12] American Welding Society. 1991. Welding Handbook, Eighth Edition, Volume 2: Welding Processes, R.L. O’Brien, Ed., Miami, FL. [13] American Welding Society. 1996. Welding Handbook, Eighth Edition, Volume 3: Materials and Applications, Part 1, W.R. Oates, Ed., Miami, FL. [14] American Welding Society. 1976. Welding Handbook: Volume One, Seventh Edition: Fundamentals of Welding, C. Weisman, Ed., Miami, FL. [15] American Welding Society. 1978. Welding Handbook: Volume Two, Seventh Edition: Welding Processes—Arc and Gas Welding and Cutting, Brazing, and Soldering, W.H. Kearns, Ed., Miami, FL. [16] American Welding Society. 1980. Welding Handbook: Volume Three, Seventh Edition: Welding Processes—Resistance and Solid-State Welding and Other Joining Processes, W.H. Kearns, Ed., Miami, FL. [17] American Welding Society. 1982. Welding Handbook: Volume Four, Seventh Edition: Metals and Their Weldability, W.H. Kearns, Ed., Miami, FL. [18] American Welding Society. 1984. Welding Handbook: Volume Five, Seventh Edition: Engineering, Costs, Quality and Safety, W.H. Kearns, Ed., Miami, FL. [19] Barsom, J.M. and Rolfe, S.T. 1987. Fracture and Fatigue Control in Structures: Applications of Fracture Mechanics, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ. [20] Blodgett, O.W. 1966. Design of Welded Structures, The James F. Lincoln Arc Welding Foundation, Cleveland, OH. [21] Blodgett, O.W. 1995. Details to Increase Ductility in SMRF Connections, The Welding Innovation Quarterly, XII(2). [22] Blodgett, O.W. 1993. The Challenge of Welding Jumbo Shapes, Part II: Increasing Ductility of Connections, The Welding Innovation Quarterly, X(1). [23] Lincoln Electric Company. 1995. The Procedure Handbook of Arc Welding, 13th ed., Cleveland, OH. 1999 by CRC Press LLC
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[24] Miller, D.K. 1996. Ensuring Weld Quality in Structural Applications, Part I: The Roles of Engineers, Fabricators & Inspectors, The Welding Innovation Quarterly, XIII(2). [25] Miller, D.K. 1996. Ensuring Weld Quality in Structural Applications, Part II: Effective Visual Inspection, The Welding Innovation Quarterly, XIII(3). [26] Miller, D.K. 1994. Northridge: The Role of Welding Clarified, The Welding Innovation Quarterly, XI(2). [27] Miller, D.K. 1996. Northridge: An Update, The Welding Innovation Quarterly, XIII(1). [28] Miller, D.K. 1994. Welding of Steel Bridges, The James F. Lincoln Arc Welding Foundation, Cleveland, OH. [29] Miller, D.K. 1988. What Structural Engineers Need to Know About Weld Metal, 1988 National Steel Construction Conference Proceedings: 35.1-35.15. American Institute of Steel Construction, Chicago, IL.
Further Reading [1] American Association of State Highway and Transportation Officials. 1978. Guide Specifications for Fracture Critical Non-Redundant Steel Bridge Members, 1978. (As revised by Interim Specifications for Bridges, 1981, 1983, 1984, 1985, 1986, and 1991.) Washington, D.C. [2] American Institute of Steel Construction. 1994. Northridge Steel Update I, Chicago, IL. [3] American Society for Metals. 1994. Hydrogen Embrittlement and Stress Corrosion Cracking, R. Gibala and R.F. Hehemann, Eds., Metals Park, OH. [4] American Welding Society. various dates. A5 Filler Metal Specifications, Miami, FL. [5] American Welding Society. 1993. Standard Methods for Determination of the Diffusible Hydro-
[6] [7] [8] [9] [10]
[11]
[12] [13]
gen Content of Martensitic, Bainitic, and Ferritic Steel Weld Metal Produced by Arc Welding: ANSI/AWS A4.3-93, Miami, FL. Bailey, N., Coe, F.R., Gooch, T.G., Hart, P.H.M., Jenkins, N., and Pargeter, R.J. 1973. Welding Steels Without Hydrogen Cracking, 2nd ed., Abington Publishing, Cambridge, England. Boniszewski, T. 1992. Self-Shielded Arc Welding. Abington Press, Cambridge, England. Masubuchi, K. 1980. Analysis of Welded Structures, 1st ed. Pergamon Press, Oxford, England. Roeder, C.W. 1985. Use of Thermal Stresses for Repair of Seismic Damage to Steel Structures. University of Washington, Seattle, WA. SAC Joint Venture. 1995. Interim Guidelines: Evaluation, Repair, Modification and Design of Welded Steel Moment Frame Structures, Report No. SAC-95-02 (FEMA 267), Sacramento, CA. Shanafelt, G.O. and Horn, W.B. 1984. Guidelines for Evaluation and Repair of Damaged Steel Bridge Members, National Cooperative Highway Research Program Report 271, Transportation Research Board, Washington, D.C. Stout, R.D. 1987. Weldability of Steels, 4th ed., Welding Research Council, New York. Wilson, A.D. 1990. Hardness Testing of Thermal Cut Edges of Steel, Eng. J., 27(3), 98-105.
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Leon, R. “Composite Connections” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Composite Connections
Roberto Leon School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA
23.1
23.1 Introduction 23.2 Connection Behavior Classification 23.3 PR Composite Connections 23.4 Moment-Rotation (M-θ ) Curves 23.5 Design of Composite Connections in Braced Frames 23.6 Design for Unbraced Frames References
Introduction
The vast majority of steel buildings built today incorporate a floor system consisting of composite beams, composite joists or trusses, stub girders, or some combination thereof [29]. Traditionally the strength and stiffness of the floor slabs have only been used for the design of simply-supported flexural members under gravity loads, i.e., for members bent in single curvature about the strong axis of the section. In this case the members are assumed to be pin-ended, the cross-section is assumed to be prismatic, and the effective width of the slab is approximated by simple rules. These assumptions allow for a member-by-member design procedure and considerably simplify the checks needed for strength and serviceability limit states. Although most structural engineers recognize that there is some degree of continuity in the floor system because of the presence of reinforcement to control crack widths over column lines, this effect is considered difficult to quantify and thus ignored in design. The effect of the floor slabs has also been neglected when assessing the strength and stiffness of frames subjected to lateral loads for four principal reasons. First, it has been assumed that neglecting the additional strength and stiffness provided by the floor slabs always results in a conservative design. Second, a sound methodology for determining the M-θ curves for these connections is a prerequisite if their effect is going to be incorporated into the analysis. However, there is scant data available in order to formulate reliable moment-rotation (M-θ ) curves for composite connections, which fall typically into the partially restrained (PR) and partial strength (PS) category. Third, it is difficult to incorporate into the analysis the non-prismatic composite cross-section that results when the member is subjected to double curvature as would occur under lateral loads. Finally, the degree of composite interaction in floor members that are part of lateral-load resisting systems in seismic areas is low, with most having only enough shear transfer capacity to satisfy diaphragm action. Research during the past 10 years [25] and damage to steel frames during recent earthquakes [22] have pointed out, however, that there is a need to reevaluate the effect of composite action in modern frames. The latter are characterized by the use of few bents to resist lateral loads, with the ratio of number of gravity to moment-resisting columns often as high as 6 or more. In these cases the 1999 by CRC Press LLC
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aggregate effect of many PR/PS connections can often add up to a significant portion of the lateral resistance of a frame. For example, many connections that were considered as pins in the analysis (i.e., connections to columns in the gravity load system) provided considerable lateral strength and stiffness to steel moment-resisting frames (MRFs) damaged during the Northridge earthquake. In these cases many of the fully restrained (FR) welded connections failed early in the load history, but the frames generally performed well. It has been speculated that the reason for the satisfactory performance was that the numerous PR/PS connections in the gravity load system were able to provide the required resistance since the input base shear decreased as the structure softened. In these PR/PS connections, much of the additional capacity arises from the presence of the floor slab which provides a moment transfer mechanism not accounted for in design. In this chapter general design considerations for a particular type of composite PR/PS connection will be given and illustrated with examples for connections in braced and unbraced frames. Information on design of other types of bolted and composite PR connections is given elsewhere [22], (Chapter 6 of [29]). The chapter begins with discussions of both the development of M-θ curves and the effect of PR connections on frame analysis and design. A clear understanding of these two topics is essential to the implementation of the design provisions that have been proposed for this type of construction [26] and which will be illustrated herein.
23.2
Connection Behavior Classification
The first step in the design of a building frame, after the general topology, the external loads, the materials, and preliminary sizes have been selected, is to carry out an analysis to determine member forces and displacements. The results of this analysis depend strongly on the assumptions made in constructing the structural model. Until recently most computer programs available to practicing engineers provided only two choices (rigid or pinned) for defining the connections stiffness. In reality connections are very complex structural elements and their behavior is best characterized by M-θ curves such as those given in Figure 23.1 for typical steel connections to an A36 W24x55 beam (Mp,beam = 4824 kip-in.). In Figure 23.1, Mconn corresponds to the moment at the column face, while θconn corresponds to the total rotation of the connection and a portion of the beam generally taken as equal to the beam depth. These curves are shown for illustrative purposes only, so that the different connection types can be contrasted. For each of the connection types shown, the curves can be shifted through a wide range by changing the connection details, i.e., the thickness of the angles in the top and seat angle case. While the M-θ curves are highly non-linear, at least three key properties for design can be obtained from such data. Figure 23.2 illustrates the following properties, as well as other relevant connection characteristics, for a composite connection: 1. Initial stiffness (kser ), which will be used in calculating deflection and vibration performance under service loads. In these analysis the connection will be represented by a linear rotational spring. Since the curves are non-linear from the beginning, and kser will be assumed constant, the latter needs to be defined as the secant stiffness to some predetermined rotation. 2. Ultimate strength (Mu,conn ), which will be used in assessing the ultimate strength of the frame. The strength is controlled either by the strength of the connection itself or that of the framing beam. In the former case the connection is defined as partial strength (PS) and in the latter as full strength (FS). 3. Maximum available rotation (θu ), which will be used in checking both the redistribution capacity under factored gravity loads and the drift under earthquake loads. The 1999 by CRC Press LLC
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FIGURE 23.1: Typical moment-rotation curves for steel connections.
FIGURE 23.2: Definition of connection properties for PR connections.
required rotational capacity depends on the design assumptions and the redundancy of the structure. It is often useful also to define a fourth quantity, the ductility (µ) of the connection. This is defined as the ratio of the ultimate rotation capacity (θu ) to some nominal “yield” rotation (θy ). It should be understood that the definition of θy is subjective and needs to account for the shape of the curve (i.e., how sharp is the transition from the service to the yield level — the sharper the transition the more valid the definition shown in Figure 23.2). In the design procedure to be discussed in this chapter, the initial stiffness, ultimate strength, maximum rotation, and ductility are properties that will need to be check by the structural engineer. 1999 by CRC Press LLC
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Figure 23.2 schematically shows that there can be a considerable range of strength and stiffness for these connections. The range depends on the specific details of the connection, as well as the normal variability expected in materials and construction practices. Figure 23.2 also shows that certain ranges of initial stiffness can be used to categorize the initial connection stiffness as either fully restrained (FR), partially restrained (PR), or simple. Because the connection behavior is strongly influenced by the strength and stiffness of the framing members, it is best to non-dimensionalize M-θ curves as shown in Figure 23.3.
FIGURE 23.3: Normalized moment-rotation curves and connection classification. (After Eurocode 3, Design of Steel Structures, Part 1: General Rules and Rules for Buildings, ENV 1993-1-1: 1992, Comite Europeen de Normalisation (CEN), Brussels, 1992.) In Figure 23.3, the vertical axis represents the ratio (m) of the moment capacity of the connection (Mu,conn ) to the nominal plastic moment capacity (Mp,beam = Zx Fy ) of the steel beam framing into it. As noted above, if this ratio is less than one then the connection is considered partial strength (PS); if it is equal or greater than one, then it is classified as a full strength (FS) connection. The horizontal axis is normalized to the end rotation of the framing beam assuming simple supports at the beam ends (θss ). This rotation depends, of course, on the loading configuration and the level of loading. Generally a factored distributed gravity load (wu ) and linear elastic behavior up to the full plastic capacity are assumed (θss = wu Lbeam3 /24EIbeam ). The resulting reference rotation (φ = Mp L/EI ), based on a Mp of wu L2 /8, is Mp L/(3EI ) = φ/3. It should be noted that the connection rotation is normalized with respect to the properties of the beam and not the column and that this normalization is meaningful only in the context of gravity loads. The column is assumed to be continuous and part of a strong column–weak beam system. For gravity loads its stiffness and strength are considered to contribute little to the connection behavior. This assumption, of course, does not account for panel zone flexibility which is important in many types of FS connections. The non-dimensional format of Figure 23.3 is important because the terms partially restrained (PR) and full restraint (FR) can only be defined with respect to the stiffness of the framing members. Thus, a FR connection is defined as one in which the ratio (α) of the connection stiffness (kser ) to the stiffness of the framing beam (EIbeam /Lbeam ) is greater than some value. For unbraced frames the recommended value ranges from 18 to 25, while for braced frames they range from 8 to 12. Figure 23.3 shows the limits chosen by the Eurocode, which are 25 for the unbraced case and 8 for 1999 by CRC Press LLC
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the braced case [15]. These ranges have been selected based on stability studies that indicate that the global buckling load of a frame with PR connections with stiffnesses above these limits is decreased by less than 5% over the case of a similar frame with rigid connections. The large difference between the braced and unbraced values stems from the P-1 and P-δ effects on the latter. PR connections are defined as those having α ranging from about 2 up to the FR limit. Connections with α less than 2 are regarded as pinned.
23.3
PR Composite Connections
Conventional steel design in the U.S. separates the design of the gravity and lateral load resisting systems. For gravity loads the floor beams are assumed to be simply supported and their section properties are based on assumed effective widths for the slab (AISC Specification I3.1 [2]) and a simplified definition of the degree of interaction (Lower Bound Moment of Inertia, Part 5 [3]). The simple supports generally represent double angle connections or single plate shear connections to the column flange. For typical floor beam sizes, these connections, tested without slabs, have shown low initial stiffness (α < 4) and moment capacity (Mu,conn < 0.1Mp,beam ) such that their effect on frame strength and stiffness can be characterized as negligible. In reality when live loads are applied, the floor slab will contribute to the force transfer at the connection if any slab reinforcement is present around the column. This reinforcement is often specified to control crack widths over the floor girders and column lines and to provide structural integrity. This results in a weak composite connection as shown in Figure 23.4. The effect of a weak PR composite connection on the behavior under gravity loads is shown in Example 23.1.
FIGURE 23.4: Weak PR composite connection.
EXAMPLE 23.1: Effect of a Weak Composite Connection
Consider the design of a simply-supported composite beam for a DL = 100 psf and a LL = 80 psf. The span is 30 ft and the tributary width is 10 ft. For this case the factored design moment (Mu ) is 3348 kip-in. and the required nominal moment (Mn ) is 3720 kip-in. From the AISC LRFD 1999 by CRC Press LLC
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Manual [3] one can select an A36 W18x35 composite beam with 92% interaction (PNA = 3, φMp = 3720 kip-in., and ILB = 1240 in.4 ). The W18x35 was selected based on optimizing the section for the construction loads, including a construction LL allowance of 20 psf. The deflection under the full live load for this beam is 0.4 in., well below the 1 in. allowed by the L/360 criterion. Thus, this section looks fine until one starts to check stresses. If we assume that all the dead load stresses from 1.2DL, which are likely to be present after the construction period, are carried by the steel beam alone, then: σDL,steel alone = MDL /Sx = 1620 kip-in. /57.6 in.3 = 28.1 ksi The stresses from live loads are then superimposed, but on the composite section. For this section Seff = 91.9 in.3 , so the additional stress due to the arbitrary point-in-time (APT) live load (0.5LL) is: σLL(AP T ) = MLL(AP T ) /Seff = 540 kip-in./91.9 in.3 = 5.9 ksi Thus, the total stress (σAP T l ) under the APT live load is: σ AP T = σDL,steel alone + σLL(AP T ) = 28.1 + 5.9 = 34.0 ksi Under the full live load (1.0LL), the stresses are: σAP T = σDL,steel alone + 2σLL(AP T ) = 28.1 + 11.8 = 39.9 ksi > Fy = 36 ksi Thus, the beam has yielded under the full live loads even though the deflection check seemed to imply that there were no problems at this level. The current LRFD provisions do not include this check, which can govern often if the steel section is optimized for the construction loads. Let us investigate next what the effect of a weak PR connection, similar to that shown in Figure 23.3, will be on the service performance of this beam. Assume that the beam frames into a column with double web angles connection and that four #3 Grade 60 bars have been specified on the slab to control cracking. These bars are located close enough to the column so that they can be considered part of the section under negative moment. The connection will be studied using the very simple model shown in Figure 23.5. In this model all deformations are assumed to be concentrated in an area very close to the connection, with the beam and column behaving as rigid bodies. The reinforcing bars are treated as a single spring (Kbars ) while the contribution to the bending stiffness of the web angles (Kshear ) is ignored. The connection is assumed to rotate about a point about 2/3 of the depth of the beam. Assuming that the angles and bolts can carry a combination of compression and shear forces without failing, at ultimate the yielding of the slab reinforcement will provide a tensile force (T) equal to: T = 4 bars ∗ 0.11 in.2 / bar ∗ 60 ksi = 26.4 kips This force acts an eccentricity (e) of at least: e = two-thirds of the beam depth + deck rib height = 12in. + 3 in. = 15 in. This results in a moment capacity for the connection (Mu,conn ) equal to: Mu,conn = T ∗ e = 26.4 ∗ 15 = 396 kip-in. The capacity of the beam (Mp,beam ) is: Mp,beam = Zx ∗ Fy = 66.5 in.3 ∗ 36 ksi = 2394 kip-in. Thus, the ratio (m) of the connection capacity to the steel beam capacity is: m = 396/2394 ∗ 100 ≈ 17% 1999 by CRC Press LLC
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FIGURE 23.5: Simple mechanistic connection model. If we assume that (1) the bars yield and transfer most of their force over a development length of 24 bar diameters from the point of inflection, (2) the strain varies linearly, and (3) the connection region extends for a length equal to the beam depth (18 in.), then the slab reinforcement can be modeled by a spring (Kbars ) equal to: Kbars = EA/L = 30,000 ksi ∗ 0.44 in.2 /(18 in.) = 733.3 kips/in. Yield will be achieved at a rotation (θy ) equal to: θy
= (T / (Kbars ∗ e)) = 26.4 kips/ 733 kips/in. × 15 in. = 0.0024 radians or 2.4 milliradians
The connection stiffness (Kser ) can be approximated as: Kser = Mu,conn /θy = 396 kip-in. /0.0024 radians = 165,000 kip-in./radian Assuming that the beam spans 30 ft, the beam stiffness is: Kbeam = EIbeam /Lbeam = 30,000 ksi ∗ 510 in.4 /360 in. = 42,500 kip-in./radian Thus, the ratio of connection to beam stiffness (α) is: α = Kser /Kbeam = 165,000/42,500 = 3.9 The relatively low values of α and m obtained for this connection, even assuming the non-composite properties in order to maximize α and m, would seem to indicate that this connection will have little effect on the behavior of the floor system. This is incorrect for two reasons. First, the rotations (0.0024 radian) at which the connection strength is achieved are within the service range, and thus much of the connection strength is activated earlier than for a steel connection. Second, the composite connections only work for live loads and thus provide substantial reserve capacity to the system. The moments at the supports (MP R conn ) due to the presence of these weak connections for the case of a uniformly distributed load (w) are: MP R conn = wL2 /12 ∗ 1/ (1 + 2/α) = wL2 /18.2 1999 by CRC Press LLC
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For the case of w being the APT live load, the moment is 238 kip-in., while for the case of the full live load it is 476 kip-in. This reduces the moments at the centerline from 540 kip-in. to 302 kip-in. for the APT live load and from 1080 kip-in. to 604 kip-in. for the full live load. The maximum additional stress is 6.6 ksi under full LL loads, so no yielding will occur. Thus, if a significant portion of the beam’s capacity has been used up by the dead loads, a weak composite connection can prevent excessive deflections at the service level. The connection illustrated in Figure 23.4 is one of the weakest variations possible when activating composite action. Figures 23.6 through 23.8 show three other variations, one with a seat angle, one with an end plate (partial or full), and one with a welded plate as the bottom connection. As compared with the simple connection in Figure 23.4, both the moment capacity and the initial stiffness of these latter connections can be increased by more slab steel, thicker web angles or end plates, and friction bolts in the seat and web connections. The selection of a bolted seat angle, end plate, or welded plate will depend on the amount of force that the designer wants to transfer at the connection and on local construction practices.
FIGURE 23.6: Seat angle composite connection.
FIGURE 23.7: End plate composite connection.
1999 by CRC Press LLC
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FIGURE 23.8: Welded bottom plate composite connection.
The behavior of these connections under gravity loads (negative moments) should be governed by gradual yielding of the reinforcing bars, and not by some brittle or semi-ductile failure mode. Examples of these latter modes are shear of the bolts and local buckling of the bottom beam flange. Both modes of failure are difficult to eliminate at large deformations due to the strength increases resulting from strain hardening of the connecting elements. The design procedures to be proposed here for composite PR connections intend to insure very ductile behavior of the connection to allow redistribution of forces and deformations consistent with a plastic design approach. Therefore, the intent in design will be to delay but not eliminate all brittle and semi-brittle modes of failure through a capacity design philosophy [22]. For the connections shown in Figures 23.6 through 23.8, if the force in the slab steel at yielding is moderate, it is likely that the bolts in a seat angle or a partial end plate will be able to handle the shear transfer between the column and the beam flanges. If the forces are high, an oversized plate with fillet welds can be used to transfer these forces. The connections in Figures 23.6 and 23.7 will probably be true PR/PS connections, while that in Figure 23.8 will likely be a PR/FS connection. In the latter case it is easy to see that considerable strength and stiffness can be obtained, but there are potential problems. These include the possibility of activating other less desirable failure mechanisms such as web crippling of the column panel zone or weld fracture. The behavior of these connections under lateral loads that induce moment reversals (positive moments) at the connections should be governed by gradual yielding of the bottom connection element (angle, partial end plate, or welded plate). Under these conditions the slab can transfer very large forces to the column by bearing if the slab contains reinforcement around the column in the two principal directions. In this case, brittle failure modes to avoid include crushing of the concrete and buckling of the slab reinforcement. The composite connections discussed here provide substantial strength reserve capacity, reliable force redistribution mechanisms (i.e., structural integrity), and ductility to frames. In addition, they provide benefits at the service load level by reducing deflection and vibration problems. Issues related to serviceability of structure with PR frames will be treated in the section on design of composite connections in braced frames.
23.4
Moment-Rotation (M-θ ) Curves
As noted earlier, a prerequisite for design of frames incorporating PR connections is a reliable knowledge of the M-θ curves for the connections being used. There are at least four ways of obtaining 1999 by CRC Press LLC
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them: 1. From experiments on full-scale specimens that represent reasonably well the connection configuration in the real structure [21]. This is expensive, time-consuming, and not practical for everyday design unless the connections are going to be reused in many projects. 2. From catalogs of M-θ curves that are available in the open literature [6, 16, 20, 27]. As discussed elsewhere [7, 22], extreme care should be used in extrapolating from the equations in these databases since they are based mostly on tests on small specimens that do not properly model the boundary conditions. 3. From advanced analysis, based primarily on detailed finite element models of the connection, that incorporate all pertinent failure modes and the non-linear material properties of the connection components. 4. From simplified models, such as that shown in Figure 23.5, in which behavioral aspects are lumped into simple spring configurations and other modes of failure are eliminated by establishing proper ranges for the pertinent variables. Ideally M-θ curves for a new type of connection should be obtained by a combination of experimentation and advanced analysis. Simplified models can then be constructed and calibrated to other tests for similar types of connections available in the literature. For the composite connections shown in Figure 23.6, which will be labeled PR-CC, Leon et al. [23] followed that approach. They developed the following M-θ equation for these connections under negative moment for rotations less than 20 milliradians: (23.1) M − = C1 ∗ 1 − e(−C2∗θ ) + C3 ∗ θ where C1 = 0.1800 ∗ 4 ∗ Arb ∗ Fyrb + 0.857 ∗ AsL ∗ FyL ∗ (d + Y 3) C2 = 0.7750 C3 = 0.0070 ∗ (AsL + AwL ) ∗ (d + Y 3) ∗ FyL θ = relative rotation (milliradians) AwL = area of web angles resisting shear (in.2 ) AsL = area of seat angle leg (in.2 ) = effective area of slab reinforcement (in.2 ) Arb d = depth of steel beam (in.) Y3 = distance from top of steel shape to center of slab force (in.) FyL = yield stress of seat and web angles (ksi) Fyrb = yield stress of slab reinforcement (ksi) Since these connections will have unsymmetric M-θ characteristics due to presence of the concrete slab, the following equation was developed for these connections under positive moments for rotations less than 10 milliradians: (23.2) M + = C1 ∗ 1 − e(−C2∗θ ) + (C3 + C4) ∗ θ where C1 = 0.2400 = ∗ [(0.48 ∗ AW l ) + ASl ] ∗ (d + Y 3) ∗ FY l C2 = 0.0210 ∗ (d + Y 3/2) C3 = 0.0100 ∗ (AwL + AsL ) ∗ (d + Y 3) ∗ FyL C4 = 0.0065 ∗ AwL ∗ (d + Y 3) ∗ FyL For preliminary design it may be necessary to model the connections as bi-linear springs only, characterized by a service stiffness (kconn ), an ultimate strength (Mu,conn ), and hardening stiffness 1999 by CRC Press LLC
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(kult ). Simplified expressions for these are as follows: Kconn Mu,conn Kult
= = =
85 4Arb Fyr + AwL FyL (d + Y 3) 0.245 4Arb Fyr + AwL FyL (d + Y 3) 12.2 4Arb Fyr + AwL FyL (d + Y 3)
(23.3) (23.4) (23.5)
For a final check, it is desirable to model the entire response using Equations 23.1 and 23.2 or some piecewise linear version of them. The author has proposed a tri-linear version for Equation 23.1 for which the three breakpoints are defined as [5]: θ 1 = the rotation at which the tangent stiffness reaches 80% of its original value M1 = moment corresponding to θ 1 θ 2 = the rotation at which the exponential term of the connection equations e−C2∗q is equal to 0.10 M2 = moment corresponding to θ2 θ 3 = equal to 0.020 radians, close to the maximum rotation required for this type of connection M3 = moment corresponding to θ 3 It is necessary in this case to differentiate Equation 23.1 and set θ equal to zero to find an initial stiffness, and then backsolve for the rotation corresponding to 80% of that initial stiffness. All the examples in this chapter are worked out in English units because metric versions of Equations 23.1 through 23.5 have not yet been properly tested.
EXAMPLE 23.2: Moment-Rotation Curves
Figure 23.9b shows the complete M-θ curve for the composite PR connection shown in Figure 23.9a. The values shown in Figure 23.9b were taken directly from substituting into Equations 23.1 through 23.4. The shaded squares show the breakpoints for the trilinear curves described in the previous section. The trilinear curve for positive moment was derived by using the same definitions as for negative moments but limiting the rotations to 10 milliradians, the limit of applicability of Equation 23.2. Tables for the preliminary and final design of this type of connection are given in a recently issued design guide [26]. The M-θ curves shown in Figure 23.9b are predicated on a certain level of detailing and some assumptions regarding Equations 23.1 through 23.5, including the following: 1. In Equations 23.1 and 23.2, the area of the seat angles (AsL ) shall not be taken as more than 1.5 times that of the reinforcing bars (Arb ). 2. In Equations 23.1 and 23.2, the area of the web angles (AwL ) resisting shear shall not be taken as more than 1.5 times that of one leg of the seat angle (AsL ) for A572 Grade 50 steel and 2.0 for Grade A36. 3. The studs shall be designed for full interaction and all provisions of Chapter I of the LRFD Specification [2] shall be met. 4. All bolts, including those to the beam web, shall be slip-critical and only standard and short-slotted holes are permitted. 5. Maximum nominal steel yield strength shall be taken as 50 ksi for the beam and 60 ksi for the reinforcing bars. Maximum concrete strength shall be taken as 5 ksi. 6. The slab reinforcement should consist of at least six longitudinal bars placed symmetrically within a total effective width of seven column flange widths. For edge beams the steel should be distributed as symmetrically as possible, with at least 1/3 of the total on the edge side. 1999 by CRC Press LLC
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FIGURE 23.9: Typical PR-CC connection and its moment-rotation curves.
7. Transverse reinforcement, consistent with a strut-and-tie model, shall be provided. In the limit the amount of transverse reinforcement will be equal to that of the longitudinal reinforcement. 8. The maximum bar size allowed is #6 and the transverse reinforcement should be placed below the top of the studs whenever possible. 9. The slab steel should extend for a distance given by the longest of Lb /4 or 24 bar diameters past the assumed inflection point. At least two bars should be carried continuously across the span. 10. All splices and reinforcement details shall be designed in accordance with ACI 318-95 [1]. 11. Whenever possible the space between the column flanges shall be filled with concrete. This aids in transferring the forces and reduces stability problems in the column flanges and web. These detailing requirements must be met because the analytical studies used to derive Equations 23.1 and 23.2 assumed this level of detailing and material performance. Only Item 11 is optional but strongly encouraged for unbraced applications Compliance with these requirements means that extensive checks for the ultimate rotation capacity will not be needed. 1999 by CRC Press LLC
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23.5
Design of Composite Connections in Braced Frames
The design of PR-CCs requires that the designer carefully understand the interaction between the detailing of the connection and the design forces. Figure 23.10 shows the moments at the end and centerline, as well as the centerline deflection, for the case of a prismatic beam under a distributed load with two equal PR connections at its ends. The graph shows three distinct, almost linear zones for each line; two horizontal zones at either end and a steep transition zone between α of 0.2 and 20. Note that the horizontal axis, which represents the ratio of the connection to the beam stiffness, is logarithmic. This means that relatively large changes in the stiffness of the connection have a relatively minor effect. For example, consider the case of a beam with PR-CCs with a nominal α of 10. This gives moments of wL2 /13.2 at the end and wL2 /20.3 at centerline, with a corresponding deflection of 1.67 wL4 /384EI. If the service stiffness (kser ) for this connection is underestimated by 25% (α = 7.5) these values change to wL2 /13.6, wL2 /19.4, and 1.84wL4 /384EI. These represent changes of 3.0%, 4.4%, and 10%, respectively, and will not affect the service or ultimate performance of the system significantly. This is why the relatively large range of moment-rotation behavior, typical of PR connections and shown schematically in Figure 23.2, does not pose an insurmountable problem from the design standpoint.
FIGURE 23.10: Moments and deflections for a prismatic beam with PR connections under a distributed load. For continuous composite floors in braced frames, where the floor system does not participate in resisting lateral loads, the design for ultimate strength can be based on elastic analysis such as that shown in Figure 23.10 or on plastic collapse mechanisms. If elastic analysis is used, it is important to recognize that both the bending resistance and the moments of inertia change from regions of negative to positive moments. The latter effect, which would be important in elastic analysis, is not considered in the calculations for Figure 23.10. In the case of the fixed ended beam with full strength connections (FR/FS), elastic analysis (α = ∞ in Figure 23.10) results in the maximum force corresponding to the area of lesser resistance. This is why it would be inefficient to design continuous composite beams with FR connections from the strength standpoint. As the connection stiffness is reduced, the ratio of the moment at the end to the centerline begins to decrease. From Figure 23.10, for a prismatic beam, the optimum connection stiffness is found to be around α = 3, where the moments at the ends and middle are equal (wL2 /16). This indicates that it takes relatively 1999 by CRC Press LLC
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little restraint to get a favorable distribution of the loads. If the effect of the changing moments of inertia is included, as it should for the case of composite beams, the sloping portions of the moment curves in Figure 23.10 will move to the right. For this case, the optimum solution will not be at the intersection of the M(end) and M (CL) lines but at the location where the ratio of Mp,ci /Mp,b equals M(end)/ M (CL). Preliminary studies indicate that the optimum connection stiffness for composite beams is generally found to be still around α of 3 to 6. This indicates that it takes relatively little restraint to get a favorable distribution of the loads. This type of simple elastic analysis, however, cannot account for the fact that the connection M-θ curves are non-linear and thus will not be useful in the analysis of PR/PS connections such as PR-CCs. Design of continuous beams with PR connections can be carried out efficiently by using plastic analysis. The collapse load factor for a beam (λb ) with a plastic moment capacity Mp,b at the center, and connection capacities Mp,c1 , and Mp,c2 (Mp,c1 > Mp,c2 ) at its ends, can be written as: λb =
d aMp,c1 + bMp,c2 + cMp,b P L or wL2
(23.6)
where the coefficients a, b, c, and d are given in Table 23.1, P and w are the point and distributed loads, and L is the beam length, respectively. For Load Cases 1 through 4, the spacing between the loads is assumed equal. TABLE 23.1
Values of Constants in Equation 23.7 for Different Loading Configurations
Connection relationship Mp,c1 = Mp,c2
1 2 3 4 5
a 1 1 1 1 1
b 0 0 0 0 0
c 1 1 1 1 1
d 4 3 2 5 3
8
Mp,c1 > Mp,c2
a 1 1 1 2
b 1 2 1 3
c 2 3 2 5
d 2 1 1 5 12
Mp,c2 = 0
a 1 1 1 2 1
b 0 0 0 0
c 2 3 2 5
0
L x
d 2 1 1 5 12 2L L−x
For the case of a distributed load (Load Case 5) with unequal end connections (Mp,c1 > Mp,c2 ), it is not possible to write a simple expression in the form of Equation 23.6 because the solution requires locating the position of the center hinge. For the case of Mp,c2 = 0, the position can be calculated by: (s ) Mp,c1 Mp,b L 1+ −1 (23.7) x= Mp,c1 Mp,b If plastic analysis is used, it is important to recognize that the flexural strength changes from the area of negative (Mp,c1 and Mp,c2 ) to positive moment (Mp,b ), and that the ratio of Mp,ci /Mp,b will often be 0.6 or less. For the service limit state, it is important again to recognize that the results shown in Figure 23.10 are valid only for a prismatic beam. In reality a continuous composite beam will be non-prismatic, with the positive moment of inertia of the cross-section (Ipos ) often being 1.5 to 2.0 times greater than the negative one (Ineg ). It has been suggested that an equivalent inertia (Ieq ), representing a 1999 by CRC Press LLC
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weighted average, should be used [5]: Ieq = 0.4Ineg + 0.6Ipos
(23.8)
The effect of accounting for the non-prismatic characteristics of the beam is far more important in calculating deflections than in calculating the required flexural resistance. For calculating deflections of beams with equal PR connections at both ends, the following expression has been proposed [5]: δP R = δF R + where δPR δFR Cθ θsym
Cθ θsym L 4
(23.9)
= = = =
the deflection of the beam with partially restrained connections the deflection of the beam with fixed-fixed connections a deflection coefficient the service load rotation corresponding to a beam with both connections equal to the stiffest connection present When the beam has equal connection stiffnesses, Cθ equals one. Values for the constant Cθ in Equation 23.9 are given in Table 23.2 for some common loading cases. TABLE 23.2
Constants for Deflection Calculations by Equation 23.9 1/(1 + α)
Kb /Ka
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1 1.05 1.11 1.18 1.27 1.39 1.54 1.76 2.09 2.63 3.70
1 1.04 1.09 1.15 1.22 1.31 1.41 1.55 1.72 1.97 2.32
1 1.04 1.08 1.13 1.18 1.25 1.32 1.41 1.52 1.66 1.83
1 1.03 1.07 1.11 1.15 1.20 1.25 1.32 1.39 1.47 1.57
1 1.03 1.05 1.09 1.12 1.16 1.20 1.24 1.29 1.34 1.40
1 1.02 1.04 1.07 1.09 1.12 1.15 1.18 1.21 1.25 1.28
1 1.02 1.03 1.05 1.07 1.08 1.10 1.12 1.15 1.17 1.19
1 1.01 1.02 1.03 1.04 1.05 1.07 1.08 1.09 1.10 1.12
1 1.01 1.01 1.02 1.02 1.03 1.03 1.04 1.04 1.05 1.05
Note: Kb = stiffness of the less stiff connection; Ka = stiffness of the stiffer connection; 1/(1 + α/2) = Mconn,PR /Mconn,fixed and; α = EI /(Ka L).
The value of θsymm is given by: θsymm =
MFEM Kser + 1 + α2
where MFEM = the fixed end moment Kconn = the stiffness of the connection α = the ratio of the connection to the beam stiffness The effect of partially restrained connections on floor vibrations is an area that has received comparatively little attention. Figure 23.11 shows the changes in natural frequency for a prismatic beam with a distributed load as the stiffness of the end connections change. The connections at both ends are assumed equal and the connection stiffness is assumed to be linear. The natural frequency (fn , H z) is given by: r Kn2 EI (23.10) fn = 2π mL4 1999 by CRC Press LLC
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FIGURE 23.11: First natural frequency of vibration for a beam with PR connections. where m is the mass per unit length, L is the length, and EI is the stiffness of the beam. Generally m is taken as the distributed load (w) given by the dead plus 25% of the live loads and divided by the acceleration of gravity (g = 386 in./s2 ). Limit values of Kn range from π 2 for the simply supported case to (1.5π)2 for the fixed case.
EXAMPLE 23.3:
Design a continuous floor system in a braced frame. The system will consist of a three-span girder with a total length of 96 ft, and will be designed for dead loads of 80 psf and live loads of 100 psf. The reduced live loads will be taken as 60 psf. This girder supports floor beams spanning 28 ft in the perpendicular direction every 8 ft, for a total of three point loads per span. In addition to the distributed loads described above, the interior span will support equipment weighing 15 kips, to be installed before the slab is cast (Figure 23.12). Cambering will be provided to offset all dead loads, including the equipment. The connections to the exterior columns will be assumed as pinned since an overhang would be required to anchor the slab reinforcement. The steel will be A572 Grade 50 and a 3-1/4 in. lightweight concrete slab (fc0 = 4 ksi) on 3 in. metal deck (Y2 = 4.5 in.) will be assumed. The construction dead loads are assumed as 60 psf and the construction live loads are taken as 15 psf. The design construction load, assuming distributed loads, is: wu,const Mu,const Zx
= = =
[1.2(0.06) + 1.6(0.015)] (28 ft.) = 2.69 k/ft wL2 /8 = (2.69)(32)2 /8 = 344 k-ft = 4129 kip-in. 4129/(0.9 × 50) = 91.8 in.3
Assuming that the beam will be supported laterally during the construction phase, the most economical steel section would be a W21x44 (Zx = 95.4 in.3 ). For the ultimate strength limit state, assuming three point loads at the location of the floor beams, for the interior span: Pu, φMu,
= =
[1.2(0.08) + 1.6(0.06)] (28 ft.)(8 ft.) + 1.2(15 kips) = 61.0 kips 15Pu L/32 = 15(61.0)(32)/32 = 915 k-ft = 10,980 kip-in.
For the ultimate strength limit state in the exterior spans: Pu, 1999 by CRC Press LLC
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=
[1.2(0.08) + 1.6(0.06)] (28 ft.)(8 ft.) = 43.0 kips
FIGURE 23.12: Design of composite floor system as simply supported beams (numbers in parenthesis are the number of shear studs). φMu,
=
15Pu L/32 = 15(61.0)(32)/32 = 645 k-ft = 7,742 kip-in.
If we assume typical current construction practice and design these girders as simply supported composite beams, for the ultimate load condition the section required will be a fully composite P W24x55 (Y2 = 4.5 in. and Qn = 810 kips). Assuming fc0 = 4 ksi and 3/4 in. headed studs, 38 shear studs per half-span, or more than two studs per flute, will be needed. This is not a very efficient design, andP thus a partially composite W24x62 will be a better choice (φMp = 930 kip-ft with Y2 = 4.0 in. and Qn = 598 kips). This results in 29 studs per half-span or roughly two studs per flute. The service load deflection in this case would be: h i δ = 19P L3 /384EI = 19 × (0.06 × 28 × 8) × (32 × 12)3 / [384 × 29000 × 2180] = 0.595 in. ≈ L/640 For the exterior spans, a W24x62 P with the minimum amount of interaction (25%, or Mp = 755 kip-ft with Y2 = 4.0 in. and Qn = 228 kips) and 21 studs total will suffice. If we were to provide a PR-CC such as the one shown in Figure 23.9, one could calculate its ultimate strength (Mu,conn ), from Equation 23.4 as: h i Mu,conn = 0.245 4 × 6 × 0.31 in.2 × 60 ksi + (4.00 × 50) × (21 + 4) = 3,959 kip-in. Note that the nominal capacity of the connection (Mu,conn = 3,959 kip-in.) has to be less than or equal to that of the steel beam φ(Mp,b = 4,293 kip-in.) in order to insure that the hinging will not occur in the beam. The author has suggested [22] that a good starting point for the strength design is to assume that the connection will carry about 70 to 80% of Mp,b . For our case the ratio is 3,959/6,888 = 0.58 which is somewhat lower but reasonable because of the heavy dead loads. For the interior span, from Equation 23.6 and assuming that Mp,c1 = Mp,c2 = φMu,conn = (0.9 × 3959) = 3563 kip-in., for a collapse load factor (λp ) of 1.00:
1999 by CRC Press LLC
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1.00
=
φMp,b
=
(2) 3563 + φMp,b (61 × 32 × 12) 8149 kip-in. = 679 kip-ft
For the exterior span, from Equation 23.6 and assuming that Mp,c1 = 0 and Mp,c2 = φMu,conn = (0.9 × 3959) = 3563 kip-in., for a collapse load factor (λp ) of 1.00: 1.00
=
φMp,b
=
(1) 3563 + 2φMp,b (43 × 32 × 12) 6474 kip-in. = 540 kip-ft
The required strength can now be provided by a fully composite W21x44 (φMn = 683 kip-in., and P QP n = 650 kips or two studs per flute) and by a partially composite W21x44 (φMn = 564 kip-in., and Qn = 260 kips or one studs per flute). Figure 23.13 shows the analysis model and the final design for this case, as well as the moment diagram for the case of DL + LL.
FIGURE 23.13: Continuous beam design with PR connections.
Figure 23.13c shows that the dead load moments are calculated on the simply supported structure (SS), while the live load ones are calculated on the continuous structure (PR). For calculation purposes, the moments of inertia were taken as 1699 in.4 for the interior span and 1399 in.4 for the exterior span, as per Equation 23.8. Figure 23.13c indicates that the maximum moment in the interior span at full service load is 647 kip-ft. This is close to the factored capacity of the section (φMn = 683 kip-ft). Thus, careful attention should be paid to the stresses and deflections at service loads when using a plastic design approach since the latter does not consider construction sequence or the onset of yielding. In this case perhaps a W21x50 section, with the same number of studs, would be a more prudent design. In computing the forces for the case of the PR system, the connection stiffness was calculated directly as a secant stiffness at 0.002 radian from Equation 23.1. The stiffness was 1.135 × 106 kip1999 by CRC Press LLC
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in./rad, which is slightly lower than the 1.398 × 106 kip-in./rad given by Equation 23.3. The α for this connection is: (1.135 × 106 )(32 × 12) Kser L = = 8.84 α= EI (29000)(1699) This puts this PR connection near the middle of the PR range for unbraced frames and near the rigid case for the case of braced frames. The deflection of the center span under the full live load is, from Equation 23.9: δP R =
P L3 Cθ MF F L = 0.161 + 0.109 = 0.270 in. ≈ L/1500 + 96EI 4Kconn 1 + α2
This deflection is considerably less than that computed for the simply supported case even when a much larger section (W24x62) was used in the latter case. An idea of the effect of this PR connection can be gleaned from inspecting Figure 23.10. Although Figure 23.10 corresponds to a different case, the moment diagrams are not substantially different and thus a meaningful comparison can be made for the elastic case. From Figure 23.10, the difference in deflection between a simple support and a PR connection with α = 8.84 is roughly a factor of 2.8 (5/1.8), while the difference in moment of inertia is only 1.28 (2180 in.4 / 1699 in.4 ). In this example, the design was governed by strength and not deflections. However, this example clearly shows the impact of a PR connection in reducing floor deflections. In addition to the strength calculation above, the design procedure requires that the following limit states and design criteria be satisfied (refer to Figure 23.9a for details): 1. Shear strength of the bolts attaching the seat angle to the beam (φVbolts ): The bolts have to be designed to transfer, through shear, a compressive force corresponding to 1.25 of the force (Tslab ) in the slab reinforcement. The 1.25 factor accounts for the typical overstrength of the reinforcement, and intends to insure that the bolts will be able to carry a force consistent with first yielding of the slab steel. Assuming 1 in. diameter A490N bolts: (φVbolts ) = 1.25Tslab = 1.25Fy Abars = 139.5 kips N bolts = (φVbolts )/35.3 = 3.95 ∼ = 4 bolts (O.K.) 2. Bearing strength at the bolt holes (φRn ): The thickness of the angle will be governed by the required flexural resistance of the angle leg connecting to the beam flange in the case of a connection in an unbraced frame, where tensile forces at the bottom of the connection are possible. It will be governed by either bearing of the bolts or compressive yielding of the angle leg in the case of a connection in a braced frame. In this case: φRn
= =
φ(2.4dtFu ) = 0.75(2.4 × 0.875 × 0.5 × 65) 51.2 kips/bolt > φVbolts , O.K.
3. Tension yield and rupture of the seat angle: This limit state is strictly applicable to the case of unbraced frames where pull-out of the angle under positive moments is possible. For the case of a connection in a braced frame, it is prudent to check the angle for yielding under compressive forces (φCn ) and possible buckling. The latter is never a problem given the short gage lengths, while the former is: φCn = φ Ag Fy = 0.9 × (8 × 0.5) × 50 = 180 kips > φVbolts , O.K. 1999 by CRC Press LLC
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4. Number and distribution of slab bars, including transverse reinforcement, to insure a proper strut-and-tie action at ultimate (see section following Example 23.2 for details). 5. Number and distribution of shear studs to provide adequate composite action (checked above as part of the flexural design). 6. Tension strength, including prying action, for the bolts connecting the beam to the column. 7. Shear capacity of the web angles. 8. Block shear capacity of the web angles. 9. Check for the need for column stiffeners Limit states (6) through (9) can be checked following the current LRFD provisions, and the details will not be provided here. However, it should be clear from the few calculations shown above that the shear capacity of the bolts is the primary mechanism limiting the forces in the connection. The structural benefits of using a PR-CC connection are clear from the results of this example. From the economic standpoint, for a PR-CC to be beneficial, the cost of the additional reinforcing bars and seat angle bars has to be offset by that of the additional studs and larger sections required for the simply supported case. In some instances the benefits may not be there from the economic standpoint, but the designer may choose to use PR-CCs anyway because of their additional redundancy and toughness. In Example 23.3, the design was controlled by strength and thus it was relatively simple to calculate forces based on plastic analysis and proportion the connection based on a simplified model similar to that shown in Figure 23.5. Since deflections did not control the design, the connection stiffness did not play an appreciable role in the preliminary design. If serviceability criteria control the design, then the proportioning of the connection can start from Equation 23.3. In this case the analysis has to be iterative, since the value of the connection stiffness will affect the moment diagram and the deflection. For applications in braced frames, however, experience indicates that it is strength and not stiffness that governs the design. This is because the steel beam size is controlled by the construction loads if the typical unshored construction process is used. In general, the steel beam selected is capable of providing the required stiffness even if it is the minimum amount of interaction (25% is recommended by AISC and 50% by this author).
23.6
Design for Unbraced Frames
As noted earlier, the design of frames with PR connections requires that the effects of the non-linear stiffness and partial strength characteristics of the connections be incorporated into the analysis. From the practical standpoint, the main difference between the design of unbraced FR and PR frames is the contribution of the connections to the lateral drift. The designer thus needs to balance not just the stiffness of the columns and beams to satisfy drift requirements, but account for the additional contribution of the concentrated rotations at the connections. There are no established practical rules on the best distribution of resistance to drift between columns, beams, and connections for PR-CCs. Trial designs indicate that distributing them about equally is reasonable (i.e., 33% to the beams, columns, and connections, respectively), and that it may be advantageous in low-rise frames to count on the columns to carry the majority of the resistance to drift (say 40 to 45% to columns, and the rest divided about equally between the beams and connections). The use of fixed column bases is imperative in the design of PR-CC frames, just as it is in the design of almost all unbraced FR frames, in order to limit drifts. Thus, designers should pay careful attention to the detailing of the foundations and the column bases. The required level of analysis for the design of unbraced frames with PR connections is currently not covered in any detail by design codes. The AISC LRFD specification [2] allows for the use of such connections by requiring that the designer provide a reliable amount of end restraint for the 1999 by CRC Press LLC
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connections by means of tests, advanced analysis, or documented satisfactory performance. The AISC LRFD specification, however, does not provide any guidance on the analysis requirements except to note that the influence of PR connections on stability and P-1 effects need to be incorporated into the design. The new NEHRP provisions and AISC seismic provisions [4, 28] will contain generic design requirements for frames with PR connections for use in intermediate and ordinary moment frames (IMF and OMF). In addition, it will contain some specific requirements for some specific types of connections, such as the PR-CCs described in this chapter. It is unlikely that there will be an attempt in the near future to codify the analysis and design of PR frames since it would be difficult to develop guidelines to cover the vast array of connection types available (Figure 23.1). Thus, design of PR frames will remain essentially the responsibility of the structural engineer with guidance, for particular types of connections, from design guidelines [8, 26], books [12, 13, 14], and other technical publications. The proposed procedures to be described next remain, therefore, only a suggestion for proportioning the entire system. Only the detailing of the connections, including checking all pertinent failure modes, should be regarded as a requirement. The design procedure to be discussed is divided into two distinct parts. For the service limit states (deflections, drift, and vibrations) the design will use a linear elastic model with elastic rotational springs at the beam ends to simulate the influence of the PR connections. For the ultimate limit states (strength and stability), a modified, second-order plastic analysis approach will be used. In this case the connections will be modeled as elastic-perfectly plastic hinges and the stability effects will be modeled through a simplified second-order approach [26]. Because the latter was calibrated to a population of regular frames with PR-CCs, the approach is only usable for PR-CC frames. The design process will be illustrated with calculations for the frame shown in Figure 23.14. For a complete design example, including all intermediate steps and design aids, the reader is referred to [26].
EXAMPLE 23.4:
Conduct the preliminary design for the frame shown in Figure 23.14. The frame is a typical interior frame, has a tributary width of 30 ft, and will be designed for an 80 mph design wind and for forces consistent with UBC 1994 seismic zone 2A. The dead loads are 55 psf for the slab and framing and 30 psf for partitions, mechanical, and miscellaneous. The weight of the facade is estimated as 700 plf. The live loads are 50 psf and 125 psf in the exterior and interior bays, respectively, and will be reduced as per ASCE 7-95. The roof dead and live loads are 30 psf and 20 psf, respectively. The floor slab will consist of a 3-1/4 in. lightweight slab on a 3 in. metal deck, resulting is a typical Y2 for the slab of 4.5 in. The design of the entire frame is beyond the scope of this chapter, so calculations for only a few key steps will be given. Part 1: Select beams and determine desired moments at the connections: Step 1: Select the beam sizes based on the factored construction loads, as illustrated in Example 23.3. For this case the exterior bays require a W21x50, while the interior bays require a W21x44. Step 2: Select moment capacity desired at the supports (Mus ) based on the live loads. A good starting point is 75% of the Mp of the steel beam selected in Step 1, but the choice is left to the designer. Once Mus has been chosen, the factored moment at the center of the span (Muc ) can be computed as the difference between the ultimate simply supported factored moment (Mu , static moment) and Mus . For the interior span, wu = 6.66 kip/ft and Mu = 1020 kip-ft of which roughly 55% corresponds to the dead loads and 45% to the live loads. Thus, select a connection capable of carrying: Mus 1999 by CRC Press LLC
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=
1020 × .45 × 0.75 = 330 kip-ft = 3965 kip-in.
FIGURE 23.14: Frame for Example 23.4. Muc
=
1020 − 330 = 690 kip-ft = 8,280 kip-in.
Step 3: Select a composite beam to carry Muc and check that it can carry the unfactored service loads without yielding of the slab reinforcement. Assume that full composite action will be required to limit vertical deflections and lateral drifts. Using the steel beams from Step 1 and following the procedure from Example 23.3, the exterior bays require a W21x50 with 58 3/4 in. diameter studs, while the interior bays require a W21x44 with 52 studs. The design procedure for lateral loads was derived assuming that the beams were fully composite. In this case that means increasing the number of studs to 66 and 58, respectively, which is a very small increase. The moments of inertia computed from Equation 23.4, and including the contribution of the reinforcement are 1843 in.4 for the W21x44 and 1899 in.4 for the W21x50. Part 2: Preliminary connection design: Step 4: Compute the amount of slab reinforcement (Arb ) required to carry Mus . Assume that the moment arm is equal to the beam depth plus the deck rib height plus 0.5 in. The nominal required moment capacity is: Mn Arb
= =
Mus /φ = 3950/0.9 = 4388 kip-in. 4388/ (60 ksi × (21 + 3 + 0.5)) = 2.98 in.2
Try 8 #5 bars (Arb = 2.48 in.2 ). It is reasonable to use less area than required by the equations above (Arb = 2.98 in.2 ) because those calculation ignore the contribution of the web angles to the ultimate capacity and the φ = 0.9 factor that has been added to the connection design. The latter accounts for the expected differences in stiffness and strength for the entire connection rather than for its individual components. Currently the LRFD Specification does not require such a factor and thus its use, while recommended, is left to the judgment of the designer. Step 5: Choose a seat angle so that the area of the angle leg (AsL ) is capable of transmitting a tensile force equal to 1.33 times the force in the slab. The 1.33 factor is used to obtain a thicker angle so that its stiffness is increased. AsL = 2.48 ∗ (60 ksi/50 ksi) ∗ 1.33 = 3.95 in.2 1999 by CRC Press LLC
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Try a L7x4x1/2x8" (AsL = 4.00 in.2 ). Step 6: For transferring the shear force consistent with the rebar reaching 1.25Fy , the bolt shear capacity required is: Vbolt = 2.48 in.2 × 60 ksi × 1.25 = 186 kips This requires four 1-in. A490X bolts. Note that if the number of bolts is taken greater than 4, they would be difficult to fit into the commonly available angle shapes. In general the number and size of bolts required to carry the shear at the bottom of the connection is the governing parameter in design. Thus, another possible way of selecting the amount of moment desired at the connection (see Step 2) is to select the size and number of bolts and determine Mus as: Mus = Vbolt beam depth + deck height + 0.5 in. Step 7: Determine the number and size of bolts required for the connection to the column flange. From typical tension capacity calculations, including prying action, two 1-in. A490X bolts are required for the connection to the column. In general, and for ease of construction, these bolts should be the same size as those determined from Step 6. Step 8: Select web angles (AwL ) assuming a bearing connection. Check bearing and block shear capacity. The factored shear (Vu ) is: Vu = 6.6(35/2) = 115.5 kips The factored shear from lateral loads is based on assuming the formation of a sidesway mechanism in which one end of the beam reaches its positive moment and the other its negative moment capacity. Since the connection has not been completely designed, assume that the negative and positive moment capacities are the same. This is conservative since the positive capacity will generally be smaller than the negative one. Vu = 2Mn,conn /L = (4,388 kip-in. + 4,388 kip-in.)/(35 × 12) = 18.7 kips From Tables 9-2 in the LRFD Manual, four 3/4 in.-diameter A325N bolts, with a pair of L4x4x1/4x12" can carry 117 kips. Note that for calculation purposes, the area of the web angles (AW l ) in Equations 23.1 and 23.2 is limited to the smallest of the gross shear area of the angles (2 × 12 × 1/4 = 6.00 in.2 ) or 1.5 times the area of the seat angle (1.5 ASl = 1.5 × 8 × 1/2 = 6.00 in.2 ). This is required because Equations 23.1 and 23.2 were derived with this limit as an assumption. Step 9: Determine connection strengths and stiffness for preliminary lateral load design. From Equations 23.3 through 23.5: kconn Mu,conn kult
= = =
85 [(4 ∗ 2.48 ∗ 60) + (6 ∗ 50)] (21 + 3.5) = 1.864 × 106 kip-in./rad 0.245 [(4 ∗ 2.48 ∗ 60) + (6 ∗ 50)] (21 + 3.5) = 5373 kip-in. 12 [(4 ∗ 2.48 ∗ 60) + (6 ∗ 50)] (21 + 3.5) = 263.2 × 103 kip-in./rad
From the more complex Equation 23.1, the ultimate moment at 0.02 radians is 5040 kip-in., the secant stiffness to 0.002 radians is 1.403 × 106 kip-in./rad, and the ultimate secant stiffness is 252 × 103 kip-in./rad. Thus, the approximate formulas seem to provide a good preliminary estimate. Whenever possible, the use of Equations 23.1 and 23.2 is recommended. The stiffness ratio for this connection is: α = 1.403 × 106 × (35 × 12)/(29000 ∗ 1699) = 11.95 1999 by CRC Press LLC
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Step 10: Check deflections under live load based on the service stiffness computed in Step 9. As for Example 23.3, the centerline deflection under full live loads is small since the α is large. The connection designed in Steps 4 through 9 is shown in Figure 23.15. In the next steps, the adequacy of the connections, designed for gravity loads, to handle the design lateral loads will be checked.
FIGURE 23.15: Connection details for Example 23.4.
Part 3: Preliminary lateral load design: Step 11: Determine column sizes based on drift requirements and/or gravity load requirements. From the gravity loads, and making a 10% allowance for second order effects, a W14x74 was selected for the exterior leaner columns and a W14x132 for the interior columns. The selection of the interior columns was checked by satisfying the interaction equations (Equations H1-1a,b in the LRFD Specification) assuming that (a) the required moment capacity will be given by the summation of the moment capacities on either side of the − + = 4193 kip-in. and Mp,conn = 3655 kip-in. from Equations 23.1 connection (Mb,conn and 23.2); (b) the axial load is given by 1.2DL + 0.5LL (Pu = 365 kips, including live load reductions); and (c) B1 = 1.0 and B2 = 1.1. The total story drift (1) can be calculated, for preliminary design purposes as: 1 1 2 P1 +P +P (23.11) 1 = VH Kc Kg Kconn X X 12EIeq (12)(29000)(2 ∗ 1843 + 1898) = 4.635 × 106 Kb = = Lb (420) X Kconn = 4(1.403 × 106 ) = 5.612 × 106 X X 12EIc (2)(12)(29000)(1530) = = 7.195 × 106 Kc = H (148) where Ic and Ieq 1999 by CRC Press LLC
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= the moments of inertia of the columns and beams
Lb = the girder length H = the story height V = the story shear = the connection stiffness kconn For the girders the effective stiffness of the exterior girders, which are pin-connected at one end and have a PR connection at the other, was taken as equal to that of one girder. For the columns, only the two interior ones are used since the exterior ones are leaners. The summations are over all beams, columns, and connections participating in the lateral load resisting system. Note that for unbraced frames subjected to lateral loads one connection at each column line will be loading and one will be unloading. Thus, their kconn should be different on either side of a column; however, for preliminary design it is sufficient to assume the kconn for negative moments. For the critical first story, the shear (H ) due to the wind loads is 26.3 kips. For drift design, this value will be checked against an allowable drift of 0.25%. From Equation 23.11, the interstory drift is 0.31 in. or H /482 which is well within the H /400 normally allowed. Note from the calculations of stiffness in Equation 23.11 that the connections actually provide about 32% of the lateral resistance. The beams provide about 27% and the columns provide the remaining 41%. This represents a well-balanced distribution of stiffness. Note again that the contribution of the columns, which is intimately tied to the assumption of base fixity, is the key to limiting drifts. The drift under seismic loads (H = 34 kips) can be checked roughly by calculating the elastic drift and multiplying it by an amplification factor (Cd ). The new ASIC Seismic and NEHRP provisions give Cd = 5.5 for PR-CCs, and allow a maximum of 1.5% drift for this type of structure. Thus: 1 = (0.31 in. × (34/26.3) × 5.5) = 2.20 in. → (2.20/148) × 100 = 1.5%, O.K. Although this frame barely meets the displacement criteria, a more refined non-linear analysis should be carried out to determine the actual drift. Step 12: The strength of a frame can be calculated based on a sidesway, plastic collapse mechanism (Figure 23.16). The first-order, rigid-plastic collapse load factor (λp ) for this type of structure is given by [26]: λp =
+ − + − (N + 1)Mp,col + ((N − 1) ∗ S) Mp, conn + Mp,conn int + (S) Mp,conn + Mp,conn ext P (Vi ∗ Hi )
(23.12)
where N = the number of bays S = the number of stories = the loads and heights at each story Vi and Hi = the column plastic capacity at the base Mp,col Mp,conn = the connection capacity at 10 milliradians ext and int refer to the exterior and interior connections + and − refer to the positive and negative moment capacities. For our case: (5)(10,530) + ((3) ∗ 2)(4193 + 3655) = 9.45 λp = P (15.6 ∗ 444 + 10.1 ∗ 296 + 4.3 ∗ 148) This is apparently a very large collapse load factor, but there is a substantial reduction in that capacity due to second-order effects (Figure 23.17). Consideration of the P-1 effects 1999 by CRC Press LLC
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results in a second-order collapse load factor λk which is a function of the rigid plastic collapse load factor (λp ) and the ratio (Sp ) of the lateral displacement at collapse (1k ) to the displacement at a load factor of one (1w ). For proportional loading this leads to a second-order collapse load factor (λk ) equal to [18]: λk = where P = θ = δ = φ = Mp =
λp P
1 + Sp λ2p
P P θδ Mp φ
(23.13)
the axial loads the story rotation the elastic interstory drift the rotation of the plastic hinges the moments at the hinges (Figure 23.16)
Values of Sp for various numbers of stories and story heights are given in Table 23.3. The ratio of θ/φ in the denominator of Equation 23.13 is equal to 1.0 when the member rotations (θ) are equal to the plastic hinge rotations (φ) as would be the case in the weak beam–strong column sway mechanism envisioned here. This results in λk = 2.99, which is a reasonable collapse load factor [17].
FIGURE 23.16: Plastic collapse mechanism and second order effects.
Step 13: Check strong column–weak beam behavior by requiring that: i X h P − + Mp,col 1 − 1.25 M p,conn + M p,conn ≤ Pmax
(23.14)
Step 14: Check stability of the columns by using AISC LRFD [2] Equation H1-1(a) and (b). Assume + plus that lateral loads will control and that the maximum moments are equal to Mp,conn − Mp,conn . These Mp,conn shall be based on a 1.25 overstrength factor for both rebar and steel angle, and a φ = 1.00. For calculating G factors, assume that the effective moment of inertia for the beams is: ! 1 (23.15) Ieff = Ieq 1 + α6 1999 by CRC Press LLC
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FIGURE 23.17: Simplified computation of plastic second-order effects. (After Horne, M.R. and Morris, L.J. Plastic Design of Low-Rise Frames, The MIT Press, Cambridge, MA, 1982.) TABLE 23.3 Number of stories 4 6 8
Values of Sp Story height (ft)
Proportional loading
12 14 16 12 14 16 12 14 16
8 7.2 5.4 5.9 4.5 3.9 3.6 2.8 2
From Hoffmann, J.J., Design Procedures and Analysis Tools for Semi-Rigid Composite Members and Frames, M.S. Thesis, University of Minnesota, Minneapolis. With permission.
Procedures for determining the stability of PR frames are still under development (see Chapter 3 of ASCE 1997 for more details). Once a preliminary design has been completed, the final checks need to be made with advanced analysis tools unless the frame is very regular and seismic forces are not a concern. The level of modeling required is left to the discretion of the designer, but should, at a minimum, include trilinear springs to model the connections and include second-order effects. The design procedures illustrated here are limited to only one type of connection. To the author’s knowledge only two other types of composite connections have received a similar level of development: those between steel beams and concrete columns and those for composite end plates. An extensive treatment of the general topic of connection design is given in Chapter 6 of Viest et al. [29]. The latest information on design of composite and PR connections can also be found in the proceedings of several international conferences [9, 10, 11, 19].
References [1] ACI 318-95. 1995. Building Code Requirements for Reinforced Concrete (ACI 318-95), American Concrete Institute, Detroit, MI. 1999 by CRC Press LLC
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[2] AISC. 1993. Load and Resistance Factor Design Specification for Structural Steel Buildings, 2nd. ed., American Institute of Steel Construction, Chicago, IL. [3] AISC. 1994. Manual of Steel Construction—Load and Resistance Factor Design, 2nd. ed., American Institute of Steel Construction, Chicago. IL. [4] AISC. 1997. Seismic Provisions for Structural Steel Building, Load and Resistance Factor Design, American Institute of Steel Construction, Chicago, IL. [5] Ammerman, D.J. and Leon, R.T. 1990. Unbraced Frames with Semi-Rigid Connections, AISC Eng. J., 27(1), 12-21. [6] Ang, K. M. and Morris, G. A. 1984. Analysis of Three-Dimensional Frames with Flexible BeamColumn Connections, Can. J. Civ. Eng., 11, 245-254. [7] ASCE. 1997. Effective Length and Notional Load Approaches for assessing Frame Stability: Implications for American Steel Design, 1st ed., ASCE, New York. [8] ASCE Task Committee on Design Criteria for Composite Structures in Steel and Concrete. 1998, Design Guide for Partially-Restrained Composite Connections (PR-CC), ASCE J. Struc. Eng., to appear in 1998. [9] Bjorhovde, R., Colson, A., and Zandonini, R., Eds. 1996. Connections in Steel Structures III: Behaviour, Strength and Design, Proceedings of the Third International Workshop on Connections held at Trento, Italy, May, 1995, Pergamon Press, London. [10] Bjorhovde, R., Colson, A., Haaijer, G., and Stark, J.W.B., Eds. 1992. Connections in Steel Structures II: Behaviour, Strength and Design, Proceedings of the Second Workshop on Connections held at Pittsburgh, April 1991, AISC, Chicago, IL. [11] Bjorhovde, R., Brozzetti, J., and Colson, A., Eds. 1988. Connections in Steel Structures: Behaviour, Strength and Design, Proceedings of the Workshop on Connections held at the Ecole Normale Superiere, Cachan, France, May 1987, Elsevier Applied Science, London. [12] Chen, W.F. and Toma, S. 1994. Advanced Analysis of Steel Frames, CRC Press, Boca Raton, FL. [13] Chen, W.F. and Lui, E. 1991. Stability Design of Steel Frames, CRC Press, Boca Raton, FL. [14] Chen, W.F., Goto, Y., and Liew, J. Y. R. 1995. Stability Design of Semi-Rigid Frames, John Wiley & Sons, New York. [15] Eurocode 3. 1992. Design of Steel Structures, Part 1: General Rules and Rules for Buildings, ENV 1993-1-1:1992, Comite Europeen de Normalisation (CEN), Brussels. [16] Goverdhan, A.V. 1984. A Collection of Experimental Moment Rotation Curves Evaluation of Predicting Equations for Semi-Rigid Connections, M.Sc. Thesis, Vanderbilt University, Nashville, TN. [17] Hoffman, J. J. 1994. Design Procedures and Analysis Tools for Semi-Rigid Composite Members and Frames, M.S. Thesis, University of Minnesota, Minneapolis. [18] Horne, M.R. and Morris, L.J. 1982. Plastic Design of Low-Rise Frames, The MIT Press, Cambridge, MA. [19] IABSE. 1989. Bolted and Special Connections, Proceedings of the International Colloquium held in Moscow, USSR, May 1989, VNIPIP, 4 Vols., IABSE, Zurich. [20] Kishi, N. and Chen, W. F. 1986. Data Base of Steel Beam-to-Column Connections, Vol. 1 & 2, Structural Engineering Report No. CE-STR-86-26, School of Civil Engineering, Purdue University, West Lafayette, IN. [21] Leon, R.T. and Deierlein, G.G. 1996. Considerations for the Use of Quasi-Static Testing, Earthquake Spectra, 12(1), 87-110. [22] Leon, R.T. 1996. Seismic Performance of Bolted and Riveted Connections, in Background
Reports on Metallurgy, Fracture Mechanics, Welding, Moment Connections and Frame System Behavior, SAC Report 95-09, SAC Joint Venture, Sacramento, CA. [23] Leon, R.T., Ammerman, D., Lin, J., and McCauley, R. 1987. Semi-Rigid Composite Steel Frames, AISC Eng. J., 24(4), 147-156.
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[24] Leon, R.T. and Hajjar, J.F. 1996. Effect of Floor Slabs on the Performance of Steel Moment Connections, Proceedings of the 11WCEE, Elsevier, London. [25] Leon, R.T. and Zandonini, R. 1992. Composite Connections, in Steel Design: An International Guide, R. Bjorhovde and P. Dowling, Eds., Elsevier Publishers, London. [26] Leon, R.T., Hoffman, J., and Staeger, T. 1996. Design of Partially-Restrained Composite Connections, AISC Design Guide 9, American Institute of Steel Construction, Chicago, IL. [27] Nethercot, D.A. 1985. Steel Beam to Column Connections — A Review of Test Data and Their Applicability to the Evaluation of the Joint Behaviour of the Performance of Steel Frames, CIRIA, London. [28] HEHRP. 1997. NEHRP Recommended Provisions for Seismic Regulations for New Buildings, BSSC, Washington, D.C. [29] Viest, I.M., Colaco, J.P., Furlong, R.W., Griffis, L.G., Leon, R.T., and Wyllie, L.A. 1996. Composite Construction Design for Buildings, McGraw-Hill, New York.
1999 by CRC Press LLC
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Dexter, R.J. and Fisher, J.W. “Fatigue and Fracture” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Fatigue and Fracture 24.1 Introduction 24.2 Design and Evaluation of Structures for Fatigue Classification of Structural Details for Fatigue • Scale Effects in Fatigue • Distortion and Multiaxial Loading Effects in Fatigue • The Effective Stress Range for Variable-Amplitude Loading • Low-Cycle Fatigue Due to Seismic Loading
24.3 Evaluation of Structural Details for Fracture
Robert J. Dexter and John W. Fisher Department of Civil Engineering, Lehigh University, Bethlehem, PA
Specification of Steel and Filler Metal • Fracture Mechanics Analysis
24.4 Summary 24.5 Defining Terms References Further Reading
24.1 Introduction This chapter provides an overview of aspects of fatigue and fracture that are relevant to design or assessment of structural components made of concrete, steel, and aluminum. This chapter is intended for practicing civil and structural engineers engaged in regulation, design, inspection, repair, and retrofit of a variety of structures ,including buildings; bridges; sign, signal, and luminaire support structures; chimneys; transmission towers ;et c. Established procedures are explained for design and in-service assessment to ensure that structures are resistant to fatigue and fracture. This chapter is not intended as a comprehensive review of the latest research results in the subject area; therefore, many interesting aspects of fatigue and fracture are not discussed. The design and assessment procedures outlined in this chapter maybe applied to other similar structures, even outside the traditional domain of civil engineers, including offshore structures, cranes, heavy vehicle frames, and ships. The mechanical engineering approach, which works well for smooth machine parts, gives an overly optimistic assessment of the fatigue strength of structural details. There are many cases of failures of these types of structures, such as the crane in Figure24.1 or the vehicle frame in Figure24.2, which would have been predicted had the structural engineering approach been applied. The possibility of fatigue must be checked for any structural member that is subjected to cyclic loading. Among the few cases where cracking has occurred in structures, the cracks are usually only a nuisance and may even go unnoticed. Only in certain truly non-redundant structural systems can cracking lead to structural collapse. The loading for most structures is essentially under fixedcan cracking lead to structural collapse. The loading for most structures is essentially under fixedconnections in redundant structures are essentially under displacement-control boundary conditions. In other words, because of the stiffness of the surrounding structure, the ends of the member have to 1999 by CRC Press LLC
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FIGURE 24.1: Fatigue cracking at welded detail in crane boom.
deform in a way that is compatible with nearby members. Under displacement control, a member can continue to provide integrity (e.g., transfer shear) after it has reached ultimate strength and is in the descending branch of the load-displacement curve. This behavior under displacement control is referred to as load shedding. In order for load shedding to be fully effective, individual critical members in tension must elongate to several times the yield strain locally without completely fracturing. Good short-term performance should not lead to complacency, because fatigue and stresscorrosion cracking may take decades to manifest. Corrosion and other structural damage can precipitate and accelerate fatigue and fracture. Also, fabrication cracks may be built into a structure and never discovered. These dormant cracks can fracture if the structure is ever loaded into the inelastic range, such as in an earthquake. Fatigue cracking in steel bridges in the U.S. has become a more frequent occurrence since the 1970s. Figure 24.3 shows a large crack that was discovered in 1970 at the end of a coverplate in one of the Yellow Mill Pond multibeam structures located at Bridgeport, Connecticut. Between 1970 and 1981, 1999 by CRC Press LLC
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FIGURE 24.2: Fatigue cracking at welded detail in vehicle frame. numerous fatigue cracks were discovered at the ends of coverplates in this bridge [19]. Fatigue cracking in bridges, such as shown in Figure 24.3, resulted from an inadequate experimental base and overly optimistic specification provisions developed from the experimental data in the 1960s. The assumption of a fatigue limit at two million cycles proved to be incorrect. As a result of extensive large-scale fatigue testing, it is now possible to clearly identify and avoid details that are expected to have low fatigue strength. The fatigue problems with the older bridges can be avoided in new construction. Fortunately, it is also possible to retrofit or upgrade the fatigue strength of existing bridges with poor details. Low-cycle fatigue is a possible failure mode for structural members or connections that are cycled into the inelastic region for a small number of cycles. For example, bracing members in a braced frame or beam-to-column connections in a welded special-moment frame (WSMF) may be subjected to low-cycle fatigue in an earthquake. In sections that are cyclically buckling, the low-cycle fatigue is linked to the buckling behavior. This emerging area of research is briefly discussed in Section 24.2.5. The primary emphasis in this chapter is on high-cycle fatigue. Truck traffic causes high-cycle fatigue of bridges. Fatigue cracking may occur in industrial buildings subjected to loads from cranes or other equipment or machinery. Although it has not been a problem in the past, fatigue cracking could occur in high-rise buildings frequently subjected to large wind loads. Wind loads have caused numerous fatigue problems in sign, signal, and luminaire support structures [32], transmission towers, and chimneys. Although cracks can form in structures cycled in compression, they arrest and are not structurally significant. Therefore, only members or connections for which the stress cycle is at least partially in tension need to be assessed. If a fatigue crack forms in one element of a bolted or riveted built-up structural member, the crack cannot propagate directly into neighboring elements. Usually, a riveted member will not fail until a second crack forms in another element. Therefore, riveted built-up structural members are inherently redundant. Once a fatigue crack forms, it can propagate directly into all elements of a continuous welded member and cause failure at service loads. The lack of 1999 by CRC Press LLC
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FIGURE 24.3: Fatigue crack originating from the weld toe of a coverplate end detail in one of the Yellow Mill Pond structures.
inherent redundancy in welded members is one reason that fatigue and fracture changed from a nuisance to a significant structural integrity problem as welding became widespread in the 1940s. Welded structures are not inferior to bolted or riveted structures; they just require more attention to design, detailing, and quality. In structures such as bridges and ships, the ratio of the fatigue-design load to the strength-design loads is large enough that fatigue may control the design of much of the structure. In long-span bridges, the load on much of the superstructure is dominated by the dead load, with the fluctuating live load relatively small. These members will not be sensitive to fatigue. However, the deck, stringers, and floorbeams of bridges are subjected to primarily live load and therefore may be controlled by fatigue. In structures controlled by fatigue, fracture is almost always preceded by fatigue cracking; therefore, the primary emphasis should be on preventing fatigue. Usually, the steel and filler metal have minimum specified toughness values (such as a Charpy V-Notch [CVN] test requirement). In this case, the cracks can grow to be quite long before fracture occurs. Fatigue cracks grow at an exponentially increasing rate; therefore, most of the life transpires while the crack is very small. Additional fracture toughness, greater than the minimum specified values, will allow the crack to grow to a larger size before sudden fracture occurs. However, the crack is growing so rapidly at the end of life that the additional toughness may increase the life only insignificantly. However, fracture is possible for buildings that are not subjected to cyclic loading. Several large tension chords of long-span trusses fractured while under construction in the 1980s. The tension chords consisted of welded jumbo shapes, i.e., shapes in groups 4 and 5, as shown in Figure 24.4 [22]. These jumbo shapes are normally used for columns, where they are not subjected to tensile stress. These sections often have low fracture toughness, particularly in the core region of the web and flange junction. The low toughness has been attributed to the relatively low rolling deformation and slow cooling in these thick shapes. The low toughness is of little consequence if the section is used as a column and remains in compression. The fractures of jumbo tension chords occurred at welded 1999 by CRC Press LLC
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FIGURE 24.4: (a) View of jumbo section used as tension chord in a roof truss and (b) closeup view of fracture in web originating from weld access holes at welded splice.
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splices at groove welds or at flame-cut edges of cope holes, as shown in Figure 24.4. In both cases the cracks formed at cope holes in the hard layer formed from thermal cutting. These cracks propagated in the core region of these jumbo sections, which has very low toughness. As a consequence of these brittle fractures, AISC (American Institute of Steel Construction) specifications now have a supplemental CVN notch toughness requirement for shapes in groups 4 and 5 and (for the same reasons) plates greater than 51 mm thick, when these are welded and subject to primary tensile stress from axial load or bending. Poorly prepared cope holes have resulted in cracks and fractures in lighter shapes as well. The detailing rules that are used to prevent fatigue are intended to avoid notches and other stress concentrations. These detailing rules are useful for the avoidance of brittle fracture as well as fatigue. For example, the detailing rules in AASHTO (American Association of State Highway Transportation Officials) bridge design specifications would not permit a backing bar to be left in place because of the unfused notch perpendicular to the tensile stress in the flange. Along with low-toughness weld metal, this type of backing bar notch was a significant factor in the brittle fracture of WSMF connections in the Northridge earthquake [33, 53, 55]. Figure 24.5 shows a cross-section of a beam-flange-tocolumn weld from a building that experienced such a fracture. It is clear that the crack emanated from the notch created by the backing bar.
FIGURE 24.5: Welded steel moment frame (WSMF) connection showing (a) location of typical fractures and (b) typical crack, which originated at the backing bar notch and propagated into the column flange.
Detailing rules similar to the AASHTO detailing rules are included in American Welding Society (AWS) D1.1 Structural Welding Code—Steel for dynamically loaded structures. Dynamically loaded has been interpreted to mean fatigue loaded. Unfortunately, most seismically loaded building frames have not been required to be detailed in accordance with these rules. Even though it is not required, it might be prudent in seismic design to follow the AWS D1.1 detailing rules for all dynamically loaded structures. 1999 by CRC Press LLC
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Design for fracture resistance in the event of an extreme load is more qualitative than fatigue design, and usually does not involve specific loads. Details are selected to maximize the strength and ductility without increasing the basic section sizes required to satisfy strength requirements. The objective is to get the yielding to spread across the cross-section and develop the reserve capacity of the structural system without allowing premature failure of an individual component to precipitate total failure of the structure. The process of design for fracture resistance involves (1) predicting conceivable failure modes due to extreme loading, then (2) correctly selecting materials for and detailing the “critical” members and connections involved in each failure mode to achieve maximum ductility. Critical members and connections are those that are required to yield, elongate, or form a plastic hinge before the ultimate strength can be achieved for these conceived failure modes. Usually, the cost to upgrade a design meeting strength criteria to also be resistant to fatigue and fracture is very reasonable. The cost may increase due to (1) details that are more expensive to fabricate, (2) more expensive welding procedures, and (3) more expensive materials. Quantitative means for assessing fracture are presented. Because of several factors, there is at best only about ± 30% accuracy in these fracture predictions, however. These factors include (1) variability of material properties; (2) changes in apparent toughness values with changes in test specimen size and geometry; (3) differences in toughness and strength of the weld zone; (4) complex residual stresses; (5) high gradients of stress in the vicinity of the crack due to stress concentrations; and (6) the behavior of cracks in complex structures of welded intersecting plates.
24.2
Design and Evaluation of Structures for Fatigue
Testing on full-scale welded members has indicated that the primary effect of constant amplitude loading can be accounted for in the live-load stress range [15, 20, 21, 34]; that is, the mean stress is not significant. The reason that the dead load has little effect on the lower bound of the results is that, locally, there are very high residual stresses. In details that are not welded, such as anchor bolts, there is a strong mean stress effect [54]. A worst-case conservative assumption (i.e., a high-tensile mean stress) is made in the testing and design of these nonwelded details. The strength and type of steel have only a negligible effect on the fatigue resistance expected for a particular detail. The welding process also does not typically have an effect on the fatigue resistance. The independence of the fatigue resistance from the type of steel greatly simplifies the development of design rules for fatigue since it eliminates the need to generate data for every type of steel. The established approach for fatigue design and assessment of metal structures is based on the S-N curve. Typically, small-scale specimen tests will result in longer apparent fatigue lives. Therefore, the S-N curve must be based on tests of full-size structural components such as girders. The reasons for these scale effects are discussed in Section 24.2.2. When information about a specific crack is available, a fracture mechanics crack growth rate analysis should be used to calculate remaining life [9, 10]. However, in the design stage, without specific initial crack size data, the fracture mechanics approach is not any more accurate than the S-N curve approach [35]. Therefore, the fracture mechanics crack growth analysis will not be discussed further. Welded and bolted details for bridges and buildings are designed based on the nominal stress range rather than the local “concentrated” stress at the weld detail. The nominal stress is usually obtained from standard design equations for bending and axial stress and does not include the effect of stress concentrations of welds and attachments. Usually, the nominal stress in the members can be easily calculated without excessive error. However, the proper definition of the nominal stresses may become a problem in regions of high stress gradients. The lower-bound S-N curves for steel in the AASHTO, AISC, AWS, and the American Railway Engineers Association (AREA) provisions are shown in Figure 24.6. These S-N curves are based on a lower bound with a 97.5% survival limit. S-N curves are presented for seven categories (A through 1999 by CRC Press LLC
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FIGURE 24.6: The AASHTO/AISC S-N curves. Dashed lines are the constant-amplitude fatigue limits and indicate the detail category. E0 ) of weld details. The effect of the welds and other stress concentrations is reflected in the ordinate of the S-N curves for the various detail categories. The slope of the regression line fit to the test data for welded details is typically in the range 2.9 to 3.1 [34]. Therefore, in the AISC and AASHTO codes as well as in Eurocode 3 [18], the slopes have been standardized at 3.0. Figure 24.6 shows the constant-amplitude fatigue limits (CAFLs) for each category as horizontal dashed lines. The CAFLs in Figure 24.6 were determined from the full-scale test data. When constantamplitude tests are performed at stress ranges below the CAFL, noticeable cracking does not occur. Note that for all but category A, the fatigue limits occur at numbers of cycles much greater than two million, and therefore the CAFL should not be confused with the fatigue strength. Fatigue strength is a term representing the nominal stress range corresponding to the lower-bound S-N curve at a particular number of cycles, usually two million cycles. Most structures experience what is known as long-life variable-amplitude loading, i.e., very large numbers of random-amplitude cycles greater than the number of cycles associated with the CAFL. For example, a structure loaded continuously at an average rate of three times per minute (0.05 Hz) would accumulate 10 million cycles in only 6 years. The CAFL is the only important property of the S-N curve for long-life variable-amplitude loading, as discussed further in Section 24.2.4. Similar S-N curves have been proposed by the Aluminum Association for welded aluminum structures. Table 24.1 summarizes the CAFLs for steel and aluminum for categories A through E 0 . The design procedures are based on associating weld details with specific categories. For both steel and aluminum, the separation of details into categories is approximately the same. Since fatigue is typically only a serviceability problem, fatigue design is carried out using service loads. The nominal stress approach is simple and sufficiently accurate, and therefore is preferred when applicable. However, for details not covered by the standard categories, or for details in the presence of secondary stresses or high-stress gradients, the “hot-spot” stress range approach may be the only alternative. The hot-spot stress range is the stress range in a plate normal to the weld axis at some small distance from the weld toe. The hot-spot stress may be determined by strain gage measurement, finite element analysis, or empirical formulas. Unfortunately, methods and locations for measuring or calculating hot-spot stress as well as the associated S-N curve vary depending on which code or recommendation is followed [56]. 1999 by CRC Press LLC
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TABLE 24.1 Constant-Amplitude Fatigue Limits for AASHTO and Aluminum Association S-N Curves Detail category
CAFL for steel (MPa)
CAFL for aluminum (MPa)
A B B0 C D E E0
165 110 83 69 48 31 18
70 41 32 28 17 13 7
In U.S. practice (i.e., the hot-spot method that has been used with the American Petroleum Institute’s API RP-2A and AWS D1.1) the hot-spot stress is determined with a strain gage located nominally 5mm from the weld toe [16]. Actually, the 5-mm distance was not specifically selected. Rather, this distance was just the closest to the weld toe that a 3-mm strain gage could be placed. This definition of hot-spot stress originated from early experimental work on pressure vessels and tubular joints and has been the working definition of hot-spot stress in the U.S. offshore industry [40]. This approach is also used for other welded tubular joints and for details in ships and other marine structures. The S-N curve used with the hot-spot stress approach is essentially the same as the nominal stress S-N curve (category C) for a transverse butt or fillet weld in a nominal membrane stress field (i.e., a stress field without any global stress concentration). The geometrical stress concentration and discontinuities associated with the local weld toe geometry are built into the S-N curve, while the global stress concentration is included in the hot-spot stress range.
24.2.1
Classification of Structural Details for Fatigue
It is standard practice in fatigue design of welded structures to separate the weld details into categories having similar fatigue resistance in terms of the nominal stress. Most common details can be idealized as analogous to one of the drawings in the specifications. The categories in Figure 24.6 range from A to E0 in order of decreasing fatigue strength. There is an eighth category, F, in the specifications, which applies to fillet welds loaded in shear. However, there have been very few if any failures related to shear, and the stress ranges are typically very low such that fatigue rarely would control the design. Therefore, the shear stress category F will not be discussed further. In fact there have been very few if any failures attributed to details that have a fatigue strength greater than category C. Most structures have many more severe details, and these will generally govern the fatigue design. Therefore, unless all connections in highly stressed elements of the structure are high-strength bolted connections rather than welded, it is usually a waste of time to check category C and better details. Therefore, only category C and more severe details will be discussed in this section. Severely corroded members should be evaluated to determine the stress range with respect to the reduced thickness and loss of section. Corrosion notches and pits may lead to fatigue cracks and should be specially evaluated. Otherwise severely corroded members may be treated as category E [44]. In addition to being used by AISC and AASHTO specifications, the S-N curves in Figure 24.6 and detail categories are essentially the same as those adopted by the AREA and AWS Structural Welding Code D1.1. The AASHTO/AISC S-N curves are also the same as 7 of the 11 S-N curves in the Eurocode 3. The British Standard (BS) 7608 has slightly different S-N curves, but these can be correlated to the nearest AISC S-N curve for comparison. The following is a brief simplified overview of the categorization of fatigue details. In all cases, the applicable specifications should also be checked. Several reports have been published that show a large number of illustrations of details and their categories in addition to those in AISC and AASHTO 1999 by CRC Press LLC
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specifications [14, 57]. Also, the Eurocode 3 and the BS 7608 have more detailed illustrations for their categorization than does the AISC or AASHTO specifications. Maddox [38] discusses categorization of many details in accordance with BS 7608, from which roughly equivalent AISC categories can be inferred. In most cases, the fatigue strength recommended in these European standards is similar to the fatigue strength in the AISC and AASHTO specifications. However, there are several cases where the fatigue strength is significantly different; usually the European specifications are more conservative. Some of these cases are discussed in the following, as well as the fatigue strength for details that are not found in the specifications. Mechanically Fastened Joints
Small holes are considered category D details. Therefore, rivetted and mechanically fastened joints (other than high-strength bolted joints) loaded in shear are evaluated as category D in terms of the net-section nominal stress. Pin plates and eyebars are designed as category E details in terms of the stress on the net section. In the AISC specifications, bolted joints loaded in direct tension are evaluated in terms of the maximum unfactored tensile load, including any prying load. Typically, these provisions are applied to hanger-type or bolted flange connections where the bolts are tensioned against the plies. If the number of cycles exceeds 20,000, the allowable load is reduced relative to the allowable load for static loading. Prying is very detrimental to fatigue, so if the number of cycles exceeds 20,000, it is advisable to minimize prying forces. When bolts are tensioned against the plies, the total fluctuating load is resisted by the whole area of the precompressed plies, so that the bolts are subjected to only a fraction of the total load [37]. The analysis to determine this fraction is difficult, and this is one reason that the bolts are designed in terms of the maximum load rather than a stress range in the AISC specifications. In BS 7608, a slightly different approach is used for bolts in tension that achieves approximately the same result as the AISC specification for high-strength bolts. The stress range, on the tensile stress area of the bolt, is taken as 20% of the total applied load, regardless of the fluctuating part of the total load. The S-N curve for bolts is proportional to Fu , so that for high-strength bolts the result is an S-N curve between category E and E0 for cycles less than two million. The tensile stress area, At , is given by π At = 4
0.9743 2 db − n
(24.1)
where db = the nominal diameter (the body or shank diameter) n = threads per inch (Note that the constant would be different if SI units were used.) In the Eurocode 3, the fatigue strength of bolts is given in terms of the actual stress range in the bolts, although it is not clear how to calculate this for pretensioned connections. The recommended fatigue strength is given in terms of the tensile stress area of the bolt and does not depend on tensile strength. The design S-N curve from Eurocode 3 is about the same as category E0 , which is consistent with BS 7608 for high-strength bolts. Anchor bolts in concrete cannot be adequately pretensioned and therefore do not behave like hanger-type or bolted flange connections. At best they are pretensioned between nuts on either side of the column base plate and the part below the bottom nut is still exposed to the full load range. Some additional test data was recently generated at the ATLSS Center at Lehigh University [54] for grade 55 and grade 105 anchor bolts. When combined with the existing data [27], the data show that the fatigue strength for anchor bolts is slightly greater than category E0 in terms of the stress range on the tensile stress area of the bolt. Some of the bolts were tested with an intentional misalignment 1999 by CRC Press LLC
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of 1:40, and these had only slightly lower fatigue strength than the aligned bolts, bringing the lower bound of the data closer to the category E0 S-N curve. The ATLSS data show the CAFL for all anchor bolts is slightly greater than the category D CAFL (48 MPa). The ATLSS data and Karl Frank’s data show that proper tightening between the double nuts had a slight beneficial effect on the CAFL, but not enough to increase it by one category. In summary, for all types of bolts, if the actual stress range on the tensile stress area can be determined, it is recommended that (1) for finite life, the category E0 S-N curve be used, and (2) for infinite life, the CAFL equivalent to that for category D be used (48 MPa). Welded Joints
Welded joints are considered longitudinal if the axis of the weld is parallel to the primary stress range. Continuous longitudinal welds are category B or B0 details. However, the termination of longitudinal fillet welds is more severe (category E). (The termination of full-penetration groove longitudinal welds requires a ground transition radius but gives a greater fatigue strength, depending on the radius.) If longitudinal welds must be terminated, it is better to terminate at a location where the stress ranges are less severe. Category C includes transverse full-penetration groove welds (butt joints) subjected to nondestructive evaluation (NDE). Experiments conducted at Lehigh University showed that groove welds containing large internal discontinuities that were not screened out by NDE had a fatigue strength comparable to category E. The BS 7608 and the British Standards Institute published document PD 6493 [12] have reduced fatigue strength curves for groove welds with defects that are generally in agreement with these experimental data. Transverse groove welds with a permanent backing bar are reduced to category D [38]. One-sided welds with melt through (without backing bars) are also classified as category D. Cope holes for weld access and to avoid intersecting welds, with edges conforming to the ANSI (American National Standards Institute) smoothness of 1000, may be considered a category D detail. Poorly executed cope holes must be treated as a category E detail. In some cases small cracks have occurred from the thermal-cut edges if martensite is developed. In those cases, crack extension will occur at lower stress ranges. Testing performed at ATLSS as well as at TNO in the Netherlands [17] has shown that the cope hole has lower fatigue strength than overlapping welds, which are less expensive but have traditionally been avoided because of the discontinuity at the overlap. There have been many fatigue-cracking problems in structures at miscellaneous and seemingly unimportant attachments to the structure for such things as racks and hand rails. Attachments are a “hard spot” on the strength member that create a stress concentration at the weld. Often, it is not realized that such secondary members become part of the girder, i.e., that these secondary members stretch with the girder and therefore are subject to large stress ranges. Consequently, problems have occurred with fatigue of such secondary members. Attachments normal to flanges or plates that do not carry significant load are rated category C if less than 51 mm long in the direction of the primary stress range, D if between 51 and 101 mm long, and E if greater than 101 mm long. (The 101-mm limit may be smaller for plates thinner than 9 mm.) If there is not at least 10 mm edge distance, then category E applies for an attachment of any length. The category E0 , slightly worse than category E, applies if the attachment plates or the flanges exceed 25 mm in thickness. Transverse stiffeners are treated as short attachments (category C). Note that the attachment to the round tube in the crane boom in Figure 24.1, the transverse attachment in the vehicle frame in Figure 24.2, and the coverplate end detail in Figure 24.3 are all category E and E0 attachments. The cruciform joint where the load-carrying member is discontinuous is considered a category C detail because it is assumed that the plate transverse to the load-carrying member does not have any stress range. A special reduction factor for the fatigue strength is provided when the load-carrying 1999 by CRC Press LLC
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plate exceeds 13 mm in thickness. This factor accounts for the possible crack initiation from the unfused area at the root of the fillet welds (as opposed to the typical crack initiation at the weld toe for thinner plates) [26]. An example of cracking through the fillet weld throat of an attachment plate is shown in Figure 24.7.
FIGURE 24.7: Cracking through the throat of fillet welds on an attachment plate.
Transverse stiffeners that are used for cross-bracing or diaphragms are also treated as category C details with respect to the stress in the main member. In most cases, the stress range in the stiffener from the diaphragm loads is not considered because these loads are typically unpredictable. In any case, the stiffener must be attached to the flanges, so even if the transverse loads were significant, most of the load would be transferred in shear to the flanges. (The web has very little out-of-plane stiffness.) In theory, the shear stress range in the fillet welds to the flanges should be checked, but shear stress ranges rarely govern design. In most other types of load-carrying attachments, there is interaction between the stress range in the transverse load-carrying attachment and the stress range in the main member. In practice, each of these stress ranges is checked separately. The attachment is evaluated with respect to the stress range in the main member and then it is separately evaluated with respect to the transverse stress range. The combined multiaxial effect of the two stress ranges is taken into account by a decrease in the fatigue strength; that is, most load-carrying attachments are considered category E details. Multiaxial effects are discussed in greater detail in Section 24.2.3. If the fillet or groove weld ends of a longitudinal attachment (load bearing or not) are ground smooth to a transition radius greater than 50 mm, the attachment can be considered category D (load bearing or not). If the transition radius of a groove-welded longitudinal attachment is increased to greater than 152 mm (with the groove-weld ends ground smooth), the detail (load bearing or not) can be considered category C. Misalignment is a primary factor in susceptibility to cracking. The misalignment causes eccentric loading, local bending, and stress concentration. If the ends of a member with a misaligned connection are essentially fixed, the stress concentration factor (SCF) associated with misalignment is (24.2) SCF = 1.0 + 3e/t 1999 by CRC Press LLC
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where e is the eccentricity and t is the smaller of the thicknesses of two opposing loaded members. The nominal stress times the SCF should then be compared to the appropriate category. Generally, such misalignment should be avoided at fatigue critical locations. Equation 24.2 can also be used where e is the distance that the weld is displaced out of plane due to angular distortion. In either case, if the ends are pinned, the SCF is twice as large. A thorough guide to the SCF for various types of misalignment and distortion, including plates of unequal thickness, can be found in the British Standards Institute published document PD 6493 [12]. Reinforced and Prestressed Concrete and Bridge Stay Cables
Concrete structures are typically less sensitive to fatigue than welded steel and aluminum structures. However, fatigue may govern the design when impact loading is involved, such as for pavement, bridge decks, and rail ties. Also, as the age of concrete girders in service increases, and as the applied stress ranges increase with increasing strength of concrete, the concern for fatigue in concrete structural members has also increased. According to ACI (American Concrete Institute) Committee Report 215R-74 in the Manual of Standard Practice [2], the fatigue strength of plain concrete at 10 million cycles is approximately 55% of the ultimate strength. However, even if failure does not occur, repeated loading may contribute to premature cracking of the concrete, such as inclined cracking in prestressed beams. This cracking can then lead to localized corrosion and fatigue of the reinforcement [30]. The fatigue strength of straight, unwelded reinforcing bars and prestressing strand can be described (in terms of the categories for steel details) with the category B S-N curve. The lowest stress range that has been known to cause a fatigue crack in a straight reinforcing bar is 145 MPa, which occurred after more than a million cycles. As expected, based on the results for steel details, minimum stress and yield strength had minimal effect on the fatigue strength of reinforcing bars. Bar size, geometry, and deformations also had minimal effect. ACI Committee 215 [2] suggested that members be designed to limit the stress range in the reinforcing bar to 138 MPa for high levels of minimum stress (possibly increasing to 161 MPa for less minimum stress). Fatigue tests show that previously bent bars had only about half the fatigue strength of straight bars, and failures have occurred down to 113 MPa [47]. Committee 215 recommends that half of the stress range for straight bars be used (i.e., 69 MPa) for the worst-case minimum stress. Equating this recommendation to the S-N curves for steel details, bent reinforcement may be treated as a category D detail. Provided the quality is good, butt welds in straight reinforcing bars do not significantly lower the fatigue strength. However, tack welds reduce the fatigue strength of straight bars about 33%, with failures occurring as low as 138 MPa. Fatigue failures have been reported in welded wire fabric and bar mats [51]. If prestressed members are designed with sufficient precompression that the section remains uncracked, there is not likely to be any problem with fatigue. This is because the entire section is resisting the load ranges and the stress range in the prestessing strand is minimal. Similarly, for unbonded prestressed members, the stress ranges will be very small. Although the fatigue strength of prestressing strand in air is about equal to category B, when the anchorages are tested as well, the fatigue strength of the system is as low as half the fatigue strength of the wire alone (i.e., about category E). However, there is reason to be concerned for bonded prestressing at cracked sections because the stress range increases locally. The concern for cracked sections is even greater if corrosion is involved. The pitting from corrosive attack can dramatically lower the fatigue strength of reinforcement [30]. The above data were generated in tests of the prestressing systems in air. When actual beams are tested, the situation is very complex, but it is clear that much lower fatigue strength can be obtained [45, 48]. Committee 215 has recommended the following for prestressed beams: 1. The stress range in prestressed reinforcement, determined from an analysis considering the section to be cracked, shall not exceed 6% of the tensile strength of the reinforcement. 1999 by CRC Press LLC
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(Note: This is approximately equivalent to category C.) 2. Without specific experimental data, the fatigue strength of unbonded reinforcement and their anchorages shall be taken as half of the fatigue strength of the prestressing steel. (Note: This is approximately equivalent to Category E.) Lesser values shall be used at anchorages with multiple elements. The Post-Tensioning Institute (PTI) has issued “Recommendations for Stay Cable Design and Testing”. The PTI recommends that uncoupled bar stay cables are category B details, while coupled (glued) bar stay cables are category D. The fatigue strengths of stay cables are verified through fatigue testing. Two types of tests are performed: (1) fatigue testing of the strand and (2) testing of relatively short lengths of the assembled cable with anchorages. The recommended test of the system is two million cycles at a stress range (158 MPa) that is 35 MPa greater than the fatigue allowable for category B at two million cycles. This test should pass with less than 2% wire breaks. A subsequent proof test must achieve 95% of the actual ultimate tensile strength of the tendons.
24.2.2
Scale Effects in Fatigue
As previously mentioned, fatigue tests on small-scale specimens will give higher apparent fatigue strength and are therefore unconservative [39, 41, 46]. There are several possible reasons for the observed scale effects. First, there is a well-known thickness effect in fatigue. This thickness effect is reflected in many places in AASHTO and AISC specifications where the fatigue strength is reduced for details with plate thickness greater than 20 or 25 mm in certain cases. For example, when coverplates exceed 25 mm in thickness or are wider than the flange, category E0 applies rather than category E. However, there may be cases where the coverplate is both wider than the flange and thicker than 25 mm. The fatigue strength in this case may be even less than category E0 . One such case is shown in Figure 24.8, which is a wind-bracing gusset attached to the bottom of a floorbeam flange. The fatigue crack began at the termination of the fillet weld (along the top weld toe) where the plates overlap gusset laps. In BS 7608, the fatigue strength of many details are keyed to plates with thickness 16 mm and less. For plates exceeding 16 mm, an equation is given that reduces the fatigue strength for thicker plates. A similar equation is used in Eurocode 3 for plates greater than 25 mm thick. These equations produce reductions in fatigue strength proportional to the 1/4 power of the ratio of the thickness to the base thickness (i.e., 16 or 25 mm). Another effect is that the applied stress range may be different in small-scale specimens. For example, the stress concentration associated with welded attachments varies with the length of the attachment in the direction of the stresses. Also, in large-scale specimens, even though the nominal stress state is uniaxial or bending, unique local multiaxial stress states may develop naturally in complex details from random stress concentrations (e.g., poor workmanship and weld shape) and eccentricities (e.g., asymmetry of the design, tolerances, misalignment, distortion from welding). These complex natural stress states may be difficult to simulate in small-scale specimens and are difficult if not impossible to simulate analytically. The state of residual stress from welding may be significantly different for small specimens due to the lack of constraint. Even if the specimens are cut from large-scale members, the residual stress will be altered. Finally, the volume of weld metal in full-scale members is sufficient to contain a structurally relevant representative sample of discontinuities (e.g., microcracks, pores, slag inclusions, hydrogen cracks, tack welds, and other notches).
1999 by CRC Press LLC
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FIGURE 24.8: Fatigue crack originating from the upper weld toe of a fillet weld where the fillet weld terminates near the overlap of thick plates.
24.2.3
Distortion and Multiaxial Loading Effects in Fatigue
In the AASHTO/AISC fatigue design provisions, the loading is assumed to be simple uniaxial loading. However, the loading may often be more complex than is commonly assumed in design. For example, fatigue design is based on the primary tension and bending stress ranges. Torsion, racking, transverse bending, and membrane action in plating are considered secondary loads and are typically not considered in fatigue analysis. However, it is clear from the type of cracks that occur in bridges that a significant proportion of the cracking is due to distortion resulting from such secondary loading [24]. The solution to the problem of fatigue cracking due to secondary loading usually relies on the qualitative art of good detailing. Often, the best solution to distortion cracking problems may be to stiffen the structure. Typically, the 1999 by CRC Press LLC
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better connections are more rigid. For example, transverse bracing and floorbeam attachment plates on welded girders should be welded directly to both flanges as well as the web. Numerous fatigue cracks have occurred due to distortion in the “web gap”, i.e., the narrow gap between the stiffener and the flange. Figure 24.9 shows an example of a longitudinal crack formed along the longitudinal fillet weld that originated in the web gap (between the top of the stiffener and the flange) when such attachment plates are not welded to the flange. There has been a tendency to avoid welding to the tension flange due to an unfounded concern about brittle fracture.
FIGURE 24.9: Typical distortion-induced cracking in the web gap of an attachment plate.
In some cases a better solution is to allow the distortion to take place over a greater area so that lower stresses are created; that is, the detail should be made more flexible. For example, if a transverse stiffener is not welded to a flange, it is important to ensure that the gap between the flange and the end of the stiffener is sufficiently large, between four and six times the thickness of the web [23]. Another example where the best details are more flexible is connection angles for simply supported beams. Despite our assumptions, such simple connections transmit up to 40% of the theoretical fixed-end moment, even though they are designed to transmit only shear forces. For a given load, the moment in the connection decreases significantly as the rotational stiffness of the connection decreases. The increased flexibility of connection angles allows the limited amount of end rotation to take place with reduced bending stresses. A criterion has been developed for the design of these angles to provide sufficient flexibility [58]. The criterion states that the angle thickness (t) must be (24.3) t < 12 g 2 /L where g is the gage in inches and L is the span length in inches. For example, for connection angles with a gage of 76 mm and a beam span of 7 m, the angle thickness should be just less than 10 mm. To solve a connection-angle cracking problem in service, the topmost rivet or bolt may be removed and replaced with a loose bolt to ensure the shear capacity. For loose bolts, steps are required to ensure that the nuts do not back off. Significant stresses from secondary loading are often in a different direction than the primary stresses. Fortunately, experience with multiaxial loading experiments on large-scale welded structural 1999 by CRC Press LLC
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details indicates the loading perpendicular to the local notch or the weld toe dominates the fatigue life. The cyclic stress in the other direction has no effect if the stress range is below 83 MPa and only a small influence above 83 MPa [16, 24]. The recommended approach for multiaxial loads is 1. Decide which loading (primary or secondary) dominates the fatigue cracking problem (typically the loading perpendicular to the weld axis or perpendicular to where cracks have previously occurred in similar details). 2. Perform the fatigue analysis using the stress range in this direction (i.e., ignore the stresses in the orthogonal directions).
24.2.4
The Effective Stress Range for Variable-Amplitude Loading
An actual service load history is likely to consist of cycles with a variety of different load ranges, i.e., variable-amplitude loading [25]. However, the fatigue design provisions are based on constantamplitude loading and do not give any guidance for variable-amplitude loading. A procedure is shown below to convert variable stress ranges to an equivalent constant-amplitude stress range with the same number of cycles. This procedure is based on the damage summation rule jointly credited to Palmgren and Miner (referred to as Miner’s rule) [42]. If the slope of the S-N curve is equal to 3, then the relative damage of stress ranges is proportional to the cube of the stress range. Therefore, the effective stress range is equal to the cube root of the mean cube of the stress ranges, i.e., h i1/3 Seffective = (ni /Ntotal ) Si3
(24.4)
The LRFD (load and resistance factor design) version of the AASHTO specification implies such an effective stress range using the straight line extension of the constant-amplitude curve. This is essentially the approach for variable-amplitude loading in BS 7608. Eurocode 3 also uses the effective stress range concept. Research on such high-cycle variable-amplitude fatigue has shown that if all but 0.01% of the stress ranges are below the CAFLs, fatigue cracking does not occur [25]. The simplified fatiguedesign procedure in the AASHTO LRFD Bridge Design Specifications [1] for structures with very large numbers of cycles is based on this observation. The objective of the AASHTO fatigue-design procedure is to ensure that the stress ranges at critical details due to a fatigue limit state load range are less than the CAFL for the particular details. The fatigue limit state load range is defined as having a probability of exceedence over the lifetime of the structure of 0.01%. A structure with millions of cycles is likely to see load ranges with this magnitude or greater hundreds of times; therefore, the fatigue limit state load range is not as large as the extreme loads used to check ultimate strength.
24.2.5
Low-Cycle Fatigue Due to Seismic Loading
Steel-braced frames and moment-resisting frames are expected to withstand cyclic plastic deformation without cracking in a large earthquake. If brittle fracture of these moment frame connections is suppressed, the connections can be cyclically deformed into the plastic range and will eventually fail by tearing at a location of strain concentration. This failure mode can be characterized as low-cycle fatigue. Low-cycle fatigue has been studied for pressure vessels and some other types of mechanical engineering structures. Since low-cycle fatigue is an inelastic phenomenon, the strain range is the key parameter rather than the stress range. However, at this time very little is understood about low-cycle fatigue in structures. For example, it is a very difficult task just to predict accurately the local strain range at a location of cyclic buckling. 1999 by CRC Press LLC
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Research performed to date indicates the feasibility of predicting curves for low-cycle fatigue from strain range vs. number of cycles in a manner analogous to high-cycle fatigue design using stressrange-based S-N curves. For example, low-cycle fatigue experiments were performed on specimens that would buckle as well as compact specimens that would not buckle but rather would fail from cracking at the welds [36]. These tests showed that the number of cycles to failure by low-cycle fatigue of welded connections could be predicted by the local strain range in a power law that is analogous to the power law (with stress range) represented by an S-N curve. They also showed that Miner’s rule could be used to predict the number of variable-amplitude cycles to failure based on constant-amplitude test data. More recently, Castiglioni [8, 13] has conducted similar experiments and plotted the results in terms of a fictitious elastic stress range that is equal to the strain range times the modulus of elasticity. In this manner he has shown that the low-cycle fatigue data plot along the same S-N curves from the Eurocode (similar to the AASHTO S-N curves) that are normally used for high-cycle fatigue. Castiglioni has equated the slenderness of the flanges with different fatigue categories, in effect treating the propensity for buckling like a “notch”. It can be hypothesized from these preliminary data that the same model used for high-cycle fatigue design, i.e., the S-N curves (converted to strain), can be used to predict fatigue behavior in the verylow-cycle regime characteristic of earthquake loading. Such a model could be very useful in seismic design of welded and bolted steel connections. Just by inspection, alternative details for a connection can be ranked in accord with their expected fatigue strength, i.e., the expected strain range that would cause cracking after a certain minimum number of cycles. After some limited verification through very-low-cycle inelastic experiments, these comparisons could rely on the existing knowledge base for the relative fatigue strength of various details in high-cycle fatigue. The detailing rules that are used to prevent high-cycle fatigue are intended to avoid notches and other stress concentrations. These detailing rules could also be useful for preventing brittle fracture and premature low-cycle fatigue cracking. The relative fatigue strength is given by the detail category and the corresponding S-N curve.
24.3
Evaluation of Structural Details for Fracture
Unlike fatigue, fracture behavior depends strongly on the type and strength level of the steel or filler metal. In general, fracture toughness has been found to decrease with increasing yield strength of a material, suggesting an inverse relationship between the two properties. In practice, however, fracture toughness is more complex than implied by this simple relationship since steels with similar strength levels can have widely varying levels of fracture toughness. Steel exhibits a transition from brittle to ductile fracture behavior as the temperature increases. For example, Figure 24.10 shows a plot of the energy required to fracture CVN impact test specimens of A588 structural steel at various temperatures. These results are typical for ordinary hot-rolled structural steel. The transition phenomena shown in Figure 24.10 is a result of changes in the underlying microstructural fracture mode. There are really at least three distinct types of fracture with distinctly different behavior. 1. Brittle fracture is associated with cleavage, which is transgranular fracture on select crystallographic planes on a microscopic scale. This type of fracture occurs at the lower end of the temperature range, although the brittle behavior can persist up to the boiling point of water in some low-toughness materials. This part of the temperature range is called the lower shelf because the minimum toughness is fairly constant up to the transition temperature. Brittle fracture is sometimes called elastic fracture because the plasticity that occurs is negligible and consequently the energy absorbed in the fracture process is also negligible. 1999 by CRC Press LLC
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FIGURE 24.10: Charpy energy transition curve for A588 grade 50 (350-MPa yield strength) structural steel.
2. Transition-range fracture occurs at temperatures between the lower shelf and the upper shelf and is associated with a mixture of cleavage and fibrous fracture on a microstructural scale. Because of the mixture of micromechanisms, transition-range fracture is characterized by extremely large variability. Fracture in the transition region is sometimes referred to as elastic-plastic fracture because the plasticity is limited in extent but has a significant impact on the toughness. 3. Ductile fracture is associated with a process of void initiation, growth, and coalescence on a microstructural scale, a process requiring substantial energy, and occurs at the higher end of the temperature range. This part of the temperature range is referred to as the upper shelf because the toughness levels off and is essentially constant for higher temperatures. Ductile fracture is sometimes called fully plastic fracture because there is substantial plasticity across most of the remaining cross-section ahead of a crack. Ductile fracture is also called fibrous fracture due to the fibrous appearance of the fracture surface, or shear fracture due to the usually large slanted shear lips on the fracture surface. Ordinary structural steel such as A36 or A572 is typically only hot rolled. To achieve very hightoughness, steels must be controlled rolled, i.e., rolled at lower temperatures, or must receive some auxiliary heat treatment such as normalization. In contrast to the weld metal, the cost of the steel is a major part of total costs. The expense of the high-toughness steels has not been found to be warranted for most building and bridges, whereas the cost of high-toughness filler metal is easily justifiable. Hot-rolled steels, which fracture in the transition region at the lowest service temperatures, have sufficient toughness for the required performance of most welded buildings and bridges.
24.3.1
Specification of Steel and Filler Metal
ASTM (American Society for Testing and Materials) specifications for bridge steel (A709) and ship steel (A131) provide for minimum CVN impact test energy levels. Structural steel specified by A36, A572, or A588, without supplemental specifications, does not require the Charpy test to be performed. If there is concern about brittle fracture and either (1) high ductility demand, (2) concern with lowtemperature exposed structures, or (3) dynamic loading, then the CVN impact test should be specified 1999 by CRC Press LLC
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by the purchaser of steel as a supplemental requirement. The results of the CVN test, impact energies, are often referred to as “notch-toughness” values. Because the Charpy test is relatively easy to perform, it will likely continue to be the measure of toughness used in steel specifications. In the range of temperatures called the brittle region, the Charpy notch toughness is approximately correlated with Kc from fracture mechanics tests. The CVN specification works by ensuring that the transition from brittle to ductile fracture occurs at some temperature less than service temperature. The notch toughness requirement ensures that brittle fracture will not occur as long as large cracks do not develop. Often 34 J (25 ft-lb), 27 J (20 ft-lb), or 20 J (15 ft-lb) are specified at a particular temperature. The intent of specifying any of these numbers is the same, i.e., to make sure that the transition starts below this temperature. Charpy toughness requirements for steel and weld metal for bridges and buildings are compared in Table 24.2. This table is simplified and does not include all the requirements. TABLE 24.2 Minimum Charpy Impact Test Requirements for Bridges and Buildings Minimum service temperature Material Steel: nonfracture critical membersa,b Steel: fracture critical membersa,b Weld metal for nonfracture criticala Weld metal for fracture criticala,b AISC: Jumbo sections and plates thicker than 50 mmb
−18◦ C Joules@◦ C
−34◦ C Joules@◦ C
−51◦ C Joules@◦ C
20@21 20@4 20@−12 34@21 34@4 34@−12 27@−18 27@−18 27@−29 34@−29◦ C for all service temperatures 27@21◦ C for all service temperature
a These requirements are for welded steel with minimum specified yield strength up to
350 MPa up to 38 mm thick. Fracture critical members are defined as those that if fractured would result in collapse of the bridge. b The requirements pertain only to members subjected to tension or tension due to bending.
Note that the bridge steel specifications require a CVN at a temperature 38◦ C greater than the minimum service temperature. This temperature shift accounts for the effect of strain rates, which are lower in the service loading of bridges (on the order of 10−3 ) than in the Charpy test (greater than 101 ). It is possible to measure the toughness using a Charpy specimen loaded at a strain rate characteristic of bridges, called an intermediate strain rate, although the test is more difficult and the results are more variable. When the CVN energies from an intermediate strain rate are plotted as a function of temperature, the transition occurs at a temperature about 38◦ C lower for materials with yield strength up to 450 MPa. As shown in Table 24.2, the AWS D1.5 Bridge Welding Code specifications for weld metal toughness are more demanding than the specifications for base metal. This is reasonable because the weld metal is always the location of discontinuities and high tensile residual stresses. However, there are no requirements for weld metal toughness in AWS D1.1. This lack of requirements was rationalized because typically the weld deposits are of higher toughness than the base metal. However, this is not always the case, e.g., the self-shielded flux-cored arc welds (FCAW-S) used in many of the WSMFs that fractured in the Northridge earthquake were reported to have very low toughness. The commentary in the AISC manual does warn that for “dynamic loading, the engineer may require the filler metals used to deliver notch-tough weld deposits”. ASTM A673 has specifications for the frequency of Charpy testing. The H frequency requires a set of three CVN specimens to be tested from one location for each heat or about 50 tons. These tests can be taken from a plate with thickness up to 9 mm different from the product thickness if it is rolled from the same heat. The P frequency requires a set of three specimens to be tested from one 1999 by CRC Press LLC
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end of every plate, or from one shape in every 15 tons of that shape. For bridge steel, the AASHTO code requires CVN tests at the H frequency as a minimum. For fracture critical members, the guide specifications require CVN testing at the P frequency. In the AISC code, CVN tests are required at the P frequency for thick plates and jumbo sections. A special test location in the core of the jumbo section is specified, as well as a requirement that the section tested be produced from the top of the ingot. Even the P testing frequency may be insufficient for as-rolled structural steel. In a recent report for the National Cooperative Highway Research Program (NCHRP) [28], CVN data were obtained from various locations on bridge steel plates. The data showed that because of extreme variability in CVN across as-rolled plates, it would be possible to miss potentially brittle areas of plates if only one location per plate is sampled. For plates that were given a normalizing heat treatment, the excessive variability was eliminated.
24.3.2
Fracture Mechanics Analysis
Fracture mechanics is based on the mathematical analysis of solids with notches or cracks. Relationships between the material toughness, the crack size, and the stress or displacement will be derived below using fracture mechanics. The objective of a fracture mechanics analysis (as outlined herein) is to ensure that brittle fracture does not occur. Even if ductile fracture occurs before local buckling or another failure mode, ductile fracture is considered to give acceptable ductility. Brittle fracture occurs with nominal net-section stresses below or just slightly above the yield point. Therefore, the relatively simple principles of linear-elastic fracture mechanics (LEFM) can be used to conservatively assess whether a welded joint is likely to fail by brittle fracture rather than in a ductile manner. It significantly simplifies the presentation and practical use of fracture mechanics if the discussion is confined to brittle fracture only. Worst-case assumptions are made regarding numerous factors that can enhance fracture toughness, e.g., temperature, strain rate, constraint, and notch acuity or sharpness. These assumptions eliminate the need for extensive discussion of these effects. If necessary, these effects can be considered and more advanced principles of fracture mechanics can be used to estimate the maximum monotonic or cyclic rotation before ductile tearing failure. Fracture mechanics can also be used to predict the subcritical propagation of cracks due to fatigue and/or stress-corrosion cracking that may precede fracture. In order to present a thorough discussion of the brittle fracture problem here, we cannot provide a detailed discussion of many other interesting fracture mechanics topics. There are several excellent books on fracture mechanics that cover these topics in detail [6, 9, 10]. Although cracks can be loaded by shear, experience shows that only the tensile stress normal to the crack is important in causing fatigue or fracture in steel structures. This tensile loading is referred to as Mode I. When the plane of the crack is not normal to the maximum principal stress, a crack that propagates subcritically or in a stable manner will generally turn as it extends, such that it becomes normal to the principal tensile stress. Therefore, it is typically recommended that a welding defect or crack-like notch that is not oriented normal to the primary stresses can be idealized as an equivalent crack with a size equal to the projection of the actual crack area on a plane that is normal to the primary stresses (see [12], for example). Brittle fracture occurs with nominal net-section stresses below or just slightly above the yield point. Therefore, the relatively simple principles of LEFM can be used to conservatively assess whether a welded joint is likely to fail by brittle fracture rather than in a ductile manner. LEFM gives a relatively straightforward method for predicting fracture, based on a parameter called the stress-intensity factor (K), which characterizes the stresses at notches or cracks [31]. The applied K is determined by the size of the crack (or crack-like notch) and the nominal cross-section stress remote from the crack. Crack-like notches and weld defects are idealized as cracks, which include crack-like notches and weld defects as well. In the case of linear elasticity, the stress-intensity factor can be considered a measure of 1999 by CRC Press LLC
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the magnitude of the crack tip stress and strain fields. Solutions for the applied stress-intensity factor, K, for a variety of geometries can be found in handbooks [43, 49, 50, 52]. Most of the solutions are variations on standard test specimens that have been studied extensively. The following discussion presents a few useful solutions and examples of their application to welded joints. In general, the applied stress-intensity factor is given as √ (24.5) K = Fc ∗ Fs ∗ Fw ∗ Fg ∗ σ π a where the F terms are modifiers on the order of 1.0, specifically: Fc = the factor for the effect of crack shape Fs = the factor, equal to 1.12, that is used if a crack originates at a free surface Fw = a correction for finite-width, which is necessary because the basic solutions were generally derived for infinite or semi-infinite bodies Fg = a factor for the effect of nonuniform stresses, such as bending stress gradient An SCF is defined as the ratio of the peak stress near the stress raiser to the nominal cross-section stress remote from the stress raiser. SCFs are often used in fracture assessments when the crack is located near a stress raiser. For example, a crack may be located at a plate edge that is badly corroded. Any SCF would also be included in Fg . The stress-intensity factor has the unusual units of MPa-m1/2 or ksi-in.1/2 . The material fracture toughness is characterized in terms of the applied K at the onset of fracture in simplified small test specimens, called K “critical” or Kc . The fracture toughness (Kc ) is considered a transferable material property; i.e., fracture of structural details is predicted if the value of the applied K in the detail exceeds Kc . Equation 24.5 relates the important factors that influence fracture: Kc represents the material, σ represents the design, and a represents the fabrication and inspection. In this section, Kc is used as any type of critical K associated with a quasi-static strain rate, derived from any one of a variety of test methods. One measure of Kc is the plane-strain fracture toughness, which is given the special subscript “I” for plane strain, KI c . KI c must be measured in specimens that are very thick and approximate plane strain. If the fracture toughness is measured in an impact test, the special designation Kd is used, where the subscript “d” is for dynamic. In practice, Kc is often estimated from correlations with the result from a CVN test because the CVN is much cheaper to perform and requires less material than a fracture mechanics test, and all test laboratories are equipped for the CVN test. A widely accepted correlation for the lower shelf and lower transition region between Kd and CVN [9]: √ (24.6) Kd = 11.5 ∗ CV N where CVN is given in joules and Kd is given in MPa-m1/2 . A different constant is used for English units. This correlation is used to construct the lower part of the curve for dynamic fracture toughness (Kd ) as a function of temperature directly from the curve of CVN vs. temperature. There is a temperature shift between the intermediate load rate values of Kc and the impact load rate values of Kd that is approximately equal to the temperature shift that occurs for CVN data. Therefore, Kc values for structural steel are obtained by shifting the Kd curve to a temperature that is 38◦ C lower. However, for brittle materials there is essentially no temperature shift and therefore Kc is approximately equal to Kd . Prior to the 1994 Northridge earthquake, the welds in the WSMF connections were commonly made with the FCAW-S process using an E7XT-4 weld wire. For the connections fractured in the Northridge earthquake that have been investigated so far, the weld metal CVN is plotted in Figure 24.11. The lower-bound impact energy is between 4 and 14 J for temperatures up to 50◦ C. If recommended weld procedures are followed, the fracture toughness increases slightly but remains inadequate. The lower bounds of the CVN and Kc data for the E7XT-4 weld metal is similar to the lower bounds from other brittle materials, such as the core region of the jumbo sections shown in Figure 24.4. This similarity in the data suggests that a there may be a lower-bound value of the fracture toughness that 1999 by CRC Press LLC
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FIGURE 24.11: Typical Charpy impact energy from E7XT-4 self-shielded flux-cored arc welding weld metal from Northridge WSMF connections.
can be assumed for brittle ferritic weld metal, structural steel, and the heat-affected zone (HAZ). The lower-bound fracture toughness reflects the worst effects of temperature and strain rate. For these materials, the lower-bound fracture toughness was between 45 and 50 MPa-m1/2 . This concept of a lower-bound fracture toughness is very useful for fracture assessment. As a consequence of the brittle fractures in jumbo sections, AISC specifications now have a supplemental Charpy requirement for shapes in groups 4 and 5 and (for the same reasons) plates greater than 51 mm thick, when these are welded and subject to primary tensile stress from axial load or bending. These jumbo shapes and thick plates must exhibit 27 J at 21◦ C. Using the correlation of Equation 24.6, 27 J will give a Kc of 60 MPa-m1/2 . There are size and constraint effects and other complications that make the LEFM fracture toughness, Kc , less than perfect as a material property. This is especially true when Kc is estimated based only on a correlation to CVN. Nevertheless, as illustrated in the following, the conservative lower-bound value of Kc can be used by structural engineers to avoid brittle fracture. Center Crack
The K solution for an infinitely wide plate with a through crack subject to uniform tensile membrane stress is √ (24.7) K = σ πa where σ = nominal cross-section stress remote from the crack 2a = the total overall crack length If the total width of the panel is given as 2W, Fw for this crack geometry can be approximated by the Fedderson or secant formula: r πa (24.8) Fw = sec 2W This formula gives a value that is close to 1.0 and can be ignored for a/W less than a third. For a/W of about 0.5, the secant formula gives Fw of about 1.2. However, the values from the secant equation go to infinity as a approaches W. The secant formula is reasonably accurate for a/W up to 0.85. The Fw may be used for other crack geometries as well. 1999 by CRC Press LLC
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Many common buried defects and notches in welded joints can be idealized as a center crack in tension. For example, Figure 24.12 shows a backing bar with a fillet that is idealized as a center crack. The unfused area of the backing bar creates a crack-like notch with one tip in the root of the fillet
FIGURE 24.12: Cross-section of a one-sided groove-welded cruciform-type connection with loaded plate discontinuous and idealization of the notches from a backing bar with a fillet weld as a centercracked tension panel.
weld and one tip at the root of the groove weld. The crack is asymmetrical but since the connection is subjected to uniform tension, the crack can be analyzed as if it were in a symmetric center-cracked panel. Of course, the applied K is higher on the crack tip that is in the root of the fillet weld because there is a high Fw for this side. Assuming the weld metal of the groove weld and the fillet weld have comparable toughness, the fillet weld side of the backing bar will govern the fracture limit state. Therefore, the panel is idealized as being symmetric with respect to the center of the backing bar. Assuming negligible weld root penetration, the crack size (2a) is taken as being equal to the backing bar thickness, say 13 mm. For a/W of 0.5, Equation 24.8 gives Fw equal to 1.2. Although this idealization seems like a gross approximation at best, the validity of the K solution for this particular weld joint was verified based on observed fatigue crack propagation rates. If this is a grade 50 steel, the yield strength could be up to 450 MPa. The notch tip could be subjected to full tensile residual stress. Therefore, Equation 24.5 is solved with the cross-section stress equal to 450 MPa, with the Fw factor of 1.2, and a of 6 mm, giving 74 MPa-m1/2 . It can be seen that this configuration could cause a brittle fracture for very brittle materials. However, weld metal and base steel with modest toughness could easily withstand this defect. Edge Crack
The stress-intensity factor for an edge crack in an infinitely wide plate is √ K = 1.12σ π a
(24.9)
where σ is the remote cross-section nominal stress and a is the depth of the edge crack or cracklike notch. It can be seen that the edge crack equation is treated like half of a center crack, i.e., Equation 24.7, where a and W are the total length and width, respectively, for the edge crack. The Fs of 1.12 is applied to account for the free edge, which is not restrained as it is in the center-cracked geometry. Equation 24.9 is also modified with the Fw as in Equation 24.8. Figure 24.5 showed a cross-section near the crack origin for a WSMF beam-flange-to-column weld that fractured in the Northridge earthquake. The fracture surfaces, such as shown in Figure 24.13, 1999 by CRC Press LLC
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FIGURE 24.13: Fracture surfaces of a typical beam-flange-to-column joint of a WSMF that fractured in the Northridge earthquake, showing the lack-of-fusion defect in the center of the weld.
indicate that the fractures originate in the root of the weld, typically at a lack-of-fusion defect. This lack-of-fusion defect is difficult to avoid when the weld must be stopped on one side of the web and started on the other side. The weld fracture surface in Figure 24.13 shows the crack-like notch formed by the combination of a lack-of-fusion defect and the unfused edge of the backing bar. On a cross-section at the deepest point of the lack-of-fusion defect, the total depth of the notch, including the unfused edge of the backing bar, is from 13 to 19 mm. The value of 45 MPa-m1/2 can be used as a lower bound to the fracture toughness of structural steel or weld metal. Equation 24.9 may be used to predict brittle fracture for the WSMF connection welds when K exceeds 45 MPa-m1/2 . For a notch depth of 13 to 19 mm, Equation 24.9 would predict that brittle fracture is likely to occur for cross-section stress between 160 and 200 MPa, well below 1999 by CRC Press LLC
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the yield point. These types of LEFM calculations, had they been performed prior to the earthquake, would have predicted that brittle fracture would occur in the WSMF connections before yielding. The propagation path of the unstable dynamic crack is seemingly chaotic, as it is influenced by dynamic stress waves and complex residual stress fields. The critical event was the initiation of the unstable crack in the brittle weld. There is little significance to whether the crack propagated in the weld or turned and entered the column. The solution to the WSMF cracking problem is elimination of the possibility of such large defects (i.e., ≈ 19 mm) that result from the backing bar and any lack-of-fusion defect. The backing bar removal and back gouging to minimize the lack of fusion will eliminate most large crack-like conditions. However, this is not enough to eliminate the problem of brittle fracture. Appropriate weld notch toughness requirements must be specified to avoid the use of low-toughness weld metal. There are certainly many other equally important design issues that influenced these fractures. The overall lack of redundancy, i.e., the reliance on only one or two massive WSMFs to resist lateral load in each direction, contributes to large forces and increases the thickness of the members and the high constraint of the connections. Even if brittle fracture is avoided, welds will typically fail at a lower level of plastic strain than base metal. Therefore, it can also be argued that it is imprudent to rely on welds for extensive plastic deformation. Several improved WSMF connections have been proposed, most of which are designed such that the plastic hinge develops in the span away from the connection. Nevertheless, in the event of unexpected loading, it is still desirable that these weld joints have a ductile failure mode. Therefore, while these improved connection designs may be worthwhile, the low-toughness weld metal and joint design with a built-in notch should still not be used under most circumstances. Buried Penny-Shaped Crack
Many internal weld defects are idealized as an ellipse or a circle that is circumscribed around the projection of the weld defect on a plane perpendicular to the stresses. Often, the increased accuracy accrued by using the relatively complex elliptical formula is not worth the effort, and the circumscribed penny-shaped or circular crack is always conservative. The stress-intensity factor for the penny-shaped crack in an infinite body is given as K=
2 √ σ πa π
(24.10)
where a is the radius of the circular crack. As for the other types of cracks, the Fw can be calculated using Equation 24.10. In terms of Equation 24.8, the crack shape factor Fc in this case is 2/π or 0.64. Using (1) the lower-bound fracture toughness of 45 MPa-m1/2 and (2) an upper-bound residual stress plus applied stress equal to the upper-bound yield strength for grade 50 steel (450 MPa), Equation 24.10 shows that a penny-shaped crack would have to have a radius exceeding 8 mm to be critical, i.e., the diameter of the allowable welding defect would be 15 mm (providing that fatigue is not a potential problem). The crack shape factor Fc is more favorable (0.64) for buried cracks as opposed to Fs of 1.12 for edge cracks, and the defect size is equal to 2a for the buried crack and only a for the edge crack. These factors explain why edge cracks of a given size are much more dangerous than buried cracks of the same size.
24.4
Summary
Structural elements for which the live load is a large percentage of the total load are potentially susceptible to fatigue. Many factors in fabrication can increase the potential for fatigue including notches, misalignment, and other geometrical discontinuities, thermal cutting, weld joint design 1999 by CRC Press LLC
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(particularly backing bars), residual stress, nondestructive evaluation and weld defects, intersecting welds, and inadequate weld access holes. The fatigue design procedures in the AASHTO and AISC specifications are based on control of the stress range and knowledge of the fatigue strength of the various details. Using these specifications, it is possible to identify and avoid details that are expected to have low fatigue strength. Welded connections and thermal-cut hole copes, blocks, or cuts are potentially susceptible to brittle fracture. Many interrelated design variables can increase the potential for brittle fracture including lack of redundancy, large forces and moments with dynamic loading rates, thick members, geometrical discontinuities, and high constraint of the connections. Low temperature can be a factor for exposed structures. The factors mentioned above that influence the potential for fatigue have a similar effect on the potential for fracture. In addition, cold work, flame straightening, weld heat input, and weld sequence can also affect the potential for fracture. The AASHTO specifications [1] require a minimum CVN notch toughness at a specified temperature for the base metal and the weld metal of members loaded in tension or tension due to bending. Almost two decades of experience with these bridge specifications has proven that they are successful in significantly reducing the number of brittle fractures. Simple, LEFM concepts can be used to predict the potential for brittle fracture in buildings.
24.5
Defining Terms
a: Crack length. At : The tensile stress area of a bolt. AASHTO: American Association of State Highway Transportation Officials. ACI: American Concrete Institute. AISC: American Institute of Steel Construction. ASTM: American Society for Testing and Materials. AWS: American Welding Society. BSI: British Standards Institute. CAFL: Constant amplitude fatigue limit, a level of stress range below which noticeable cracking does not occur in constant amplitude fatigue tests. CVN: Charpy V-Notch impact test energy. db : The nominal diameter of a bolt (the body or shank diameter). Fc : Factor for the effect of crack shape. Fs : Factor, equal to 1.12, that is used if a crack originates at a free surface. Fw : Correction for finite-width. Fg : Factor for the effect of non-uniform stresses, such as bending stress gradient. Fu : Ultimate tensile strength. K: Stress-intensity factor. Kc : Fracture toughness. Kd : Dynamic fracture toughness. L: Span length. LFRD: Load and resistance factor design. n: Threads per inch in bolts. ni : Number of cycles in an interval of stress range i. Ntotal : Total number of stress ranges. 1999 by CRC Press LLC
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PTI: Post-Tensioning Institute. Seffective : The effective stress range which is equal to the cube root of the mean cube of the all stress ranges. Si : Stress range for interval i. SCF: Stress concentration factor. t: Plate thickness. g: Gage.
References [1] American Association of State Highway Transportation Officials. 1994. AASHTO LRFD Bridge Design Specifications, First edition, Washington, D.C. [2] American Concrete Institute Committee 215. 1996. Considerations for Design of Concrete Structures Subjected to Fatigue Loading, ACI 215R-74 (Revised 1992), ACI Manual of Standard Practice, Vol. 1. [3] Ad-hoc Committee on Cable-Stayed Bridges. 1986. Recommendations for Stay Cable Design and Testing, Post-Tensioning Institute, Phoenix, AZ. [4] American Institute of Steel Construction. 1994. Load and Resistance Factor Design Specification for Structural Steel Buildings, Second edition, Chicago, IL. [5] American Petroleum Institute. 1989. Recommended Practice for Planning, Designing, and Constructing Fixed Offshore Platforms, API RP2A, 18th edition. [6] Anderson, T.L. 1995. Fracture Mechanics—Fundamentals and Applications, Second edition, CRC Press, Boca Raton, FL. [7] American Welding Society. 1996. Structural Welding Code—Steel, ANSI/AWS D1.1, Miami, FL. [8] Ballio, G. and Castiglioni, C.A. 1994. A Unified Approach for the Design of Steel Structures Under Low and/or High Cycle Fatigue, J. Constructional Steel Res. [9] Barsom, J.M. and Rolfe, S.T. 1987. Fracture and Fatigue Control in Structures, Second edition, Prentice-Hall, Englewood Cliffs, NJ. [10] Broek, D. 1987. Elementary Fracture Mechanics, Fourth edition, Martinus Nijhoff Publishers, Dordrecht, the Netherlands. [11] British Standards Institute. 1994. Code of Practice for Fatigue Design and Assessment of Steel Structures, BS 7608, London. [12] British Standards Institute. 1991. Guidance on Some Methods for the Derivation of Acceptance Levels for Defects in Fusion Welded Joints, BSI PD 6493, London. [13] Castiglioni, C.A. 1995. Cumulative Damage Assessment in Structural Steel Details, Extending the Lifespan of Structures, IABSE Symposium, San Francisco, CA, pp. 1061-1066. [14] Demers, C. and Fisher, J.W. 1990. Fatigue Cracking of Steel Bridge Structures, Volume I: A Survey of Localized Cracking in Steel Bridges—1981 to 1988, Report No. FHWA-RD-89-166; Volume II, A Commentary and Guide for Design, Evaluation, and Investigating Cracking, Report No. FHWA-RD-89-167, Federal Highway Administration, McLean, VA. [15] Dexter, R.J., Fisher, J.W., and Beach, J.E. 1993. Fatigue Behavior of Welded HSLA-80 Members,
Proceedings, 12th International Conference on Offshore Mechanics and Arctic Engineering, Vol. III, Part A, Materials Engineering, American Society of Mechanical Engineers, New York, pp. 493-502. [16] Dexter, R.J., Tarquinio, J.E., and Fisher, J.W. 1994. An Application of Hot-Spot Stress Fatigue Analysis to Attachments on Flexible Plate, Proceedings of the 13th International Conference 1999 by CRC Press LLC
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[17]
[18] [19] [20]
[21]
[22] [23] [24]
[25]
[26] [27] [28] [29] [30]
[31] [32]
[33] [34] [35]
on Offshore Mechanics and Arctic Engineering, American Society of Mechanical Engineers, New York. Djikstra, O.D., Wardenier, J., and Hertogs, A.A. 1988. The Fatigue Behavior of Welded Splices With and Without Mouse Holes in IPE 400 and HEM 320 Beams, Proceedings of the International Conference on Weld Failures—Weldtech ’88, The Welding Institute, Abington Hall, Cambridge, U.K. European Committee for Standardization. 1992. Eurocode 3: Design of Steel Structures—Part 1.1: General Rules and Rules for Buildings, ENV 1993-1-1, Brussels. Fisher, J.W. 1984. Fatigue and Fracture in Steel Bridges, ISBNO-471-80469-X, John Wiley & Sons, New York. Fisher, J.W., Frank, K.H., Hirt, M.A., and McNamee, B.M. 1970. Effect of Weldments on the Fatigue Strength of Steel Beams, National Cooperative Highway Research Program Report 102, Highway Research Board, Washington, D.C. Fisher, J.W., Albrecht, P.A., Yen, B.T., Klingerman, D.J., and McNamee, B.M. 1974. Fatigue Strength of Steel Beams with Welded Stiffeners and Attachments, National Cooperative Highway Research Program Report 147, Transportation Research Board, Washington, D.C. Fisher, J.W. and Pense, A.W. 1987. Experience with Use of Heavy W Shapes in Tension, Eng. J., American Institute of Steel Construction, 24(2). Fisher, J.W. and Keating, P.B. 1989. Distortion-Induced Fatigue Cracking of Bridge Details with Web Gaps, J. Constructional Steel Res., (12), 215-228. Fisher, J.W., Jian, J., Wagner, D.C., and Yen, B.T. 1990. Distortion-Induced Fatigue Cracking in Steel Bridges, National Cooperative Highway Research Program Report 336, Transportation Research Board, Washington, D.C. Fisher, J.W., et al. 1993. Resistance of Welded Details Under Variable Amplitude Long-Life Fatigue Loading, National Cooperative Highway Research Program Report 354, Transportation Research Board, Washington, D.C. Frank, K.H. and Fisher, J.W. 1979. Fatigue Strength of Fillet Welded Cruciform Joints, J. Structural Div., ASCE, 105(ST9), 1727-1740. Frank, K.H. 1980. Fatigue Strength of Anchor Bolts, J. Structural Div., ASCE, 106 (ST). Frank, K.H., et al. 1993. Notch Toughness Variability in Bridge Steel Plates, National Cooperative Highway Research Program Report 355, Washington, D.C. American Association of State Highway and Transportation Officials. 1978. Guide Specifications for Fracture Critical Non-Redundant Steel Bridge Members, Washington, D.C. Hahin, C. 1994. Effects of Corrosion and Fatigue on the Load-Carrying Capacity of Structural Steel and Reinforcing Steel, Illinois Physical Research Report No. 108, Illinois Department of Transportation, Springfield, IL. Irwin, G.R. 1957. Analysis of Stresses and Strains Near the End of Crack Traversing a Plate, Trans., ASME, J. Appl. Mech., Vol. 24. Also reprinted in ASTM volume on classic papers. Kaczinski, M.R., Dexter, R.J., and Van Dien, J.P. 1996. Fatigue-Resistant Design of Cantilevered Signal, Sign, and Light Supports, Final Report for NCHRP Project 10-38, ATLSS Engineering Research Center, Lehigh University. Kaufmann, E.J., Xue, M., Lu, L.-W., and Fisher, J.W. 1996. Achieving Ductile Behavior of Moment Connections, Modern Steel Construction, 36(1), 30-39. Keating, P.B. and Fisher, J.W. 1986. Evaluation of Fatigue Tests and Design Criteria on Welded Details, National Cooperative Highway Research Program, Report 286, Washington, D.C. Kober, Dexter, R.J., Kaufmann, E.J., Yen, B.T., and Fisher, J.W. 1994. The Effect of Welding Discontinuities on the Variability of Fatigue Life, Fracture Mech., Twenty-Fifth Volume, ASTM STP 1220, F. Erdogan and Ronald J. Hartranft, Eds., American Society for Testing and Materials, Philadelphia, PA.
1999 by CRC Press LLC
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[36] Krawinkler, H. and Zohrei, M. 1983. Cumulative Damage in Steel Structures Subjected to Earthquake Ground Motion, Computers and Structures, 16(1-4), 531-541. [37] Kulak, G.L., Fisher, J.W., and Struick, J.H. 1987. Guide to Design Criteria for Bolted and Riveted Joints, Second Edition, Prentice-Hall, Englewood Cliffs, NJ. [38] Maddox, S.J. 1991. Fatigue Strength of Welded Structures, Second Edition, Abington Publishing, Cambridge, UK. [39] Marsh, K.J., Ed. 1988. Full-Scale Fatigue Testing of Components and Structures, Butterworths, London. [40] Marshall, P.W. 1992. Design of Welded Tubular Connections, Elsevier, New York. [41] Miki, C., Nishimura, T., Tajima, J., and Okukawa, A. 1980. Fatigue Strength of Steel Members Having Longitudinal Single-Bevel Groove Welds, Trans. Japan Weld. Soc., 11(1), 43-56. [42] Miner, M.A. 1945. Cumulative Damage in Fatigue, J. Appl. Mech., 12, A-159. [43] Murakami, Y., et al., Eds. 1987. Stress Intensity Factors Handbook, Vols. 1 and 2, Pergamon Press, Oxford, U.K. [44] Outt, J.M.M., Fisher, J.W., and Yen, B.T. 1984. Fatigue Strength of Weathered and Deteriorated Riveted Members, Report DOT/OST/P-34/85/016, Department of Transportation, Federal Highway Administration, Washington, D.C. [45] Overnman, T.R., Breen, J.E., and Frank, K.H. 1984. Fatigue Behavior of Pretensioned Concrete Girders, Research Report 300-2F, Center for Transportation Research, The University of Texas at Austin. [46] Petershagen, H. 1986. Fatigue Problems in Ship Structures, Advances in Marine Structures, Elsevier Applied Science, London, pp. 281-304. [47] Pfister, J.F. and Hognestad, E. 1964. High Strength Bars as Concrete Reinforcement, Part 6, Fatigue Tests, J. PCA Res. Dev. Lab., 6(1), 65-84. [48] Rabbat, B.G., et al. 1979. Fatigue Tests of Pretensioned Girders with Blanketed and Draped Strands, J. Prestressed Concrete Inst., 24(4), 88-115. [49] Rooke, D.P. and Cartwright, D.J. 1974. Compendium of Stress Intensity Factors, Her Majesty’s Stationery Office, London. [50] Rooke, D.P. 1986. Compounding Stress Intensity Factors, Research Reports in Materials Science (Series One), The Parthenon Press, Lancashire, U.K. [51] Sternberg, F. 1969. Performance of Continuously Reinforced Concrete Pavement, I-84 Southington, Connecticut State Highway Department. [52] Tada, H. 1985. The Stress Analysis of Cracks Handbook, Paris Productions, Inc., St. Louis, MO. [53] Tide, R.H.R., Fisher, J.W., and Kaufmann, E.J. 1996. Substandard Welding Quality Exposed: Northridge, California Earthquake, January 17, 1994, IIW Asian Pacific Welding Congress, Auckland, New Zealand, February 4-9. [54] VanDien, J.P., Kaczinski, M.R., and Dexter, R.J. 1996. Fatigue Testing of Anchor Bolts, Building an International Community of Structural Engineers, Vol. 1, Proceedings of Structures Congress XIV, Chicago, pp. 337-344. [55] Xue, M., Kaufmann, E.J., Lu, L.W., and Fisher, J.W. 1996. Achieving Ductile Behavior of Moment Connections—Part II, Modern Steel Construction, 36(6), 38-42. [56] Yagi, J., Machida, S., Tomita, Y., Matoba, M., and Kawasaki, T. 1991. Definition of Hot-Spot Stress in Welded Plate Type Structure for Fatigue Assessment, International Institute of Welding, IIW-XIII-1414-91. [57] Yen, B.T., Huang, T., Lai, L.Y., and Fisher, J.W. 1990. Manual for Inspecting Bridges for Fatigue Damage Conditions, Report No. FHWA-PA-89-022 + 85-02, Fritz Engineering Laboratory Report No. 511.1, Pennsylvania Department of Transportation, Harrisburg, PA. [58] Yen, B.T., et al. 1991. Fatigue Behavior of Stringer-Floorbeam Connections, Proceedings of the Eighth International Bridge Conference, IBC-91-19, Engineers’ Society of Western Pennsylvania, pp. 149-155. 1999 by CRC Press LLC
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Further Reading [1] Anderson, T.L. 1995. Fracture Mechanics — Fundamentals and Applications, 2nd ed., CRC Press, Boca Raton, FL. [2] Barsom, J.M. and Rolfe, S.T. 1987. Fracture and Fatigue Control in Structures, 2nd ed., PrenticeHall, Englewood Cliffs, NJ. [3] Broek, D. 1987. Elementary Fracture Mechanics, 4th ed., Martinus Nijhoff Publishers, Dordrecht, Netherlands. [4] Fisher, J.W. 1984. Fatigue and Fracture in Steel Bridges, John Wiley & Sons, New York. [5] Maddox, S.J. 1991. Fatigue Strength of Welded Structures, 2nd ed., Abington Publishing, Cambridge, U.K.
1999 by CRC Press LLC
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J. M. Doyle, J.M. and Fang, S.J. “Underground Pipe” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Underground Pipe 25.1 Introduction 25.2 External Loads
Overburden • Surcharge at Grade • Live Loads • Seismic Loads
25.3 Internal Loads
Internal Pressure and Vacuum • Pipe and Contents
25.4 Design Methods
General • Flexible Design • Rigid Design
25.5 Joints
General • Joint Types • Hydrostatic Testing
J. M. Doyle and S.J. Fang
25.6 Corrosion Protection
Sargent & Lundy, LLC Chicago, IL
References
25.1
Coatings • Cathodic Protection
Introduction
Throughout recorded history, works have been constructed for conveying water from one place to another. The Roman aqueducts are often mentioned as examples of great technical achievement; indeed, some of these early structures are still in use today. Although most of the early water carrying structures were open channels, conduits and pipes of various materials were also used in Roman times. It appears, though, that the effectiveness of the early pipes was limited because their materials were weak in tensile capacity. Therefore, the pipes could not carry fluid under any appreciable pressure. Beginning in the 17th century, wood and cast iron were used in water pipe applications in order to carry water under pressure from pumping, which was introduced about the same time. Since then, many materials have evolved for use in pipes. As a general rule, the goals for new pipe material development has been increased tensile strength, reduced weight, and, of course, reduced cost. Pipe that is buried underground must sustain other loads besides the internal fluid pressure. That is, it must support the soil overburden, groundwater, loads applied at the ground surface, such as vehicular traffic, and forces induced by seismic motion. Buried pipe is, therefore, a structure as well as a conduit for conveying fluid. That being the case, special design procedures are required to insure that both functions are fulfilled. It is the purpose of this chapter to present techniques that are currently in use for the design of underground pipelines. Such lines are used for public water systems, sewers, drainage facilities, and many industrial processes. Pipe materials to be considered include steel, concrete, and fiberglass reinforced plastic. This selection provides examples of both flexible and rigid behavior. The methodologies presented here can be applied to other materials as well. Design procedures given are, for the most part, based on material contained in U.S. national standards or recommended practices developed by industry organizations. It is our intention to provide an exposition of the essential elements of the various design procedures. No claim is made to 1999 by CRC Press LLC
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total inclusiveness for the methodologies discussed. Readers interested in the full range of refinements and subtleties of any of the approaches are encouraged to consult the cited works. For convenience when comparing references, the notations used in work by others will be maintained here. Attention is focused on large-diameter lines, generally greater than 24 in. Worked sample problems are included to illustrate the material presented.
25.2
External Loads
25.2.1
Overburden
The vertical load that the pipe supports consists of a block of soil extending from the ground surface to the top of the pipe plus (or minus) shear forces along the edges of the block. The shear forces are developed when the soil prism above the pipe or the soil surrounding the prism settle relative to each other. For example, the soil prism above the pipe in an excavated trench would tend to settle relative to the surrounding soil. The shear forces between the backfill and the undisturbed soil would resist the settlement, thus reducing the prism load to be carried by the pipe. For a pipe placed on the ground and covered by a new fill, the effect may be the same or opposite, in which case the load to be supported by the pipe would be greater than the soil prism. The difference in behavior depends on the difference in settlement between the pipe itself and the fill material. Sketches of typical methods of buried pipe installation are shown in Figure 25.1. Methods developed by Marston and Spangler, and their co-workers, at Iowa State University [28, 29, 34, 35, 36, 39] over a period of about 50 years, are the accepted tools for evaluating overburden loads on buried conduits and are widely used in design practice. The general form of the expression, developed by this group, used to calculate the overburden load carried by the pipe is given as Wc = CwB 2
(25.1)
where: Wc = total load on pipe, per unit of length C = load coefficient, dependent on type of installation, trench or fill, on the soil type, and on relative rates of settlement of the pipe and surrounding soil w = unit weight of soil supported by pipe B = width of trench of outer diameter of pipe Values for the load coefficient, C, for varying conditions of installation, are given in several standard references (see, e.g., [20]). The American Water Works Association (AWWA) [21], in its design manual for steel pipe, recommends that the total overburden load on buried steel pipes be assumed equal to a soil prism with width equal to the outer diameter of the pipe and height equal to the cover depth. That is, Wc = wBc h
(25.2)
where Bc = external pipe diameter h = depth from ground surface to top of pipe
25.2.2
Surcharge at Grade
Besides the direct loads imposed by the soil overburden, underground pipes must also sustain loads applied on the ground surface. Typically, such loads occur as a result of vehicular traffic passing over the route of the pipe. However, they can be caused by static objects placed directly, or nearly so, above the pipe as well. 1999 by CRC Press LLC
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FIGURE 25.1: Typical underground pipe installations. (Reprinted from Concrete Pressure Pipe, M9, c by permission. Copyright 1995, American Water Works Association.) Experimental results, by the Iowa State University researchers and others [33, 37], have confirmed that the load intensity at the pipe depth, due to surface loads, can be predicted on the basis of the theory of elasticity. The effects of an arbitrary spatial distribution of surface load can be obtained by utilizing the well-known Boussinesq solution [41], for a point load on an elastic half space, as an influence function. Since the Boussinesq solution provides a stress distribution for which magnitudes decay with distance from the load, it follows that the intensity of surface loads decreases with increased depth. Therefore, the consequence of traffic, or other surface loads, on deeply buried pipes is relatively minor. Conversely, surface loads applied over pipes with shallow cover can be quite serious. For this reason, a minimum cover is usually required in any place where vehicular traffic will operate over underground conduits. Prior to development of present day computational tools, the evaluation of the Boussinesq equations to determine the total load on a buried pipe due to an arbitrary surface load was beyond the capability of most practitioners. For that reason, tables were developed, based on simple surface load distributions, and have been included in most design literature for buried pipe for many years. See, for example, the tables of values in the AWWA Manual M11 [21]. Loading configurations not 1999 by CRC Press LLC
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covered by the previously developed tables can be investigated using available software programs. Mathcad [30], for example, can be utilized to carry out the analysis necessary to evaluate the effect of arbitrary surface loads on buried structures, including pipes.
25.2.3
Live Loads
The main source of design live loads on buried pipes is wheeled traffic from highway trucks, railroad locomotives, and aircraft. Loads transmitted to buried structures by the standard HS-20 truck loading [1] and the Cooper E-80 railroad loading have been evaluated using the Boussinesq solution and engineering judgment, for varying depths of cover, and are available, in different forms, in several publications (see, e.g., [6, 20]). Due to the wide variation in aircraft wheel loadings, it is usually necessary to evaluate each case separately. FAA Advisory Circular 150/5320-5B provides information on aircraft wheel loads. The load intensity at the depth of the pipe has been reported in numerous references. Simple load intensities for the HS-20 truck loads and for the Cooper E-80 locomotive loads, at varying depths, are given in Tables 25.1 and 25.2, respectively [6]. More comprehensive tables for truck and railroad loads have been published [20, 27]. In general, the intensities given in Tables 25.1 and 25.2 are close to the intensities given in the other tables, though some differences do exist. For examples in this chapter, live loads will be based on the intensities given in Tables 25.1 and 25.2. In case of doubt as to appropriate live load values to use in design of buried pipe, the advice of a geotechnical engineer should be obtained. TABLE 25.1
HS-20 Live Load
Height of cover, ft
Live load, lb/ft2
1 2 3 4 5 6 7 8 Over 8
1800 800 600 400 250 200 175 100 Neglect
From American Society for Testing and Materials. 1994. A796. Standard Practice for Structural Design of Corrugated Steel Pipe, Pipe-Arches and Arches for Storm and Sanitary Sewers and Other Buried Applications. With permission.
25.2.4
Seismic Loads
In zones of high seismicity, buried conduits must be designed for the stresses imposed by earthquake ground motions. The American Society of Civil Engineers (ASCE) has developed procedures for evaluation of the magnitude of axial and flexural strains induced in underground lines by seismic motions [24]. The document reflects the research efforts of many of the leading seismic engineers in the country and the methodology is widely used for design of underground conduits of all kinds. As a general rule, the stresses in pipe walls due to seismic motion–induced strains are quite small and do not adversely affect the design. Since most design codes allow for an increase in allowable stress, or a decrease in load factors, when seismic loads are included in a load combination, buried pipes that are sized to sustain other design loads usually have sufficient strength to resist seismic-imposed stresses. 1999 by CRC Press LLC
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TABLE 25.2
Cooper E-80 Live Load
Height of cover, ft
Live load, lb/ft2
2 5 8 10 12 15 20 30 Over 30
3800 2400 1600 1100 800 600 300 100 Neglect
From American Society for Testing and Materials. 1994. A796. Standard Practice for Structural Design of Corrugated Steel Pipe, PipeArches and Arches for Storm and Sanitary Sewers and Other Buried Applications. With permission.
Consequently, the major consideration to be addressed in design of underground pipe is not strength but excessive relative movement. Unrestrained slip joints in buried pipe may be subject to relative movement, between the two segments meeting at a joint, that exceeds the limit of the joint’s capacity to function. For that reason, slip joint pipe must be investigated for maximum relative movement when subject to seismic motion. Types of pipe commonly utilizing slip joints include ductile iron, reinforced and prestressed concrete, and fiberglass reinforced plastic.
25.3
Internal Loads
25.3.1
Internal Pressure and Vacuum
Underground pipe systems operate under varying levels of internal pressure. Gravity sewer lines normally operate under fairly low internal pressure whereas water supply mains and industrial process pipes may be subject to internal pressures of several hundred pounds per square inch. High-pressure pipelines are often designed for a continuous operating pressure and for a short-term transient pressure. Certain operational events may cause a temporary vacuum in buried conduits. In most cases the duration of application of vacuum loading is extremely short and its effects can usually be examined separately from other live loads. For design, a hydraulic analysis of the system may be used to predict the magnitude and time variation of transients in both the positive and negative internal pressure.
25.3.2
Pipe and Contents
The effects of dead weight of the pipe wall and the fluid carried must be resisted by the structural capacity of the pipe. Neither of these loads contribute significantly to the overall stress state in most circumstances. In practice, loads from these two sources are often neglected in design of steel or plastic pipe, but they are usually included in design of prestressed and reinforced concrete pressure pipe and can be included in design of concrete nonpressure pipe as well. Formulas for determination of pipe wall bending moments and thrust forces, due to self-weight and fluid loads, are available in standard stress analysis references [43]. Since these loads are usually small compared to the overburden, they can be added to the vertical soil loads for simplicity and with conservatism. 1999 by CRC Press LLC
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25.4
Design Methods
25.4.1
General
The principal structural consideration in design of buried pipe is the ability to support all imposed loads. Other important items include the type of joints to be used and protection against environmental exposure. There are two fundamental approaches to design of buried pipe, based on the pipe’s behavior under load [32, 40]. Pipe that undergoes relatively large deformations under its gravity loads, and obtains a large part of its supporting capacity from the passive pressure of the surrounding soil, is referred to as “flexible”. As will be observed, the evaluation of the contribution of the soil to pipe strength is difficult due to varying conditions of pie installation. For that reason, prudence in design must be followed. However, as with most design problems, the engineer must, ultimately, balance conservatism with economic considerations. Pipes with stiffer walls that resist most of the imposed load without much benefit of engagement of passive soil pressure, because deformation under load is restricted, are called “rigid”. Steel, both corrugated and plain plate, ductile iron, and fiberglass reinforced plastic pipes are considered flexible; concrete pipe is considered rigid. Different methodologies are employed in assessing the strength of each type.
25.4.2
Flexible Design
Plain Steel
The structural capacity of flexible pipes is evaluated on the basis of resistance to buckling (compressive yield) and vertical diametrical deflection under load. Additionally, for flexible pipes, a nonstructural requirement in the form of a minimum stiffness to ensure that the pipe is not damaged during shipping and handling is normally imposed. In the case of steel pipes designed according to the recommendations of AWWA Manual M11 [21], the following two equations are used to choose pipe wall thickness sufficient to satisfy the handling requirement: t t
D 288 D+20 400
≥ ≥
for diameter up to 54 in. for diameter greater than 54 in.
(25.3)
It is of interest to note that for many years, a minimum thickness of D/200 was used by pipe designers. In our experience, wall thicknesses meeting this ratio will usually result in designs that also satisfy the strength and deflection criteria discussed below. Tensile stresses due to internal pressure must be limited to a fraction of the tensile yield of the material. AWWA recommends limiting the tensile stress to 50% of yield. Collapse, or buckling, of flexible pipes is difficult to predict theoretically because of the indeterminate nature of the load pattern. AWWA has published an expression for the determination of capacity of a given pipe to support imposed loads. The equation, given as Equation 6-7 in AWWA Manual M11 [21], incorporates the effects of the passive soil resistance, the buoyant effect of groundwater, and the stiffness of the pipe itself. Allowable buckling pressure is given by: qa =
1 FS
where qa = allowable buckling pressure (psi) F S = factor of safety = 2.5 for (12h/D) ≥ 2 and 1999 by CRC Press LLC
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1/2 0 0 EI 32Rw B E 3 D
(25.4)
= 3.0 for (12h/D) < 2 Rw = water buoyancy factor = 1-0.33 (hw / h) h = height of ground surface above top of pipe (ft) hw = height of groundwater surface above top of pipe (ft) D = diameter of pipe (in.) B 0 = coefficient of elastic support 1 = 1+4e−0.065h 0 E = modulus of soil reaction (psi) E = modulus of elasticity of pipe wall (psi) I = moment of inertia per inch length of pipe wall (in.3 ) In case vacuum load and surface live load are both included in the design conditions, AWWA recommends that separate load combinations be considered for each. That is because vacuum loads usually occur only for a short time and the probability of vacuum and maximum surface load occurring simultaneously is very small. In particular the following two load cases should be considered. For traffic live load: qa ≥ γw hw + Rw
Wc WL + D D
where γw = specific weight of water (0.0361 lb/in.3 ) WL = live load on pipe (lb/in. length of pipe) Wc = vertical soil load on pipe (lb/in. length of pipe) For vacuum load: Wc + Pν qa ≥ γw hw + Rw D
(25.5)
(25.6)
where Pν = internal vacuum pressure (psi) Deflection is determined by the Spangler formula: 1y = Dl
KWc r 3 EI + 0.061E 0 r 3
(25.7)
where 1y = deflection of pipe (in.) Dl = deflection lag factor (1.0 to 1.5) K = bedding constant (0.1) r = pipe radius (in.) This form of the deflection equation was obtained by ordinary bending theory of a ring subject to an assumed pattern of applied vertical load, width of vertical reaction, and distribution of horizontal passive pressure [38, 42] and has been used in pipe design for over 50 years. According to the formula, deflection is limited by the stiffness of the pipe wall itself and by the effect of the passive pressure. It is significant to note that in the sizes of steel pipes often encountered, the ratio of the two components of resistance is on the order of 1:20, with the pipe wall stiffness being the smaller. Therefore, it is obvious that the passive resistance, which is closely related to the type of backfill and its degree of compaction, is the dominant influence on the vertical deflection of flexible pipes. That being the case, it becomes apparent that increasing the strength of a flexible pipe will probably be an inefficient way to properly limit deflection of underground pipe in most cases. The pipe installation must be completed as specified in order for this to be achieved. 1999 by CRC Press LLC
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Efforts to quantify the modulus of soil reaction, E 0 , have continued since the initial development of the deflection equation. Suggested values are published in numerous references, including AWWA Manual M11 [21]. Values given there range from 200 to 3000 psi. The values depend on the type and level of compaction of the surrounding soil. Since pipe designers often have little control over the installation of pipe, historically, a value of E 0 in the range of 700 to 1000 psi has been assumed representative of average installations for estimating deflection at time of design. In a recent work, engineers at the U.S. Bureau of Reclamation addressed the question of deflection of flexible pipe [27]. Their work, which is based on the wide experience of the Bureau of Reclamation in construction of all kinds of underground pipes, discusses appropriate values of E 0 based on not only the backfill and compaction used, but also the native soil. In addition to the soil modulus values, the authors also give a modified form of the deflection equation that includes a factor to account for long-term deflection, Tf (which replaces the factor Dl in Equation 25.7), and an additional multiplier on the soil modulus, called a design factor, Fd , with values ranging from 0.3 to 1.0. The combined effect of these two changes is, generally, to predict larger deflections than with the original Spangler equation. The revised equation becomes: 1y = Tf
KWc r 3 EI + 0.061Fd E 0 r 3
(25.8)
where Tf = time lag factor Fd = design factor Values for Spangler’s deflection lag factor, Dl , of 1.0 to 1.5 are recommended; designers usually use the 1.5 value for conservatism. Since the minimum recommended value for Tf is 1.5, the deflections by the modified equation will be higher. Values of the design factor, Fd , are presented for three cases, A, B, and C. The value for case A is 1.0; case B values, which are recommended for design, vary from 0.5 to 1.0; and case C values, which are recommended for designs in which deflection is critical, range from 0.3 to 0.75. In all cases the values of Fd increase with quality and level of compaction of the backfill. It follows that control of bedding and backfill of flexible pipes during construction is critical to their performance. The required passive pressure can be developed only in high-quality fill material, compacted to the proper density. The material surrounding the pipe and extending above the pipe for at least 12 in. should be a well-graded granular stone. Coarse-grained material provides much higher passive resistance and, therefore, limits pipe deflection, in flexible pipe systems, more than fine-grained soil types. Compaction in the lower levels of the pipe is critical. Hand tampers or similar equipment are necessary to ensure that adequate density is obtained in the region below the lower haunches of the pipe. Historically, failure to achieve the proper level of compaction in this area of difficult accessibility has been identified as a major contributing cause to excessive deformations in flexible pipe construction. It is common practice to limit the final vertical deflection of unlined pipes to less than 5% of the diameter. Deflection of pipes with cement mortar coatings should be limited to 2% of the diameter. Field observations of steel pipes in service indicate that once the deflection reaches 20% of the diameter, collapse is imminent.
EXAMPLE 25.1:
A 96-in.-diameter steel pipe with a 1/2-in. wall is installed with its top 15 ft below the ground surface. The local water table is located 7 ft below the surface. Assume that the soil has a modulus of reaction, E 0 , of 1000 psi, and that it has a unit weight of 120 pcf. 1999 by CRC Press LLC
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1. Verify that the pipe will satisfy the buckling and deflection criteria given in AWWA Manual M11 [21]. 2. Determine the amount of vacuum load that can be supported by the pipe. Solution
1. The weight of soil bearing on the pipe is calculated from the prism of soil from the top of the pipe to the ground surface: Wc
= γs hD = 120 × 15 × (96/12) = 14,400 lb/ft
Determine the h/D ratio to obtain the appropriate factor of safety: 15 h = = 1.875 < 2; therefore F S = 3 D 8 The groundwater surface is 15 − 7 = 8 ft above the top of the pie. The water buoyancy factor (Rw ) and the coefficient of elastic support (E 0 ) are calculated on the basis of the depth of cover and groundwater: hw
=
8f t
Rw
=
1 − 0.33
B0
=
hw 8 = 1 − 0.33 × = 0.824 h 15
1 1 + 4e−0.065×15
= 0.399
The modulus of elasticity for steel is 29 × 106 psi; the moment of inertia per inch length of pipe is 0.53 t3 = = 0.0104 in.3 ; hence the product EI = 302,083 in.-lb 12 12 Therefore, by Equation 25.4, the allowable buckling pressure is 302,083 1/2 1 32 × 0.824 × 0.399 × 1,000 × = 19.968 psi qa = 3 963 I=
The total applied load intensity, Q, is given by WL 14,400 Wc + = 0.0361 × 96 + 0.824 × + 0 = 13.766 psi D D 12 × 96 Since Q < qa , the pipe is safe against buckling. Check deflection: ! 3 0.1 × 14,400 12 × 48 = 2.824 in. 1y = 1.5 302,083 + 0.061 × 1,000 × 483 Q = γw hw + Rw
The calculated deflection is approximately 3% of the diameter, less than the 5% usually specified as the limit for unlined pipe. 2. The vacuum pressure that can be supported within the buckling capacity of the pipe is the difference between the calculated critical buckling capacity, qa , and the applied load intensity, Q: Pν = qa − Q = 19.968 − 13.766 = 6.202 psi 1999 by CRC Press LLC
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Corrugated Steel
Corrugated steel has the advantage of greater flexural strength per unit weight of material than plain steel, and has been widely used in surface drainage systems and to a lesser extent in process water systems. In this form, the pipe is assembled from corrugated sheets, rolled to radius and bolted or riveted together. Corrugated steel pipes can be designed according to ASTM standard practice A796 (American Society for Testing and Materials). The practice covers both curve and tangent (“sinusoidal”) walls and smooth walls with helical ribs of rectangular section at regular intervals for increased strength. As with plain steel pipe, this design procedure requires a minimum stiffness in the pipe wall for shipping and handling. To make a quantitative evaluation of the degree of stiffness, a flexibility factor, defined for all wall configurations as D2 (25.9) FF = EI where F F = flexibility factor (in.-lb−1 ) D = pipe diameter (in.) E = modulus of elasticity (psi) I = moment of inertia of wall cross-section per inch (in.3 ) is subject to limits depending on the corrugation configuration and the type of installation. For example, in configurations of sinusoidal corrugations, specified in ASTM A760 and A761 [4, 5], values of the flexibility factor are restricted to 0.020 to 0.060. The phenomenon of buckling of buried corrugated pipes has been investigated, through prototype testing, by Watkins [3]. Design curves utilizing the results of that research were originally published in an American Iron and Steel Institute (AISI) design manual [3] and have been continued into the current edition of the book. The curves provide buckling loads for corrugated steel–walled pipes as a function of diameter-to-radius of gyration ratio. The design equations given in ASTM A796 [6] are of the same general form as the design curves developed by AISI. That is, there are three ranges of behavior—elastic buckling, inelastic buckling, and yield—and the dependence of the expressions on the independent variable, D/r, is the same in the two regimes of the formulas in both documents. The principal difference between the two approaches is the inclusion of an explicit dependence on soil stiffness in the ASTM A796 equations. The AISI formulas, on the other hand, account for soil stiffness by reduction in applied load for well-compacted backfills. The applicable formulas for critical buckling stress, as given in ASTM A796 [6], and their applicable ranges of diameter-to-radius of gyration ratio are given below: s fu2 kD 2 kD 24E for ≤ fc = fu − 48E r r fu s 12E kD 24E fc = for (25.10) > kD 2 r fu r
subject to the provision that: fc ≤ fy where fc = fy = fu = k =
critical buckling stress (psi) specified minimum yield stress (psi) specified minimum ultimate stress (psi) soil stiffness factor = 0.22 for material at 90% density
1999 by CRC Press LLC
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(25.11)
E D r
= modulus of elasticity (psi) = pipe diameter (in.) = radius of gyration of pipe wall (in.) The buckling formulas in ASTM A796 [6] also are given in the AASHTO (American Association of State Highway and Transportation Officials) specification for design of highway bridges [1]. The corresponding equations provided in the AISI handbook [3] are 2 D D for 294 < < 500 fc = 40,000 − 0.081 r r fc
=
4.93 × 109 D 2
for
r
D > 500 r
(25.12)
with, again, the provision that the critical buckling stress cannot exceed the yield stress, fy . A comparison of the two sets of formulas can be made to determine the maximum variation. The soil parameter, k, obviously affects the results. In ASTM A796 [6], a value of k = 0.22 is recommended for good fill material compacted to 90% of standard density; no suggestions are provided for other backfill conditions. The AISI expressions for critical buckling stress (Equation 25.12) do not contain any dependence on the degree of compaction of the surrounding soil. However, the handbook does recommend load factors, which multiply the applied loads, that are related to the degree of compaction. For example, the recommended load factor for 90% compaction is 0.75. Use of a load factor of 0.75 has the same effect as increasing the allowable stress by 1.33. If the results from the ASTM and AISI equations are compared on that basis, the values are within 10%. On the other hand, if a load factor of 1.0, which corresponds to a density of only 80% standard, is used, the soil stiffness, k, must be increased to 0.26 for the two sets of formulas to give approximately the same results. Clearly, use of the higher value for k results in a slightly more conservative design, and that may be desirable, in view of the normally unknown character of the actual installed backfill. In either case, the appropriate wall cross-section must be selected to satisfy fc Wc ≤ 2A (SF )
(25.13)
where A = cross-section area of wall per unit length SF = safety factor = 2 Wc = vertical load per unit length of pipe (Note: Wc must be multiplied by the appropriate load factor if the AISI equations are used) It is of interest to note the D/r ratios that form the transition between elastic and inelastic and between buckling and yield behavior in the ASTM A796 equations. For pipe meeting ASTM A760, maximum thickness of 0.168 in., the specified minimum yield and ultimate stress are 33 ksi and 45 ksi, respectively. For those values, elastic buckling controls design for D/r ratios greater than 478 and yield controls for D/r ratios less than 350, for k equal to 0.26. These values correspond fairly closely with the AISI limits of 500 and 294, respectively.
EXAMPLE 25.2:
Consider the pipe in Example 25.1. Determine the minimum wall thickness of 3 in. x 1 in. corrugated pipe that will satisfy the buckling expressions of Equation 25.10 and the handling requirement of Equation 25.9 with F F limited to a maximum value of 0.033. Assume a value of k of 0.26. Also, the minimum specified yield (fy ) and ultimate (fu ) stresses for the material are 33,000 and 45,000 psi, respectively. 1999 by CRC Press LLC
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Solution The radius of gyration for all thicknesses in the 3 × 1 series range from 0.3410 to 0.3499 in. (see Table 25.3). Therefore, in the calculations use r = 0.34 in.
It follows that
Since
kD r
fy
Since the calculated buckling stress exceeds the yield, the critical stress is the yield stress, 33,000 psi. The required wall area per foot of length can be determined by rearrangement of Equation 25.13: A=
Wc
fc 2 (SF )
=
14,400 2×
33,000 2
= 0.437 in.2
The lightest section, thickness of 0.052 in., will satisfy the stress requirement. Check the handling requirement. The moment of inertia per inch of wall is 6.892 × 10−3 in.3 : FF =
962 D2 = = 0.046 > 0.033 EI (29 × 106 ) × (6.892 × 10−3 )
Since the flexibility factor is too large, try the next section in the series, the 0.064-in. thickness, and I = 8.658 × 10−3 in.3 : FF =
962 D2 = 0.037 > 0.033 = EI (29 × 106 ) × (8.658 × 10−3 )
Since the flexibility is still too great, the next section in the series, with a thickness of 0.079 in. and flexibility factor of 0.029 is chosen. This example demonstrates a condition that occurs quite often in the design of flexible pipes; if the handling and installation minimum stiffness requirements are met, the strength requirements are automatically taken care of. Fiberglass Reinforced Plastic
Fiberglass reinforced plastic (FRP) pipe is fabricated by winding glass strands into a matrix of organic resin on a mandrel of the desired diameter. A variation on the fiberglass-resin matrix utilizes cement of polymer mortar incorporated into the structure to add stiffness and reduce cost of materials. ASTM standards D3262 [9], D3517 [10], and D3754 [11], and AWWA standard C950 [19] provide requirements for manufacture of both the fiberglass-resin and the mortar pipe in diameters up to 144 in. Structural strength and rigidity against external loads for this type of pipe are established by load tests performed as specified by ASTM D2412 [8]. In the load test, equal and opposite concentrated loads are applied on opposite ends of a diameter. Load deflection data are obtained from which stiffness and related buckling strength of the pipe can be determined. Each of the mentioned pipe specifications provides for levels of pipe stiffness (P S) of 9, 18, 36, and 72 psi. These values represent applied force per unit length of pipe divided by deflection. Use of 1999 by CRC Press LLC
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TABLE 25.3 Sectional Properties for 3 × 1 in. Corrugated Sheets Thickness in.
Area in.2 /ft
Moment of inertia in.3 × 10−3
Radius of gyration in.
0.052 0.064 0.079 0.109 0.138 0.168
0.711 0.890 1.113 1.560 2.008 2.458
6.892 8.658 10.883 15.458 20.175 25.083
0.3410 0.3417 0.3427 0.3448 0.3472 0.3499
From American Society for Testing and Materials. 1994. A796. Standard Practice for Structural Design of Corrugated Steel Pipe, Pipe-Arches and Arches for Storm and Sanitary Sewers and Other Buried Applications. With permission.
the pipe stiffness and the formula for deflection of a point-loaded circular ring allows determination of the product, EI , of the composite pipe wall. In FRP construction, the modulus of elasticity (E) depends on several variables: the moduli of the resin and the glass reinforcement, the relative amounts of glass 20 and resin, and the angle of the filament winding. For that reason, it is convenient to utilize the experimentally determined overall pipe stiffness in design rather than to base calculations on the composite modulus of elasticity of the material. In particular, the buckling formula (Equation 25.4) and Spangler’s equation for deflection (Equation 25.7) can be recast in terms of the pipe stiffness, as shown in the following steps. The formula for deflection of a concentrated loaded pipe can be rearranged to provide an expression for EI : EI = 0.149 where F = concentrated load per inch of pipe (lb/in.) D = pipe deflection (in.) r = pipe radius (in.) Since the pipe stiffness is defined as PS = the relationship between EI and P S becomes
F 3 r 1
(25.14)
F 1
(25.15)
EI = 0.149(P S)r 3
(25.16)
The allowable buckling stress expression, therefore, can be rewritten: 1/2 1 0.596Rw B 0 E 0 P S qa = FS and the deflection formula is
1y = Dl
KWc 0.149P S + 0.061E 0
(25.17)
(25.18)
For underground installations, many fiberglass pipe manufacturers recommend a minimum pipe stiffness of 36 psi in order to ensure sufficient stiffness to perform backfilling properly. Deflections are normally restricted to 5% of the diameter. In contrast to design of steel pipe, it is normal practice to consider the bending stresses induced in the wall by deflection of the pipe. Methods for evaluating these stresses in combination with the stresses due to internal pressure have been developed by a committee of AWWA and will be included in one chapter of a manual on design of fiberglass reinforced water pipe, scheduled for publication in the near future [22].
1999 by CRC Press LLC
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25.4.3
Rigid Design
Concrete Pressure Pipe
Concrete pipe can be used for both pressure and nonpressure applications. It offers the advantage of being corrosion resistant in conditions where steel might be attacked, and in some instances it may be a more cost-effective solution than steel or plastic. When concrete pipe is used in high-pressure systems, prestressed concrete pipe is the type most often selected. The pipe is manufactured in diameters from 24 to 144 in. and is fabricated with walls from 4 to 12 in. thick. A steel cylinder, usually of 16-gal thickness, is embedded in the wall for leak protection. The outer surface of the wall is wrapped with high-strength wire under tensile stresses in the 175–200 ksi range. The prestressing places the concrete wall into compression of sufficient magnitude so that it will not be fully relieved under design internal pressure loadings. Finally, a coat of sand-cement mortar is applied over the prestressing wires to provide corrosion protection. A comprehensive design procedure for this type of pipe is contained in AWWA C304 [17]. Prestressed pipe is normally designed only by pipe manufacturers. The design provisions meet AWWA C304 and the actual pipe is fabricated according to AWWA C301 [14]. Pipe purchasers must indicate the design pressures, including transients, installation conditions, and surface loads. Reinforced concrete pipe can be designed to sustain internal pressure loads, but the maximum pressures that can be carried are significantly less than with prestressed pipe and its use in such applications is limited. AWWA C300, C302, and C303 [13, 15, 16] are all specifications covering the design and fabrication of reinforced concrete pressure pipe in differing configurations of reinforcing and with or without the steel cylinder pressure boundary. Design procedures for all three specifications are presented in AWWA Manual M9 [20] and the interested reader is encouraged to review that manual for details. As with prestressed pipe, the pipe specifier usually supplies only the performance attributes and the pipe fabricator performs the design to meet the appropriate specification. Concrete Nonpressure Pipe
ASTM C76 [7] contains specification requirements for reinforced concrete pipe not intended for pressure applications. Five classes of pipe, classes I–V, respectively, representing five levels of structural strength, are specified. The strength is characterized by the concentrated load required to cause a crack of 0.01 in. width and the ultimate concentrated load. Load values are determined experimentally by the three-edge-bearing test. The test simulates concentrated loads applied at opposite ends of a pipe diameter. These loads are referred to as D-loads (D0.01 and Dult ): the concentrated force per unit length of pipe per unit length of diameter necessary to cause either the 10-mil-width crack or ultimate failure of the pipe. D-load values for the five pipe classes included in ASTM C76 are shown in Table 25.4. In determination of the strength required to resist external loads, the total pipe load is estimated by standard methods. Bedding factors, based on the type of installation, the soil type, and its level of compaction, have been developed by the American Concrete Pipe Association [2]. These factors represent the ratio of the maximum bending moment due to a concentrated load to the moment caused by the actual live and dead load of the same magnitude as the concentrated load. ACPA has defined four standard installation types, for which relevant information is shown in Tables 25.5 and 25.6. Bedding factors for embankment installations are given in Table 25.7. Other bedding factors, for trench installations and for live load effects, have also been obtained by ACPA but are not reproduced here. It is noted that ACPA recommends using the dead load factor for live load contributions as well, if the tabulated live load factor is larger than the dead load factor. The calculation methodology used to obtain the various factors is described in the ACPA design data [2]. Use of the appropriate bedding factor allows the conversion of the actual load to an equivalent point load. Comparison of that equivalent load with standard D-loads is used to establish the appropriate class of pipe with sufficient capacity to support the design loads. Normal procedure is to utilize a 1999 by CRC Press LLC
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TABLE 25.4 D -Loads for ASTM C76 Concrete Pipe Pipe class
D0.01 load
Dult load
I II III IV V
800 1000 1350 2000 3000
1200 1500 2000 3000 3750
From American Society for Testing and Materials. 1994. C76. Standard Specification for Reinforced Concrete Culvert, Storm Drain, and Sewer Pipe. With permission.
design factor of safety of 1.0 against the D-load required to cause a 0.01-in. crack. Details of the procedure are illustrated in the following example problem for embankment installation. The design procedure is similar for trench installation.
EXAMPLE 25.3:
Consider a 48-in.-diameter reinforced concrete pipe to be installed beneath a railroad for surface drainage. The pipe is to be installed in an embankment with a depth of cover of 5 ft. For the purpose of this example, assume that the overburden load is equal to the prism of soil above the pipe. Soil unit weight is 120 pcf and the backfill conditions are such that a standard installation type 3 exists. Solution For a 48-in. pipe, the wall thickness of a pipe meeting ASTM C76 is 5 in. The soil dead weight is given by WE = whBc = 120 × 5 ×
48 + 10 = 2900 lb/ft 12
The live load intensity is obtained from Table 25.2: WL = wLL Bc = 2400 ×
48 + 10 12
= 11,600 lb/ft
The total overburden plus live load is WE + WL = 2,900 + 11,600 = 14,500 lb/ft From Table 25.7, the bedding factor, Bf e , is found to be 2.2; the live load bedding factor (not tabulated here) is also 2.2 for this installation. Therefore, use a bedding factor of 2.2 for the total load: 14,500 Total load = = 6,591 Bedding factor 2.2 To obtain required D-load, divide this result by the pipe diameter: D0.01 required = Using class IV pipe, D0.01 = 2000, 1999 by CRC Press LLC
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Dult = 3000.
6591 = 1648 4
TABLE 25.5 Equivalent USCS and AASHTO Soil Classifications for SIDD Soil Designations (ACPA) Representative Soil Types SIDDa soil
USCS
AASHTO
Gravelly sand (category I) Sandy silt (category II)
SW, SP, GW, GP
A1, A3
GM, SM, ML Also GC, SC with less then 20% passing #200 sieve CL, MH, GC, SC CH
A2, A4
Silty clay (category III)
A5, A6 A7
a Standard Installations Direct Design, ACPA From American Concrete Pipe Association. 1995. Design Data 40. Standard Installations and Bedding Factors for the Indirect Design Method. With permission.
TABLE 25.6 Standard Embankment Installation Soils and Minimum Compaction Requirements (ACPA) Installation type
Haunch and outer bedding
Type 1
95% Category I
Type 2
90% Category I or 95% category II 85% Category I, 90% category II, or 95% category III No compaction required, except if category III, use 85% category III
Type 3 Type 4
Lower side 90% Category I, 95% category II, or 100% category III 85% Category I, 90% category II, or 95% category III 85% Category I, 90% category II, or 95% category III No compaction required except if category III, use 85% category III
Note: Bedding thickness for all types: Do /24 minimum, not less than 3 in. If rock foundation, use Do /12 minimum, not less than 6 in. Compaction is standard Proctor. From American Concrete Pipe Association. 1995. Design Data 40. Standard Installations and Bedding Factors for the Indirect Design Method. With permission.
TABLE 25.7 Bedding Factors, Embankment Condition, Bf e (ACPA) Pipe diameter, (in.)
Type 2
12 24 36 72 144
3.2 3.0 2.9 2.8 2.8
Standard installation Type 1 Type 3 4.4 4.2 4.0 3.8 3.6
2.5 2.4 2.3 2.2 2.2
Type 4 1.7 1.7 1.7 1.7 1.7
From American Concrete Pipe Association. 1995. Design Data 40. Standard Installations and Bedding Factors for the Indirect Design Method. With permission.
25.5
Joints
25.5.1
General
In order to form a continuous conduit from the individual pipe sections, it is necessary to connect the sections together in such a way that the pressure-containing and load-resisting capability is preserved in the completed assembly. Each type of pipe discussed previously utilizes special types of joints as explained in the following. 1999 by CRC Press LLC
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25.5.2
Joint Types
Plain Steel
In plain steel plate pipe, the individual pipe sections are fabricated from plate, rolled to the proper radius, and welded together. Joints, in fabricated sections, are either continuous helical or longitudinal. When installed, the sections are welded together using either bell-and-spigot or butt joints. In plain steel pipe, full penetration butt welds are used extensively for field joints. In water works construction, welding of pipelines is covered by AWWA C206 [12]. That standard requires that welding procedures and welding operators be prequalified before use on a job. In addition, tolerances on fit-up are specified and inspection requirements are set out. Strict adherence to project specifications is necessary to guarantee that the desired continuity is obtained at the junction. Although connecting pipe segments by use of full-penetration butt welds has enjoyed wide acceptance in pipeline construction, bell-and-spigot joints, fillet welded, may also be used. These joints require only fillet welds and are generally considered to be less expensive to install than the full-penetration butt weld. However, due to the inherent eccentricity in such joints, a potential for failure exists under certain temperature conditions when longitudinal tensile stresses are developed. Some failures of welded bell-and-spigot joints were reported in the technical literature a few years ago [26, 31]. Since that time, requirements for welding of bell-and-spigot joints in steel pipe in AWWA C206 [12] have been revised to minimize the potential for failure in this type of joint. Corrugated Steel
The field joints used in corrugated steel are usually made by bolting, either in lap joints or with coupling bands that fit over two adjacent sections. In most cases, gaskets should be used at joints to provide leak tightness. Fiberglass Reinforced Plastic
Several types of joints are used in fiberglass pipe. Coupling or bell-and-spigot joints with O-ring gaskets (see Figure 25.2) and mechanical couplings, for unrestrained joints, are specified by the ASTM fiberglass pipe specifications mentioned previously. These joints can be used with restraining devices, such as tie rods, if necessary. In addition, continuous hand lay-up joints consisting of alternating layers of glass fabric and resin or adhesive-bonded bell-and-spigot joints are used for joints that must resist longitudinal force as well as contain the pressure exerted by the fluid carried.
FIGURE 25.2: Bell-and-spigot and coupling joints for fiberglass pipe. (From American Society for Testing and Materials. 1991. D3517. Standard Specification for “Fiberglass” Glass-Fiber-Reinforced Thermosetting-Resin Pressure Pipe. With permission.)
Prestressed Concrete
In straight runs of prestressed concrete pressure pipe, the most common joint type is the belland-spigot slip-on joint with a rubber O-ring gasket (see Figure 25.3). When making the joint, care should be used to ensure that the gasket is in its proper place and that the mating ends are properly located with respect to each other. The exterior of the joint should be filled with flowable sand-cement 1999 by CRC Press LLC
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grout, contained by a suitable appliance. Grouting of the inside joint gap may be required, depending on water chemistry. When it is, the grouting should be completed after the backfill is compacted. The interior surface of each joint must be smoothed to allow unrestricted flow. When axial tension forces must be transmitted across a joint, locking variations of the basic slip-on joint are available.
FIGURE 25.3: Prestressed concrete joint details. (Reprinted from Concrete Pressure Pipe, M9, by c permission. Copyright 1995, American Water Works Association.)
Reinforced Concrete
Typical joints for reinforced concrete pressure pipe are shown in Figure 25.4. Joints for concrete nonpressure pipe are similar to the concrete-only joints in Figure 25.4. In some cases, gaskets are not used in nonpressure pipe.
1999 by CRC Press LLC
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FIGURE 25.4: Joint details for reinforced concrete pipe. (Reprinted from Concrete Pressure Pipe, M9, c by permission. Copyright 1995, American Water Works Association.)
25.5.3
Hydrostatic Testing
A field hydrostatic test is usually performed to verify that all joints are watertight. Test pressure should exceed the maximum design pressure, including transients, by at least 25%. Leakage through welded joints should be virtually nonexistent. It is common to allow a slight leakage rate for O-ring gasketed joints (AWWA, C600).
25.6
Corrosion Protection
There are three environmental agents that exert strong influence on corrosion of the pipe wall material in buried installations. These are the water, or other fluid, carried, the soil in contact with the pipe, and the groundwater. In the case of certain process water systems, such as power plant condenser cooling systems, the water may be circulated continuously within a closed loop using cooling towers, lakes, or other means of exhausting heat. When closed systems are used, even in fresh water environments, the concentrations of certain compounds in the water may increase and cause elevated corrosion rates in steel pipes. Chlorides are generally believed to be the most aggressive compounds, normally found in water sources, in regard to corrosion of steel. Chlorides can also be harmful to concrete pipes, posing threats to the concrete itself and to steel reinforcing. Sulfates are not usually associated with steel corrosion, but they can be detrimental to concrete. Once-through systems, on the other hand, are usually less corrosive for steel pipes and less harmful to concrete than the closed-cycle systems. When brackish water is used for cooling, positive steps must be taken to ensure that corrosion is controlled. While the process water carried may promote corrosion or other damage on the inside of the pipe, the outside surface may be attacked by the surrounding soil, the groundwater, or both. Soils with low 1999 by CRC Press LLC
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electrical resistivity may help advance corrosion in steel. Soils that have sulfate compounds above certain critical levels can cause damage to concrete pipe. The groundwater can have the same effects on the exterior of the pipe as the process fluid has on the inside. Specifically, groundwater with high chloride or sulfate contents may be harmful to the pipe material. Because of the wide range of possibilities for the existence of detrimental chemical action, it is essential that the nature of the external and internal environments of the pipe be evaluated in the design process. Chemical analyses of the process water and the groundwater are essential. Also, chemical analysis of the soil and resistivity survey results must be available in order to make the best choice of pipe system to withstand the exposure throughout the design life of the facility.
25.6.1
Coatings
Coatings can be used to inhibit corrosion or other forms of deterioration in both concrete and steel pipes. Type and extent of coatings depends on the service environment. Steel pipes are almost always coated externally. Coal tar enamel and wrapping has been used successfully in the U.S. for decades. Epoxies and urethanes, among others, have become popular in more recent times. Cement mortar coatings may be applied to both the interior and exterior of steel pipes. This type of coating offers several advantages and has an extensive record of satisfactory service. When exterior coating of prestressed concrete pipes is desired, certain epoxies are acceptable. Because prevention of corrosion in the prestressing wires is so important, pipe designers sometimes specify an additional coating to supplement the protection furnished by the cement mortar coating.
25.6.2
Cathodic Protection
Protection against corrosion may, in certain circumstances, require a cathodic protection system. For example, cathodic protection has proven to be very successful in providing leak-free high-pressure oil and natural gas pipelines throughout the U.S. Power plant sites have widely dispersed grounding systems, which can cause unpredictable stray currents that may promote steel corrosion. Cathodic protection must be designed by competent engineers based on information regarding the extent of buried facilities, the soil resistivity measurements, and the plant grounding system. Electrical continuity should be provided on prestressed concrete pipe if present or future installation of cathodic protection is a possibility.
References [1] American Association of State Highway and Transportation Officials (AASHTO). 1992. Standard Specifications for Highway Bridges. 15th ed. [2] American Concrete Pipe Association (ACPA). 1995. Design Data 40. Standard Installations
and Bedding Factors for the Indirect Design Method. [3] American Iron and Steel Institute (AISI). 1977. Handbook of Steel Drainage & Highway
Construction Products. [4] American Society for Testing and Materials (ASTM). 1993. A760. Standard Specification for
Corrugated Steel Pipe, Metallic Coated for Sewers and Drains. [5] American Society for Testing and Materials (ASTM). 1990. A761. Standard Specification for
Corrugated Steel Structural Plate, Zinc-Coated, for Field-Bolted Pipe, Pipe-Arches, and Arches. [6] American Society for Testing and Materials (ASTM). 1994. A796. Standard Practice for Structural Design of Corrugated Steel Pipe, Pipe-Arches and Arches for Storm and Sanitary Sewers and Other Buried Applications. 1999 by CRC Press LLC
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[7] American Society for Testing and Materials (ASTM). 1994. C76. Standard Specification for
Reinforced Concrete Culvert, Storm Drain, and Sewer Pipe. [8] American Society for Testing and Materials (ASTM). 1993. C2412. Standard Test Method for
Determination of External Loading Characteristics of Plastic Pipe by Parallel-Plate Loading. [9] American Society for Testing and Materials (ASTM). 1993. D3262. Standard Specification for
“Fiberglass” (Glass-Fiber-Reinforced Thermosetting-Resin Sewer Pipe). [10] American Society for Testing and Materials (ASTM). 1991. D3517. Standard Specification for
“Fiberglass” (Glass-Fiber-Reinforced Thermosetting-Resin Pressure Pipe). [11] American Society for Testing and Materials (ASTM). 1991. D3754. Standard Specification
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
[24]
[25] [26] [27] [28]
[29] [30]
for “Fiberglass” (Glass-Fiber-Reinforced Thermosetting-Resin Sewer and Industrial Pressure Pipe). American Water Works Association (AWWA). 1991. C206. Standard for Field Welding of Steel Water Pipe. American Water Works Association (AWWA). 1989. C300. Standard for Reinforced Concrete Pressure Pipe, Steel-Cylinder Type, for Water and Other Liquids. American Water Works Association (AWWA). 1992. C301. Standard for Prestressed Concrete Pressure Pipe, Steel-Cylinder Type, for Water and Other Liquids. American Water Works Association (AWWA). 1987. C302. Standard for Reinforced Concrete Pressure Pipe, Noncylinder Type, for Water and Other Liquids. American Water Works Association (AWWA). 1987. C303. Standard for Reinforced Concrete Pressure Pipe, Steel Cylinder Type, Pretensioned, for Water and Other Liquids. American Water Works Association (AWWA). 1992. C304. Standard for Design of Prestressed Concrete Cylinder Pipe. American Water Works Association (AWWA). 1987. C303. Standard for Installation of DuctileIron Water Mains and Their Appurtenances. American Water Works Association (AWWA). 1988. C950. Standard for Fiberglass Pressure Pipe. American Water Works Association (AWWA). 1995. Manual M9. Concrete Pressure Pipe. American Water Works Association (AWWA). 1989. Manual M11. Steel Pipe-A Guide for Design and Installation. American Water Works Association (AWWA). 1997 (projected). Manual M45. Fiberglass Pressure Pipe. Anton, W.F., J.E. Herold, R.T. Dailey, and W.J. Cichanski. 1990. Investigation & Rehabilitation of Seattle’s Tolt Pipeline. Proceedings of the International Conference on Pipeline Design and Installation. ASCE. Las Vegas. pp. 213–229. Committee on Seismic Analysis of the ASCE Structural Division Committee on Nuclear Structures and Materials. 1983. Seismic Response of Buried Pipes and Structural Components. American Society of Civil Engineers (ASCE). Doyle, J.M. and S.L. Chu. 1968. Plastic Design of Flexible Conduits. J. of Structural Division (ASCE), 94, 1935–1944. Eberhardt, A. 1990. 108-in. Diameter Steel Water Conduit Failure and Assessment of AWWA Practice. J. of Performance of Constructed Facilities (ASCE), 4, 30–50. Howard, A.K., L.A. Kinney, and R.P. Fuerst. 1995. Method for Prediction of Flexible Pipe Deflection. Report M-25 (M0250000.995). U.S. Bureau of Reclamation. Denver, CO. Marston, A. and A.O. Anderson. 1913. The Theory of Loads on Pipes in Ditches and Tests of Cement and Clay Drain Tile and Sewer Pipe. Bull. 31. Iowa Engineering Experiment Station. Ames, IA. Marston, A. 1930. The Theory of External Loads on Closed Conduits in the Light of the Latest Experiments. Bull. 96. Iowa Engineering Experiment Station. Ames, IA. MathSoft Inc. 1994. Mathcad PLUS 5.0. MathSoft Inc. Cambridge, MA.
1999 by CRC Press LLC
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[31] Moncarz, P.D., J.C. Shyne, and G.K. Derbalian. 1987. Failures of 108-inch Steel Pipe Water Main. J. of Performance of Constructed Facilities (ASCE), 1, 168–187. [32] Netzel, R.J. and J.M. Doyle. 1983. Design of Underground Pipelines. Proc. American Power Conference, 45, 792. [33] Newmark, N.M. 1935. Simplified Computation of Vertical Pressures in Elastic Foundations. Circular No. 24. Engineering Experiment Station. University of Illinois. Urbana, IL. [34] Schlick, W.J. 1932. Loads on Pipe in Wide Ditches. Bull. 108. Iowa Engineering Station. Ames, IA. [35] Schlick, W.J. 1952. Loads on Negative Projecting Conduits. Proc. Highway Research Board, 31, 308. [36] Spangler, M.G., R. Winfrey, and C. Mason. 1926. Experimental Determination of Static and Impact Loads Transmitted to Culverts. Bull. 76. Iowa Engineering Experiment Station. Ames, IA. [37] Spangler, M.G. and R.L. Hennessy. 1946. A Method of Computing Live Loads Transmitted to Underground Conduits. Proc. Highway Research Board, 26, 83. [38] Spangler, M.G. 1941. The Structural Design of Flexible Pipe Culverts. Bull. 153. Iowa Engineering Experiment Station. Ames, IA. [39] Spangler, M.G. 1950. A Theory of Loads on Negative Projecting Conduits. Proc. Highway Research Board, 29, 153. [40] Spangler, M.G. and R.L. Handy. 1982. Soil Engineering. 4th ed. Harper & Row. [41] Timoshenko, S. and J.N. Goodier. 1951. Theory of Elasticity. McGraw-Hill, New York. [42] Watkins, R.K. and M.G. Spangler. 1958. Some Characteristics of the Modulus of Passive Resistance of Soil: A Study in Similitude. Proc. Highway Research Board, 37, 576. [43] Young, W.C.1989. Roark’s Formulas for Stress & Strain. 6th ed. McGraw-Hill, New York.
1999 by CRC Press LLC
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Rosowsky, D. V. “Structural Reliability” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Structural Reliability
1
26.1 Introduction
Definition of Reliability • Introduction to Reliability-Based Design Concepts
26.2 Basic Probability Concepts
Random Variables and Distributions • Moments • Concept of Independence • Examples • Approximate Analysis of Moments • Statistical Estimation and Distribution Fitting
26.3 Basic Reliability Problem
Basic R−S Problem • More Complicated Limit State Functions Reducible to R − S Form • Examples
26.4 Generalized Reliability Problem
Introduction • FORM/SORM Techniques • Monte Carlo Simulation
26.5 System Reliability
Introduction • Basic Systems • Introduction to Classical System Reliability Theory • Redundant Systems • Examples
26.6 Reliability-Based Design (Codes)
Introduction • Calibration and Selection of Target Reliabilities • Material Properties and Design Values • Design Loads and Load Combinations • Evaluation of Load and Resistance Factors
D. V. Rosowsky Department of Civil Engineering, Clemson University, Clemson, SC
26.1
26.7 Defining Terms Acknowledgments References Further Reading Appendix
Introduction
26.1.1 Definition of Reliability Reliability and reliability-based design (RBD) are terms that are being associated increasingly with the design of civil engineering structures. While the subject of reliability may not be treated explicitly in the civil engineering curriculum, either at the graduate or undergraduate levels, some basic knowledge of the concepts of structural reliability can be useful in understanding the development and bases for many modern design codes (including those of the American Institute of Steel Construction [AISC],
1 Parts of this chapter were previously published by CRC Press in The Civil Engineering Handbook, W.F. Chen, Ed., 1995.
1999 by CRC Press LLC
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the American Concrete Institute [ACI], the American Association of State Highway Transportation Officials [AASHTO], and others). Reliability simply refers to some probabilistic measure of satisfactory (or safe) performance, and as such, may be viewed as a complementary function of the probability of failure. Reliability = f cn (1 − Pfailure )
(26.1)
When we talk about the reliability of a structure (or member or system), we are referring to the probability of safe performance for a particular limit state. A limit state can refer to ultimate failure (such as collapse) or a condition of unserviceability (such as excessive vibration, deflection, or cracking). The treatment of structural loads and resistances using probability (or reliability) theory, and of course the theories of structural analysis and mechanics, has led to the development of the latest generation of probability-based, reliability-based, or limit states design codes. If the subject of structural reliability is generally not treated in the undergraduate civil engineering curriculum, and only a relatively small number of universities offer graduate courses in structural reliability, why include a basic (introductory) treatment in this handbook? Besides providing some insight into the bases for modern codes, it is likely that future generations of structural codes and specifications will rely more and more on probabilistic methods and reliability analyses. The treatment of (1) structural analysis, (2) structural design, and (3) probability and statistics in most civil engineering curricula permits this introduction to structural reliability without the need for more advanced study. This section by no means contains a complete treatment of the subject, nor does it contain a complete review of probability theory. At this point in time, structural reliability is usually only treated at the graduate level. However, it is likely that as RBD becomes more accepted and more prevalent, additional material will appear in both the graduate and undergraduate curricula.
26.1.2
Introduction to Reliability-Based Design Concepts
The concept of RBD is most easily illustrated in Figure 26.1. As shown in that figure, we consider the
FIGURE 26.1: Basic concept of structural reliability.
acting load and the structural resistance to be random variables. Also as the figure illustrates, there is the possibility of a resistance (or strength) that is inadequate for the acting load (or conversely, that the load exceeds the available strength). This possibility is indicated by the region of overlap on Figure 26.1 in which realizations of the load and resistance variables lead to failure. The objective 1999 by CRC Press LLC
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of RBD is to ensure the probability of this condition is acceptably small. Of course, the load can refer to any appropriate structural, service, or environmental loading (actually, its effect), and the resistance can refer to any limit state capacity (i.e., flexural strength, bending stiffness, maximum tolerable deflection, etc.). If we formulate the simplest expression for the probability of failure (Pf ) as (26.2) Pf = P [(R − S) < 0] we need only ensure that the units of the resistance (R) and the load (S) are consistent. We can then use probability theory to estimate these limit state probabilities. Since RBD is intended to provide (or ensure) uniform and acceptably small failure probabilities for similar designs (limit states, materials, occupancy, etc.), these acceptable levels must be predetermined. This is the responsibility of code development groups and is based largely on previous experience (i.e., calibration to previous design philosophies such as allowable stress design [ASD] for steel) and engineering judgment. Finally, with information describing the statistical variability of the loads and resistances, and the target probability of failure (or target reliability) established, factors for codified design can be evaluated for the relevant load and resistance quantities (again, for the particular limit state being considered). This results, for instance, in the familiar form of design checking equations: X γi Qn,i (26.3) φRn ≥ i
referred to as load and resistance factor design (LRFD) in the U.S., and in which Rn is the nominal (or design) resistance and Qn are the nominal load effects. The factors γi and φ in Equation 26.3 are the load and resistance factors, respectively. This will be described in more detail in later sections. Additional information on this subject may be found in a number of available texts [3, 21].
26.2
Basic Probability Concepts
This section presents an introduction to basic probability and statistics concepts. Only a sufficient presentation of topics to permit the discussion of reliability theory and applications that follows is included herein. For additional information and a more detailed presentation, the reader is referred to a number of widely used textbooks (i.e., [2, 5]).
26.2.1
Random Variables and Distributions
Random variables can be classified as being either discrete or continuous. Discrete random variables can assume only discrete values, whereas continuous random variables can assume any value within a range (which may or may not be bounded from above or below). In general, the random variables considered in structural reliability analyses are continuous, though some important cases exist where one or more variables are discrete (i.e., the number of earthquakes in a region). A brief discussion of both discrete and continuous random variables is presented here; however, the reliability analysis (theory and applications) sections that follow will focus mainly on continuous random variables. The relative frequency of a variable is described by its probability mass function (PMF), denoted pX (x), if it is discrete, or its probability density function (PDF), denoted fX (x), if it is continuous. (A histogram is an example of a PMF, whereas its continuous analog, a smooth function, would represent a PDF.) The cumulative frequency (for either a discrete or continuous random variable) is described by its cumulative distribution function (CDF), denoted FX (x). (See Figure 26.2.) There are three basic axioms of probability that serve to define valid probability assignments and provide the basis for probability theory.
1999 by CRC Press LLC
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FIGURE 26.2: Sample probability functions. 1. The probability of an event is bounded by zero and one (corresponding to the cases of zero probability and certainty, respectively). 2. The sum of all possible outcomes in a sample space must equal one (a statement of collectively exhaustive events). 3. The probability of the union of two mutually exclusive events is the sum of the two individual event probabilities, P [A ∪ B] = P [A] + P [B]. The PMF or PDF, describing the relative frequency of the random variable, can be used to evaluate the probability that a variable takes on a value within some range. P [a < Xdiscr ≤ b] =
b X
pX (x)
(26.4)
fX (x)dx
(26.5)
a
Z P [a < Xcts ≤ b] =
b
a
The CDF is used to describe the probability that a random variable is less than or equal to some value. Thus, there exists a simple integral relationship between the PDF and the CDF. For example, for a continuous random variable, Z a fX (x)dx (26.6) FX (a) = P [X ≤ a] = −∞
There are a number of common distribution forms. The probability functions for these distribution forms are given in Table 26.1. 1999 by CRC Press LLC
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TABLE 26.1 Distribution Binomial Geometric Poisson Exponential
Gamma
Common Distribution Forms and Their Parameters PMF or PDF
Parameters
n px (1 − p)n−x x x = 0, 1, 2, . . . , n pX (x) = p (1 − p)x−1 x = 0, 1, 2, . . . pX (x) =
p
Var[X] = np(1 − p) E[X] = 1/p Var[X] = (1 − p)/p2 E[X] = υt Var[X] = υt E[X] = 1/λ Var[X] = 1/λ2
p
x
−υt pX (x) = (υt) x! e x = 0, 1, 2, . . . fX (x) = λe−λx x≥0
υ λ
k−1 −υx fX (x) = υ(υx)0(k)e
υ, k
E[X] = k/υ
Var[X] = k/υ 2
x≥0
Normal
fX (x) = √ 1
2πσ
exp − 21
x−µ σ
2
µ, σ
E[X] = µ
Var[X] = σ 2
−∞ < x < ∞
Lognormal
fX (x) = √ 1
2πζ x
2 exp − 21 ln x−λ ζ
E[X] = exp λ + 21 ζ 2
λ, ζ
Var[X] = E 2 [X] exp ζ 2 − 1
x≥0
Uniform
1 fX (x) = b−a
E[X] = (a+b) 2
a, b
1 (b − a)2 Var[X] = 12
a 2)
E[X] = ε + (u − ε)0 1 + k1
k, w, ε
h i Var[X] = (u − ε)2 0 1 + 2k − 0 2 1 + k1
An important class of distributions for reliability analysis is based on the statistical theory of extreme values. Extreme value distributions are used to describe the distribution of the largest or smallest of a set of independent and identically distributed random variables. This has obvious implications for reliability problems in which we may be concerned with the largest of a set of 50 annual-extreme snow loads or the smallest (lowest) concrete strength from a set of 100 cylinder tests, for example. There are three important extreme value distributions (referred to as Type I, II, and III, respectively), which are also included in Table 26.1. Additional information on the derivation and application of extreme value distributions may be found in various texts (e.g., [3, 21]). In most cases, the solution to the integral of the probability function (see Equations 26.5 and 26.6) is available in closed form. The exceptions are two of the more common distributions, the normal and lognormal distributions. For these cases, tables are available (i.e., [2, 5, 21]) to evaluate the integrals. To simplify the matter, and eliminate the need for multiple tables, the standard normal distribution is most often tabulated. In the case of the normal distribution, the probability is evaluated:
b − µx P [a < X ≤ b] = FX (b) − FX (a) = 8 σx 1999 by CRC Press LLC
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a − µx −8 σx
(26.7)
where FX (·) = the particular normal distribution, 8(·) = the standard normal CDF, µx = mean of random variable X, and σx = standard deviation of random variable X. Since the standard normal variate is therefore the variate minus its mean, divided by its standard deviation, it too is a normal random variable with mean equal to zero and standard deviation equal to one. Table 26.2 presents the standard normal CDF in tabulated form. In the case of the lognormal distribution, the probability is evaluated (also using the standard normal probability tables):
ln b − λy P [a < Y ≤ b] = Fy (b) − FY (a) = 8 ξy
ln a − λy −8 ξy
(26.8)
where FY (·) = the particular lognormal distribution, 8(·) = the standard normal CDF, and λy and ξy are the lognormal distribution parameters related to µy = mean of random variable Y and Vy = coefficient of variation (COV) of random variable Y , by the following: λy
=
ξy2
=
1 ln µy − ξy2 2 ln Vy2 + 1
(26.9) (26.10)
Note that for relatively low coefficients of variation (Vy ≈ 0.3 or less), Equation 26.10 suggests the approximation, ξ ≈ Vy .
26.2.2
Moments
Random variables are characterized by their distribution form (i.e., probability function) and their moments. These values may be thought of as shifts and scales for the distribution and serve to uniquely define the probability function. In the case of the familiar normal distribution, there are two moments: the mean and the standard deviation. The mean describes the central tendency of the distribution (the normal distribution is a symmetric distribution), while the standard deviation is a measure of the dispersion about the mean value. Given a set of n data points, the sample mean and the sample variance (which is the square of the sample standard deviation) are computed as mx = σˆ x2 =
1 n−1
1X Xi n i X (Xi − mx )2
(26.11) (26.12)
i
Many common distributions are two-parameter distributions and, while not necessarily symmetric, are completely characterized by their first two moments (see Table 26.1). The population mean, or first moment of a continuous random variable, is computed as Z µx = E[X] =
+∞
−∞
xfX (x)dx
(26.13)
where E[X] is referred to as the expected value of X. The population variance (the square of the population standard deviation) of a continuous random variable is computed as σx2 1999 by CRC Press LLC
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h
= Var[X] = E (X − µx )
2
i
Z =
+∞
−∞
(x − µx )2 fX (x)dx
(26.14)
TABLE 26.2
Complementary Standard Normal Table,
8(−β) = 1 − 8(β)
1999 by CRC Press LLC
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β
8(−β)
β
8(−β)
β
8(−β)
.00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .21 .22 .23 .24 .25 .26 .27 .28 .29 .30 .31 .32 .33 .34 .35 .36 .37 .38 .39 .40 .41 .42 .43 .44 .45 .46 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63
.50000 + 00 .4960E + 00 .4920E + 00 .4880E + 00 .4840E + 00 .4801E + 00 .4761E + 00 .4721E + 00 .4681E + 00 .4641E + 00 .4602E + 00 .4562E + 00 .4522E + 00 .4483E + 00 .4443E + 00 .4404E + 00 .4364E + 00 .4325E + 00 .4286E + 00 .4247E + 00 .4207E + 00 .4168E + 00 .4129E + 00 .4090E + 00 .4052E + 00 .4013E + 00 .3974E + 00 .3936E + 00 .3897E + 00 .3859E + 00 .3821E + 00 .3783E + 00 .3745E + 00 .3707E + 00 .3669E + 00 .3632E + 00 .3594E + 00 .3557E + 00 .3520E + 00 .3483E + 00 .3446E + 00 .3409E + 00 .3372E + 00 .3336E + 00 .3300E + 00 .3264E + 00 .3228E + 00 .7927E − 01 .7780E − 01 .7636E − 01 .7493E − 01 .7353E − 01 .7215E − 01 .7078E − 01 .6944E − 01 .6811E − 01 .6681E − 01 .6552E − 01 .6426E − 01 .6301E − 01 .6178E − 01 .6057E − 01 .5938E − 01 .5821E − 01 .5705E − 01 .5592E − 01 .5480E − 01 .5370E − 01 .5262E − 01 .5155E − 01
.47 .48 .49 .50 .51 .52 .53 .54 .55 .56 .57 .58 .59 .60 .61 .62 .63 .64 .65 .66 .67 .68 .69 .70 .71 .72 .73 .74 .75 .76 .77 .78 .79 .80 .81 .82 .83 .84 .85 .86 .87 .88 .89 .90 .91 .92 .93 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10
.3192E + 00 .3156E + 00 .3121E + 00 .3085E + 00 .3050E + 00 .3015E + 00 .2981E + 00 .2946E + 00 .2912E + 00 .2877E + 00 .2843E + 00 .2810E + 00 .2776E + 00 .2743E + 00 .2709E + 00 .2676E + 00 .2643E + 00 .2611E + 00 .2578E + 00 .2546E + 00 .2514E + 00 .2483E + 00 .2451E + 00 .2420E + 00 .2389E + 00 .2358E + 00 .2327E + 00 .2297E + 00 .2266E + 00 .2236E + 00 .2207E + 00 .2177E + 00 .2148E + 00 .2119E + 00 .2090E + 00 .2061E + 00 .2033E + 00 .2005E + 00 .1977E + 00 .1949E + 00 .1922E + 00 .1894E + 00 .1867E + 00 .1841E + 00 .1814E + 00 .1788E + 00 .1762E + 00 .3005E − 01 .2938E − 01 .2872E − 01 .2807E − 01 .2743E − 01 .2680E − 01 .2619E − 01 .2559E − 01 .2500E − 01 .2442E − 01 .2385E − 01 .2330E − 01 .2275E − 01 .2222E − 01 .2169E − 01 .2118E − 01 .2068E − 01 .2018E − 01 .1970E − 01 .1923E − 01 .1876E − 01 .1831E − 01 .1786E − 01
.94 .95 .96 .97 .98 .99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 2.35 2.36 2.37 2.38 2.39 2.40 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.50 2.51 2.52 2.53 2.54 2.55 2.56 2.57
.1736E + 00 .1711E + 00 .1685E + 00 .1660E + 00 .1635E + 00 .1611E + 00 .1587E + 00 .1562E + 00 .1539E + 00 .1515E + 00 .1492E + 00 .1469E + 00 .1446E + 00 .1423E + 00 .1401E + 00 .1379E + 00 .1357E + 00 .1335E + 00 .1314E + 00 .1292E + 00 .1271E + 00 .1251E + 00 .1230E + 00 .1210E + 00 .1190E + 00 .1170E + 00 .1151E + 00 .1131E + 00 .1112E + 00 .1093E + 00 .1075E + 00 .1056E + 00 .1038E + 00 .1020E + 00 .1003E + 00 .9853E − 01 .9680E − 01 .9510E − 01 .9342E − 01 .9176E − 01 .9012E − 01 .8851E − 01 .8691E − 01 .8534E − 01 .8379E − 01 .8226E − 01 .8076E − 01 .9387E − 02 .9138E − 02 .8894E − 02 .8656E − 02 .8424E − 02 .8198E − 02 .7976E − 02 .7760E − 02 .7549E − 02 .7344E − 02 .7143E − 02 .6947E − 02 .6756E − 02 .6569E − 02 .6387E − 02 .6210E − 02 .6037E − 02 .5868E − 02 .5703E − 02 .5543E − 02 .5386E − 02 .5234E − 02 .5085E − 02
TABLE 26.2
Complementary Standard Normal Table,
8(−β) = 1 − 8(β) (continued)
1999 by CRC Press LLC
c
β
8(−β)
β
8(−β)
β
8(−β)
1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 2.82 2.83 2.84 2.85 2.86 2.87 2.88 2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28
.5050E − 01 .4947E − 01 .4846E − 01 .4746E − 01 .4648E − 01 .4551E − 01 .4457E − 01 .4363E − 01 .4272E − 01 .4182E − 01 .4093E − 01 .4006E − 01 .3920E − 01 .3836E − 01 .3754E − 01 .3673E − 01 .3593E − 01 .3515E − 01 .3438E − 01 .3363E − 01 .3288E − 01 .3216E − 01 .3144E − 01 .3074E − 01 .2401E − 02 .2327E − 02 .2256E − 02 .2186E − 02 .2118E − 02 .2052E − 02 .1988E − 02 .1926E − 02 .1866E − 02 .1807E − 02 .1750E − 02 .1695E − 02 .1641E − 02 .1589E − 02 .1538E − 02 .1489E − 02 .1441E − 02 .1395E − 02 .1350E − 02 .1306E − 02 .1264E − 02 .1223E − 02 .1183E − 02 .1144E − 02 .1107E − 02 .1070E − 02 .1035E − 02 .1001E − 02 .9676E − 03 .9354E − 03 .9042E − 03 .8740E − 03 .8447E − 03 .8163E − 03 .7888E − 03 .7622E − 03 .7363E − 03 .7113E − 03 .6871E − 03 .6636E − 03 .6409E − 03 .6189E − 03 .5976E − 03 .5770E − 03 .5570E − 03 .5377E − 03 .5190E − 03
2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.50 3.51 3.52 3.53 3.54 3.55 3.56 3.57 3.58 3.59 3.60 3.61 3.62 3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.70 3.71 3.72 3.73 3.74 3.75
.1743E − 01 .1700E − 01 .1659E − 01 .1618E − 01 .1578E − 01 .1539E − 01 .1500E − 01 .1463E − 01 .1426E − 01 .1390E − 01 .1355E − 01 .1321E − 01 .1287E − 01 .1255E − 01 .1222E − 01 .1191E − 01 .1160E − 01 .1130E − 01 .1101E − 01 .1072E − 01 .1044E − 01 .1017E − 01 .9903E − 02 .9642E − 02 .5009E − 03 .4834E − 03 .4664E − 03 .4500E − 03 .4342E − 03 .4189E − 03 .4040E − 03 .3897E − 03 .3758E − 03 .3624E − 03 .3494E − 03 .3369E − 03 .3248E − 03 .3131E − 03 .3017E − 03 .2908E − 03 .2802E − 03 .2700E − 03 .2602E − 03 .2507E − 03 .2415E − 03 .2326E − 03 .2240E − 03 .2157E − 03 .2077E − 03 .2000E − 03 .1926E − 03 .1854E − 03 .1784E − 03 .1717E − 03 .1653E − 03 .1591E − 03 .1531E − 03 .1473E − 03 .1417E − 03 .1363E − 03 .1311E − 03 .1261E − 03 .1212E − 03 .1166E − 03 .1121E − 03 .1077E − 03 .1036E − 03 .9956E − 04 .9569E − 04 .9196E − 04 .8837E − 04
2.58 2.59 2.60 2.61 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.70 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.80 2.81 3.76 3.77 3.78 3.79 3.80 3.81 3.82 3.83 3.84 3.85 3.86 3.87 3.88 3.89 3.90 3.91 3.92 3.93 3.94 3.95 3.96 3.97 3.98 3.99 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00
.4940E − 02 .4799E − 02 .4661E − 02 .4527E − 02 .4396E − 02 .4269E − 02 .4145E − 02 .4024E − 02 .3907E − 02 .3792E − 02 .3681E − 02 .3572E − 02 .3467E − 02 .3364E − 02 .3264E − 02 .3167E − 02 .3072E − 02 .2980E − 02 .2890E − 02 .2803E − 02 .2718E − 02 .2635E − 02 .2555E − 02 .2477E − 02 .8491E − 04 .8157E − 04 .7836E − 04 .7527E − 04 .7230E − 04 .6943E − 04 .6667E − 04 .6402E − 04 .6147E − 04 .5901E − 04 .5664E − 04 .5437E − 04 .5218E − 04 .5007E − 04 .4804E − 04 .4610E − 04 .4422E − 04 .4242E − 04 .4069E − 04 .3902E − 04 .3742E − 04 .3588E − 04 .3441E − 04 .3298E − 04 .3162E − 04 .2062E − 04 .1332E − 04 .8524E − 05 .5402E − 05 .3391E − 05 .2108E − 05 .1298E − 05 .7914E − 06 .4780E − 06 .2859E − 06 .1694E − 06 .9935E − 07 .5772E − 07 .3321E − 07 .1892E − 07 .9716E − 09 .3945E − 10 .1254E − 11 .3116E − 13 .6056E − 15 .9197E − 17 .1091E − 18
The population variance can also be expressed in terms of expectations as Z σx2 = E[X2 ] − E 2 [X] =
+∞
−∞
Z x 2 fX (x)dx −
+∞
−∞
2 xfX (x)dx
(26.15)
The COV is defined as the ratio of the standard deviation to the mean, and therefore serves as a nondimensional measure of variability. COV = VX =
σx µx
(26.16)
In some cases, higher order (> 2) moments exist, and these may be computed similarly as µ(n) x
n
= E (X − µx )
Z =
+∞
−∞
(x − µx )n fX (x)dx
(26.17)
(n)
where µx = the nth central moment of random variable X. Often, it is more convenient to define the probability distribution in terms of its parameters. These parameters can be expressed as functions of the moments (see Table 26.1).
26.2.3
Concept of Independence
The concept of statistical independence is very important in structural reliability as it often permits great simplification of the problem. While not all random quantities in a reliability analysis may be assumed independent, it is certainly reasonable to assume (in most cases) that loads and resistances are statistically independent. Often, the assumption of independent loads (actions) can be made as well. Two events, A and B, are statistically independent if the outcome of one in no way affects the outcome of the other. Therefore, two random variables, X and Y , are statistically independent if information on one variable’s probability of taking on some value in no way affects the probability of the other random variable taking on some value. One of the most significant consequences of this statement of independence is that the joint probability of occurrence of two (or more) random variables can be written as the product of the individual marginal probabilities. Therefore, if we consider two events (A = probability that an earthquake occurs and B = probability that a hurricane occurs), and we assume these occurrences are statistically independent in a particular region, the probability of both an earthquake and a hurricane occurring is simply the product of the two probabilities: P A “and” B = P [A ∩ B] = P [A]P [B]
(26.18)
Similarly, if we consider resistance (R) and load (S) to be continuous random variables, and assume independence, we can write the probability of R being less than or equal to some value r and the probability that S exceeds some value s (i.e., failure) as P [R ≤ r ∩ S > s]
= P [R ≤ r]P [S > s] = P [R ≤ r] (1 − P [S ≤ s]) = FR (r) (1 − FS (s))
(26.19)
Additional implications of statistical independence will be discussed in later sections. The treatments of dependent random variables, including issues of correlation, joint probability, and conditional probability are beyond the scope of this introduction, but may be found in any elementary text (e.g., [2, 5]). 1999 by CRC Press LLC
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26.2.4
Examples
Three relatively simple examples are presented here. These examples serve to illustrate some important elements of probability theory and introduce the reader to some basic reliability concepts in structural engineering and design.
EXAMPLE 26.1:
The Richter magnitude of an earthquake, given that it has occurred, is assumed to be exponentially distributed. For a particular region in Southern California, the exponential distribution parameter (λ) has been estimated to be 2.23. What is the probability that a given earthquake will have a magnitude greater than 5.5? = 1 − P [M ≤ 5.5] = 1 − FX (5.5) i h = 1 − 1 − e−5.5λ
P [M > 5.5]
= e−2.23×5.5 = e−12.265 ≈ 4.71 × 10−6 Given that two earthquakes have occurred in this region, what is the probability that both of their magnitudes were greater than 5.5? P [M1 > 5.5 ∩ M2 > 5.5]
= P [M1 > 5.5]P [M2 > 5.5] (assumed independence) = (P [M > 5.5])2 (identically distributed) 2 = 4.71 × 10−6 ≈
2.22 × 10−11 (very small!)
EXAMPLE 26.2:
Consider the cross-section of a reinforced concrete column with 12 reinforcing bars. Assume the load-carrying capacity of each of the 12 reinforcing bars (Ri ) is normally distributed with mean of 100 kN and standard deviation of 20 kN. Further assume that the load-carrying capacity of the concrete itself is rc = 500 kN (deterministic) and that the column is subjected to a known load of 1500 kN. What is the probability that this column will fail? First, we can compute the mean and standard deviation of the column’s total load-carrying capacity. E[R]
=
mR = rc +
12 X
E[Ri ] = 500 + 12(100) = 1700 kN
i=1
Var[R]
=
σR2 =
12 X i=1
σR2i = 12 (20)2 = 4800 kN2
... σR = 69.28 kN
Since the total capacity is the sum of a number of normal variables, it too is a normal variable (central limit theorem). Therefore, we can compute the probability of failure as the probability that the load-carrying capacity, R, is less than the load of 1500 kN. 1500 − 1700 = 8(−2.89) ≈ 0.00193 P [R < 1500] = FR (1500) = 8 69.28 1999 by CRC Press LLC
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EXAMPLE 26.3:
The moment capacity (M) of the simply supported beam (l = 10 ft) shown in Figure 26.3 is assumed to be normally distributed with mean of 25 ft-kips and COV of 0.20. Failure occurs if the
FIGURE 26.3: Simply supported beam (for Example 26.3). maximum moment exceeds the moment capacity. If only a concentrated load P = 4 kips is applied at midspan, what is the failure probability? 4 100 Pl = = 10 ft-kips Mmax = 4 4 10 − 25 = 8(−3.0) ≈ 0.00135 Pf = P [M < Mmax ] = FM (10) = 8 5 If only a uniform load w = 1 kip/ft is applied along the entire length of the beam, what is the failure probability? 2
Mmax
=
1 100 wl 2 = 8 8
Pf
=
P [M < Mmax ] = FM (12.5) = 8
= 12.5 ft-kips
12.5 − 25 5
= 8(−2.5) ≈ 0.00621
If the beam is subjected to both P and w simultaneously, what is the probability the beam performs safely? Mmax Pf
wl 2 Pl + = 10 + 12.5 = 22.5 ft-kips 4 8 22.5 − 25 = 8(−0.5) ≈ 0.3085 = P [M < Mmax ] = FM (22.5) = 8 5 ... P “safety” = PS = (1 − Pf ) = 0.692 =
Note that this failure probability is not simply the sum of the two individual failure probabilities computed previously. Finally, for design purposes, suppose we want a probability of safe performance Ps = 99.9%, for the case of the beam subjected to the uniform load (w) only. What value of wmax 1999 by CRC Press LLC
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(i.e., maximum allowable uniform load for design) should we specify? 2 wmax l 2 10 = wmax = 12.5 (wmax ) Mallow. = 8 8 goal : P [M > 12.5wmax ] = 0.999 1 − FM (12.5wmax ) = 0.999 12.5wmax − 25 = 0.999 1−8 5 12.5wmax − 25 ... 8−1 (1.0 − 0999) = 5 (−3.09)(5) + 25 ≈ 0.76 kips/ft ... wmax = 12.5
EXAMPLE 26.4:
The total annual snowfall for a particular location is modeled as a normal random variable with mean of 60 in. and standard deviation of 15 in. What is the probability that in any given year the total snowfall in that location is between 45 and 65 in.? 65 − 60 45 − 60 −8 P [45 < S ≤ 65] = FS (65) − FS (45) = 8 15 15 = 8(0.33) − 8(−1.00) = 8(0.33) − [1 − 8(1.00)] = 0.629 − (1 − 0.841) ≈ 0.47 (about 47%) What is the probability the total annual snowfall is at least 30 in. in this location? 30 − 60 1 − FS (30) = 1 − 8 15 = 1 − 8(−2.0) = 1 − [1 − 8(2.0)] = 8(2.0) ≈ 0.977 (about 98%) Suppose for design we want to specify the 95th percentile snowfall value (i.e., a value that has a 5% exceedence probability). Estimate the value of S.95 . P [S < S.95 ] = .95 P [S > S.95 ] ≡ 0.05 S.95 − 60 = 0.95 8 15 h i ... S.95 = 15 × 8−1 (.95) + 60 = (15)(1.64) + 60 = 84.6 in. (so, specify 85 in.) Now, assume the total annual snowfall is lognormally distributed (rather than normally) with the same mean and standard deviation as before. Recompute P [45 in. ≤ S ≤ 65 in.]. First, we obtain the lognormal distribution parameters: ! 2 15 2 2 = ln(VS + 1) = ln + 1 = 0.061 ξ 60 o.k. for V ≈ 0.3 or less) ξ = 0.246 (≈ 0.25 = VS ; 2 λ = ln(mS ) − 0.5ξ = ln(60) − 0.5(0.61) = 4.064 1999 by CRC Press LLC
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Now, using these parameters, recompute the probability: ln(45) − 4.06 ln(65) − 4.06 −8 P [45 < SLN ≤ 65] = FS (65) − FS (45) = 8 0.25 0.25 = 8(0.46) − 8(−1.01) = 8(0.46) − [1 − 8(1.01)] = 0.677 − (1 − 0.844) ≈ 0.52 (about 52%) Note that this is slightly higher than the value obtained assuming the snowfall was normally distributed (47%). Finally, again assuming the total annual snowfall to be lognormally distributed, recompute the 5% exceedence limit (i.e., the 95th percentile value): P [S < S.95 ] = .95 ln(S.95 ) − 4.06 = 0.95 8 0.25 h i ... ln(S.95 ) = .25 × 8−1 (.95) + 4.06 = (.25)(1.64) + 4.06 = 4.47 ... S.95 = exp(4.47) ≈ 87.4 in. (specify 88 in.) Again, this value is slightly higher than the value obtained assuming the total snowfall was normally distributed (about 85 in.).
26.2.5
Approximate Analysis of Moments
In some cases, it may be desired to estimate approximately the statistical moments of a function of random variables. For a function given by Y = g (X1 , X2 , . . . , Xn )
(26.20)
approximate estimates for the moments can be obtained using a first-order Taylor series expansion of the function about the vector of mean values. Keeping only the 0th- and 1st-order terms results in an approximate mean (26.21) E[Y ] ≈ g (µ1 , µ2 , . . . , µn ) in which µi = mean of random variable Xi , and an approximate variance Var[Y ] ≈
n X i=1
ci2 Var[Xi ] +
n n X X
ci cj Cov[Xi , Xj ]
(26.22)
i6 =j
in which ci and cj are the values of the partial derivatives ∂g/∂Xi and ∂g/∂Xj , respectively, evaluated at the vector of mean values (µ1 , µ2 , . . . , µn ), and Cov[Xi , Xj ] = covariance function of Xi and Xj . If all random variables Xi and Xj are mutually uncorrelated (statistically independent), the approximate variance reduces to n X ci2 Var[Xi ] (26.23) Var[Y ] ≈ i=1
These approximations can be shown to be valid for reasonably linear functions g(X). For nonlinear functions, the approximations are still reasonable if the variances of the individual random variables, Xi , are relatively small. 1999 by CRC Press LLC
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The estimates of the moments can be improved if the second-order terms from the Taylor series expansions are included in the approximation. The resulting second-order approximation for the mean assuming all Xi , Xj uncorrelated is ! n 1 X ∂ 2g (26.24) E[Y ] ≈ g (µ1 , µ2 , . . . , µn ) + Var[Xi ] 2 ∂Xi2 i=1 For uncorrelated Xi , Xj , however, there is no improvement over Equation 26.23 for the approximate variance. Therefore, while the second-order analysis provides additional information for estimating the mean, the variance estimate may still be inadequate for nonlinear functions.
26.2.6
Statistical Estimation and Distribution Fitting
There are two general classes of techniques for estimating statistical moments: point-estimate methods and interval-estimate methods. The method of moments is an example of a point-estimate method, while confidence intervals and hypothesis testing are examples of interval-estimate techniques. These topics are treated generally in an introductory statistics course and therefore are not covered in this chapter. However, the topics are treated in detail in Ang and Tang [2] and Benjamin and Cornell [5], as well as many other texts. The most commonly used tests for goodness-of-fit of distributions are the Chi-Squared (χ 2 ) test and the Kolmogorov-Smirnov (K-S) test. Again, while not presented in detail herein, these tests are described in most introductory statistics texts. The χ 2 test compares the observed relative frequency histogram with an assumed, or theoretical, PDF. The K-S test compares the observed cumulative frequency plot with the assumed, or theoretical, CDF. While these tests are widely used, they are both limited by (1) often having only limited data in the tail regions of the distribution (the region most often of interest in reliability analyses), and (2) not allowing evaluation of goodness-of-fit in specific regions of the distribution. These methods do provide established and effective (as well as statistically robust) means of evaluating the relative goodness-of-fit of various distributions over the entire range of values. However, when it becomes necessary to assure a fit in a particular region of the distribution of values, such as an upper or lower tail, other methods must be employed. One such method, sometimes called the inverse CDF method, is described here. The inverse CDF method is a simple, graphical technique similar to that of using probability paper to evaluate goodness-of-fit. It can be shown using the theory of order statistics [5] that E [FX (yi )] =
i n+1
(26.25)
where FX (·) = cumulative distribution function, yi = mean of the ith order statistic, and n = number of independent samples. Hence, the term i/(n + 1) is referred to as the ith rank mean plotting position. This well-known plotting position has the properties of being nonparametric (i.e., distribution independent), unbiased, and easy to compute. With a sufficiently large number of observations, n, a cumulative frequency plot is obtained by plotting the rank-ordered observation xi versus the quantity i/(n + 1). As n becomes large, this observed cumulative frequency plot approaches the true CDF of the underlying phenomenon. Therefore, the plotting position is taken to approximate the CDF evaluated at xi : i i = 1, . . . , n (26.26) FX (xi ) ≈ n+1 Simply examining the resulting estimate for the CDF is limited as discussed previously. That is, assessing goodness-of-fit in the tail regions can be difficult. Furthermore, relative goodness-of-fit 1999 by CRC Press LLC
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over all regions of the CDF is essentially impossible. To address this shortcoming, the inverse CDF is considered. For example, taking the inverse CDF of both sides of Equation (26.26) yields
FX−1 [FX (xi )] ≈ FX−1
i n+1
(26.27)
where the left-hand side simply reduces to xi . Therefore, an estimate for the ith observation can be obtained provided the inverse of the assumed underlying CDF exists (see Table 26.5). Finally, if the ith (rank-ordered) observation is plotted against the inverse CDF of the rank mean plotting position, which serves as an estimate of the ith observation, the relative goodness-of-fit can be evaluated over the entire range of observations. Essentially, therefore, one is seeking a close fit to the 1:1 line. The better this fit, the better the assumed underlying distribution FX (·). Figure 26.4 presents an example of a relatively good fit of an Extreme Type I largest (Gumbel) distribution to annual maximum wind speed data from Boston, Massachusetts.
FIGURE 26.4: Inverse CDF (Extreme Type I largest) of annual maximum wind speeds, Boston, MA (1936–1977).
Caution must be exercised in interpreting goodness-of-fit using this method. Clearly, a perfect fit will not be possible, unless the phenomenon itself corresponds directly to a single underlying distribution. Furthermore, care must be taken in evaluating goodness-of-fit in the tail regions, as often limited data exists in these regions. A poor fit in the upper tail, for instance, may not necessarily mean that the distribution should be rejected. This method does have the advantage, however, of permitting an evaluation over specific ranges of values corresponding to specific regions of the distribution. While this evaluation is essentially qualitative, as described herein, it is a relatively simple extension to quantify the relative goodness-of-fit using some measure of correlation, for example. Finally, the inverse CDF method has advantages over the use of probability paper in that (1) the method can be generalized for any distribution form without the need for specific types of plotting paper, and (2) the method can be easily programmed. 1999 by CRC Press LLC
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26.3
Basic Reliability Problem
A complete treatment of structural reliability theory is not included in this section. However, a number of texts are available (in varying degrees of difficulty) on this subject [3, 10, 21, 23]. For the purpose of an introduction, an elementary treatment of the basic (two-variable) reliability problem is provided in the following sections.
26.3.1
Basic R − S Problem
As described previously, the simplest formulation of the failure probability problem may be written: Pf = P [R < S] = P [R − S < 0]
(26.28)
in which R = resistance and S = load. The simple function, g(X) = R − S, where X = vector of basic random variables, is termed the limit state function. It is customary to formulate this limit state function such that the condition g(X) < 0 corresponds to failure, while g(X) > 0 corresponds to a condition of safety. The limit state surface corresponds to points where g(X) = 0 (where the term “surface” implies it is possible to have problems involving more than two random variables). For the simple two-variable case, if the assumption can be made that the load and resistance quantities are statistically independent, and that the population statistics can be estimated by the sample statistics, the failure probabilities for the cases of normal or lognormal variates (R, S) are given by Pf (N)
=
Pf (LN )
=
m − m S R = 8 q 2 σˆ S + σˆ R2 0 − mM λ − λ S R 8 = 8 q σˆ M ξS2 + ξR2
0 − mM 8 σˆ M
(26.29)
(26.30)
where M = R − S is the safety margin (or limit state function). The concept of a safety margin and the reliability index, β, is illustrated in Figure 26.5. Here, it can be seen that the reliability index, β, corresponds to the distance (specifically, the number of standard deviations) the mean of the
FIGURE 26.5: Safety margin concept, M = R − S. 1999 by CRC Press LLC
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safety margin is away from the origin (recall, M = 0 corresponds to failure). The most common, generalized definition of reliability is the second-moment reliability index, β, which derives from this simple two-dimensional case, and is related (approximately) to the failure probability by β ≈ 8−1 (1 − Pf )
(26.31)
where 8−1 (·) = inverse standard normal CDF. Table 26.2 can also be used to evaluate this function. (In the case of normal variates, Equation 26.31 is exact. Additional discussion of the reliability index, β, may be found in any of the texts cited previously.) To gain a feel for relative values of the reliability index, β, the corresponding failure probabilities are shown in Table 26.3. Based on the above discussion (Equations 26.29 through 26.31), for the case of R and S both distributed normal or lognormal, expressions for the reliability index are given by β(N )
=
mM mR − mS =q σˆ M σˆ R2 + σˆ S2
(26.32)
β(LN )
=
mM λR − λS =q σˆ M ξR2 + ξS2
(26.33)
For the less generalized case where R and S are not necessarily both distributed normal or lognormal TABLE 26.3 Failure Probabilities and Corresponding Reliability Values Probability of failure, Pf
Reliability index, β
.5 .1 .01 .001 10−4 10−5 10−6
0.00 1.28 2.32 3.09 3.71 4.75 5.60
(but are still independent), the failure probability may be evaluated by solving the convolution integral shown in Equation 26.34a or 26.34b either numerically or by simulation: Z Pf = P [R < S] = Z Pf = P [R < S] =
+∞
FR (x)fS (x)dx
(26.34a)
[1 − FS (x)] fR (x)dx
(26.34b)
−∞ +∞
−∞
Again, the second-moment reliability is approximated as β = 8−1 (1 − Pf ). Additional methods for evaluating β (for the case of multiple random variables and more complicated limit state functions) are presented in subsequent sections.
26.3.2
More Complicated Limit State Functions Reducible to R − S Form
It may be possible that what appears to be a more complicated limit state function (i.e., more than two random variables) can be reduced, or simplified, to the basic R − S form. Three points may be useful in this regard: 1999 by CRC Press LLC
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1. If the COV of one random variable is very small relative to the other random variables, it may be able to be treated as a it deterministic quantity. 2. If multiple, statistically independent random variables (Xi ) are taken in a summation function (Z = aX1 + bX2 + . . .), and the random variables are assumed to be normal, the summation can be replaced with a single normal random variable (Z) with moments: E[Z] = aE[X1 ] + bE[X2 ] + . . . Var[Z] = σz2 = a 2 σx21 + b2 σx22 + . . .
(26.35) (26.36)
3. If multiple, statistically independent random variables (Yi ) are taken in a product function (Z 0 = Y1 Y2 . . .), and the random variables are assumed to be lognormal, the product can be replaced with a single lognormal random variable (Z 0 ) with moments (shown here for the case of the product of two variables): E[Z 0 ] = E[Y1 ]E[Y2 ] Var[Z 0 ] = µ2Y1 σY22 + µ2Y2 σY21 + σY21 σY22
(26.37) (26.38)
Note that the last term in Equation 26.38 is very small if the coefficients of variation are small. In this case, and more generally, for the product of n random variables, the COV of the product may be expressed: q VZ ≈
VY21 + VY22 + . . . + VY2n
(26.39)
When it is not possible to reduce the limit state function to the simple R − S form, and/or when the random variables are not both normal or lognormal, more advanced methods for the evaluation of the failure probability (and hence the reliability) must be employed. Some of these methods will be described in the next section after some illustrative examples.
26.3.3
Examples
The following examples all contain limit state functions that are in, or can be reduced to, the form of the basic R − S problem. Note that in all cases the random variables are all either normal or lognormal. Additional information suggesting when such distribution assumptions may be reasonable (or acceptable) is also provided in these examples.
EXAMPLE 26.5:
Consider the statically indeterminate beam shown in Figure 26.6, subjected to a concentrated load, P . The moment capacity, Mcap , is a random variable with mean of 20 ft-kips and standard deviation of 4 ft-kips. The load, P , is a random variable with mean of 4 kips and standard deviation of 1 kip. Compute the second-moment reliability index assuming P and Mcap are normally distributed and statistically independent. Mmax
=
Pl 2
Pf
=
P Mcap
1999 by CRC Press LLC
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Pl Pl = P Mcap − < 0 = P Mcap − 2P < 0 < 2 2
FIGURE 26.6: Cantilever beam subject to point load (Example 26.5). Here, the failure probability is expressed in terms of R − S, where R = Mcap and S = 2P . Now, we compute the moments of the safety margin given by M = R − S: mM
=
E[M] = E[R − S] = E[R] − E[S] = mMcap − 2mp = 20 − 2(4) = 12 ft-kips
2 σˆ M
=
2 Var[M] = Var[R] + Var[S] = σˆ M + (2)2 σˆ p2 = (4)2 + 4(1)2 = 20 (ft-kips)2 cap
Finally, we can compute the second-moment reliability index, β, as β=
mR − mS 12 mM =q = √ ≈ 2.68 σˆ M 2 2 20 σˆ R + σˆ S
(The corresponding failure probability is therefore Pf ≈ 8(−β) = 8(−2.68) ≈ 0.00368.)
EXAMPLE 26.6:
When designing a building, the total force acting on the columns must be considered. For a particular design situation, the total column force may consist of components of dead load (selfweight), live load (occupancy), and wind load, denoted D, L, and W , respectively. It is reasonable to assume these variables are statistically independent, and here we will further assume them to be normally distributed with the following moments: Variable
Mean(m)
SD(σ )
D L W
4.0 kips 8.0 kips 3.4 kips
0.4 kips 2.0 kips 0.7 kips
If the column has a strength that is assumed to be deterministic, R = 20 kips, what is the probability of failure and the corresponding second-moment reliability index, β? First, we compute the moments of the combined load, S = D + L + W : mS
=
σˆ S
=
mD + mL + mW = 4.0 + 8.0 + 3.4 = 15.4 kips q p 2 = (0.4)2 + (2.0)2 + (0.7)2 = 2.16 kips σˆ D2 + σˆ L2 + σˆ W
Since S is the sum of a number of normal random variables, it is itself a normal variable. Now, since the resistance is assumed to be deterministic, we can simply compute the failure probability directly in terms of the standard normal CDF (rather than formulating the limit state function). Pf
=
1999 by CRC Press LLC
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P [S > R] = 1 − P [S < R] = 1 − FS (20)
=
20 − 15.4 2.16 (... β = 2.13)
1−8
= 1 − 8(2.13) ≈ 1 − (.9834) = .0166
If we were to formulate this in terms of a limit state function (of course, the same result would be obtained), we would have g(X) = R − S, where the moments of S are given above and the moments of R would be mR = 20 kips and σR = 0. Now, if we assume the resistance, R, is a random variable (rather than being deterministic), with mean and standard deviation given by mR = 20 kips and σR = 2 kips (i.e., COV = 0.10), how would this additional uncertainty affect the probability of failure (and the reliability)? To answer this, we analyze this as a basic R − S problem, assuming normal variables, and making the reasonable assumption that the loads and resistance are independent quantities. Therefore, from Equation 26.29: ! 15.4 − 20 m − m S R =8 p Pf = P [R − S < 0] = 8 q (2.16)2 + (2)2 σˆ S2 + σˆ R2 −4.6 = 8(−1.56) ≈ 0.0594 = 8 √ 8.67 (... β = 1.56) As one would expect, the uncertainty in the resistance serves to increase the failure probability (in this case, fairly significantly), thereby decreasing the reliability.
EXAMPLE 26.7:
The fully plastic flexural capacity of a steel beam section is given by the product Y Z, where Y = steel yield strength and Z = section modulus. Therefore, for an applied moment, M, we can express the limit state function as g(X) = Y Z − M, where failure corresponds to the condition g(X) < 0. Given the statistics shown below and assuming all random variables are lognormally distributed (this ensures non-negativity of the load and resistance variables), reduce this to the simple R − S form and estimate the second-moment reliability index. Variable
Distribution
Mean
COV
Y Z M
Lognormal Lognormal Lognormal
40 ksi 50 in.3 1000 in.-kip
0.10 0.05 0.20
First, we obtain the moments of R and S as follows: “R” = Y Z: E[R] = mR = mY mZ = (40)(50) = 2000 in.-kips q VR = COV ≈ VY2 + VZ2 = 0.112 (since COVs are “small”) “S” = M: E[S] = mM = 1000 in.-kips VS = COV = VM = 0.20 Now, we can compute the lognormal parameters (λ and ξ ) for R and S: ξR 1999 by CRC Press LLC
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≈
VR = 0.112 (since small COV)
λR
=
ξS
≈
λS
=
1 1 ln mR − ξR2 = ln(2000) − (.112)2 = 7.595 2 2 VS = 0.20 (since small COV) 1 1 ln mS − ξS2 = ln(1000) − (.2)2 = 6.888 2 2
Finally, the second-moment reliability index, β, is computed: λR − λS 7.595 − 6.888 =p ≈ 3.08 βLN = q 2 2 (.112)2 + (.2)2 ξR + ξ S Since the variability in the section modulus, Z, is very small (VZ = 0.05), we could choose to neglect it in the reliability analysis (i.e., assume Z deterministic). Still assuming variables Y and M to be lognormally distributed, and using Equation 26.33 to evaluate the reliability index, we obtain β = 3.17. If we further assumed Y and M to be normal (instead of lognormal) random variables, the reliability index computed using Equation 26.32 would be β = 3.54. This illustrates the relative error one might expect from (a) assuming certain variables with low COVs to be essentially deterministic (i.e., 3.17 vs. 3.08), and (b) assuming the incorrect distributions, or simply using the normal distribution when more statistical information is available suggesting another distribution form (i.e., 3.54 vs. 3.08).
EXAMPLE 26.8:
Consider again the simply supported beam shown in Figure 26.3, subjected to a uniform load, w (only), along its entire length. Assume that, in addition to w being a random variable, the member properties E and I are also random variables. (The length, however, may be assumed to be deterministic.) Formulate the limit state function for excessive deflection (assume a maximum allowable deflection of l/360, where l = length of the beam) and then reduce it to the simple R − S form. (Set-up only.) δmax
=
Pf
=
5wl 4 384EI P [δallow. − δmax < 0]
The failure probability is in the R − S form (R = δallow. and S = δmax ); however, we still must express the limit state function in terms of the basic variables. 5wl 4 l − < 0 (for failure) 360 384EI 5wl 3 EI − 0.8) can be considered to imply fully dependent variables. Additional discussion of correlated variables in FORM/SORM may be found in the references [21, 23]. The limit state function, expressed in terms of the basic variables, Xi , is first transformed to reduced variables, ui , having zero mean and unit 1999 by CRC Press LLC
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standard deviation: ui =
Xi − µXi σXi
(26.41)
A transformed limit state function can then be expressed in terms of the reduced variables: g1 (u1 , . . . , un ) = 0
(26.42)
with failure now being defined as g1 (u) < 0. The space corresponding to the reduced variables can be shown to have rotational symmetry, as indicated by the concentric circles of equiprobability shown on Figure 26.7. The reliability index, β, is now defined as the shortest distance between the
FIGURE 26.7: Formulation of reliability analysis in reduced variable space. (Adapted from Ellingwood, B., Galambos, T.V., MacGregor, J. G. and Cornell, C. A. 1980. Development of a Probability Based Load Criterion for American National Standard A58, NBS Special Publication SP577, National Bureau of Standards, Washington, D.C.) limit state surface, g1 (u) = 0, and the origin in reduced variable space (see Figure 26.7). The point (u∗1 , . . . , u∗n ) on the limit state surface that corresponds to this minimum distance is referred to as the checking (or design) point and can be determined by simultaneously solving the set of equations: ∂g1 ∂ui
αi = r P ∂g1 2 i
(26.43)
∂ui
u∗i = −αi β g1 u∗1 , . . . , u∗n
(26.44)
=0
(26.45)
and searching for the direction cosines, αi , that minimize β. The partial derivatives in Equation 26.43 are evaluated at the reduced space design point (u∗1 , . . . , u∗n ). This procedure, and Equations 26.43 through 26.45, result from linearizing the limit state surface (in reduced space) and computing the reliability as the shortest distance from the origin in reduced space to the limit state hyperplane. It 1999 by CRC Press LLC
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may be useful at this point to compare Figures 26.5 and 26.7 to gain some additional insight into this technique. Once the convergent solution is obtained, it can be shown that the checking point in the original random variable space corresponds to the points: (26.46) Xi∗ = µXi 1 − αi βVXi such that g(X1∗ , . . . , Xn∗ ) = 0. These variables will correspond to values in the upper tails of the probability distributions for load variables and the lower tails for resistance (or geometric) variables. The formulation described above provides an exact estimate of the reliability index, β, for cases in which the basic variables are normal and in which the limit state function is linear. In other cases, the results are only approximate. As many structural load and resistance quantities are known to be non-normal, it seems reasonable that information on distribution type be incorporated into the reliability analysis. This is especially true since the limit state probabilities can be affected significantly by different distributions’ tail behaviors. Methods that include distribution information are known as full-distribution methods or advanced FOSM methods. One commonly used technique is described below. Because of the ease of working with normal variables, the objective here is to transform the nonnormal random variables into equivalent normal variables, and then to perform the analysis for a solution of the reliability index, as described previously. This transformation is accomplished by approximating the true distribution by a normal distribution at the value corresponding to the design point on the failure surface. By fitting an equivalent normal distribution at this point, we are forcing the best approximation to be in the tail of interest of the particular random variable. The fitting is accomplished by determining the mean and standard deviation of the equivalent normal variable such that, at the value corresponding to the design point, the cumulative probability and the probability density of the actual (non-normal) and the equivalent normal variable are equal. (This is the basis for the so-called Rackwitz-Fiessler algorithm.) These moments of the equivalent normal variable are given by φ 8−1 Fi Xi∗ (26.47) σiN = fi Xi∗ µN = Xi∗ − 8−1 Fi Xi∗ σiN (26.48) i in which Fi (·) and fi (·) are the non-normal CDF and PDF, respectively, φ(·) = standard normal PDF, and 8−1 (·) = inverse standard normal CDF. Once the equivalent normal mean and standard deviation given by Equations 26.47 and 26.48 are determined, the solution proceeds exactly as described previously. Since the checking point, Xi∗ , is updated at each iteration, the equivalent normal mean and standard deviation must be updated at each iteration cycle as well. While this can be rather laborious by hand, the computer handles this quite efficiently. Only in the case of highly nonlinear limit state functions does this procedure yield results that may be in error. One possible procedure for computing the reliability index, β, for a limit state with non-normal basic variables is shown below: 1. 2. 3. 4. 5. 6.
Define the appropriate limit state function. Make an initial guess at the reliability index, β. Set the initial checking point values, Xi∗ = µi , for all i variables. Compute the equivalent normal mean and standard deviation for non-normal variables. Compute the partial derivatives (∂g/∂Xi ) evaluated at the design point Xi∗ . Compute the direction cosines, αi , as
1999 by CRC Press LLC
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∂g N ∂Xi σi
αi = r P i
∂g N ∂Xi σi
2
(26.49)
7. Compute the new values of design point Xi∗ as N Xi∗ = µN i − αi βσi
(26.50)
8. Repeat steps 4 through 7 until estimates of αi stabilize (usually fast). 9. Compute the value of β such that g(X1∗ , . . . , Xn∗ ) = 0. 10. Repeat steps 4 through 9 until the value for β converges. (This normally occurs within five cycles or less, depending on the nonlinearity of the limit state function.) As with the previous procedure, this method is easily programmed on the computer. Many spreadsheet programs and other numerical analysis software packages also have considerable statistical capabilities, and therefore can be used to perform these types of analyses. This procedure can also be modified to estimate a design parameter (i.e., a section modulus) such that a specific target reliability is achieved. Other procedures are presented elsewhere in the literature [3, 12, 21, 23] including a somewhat different technique in which the equivalent normal mean and standard deviation are used directly in the reduction of the variables to standard normal form (i.e., ui space). Additional information on SORM techniques may be found in the literature [8, 9].
26.4.3
Monte Carlo Simulation
An alternative to integration of the relevant joint probability equation over the domain of random variables corresponding to failure is to use Monte Carlo simulation (MCS). While FORM/SORM techniques are approximate in the case of nonlinear limit state functions, or with non-normal random variables (even when advanced FORM/SORM techniques are used), MCS offers the advantage of providing an exact solution to the failure probability. The potential disadvantage of MCS is the amount of computing time needed, especially when very small probabilities of failure are being estimated. Still, as computing power continues to increase and with the development and refinement of variance reduction techniques (VRTs) MCS is becoming more accepted and more utilized, especially for the analysis of increasingly complicated structural systems. VRTs such as importance sampling, stratified sampling, and Latin hypercube sampling can often be used to significantly reduce the number of simulations required to obtain reliable estimates of the failure probability. A brief description of MCS is presented here. Additional information may be found elsewhere [21, 22]. The concept behind MCS is to generate sets of realizations of the random variables in the limit state function (with the assumed known probability distributions) and to record the number of times the resulting limit state function is less than zero (i.e., failure). The estimate of the probability of failure (Pf ) then is simply the number of failures divided by the total number of simulations (N ). Clearly, the accuracy of this estimate increases as N increases, and a larger number of simulations are required to reliably estimate smaller failure probabilities. Table 26.4 presents the number of simulations required to obtain three different confidence intervals on the estimate of Pf for some typical values in structural reliability analyses. The generation of random variates is a relatively simple task (provided the random variables may be assumed independent) and requires only (1) that the relevant CDF is invertable (or in the case of normal and lognormal variates, numerical approximations exist for the inverse CDF), and (2) that a uniform random number generator is available. (See the Appendix for two examples of uniform 1999 by CRC Press LLC
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TABLE 26.4 Approximate Number of Simulations Required for Given Confidence Intervals (α × 100%) on Reliability Index β ±ε
α = 0.90 (k = 1.64)
α =0.95 (k = 1.96)
α =0.99 (k = 2.58)
1.5 ± .10 1.5 ± .05 1.5 ± .01 2.0 ± .10 2.0 ± .05 2.0 ± .01 3.0 ± .10 3.0 ± .05 3.0 ± .01
1,000 4,000 100,000 2,000 8,200 240,000 18,000 75,000 2,270,000
1,400 5,700 142,000 3,000 12,000 342,000 25,600 107,000 3,240,000
2,500 9,800 246,000 5,100 20,500 592,000 44,300 186,000 5,610,000
random number generators. Random number generators for other distributions may be available to you, and would further simplify the simulation analysis.) The generation of correlated variates is not described here, but information may be found in the literature [9, 21, 23] As shown in Figure 26.8, the value of the CDF for random variable X is (by definition) uniformly distributed on {0, 1}. Therefore,
FIGURE 26.8: Random variable simulation. if we generate a uniform {0, 1} deviate and substitute this into the inverse of the CDF of interest (with the relevant parameters or moments), we obtain a realization of a variate with this CDF. For example, consider the generation of an exponential variate with parameter λ. The CDF is expressed: FX (x) = 1 − exp (−λx)
(26.51)
If we substitute ui (a uniform {0, 1} deviate; see the Appendix) for FX (x) and invert the CDF to solve for xi , we obtain 1 (26.52) xi = − ln(1 − ui ) λ Here, xi is an exponential variate with parameter λ. As another example, consider the normal 1999 by CRC Press LLC
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distribution, for which no closed-form expression exists for the CDF or its inverse. The generalized normal CDF can be written as a function of the standard normal CDF as x − µx (26.53) FX (x) = 8 σx Therefore, an expression for a generalized normal variate would be: xi = µx + σx 8−1 (ui )
(26.54)
where µx and σx are the mean and standard deviation, respectively, ui = uniform {0, 1} deviate, and 8−1 (·) = inverse standard normal CDF. While not available in closed form, numerical approximations for 8−1 (·) (i.e., in the form of algorithms or subroutines) are available (e.g., [12]). The Appendix presents approximate functions for both 8(·) and 8−1 (·). Table 26.5 presents the inverse CDFs for a number of common distribution types. TABLE 26.5
Common Distributions, CDFs, and Inverse CDFs
Distribution
CDF( = ui )
Normal
FX (x) = 8
Lognormal
FX (x) = 8 ln x−λ ξ
Inverse CDF xi = 8−1 (ui ) × σ + µ h i xi = exp 8−1 (ui ) × ξ + λ
Uniform
FX (x) = x−a b−a
xi = a + (b − a)ui
Exponential
FX (x) = 1 − exp(−λx)
xi = − λ1 ln(1 − ui )
Extreme Type I (largest), “Gumbel” Extreme Type II (largest)
FX (x) = exp (− exp (−α(x − u))) FX (x) = exp − (u/x)k
xi = u − ln ui
x−ε k FX (x) = 1 − exp − w−ε
xi = − ln 1 − ui
Extreme Type III
x−µ σ
xi = − α1 ln − ln ui + u −1/k 1/k
(w − ε) + ε
(smallest), “Weibull”
MCS can provide a very powerful tool for the solution of a wide variety of problems. Improvements in efficiency over crude or direct MCS can be realized by improved algorithmic design (programming) and by the utilization of VRTs. Monte Carlo techniques can also be used for the simulation of discrete and continuous random processes.
26.5
System Reliability
26.5.1
Introduction
While most structural codes in the U.S. treat design on a member-by-member basis, most elements within a structure are actually performing as part of an often complicated structural system. Interest in characterizing the performance and safety of structural systems has led to an increased interest in the area of system reliability. The classical theories of series and parallel system reliability are well developed and have been applied to the analysis of such complicated structural systems as nuclear power plants and offshore structures. In the following sections, a brief introduction to system reliability is presented along with some examples. This subject within the broad field of structural reliability is relatively new, and advances both in the theory and application of system reliability concepts to civil engineering design can be expected in the coming years. 1999 by CRC Press LLC
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26.5.2
Basic Systems
The two types of systems in classical theory are the series (or weakest link) system and the parallel system. The literature is replete with formulations for the reliability of these systems, including the possibility of correlated element strengths (e.g., [23, 24]). The relevant limit state is defined by the system type. For a series system, the system limit state is taken by definition to correspond to the first member failure, hence the name “weakest link.” In the case of the strictly parallel system, the system limit state is taken by definition to correspond to the failure of all members. Formulations for the system reliability of a parallel system in which the load-deformation behavior of the members is assumed to be ductile or brittle are both well developed and presented in the literature (see [24], for example). In all cases, the system reliabilities are expressed in terms of the component (or member) reliabilities. Classical system reliability theory has been able to be extended somewhat to model more complicated systems using combinations of series and parallel systems. These formulations, however, are still subject to limitations with regard to possible load sharing (distribution of load among components of the system) and time-dependent effects, such as degrading member resistances.
26.5.3
Introduction to Classical System Reliability Theory
For a system limit state defined by g(x1 , . . . , xm ) = 0, where xi are the basic variables, the failure probability is computed as the integral over the failure domain (g(X) < 0) of the joint probability density function of X. In general, the failure of any system can be expressed as a union and/or intersection of events. For example, the failure of an ideal series (or weakest link) system may be expressed, (26.55) Fsys = F1 ∪ F2 ∪ . . . ∪ Fm in which ∪ denotes the Boolean OR operator and Fi = ith component (element) failure event. A statically determinate truss is modeled as a series system since the failure of the truss corresponds to the failure of any single member. Both first-order and second-order (which includes information on the joint probability behavior) bounds have been developed to express the system failure probability as a function of the individual element failure probabilities. These formulations are well developed and presented in the literature [3, 10, 21, 24]. The failure of a strictly parallel system may be expressed, Fsys = F1 ∩ F2 ∩ . . . ∩ Fm
(26.56)
in which ∩ denotes the Boolean AND operator. Such is the case for the classical “Daniels” system of parallel, ductile rods or cables subject to equal deformation. In this case, system failure corresponds to the failure of all members or elements. First- and second-order bounds are also available for this system idealization (e.g., [17]). Furthermore, bounds that account for possible dependence of failure modes (modal correlation) have been developed [3]. If the parallel system is composed of brittle elements, the analysis may be further complicated by having to account for load redistribution following member failure. This total failure may therefore be the result of progressive element failures. Returning again to the two fundamental system types, series and parallel, we can examine the probability distributions for the strength of these systems as functions of the distributions of the strengths of the individual members (elements). In the simple structural idealization of a series system of n elements (for which the characterization of the member failures as brittle or ductile is irrelevant since system failure corresponds to first-member failure), the distribution function for the system strength, Rsys , can be expressed: FRsys (r) = 1 −
n Y i=1
1999 by CRC Press LLC
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1 − FRi (r)
(26.57)
where the individual member strengths are assumed independent. In Equation 26.57, FRi (r) = distribution function (CDF) for the individual member resistance. If the n individual member strengths are also identically distributed (i.e., have the same parent distribution, FR (r), with the same moments), Equation 26.57 can be simplified to FRsys (r) = 1 − (1 − FR (r))n
(26.58)
In the case of the idealized parallel system of n elements, the system failure is dependent on whether the member behavior is perfectly brittle or perfectly ductile. In the simple case of the parallel system with n perfectly ductile elements, the system strength is given by Rsys =
n X
Ri
(26.59)
i=1
where Ri = strength of element i. The central limit theorem (see [2, 5]) suggests that as the number of members in this system gets large, the system strength approaches a normal random variable, regardless of the distributions of the individual member strengths. When the member behavior is perfectly brittle, the system behavior is dependent on the degree of indeterminacy (redundancy) of the system and the ability of the system to redistribute loads to other members. For some applications, it may be reasonable to model structures idealized as parallel systems with brittle members as series systems, if the brittle failure of one member is likely to overload the remaining members. The issue of correlated member strengths (and correlated failure modes) is beyond the scope of this introduction, but information may be found in [3, 23, 24]. It is appropriate at this point to present the simple first-order bounds for the two fundamental systems. Additional information on the development and application of these as well as the secondorder bounds may be found in the literature cited previously. The first-order bounds for a series system are given by ! n Y n max Pfi ≤ Pfsys ≤ 1 − (1 − Pfi ) (26.60) i=1
i=1
where Pfi = failure probability for member (element) i. The first-order bounds for a parallel system are given by n Y n Pfi ≤ Pfsys ≤ min Pfi (26.61) i=1
i=1
Improved (second-order) bounds (the first-order bounds are often too broad to be of practical use) that include information on the joint probability behavior (i.e., member or modal correlation) have been developed and are described in the literature (e.g., [3, 10, 24]). Classical system reliability theory, as briefly introduced above, is limited in that it cannot account for more complicated load-deformation behavior and the time dependencies associated with load redistribution following (brittle) member failure. Generalized formulations for the reliability of systems that are neither strictly series nor strictly parallel type systems are not available. Analyses of these systems are often based on combined series and parallel system models in which the complete system is modeled as some arrangement of these classical subsystems. These solutions tend to be problem specific and still do not address any possible time-dependent or load-sharing issues.
26.5.4
Redundant Systems
A redundant (indeterminate) system may be defined as having some overload capacity following the failure of an element. The level of redundancy (or degree of indeterminacy) refers to the number of 1999 by CRC Press LLC
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element failures that can be tolerated without the system failing. The reliability of such a structure is dependent on the nature (type) of redundancy. The level of redundancy dictates how many members can fail prior to collapse, and therefore answers the question, “Would the failure of member j lead to impending collapse?” Furthermore, load-deformation behavior of the individual members specifies whether or not the limit states are load-path dependent. For ductile element behavior (i.e., the Daniels system), the limit state is effectively load-path independent, implying the order of member failures is not significant. For a system of brittle elements, however, the limit state may be load-path dependent. In this case, the performance of the system is related to the load redistribution behavior following member failure, and hence the order (or relative position) of member failures becomes important. The parallel-member system model with brittle elements (i.e., perfectly elastic loaddeformation behavior) is appropriate for (and has been used to model) a wide range of redundant structural systems including floors, roofs, and wall systems.
26.5.5
Examples
Three examples are described in this section. The first example considers a series system in which the elements are considered to represent different modes of failure. Modal failure analysis is often treated using the concepts of system reliability (i.e [3]). Here, the structure being considered (actually, the simply supported beam element, i.e., Figure 26.3) may fail in any one of three different modes: flexure, shear, and excessive deflection. (The last mode corresponds to a serviceability-type limit state rather than an ultimate strength type.) The “failure” of the structural element is assumed to occur when any of these limit states is violated. For simplicity, the modal failure probabilities are assumed to be uncorrelated. (For information on handling correlated failure modes, see [3]). In other words, the element (system) fails when it fails in flexure, or it fails in shear, or it experiences excessive deflection: (26.62) Fsys = FM ∪ FV ∪ Fδ If, for example, the probabilities of moment, shear, and deflection failure, respectively, are given by FM = 0.0015, FV = 0.002, and Fδ = 0.005, the first-order bounds shown in Equation 26.60 result in 0.005 ≤ Pfsys ≤ 1 − (1 − 0.0015)(1 − 0.002)(1 − 0.005) 0.005 ≤ Pfsys ≤ 0.0085
(26.63)
This corresponds to a range for β of 2.39 ≤ βsys ≤ 2.58. The second example considers a strictly parallel system of five cables supporting a load (see Figure 26.9). In this case, the system failure corresponds to the condition where the cable system can no longer carry any load. Therefore, all of the cables must have failed for the system to have failed. In this simple example, the issue of load redistribution following the failure of one of the cables is not addressed; however, this problem has been studied extensively (e.g., [19]). Here, the five cable strengths are assumed to be statistically independent, and the system failure probability is the probability that P is large enough to fail all of the cables simultaneously: Fsys = F1 ∩ F2 ∩ . . . ∩ F5
(26.64)
If, for example, the probability of failure of an individual cable is 0.001, and the cable strengths are assumed to be independent, identically distributed random variables, the first-order bounds on the system failure probability given by Equation 26.61 become (0.001)5 ≤ Pfsys ≤ 0.001 1999 by CRC Press LLC
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(26.65)
FIGURE 26.9: Five-element parallel system.
Here, the lower bound corresponds to the case of perfectly uncorrelated member strengths (i.e., independent cable failures), while the upper bound corresponds to the case of perfect correlation. These first-order bounds, as indicated by Equation 26.65, become very wide with increasing n. Here, information on correlation can be important in computing narrower and more useful bounds. Finally, as a third example, a combined (series and parallel) system is considered. In this case, the event probabilities correspond to the failure probabilities of different components required for a safe shutdown of a nuclear power plant. While these events are assumed to be independent, their arrangement describing safe system performance (see Figure 26.10) forms a combined series-parallel system. In this case, the three subsystems are arranged in series: subsystem A is a series system and
FIGURE 26.10: Safe shutdown of a nuclear power plant. 1999 by CRC Press LLC
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subsystems B and C are parallel systems. In this case, the system failure probability is given by Fsys = FA ∪ FB ∪ FC or, expressed in terms of the individual component failure probabilities: Fsys = FA1 ∪ FA2 ∪ FB1 ∩ FB2 ∩ FB3 ∪ FC1 ∩ FC2
26.6
Reliability-Based Design (Codes)
26.6.1
Introduction
(26.66)
(26.67)
This section will provide a brief introduction to reliability-based design concepts in civil engineering, with specific emphasis on structural engineering design. Since the 1970s, the theories of probability and statistics and reliability have provided the bases for modern structural design codes and specifications. Thus, probabilistic codes have been replacing previous deterministic-format codes in recent years. RBD procedures are intended to provide more predictable levels of safety and more risk-consistent (i.e., design-to-design) structures, while utilizing the most up-to-date statistical information on material strengths, as well as structural and environmental loads. An excellent discussion of RBD in the U.S. as well as other countries is presented in [21]. Other references are also available that deal specifically with probabilistic code development in the U.S. [12, 13, 15]. The following sections provide some basic information on the application of reliability theory to aspects of RBD.
26.6.2 Calibration and Selection of Target Reliabilities Calibration refers to the linking of new design procedures to previous existing design philosophies. Much of the need for calibration arises from making any new code changes acceptable to the engineering and design communities. For purely practical reasons, it is undesirable to make drastic changes in the procedures for estimating design values, for example, or in the overall formats of design checking equations. If such changes are to be made, it is impractical and uneconomical to make them often. Hence, code development is an often slow process, involving many years and many revisions. The other justification for code calibration has been the notion that previous design philosophies (i.e., ASD) have resulted in safe designs (or designs with acceptable levels of performance), and that therefore these previous levels of safety should serve as benchmarks in the development of new specifications or procedures (i.e., LRFD). The actual process of calibration is relatively simple. For a given design procedure (i.e., ASD for steel beams in flexure), estimate the reliability based on the available statistical information on the loads and resistances and the governing checking equation. This becomes the target reliability and is used to develop the appropriate load and resistance factors, for example, for the new procedure (i.e., LRFD). In the development of LRFD for both steel and wood, for example, the calibration process revealed an inconsistency in the reliability levels for different load combinations. As this was undesirable, a single target reliability was selected and the new LRFD procedures were able to correct this problem. For more information on code calibration, the reader is referred to the literature [12, 15, 21].
26.6.3
Material Properties and Design Values
The basis for many design values encountered in structural engineering design is now probabilistic. Earlier design values were often based on mean values of member strength, for example, with the 1999 by CRC Press LLC
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factor of safety intended to account for all forms of uncertainty, including material property variability. Later, as more statistical information became available, as people became more aware of the concept of relative uncertainty, and with the use of probabilistic methods in code development, characteristic values were selected for use in design. The characteristic values (referred to as nominal or design values in most specifications) are generally selected from the lower tail of the distribution describing the material property (see Figure 26.11). Typically, the 5th percentile value (that value
FIGURE 26.11: Typical specification of design (nominal) load and resistance values.
below which 5% of the probability density lies) is selected as the nominal resistance (i.e., nominal strength), though in some cases, a different percentile value may be selected. While this value may serve as the starting point for establishing the design value, modifications are often needed to account for such things as size effects, system effects, or (in the case of wood) moisture content effects, etc. The bases for the design resistance values for specifications in the U.S. are described in the literature (e.g., [12, 16, 20]). An excellent review of resistance modeling and a summary of statistical properties for structural elements is presented in [21]. Table 26.6 presents some typical resistance statistics for concrete and steel members. Additional statistics are available, along with statistics for masonry, aluminum, and wood members in [12] as well. The mean values are presented in ratio to their nominal (or design) values, mR /Rn . In addition, the coefficient of variation, VR , and the PDF are
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listed in Table 26.6. TABLE 26.6
Typical Resistance Statistics for Concrete and Steel Members Type of member
Concrete elements Flexure, reinforced concrete
Flexure, prestressed concrete Axial load and flexure
Shear
Continuous one-way slabs Two-way slabs One-way pan joists Beams, grade 40, fc0 = 5 ksi Beams, grade 60, fc0 = 5 ksi Overall values Plant precast pretensioned Cast-in-place post-tensioned Short columns, compression Short columns, tension Slender columns, compression Slender columns, tension Beams with a/d < 2.5, ρw = 0.008: No stirrups Minimum stirrups Moderate stirrups
mR /Rn
VR
1.22 1.12-1.16 1.13 1.14-118 1.01-1.09 1.05 1.06 1.04 0.95-1.05 1.05 1.10 0.95
0.16 0.15 0.14 0.14 0.08-0.12 0.11 0.08 0.10 0.14-0.16 0.12 0.17 0.12
0.93 1.00 1.09
0.21 0.19 0.17
1.05 1.10 1.07 1.11 1.03 1.11 1.07 1.08 1.14 1.04 0.88 1.20 1.07 0.60 0.52
0.11 0.11 0.13 0.13 0.12 0.14 0.15 0.12 0.16 0.14 0.18 0.09 0.05 0.10 0.07
Hot-rolled steel elements Tension member, yield Tension member, ultimate Compact beam, uniform moment Compact beam, continuous Elastic beam, LTB Inelastic beam, LTB Beam columns Plate-girders, flexure Plate girders, shear Compact composite beams Fillet welds ASS bolts in tension, A325 ASS bolts in tension, A490 HSS bolts in shear, A325 HSS bolts in shear, A490
Adapted from Ellingwood, B., Galambos, T.V., MacGregor, J.G., and Cornell, C.A. 1980. “Development of a Probability Based Load Criterion for American National Standard A58,” NBS Special Publication SP577, National Bureau of Standards, Washington, D.C.
26.6.4
Design Loads and Load Combinations
The selection of design load values, such as those found in the ASCE 7-95 standard [4] (formerly the ANSI A58.1 standard), Minimum Design Loads for Buildings and Other Structures, is also largely probability based. Though somewhat more complicated than the selection of design resistance values as described above, the concept is quite similar. Of course, greater complexity is introduced since we may be concerned with both spatial and temporal variations in the load effects. In addition, because of the difficulties in conducting load surveys, and the large amount of variability associated with naturally occurring phenomena giving rise to many structural and environmental loadings, there is a high degree of uncertainty associated with these quantities. A number of load surveys have been conducted, and the valuable data collected have formed the basis for many of our design values (e.g., [7, 11, 14, 18]). When needed, such as in the case where data simply are not available or able to be collected with any reasonable amount of effort, this information is supplemented by engineering judgment and expert opinion. Therefore, design load values are based on (1) statistical information, such as load survey data, and (2) engineering judgment, including past experience, and scenario analysis. As shown in Figure 26.11, the design load value can be visualized as some characteristic value in the upper tail of the distribution describing the load. For example, the 95th percentile wind 1999 by CRC Press LLC
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speed is that value of wind speed that has a 5% (1 – .95) exceedence probability. Probabilistic load modeling represents an extensive area of research, and a significant amount of work is reported on in the literature [21, 28]. A summary of load statistics is presented in Table 26.7. TABLE 26.7
Typical Load Statistics
Load type
Mean-to-nominal
COV
1.05
0.10
Normal
0.30 0.50 1.00
0.60 0.87 0.25
Gamma Gamma Type I
0.20
0.87
Lognormal
0.78 0.33 0.5-1.0
0.37 0.59 0.5-1.4
Type I Type I Type II
Dead load Live load Sustained component Extraordinary component Total (max., 50 years) Snow load (annual max.) General site (northeast U.S.) Wind load 50-year maximum Annual maximum Earthquake load
Distribution
In most codes, a number of different load combinations are suggested for use in the appropriate checking equation format. For example, the ASCE 7-95 standard recommends the following load combinations [4]: U U U U U U
= = = = = =
1.4Dn 1.2Dn + 1.6Ln 1.2Dn + 1.6Sn + (0.5Ln or 0.8Wn ) 1.2Dn + 1.3Wn + 0.5Ln 1.2Dn + 1.0En + 0.5Ln + 0.2Sn 0.9Dn + (−1.3Wn or 1.0En )
(26.68)
where Dn , Ln , Sn , Wn , and En are the nominal (design) values for dead load, live load, snow load, wind load, and earthquake load, respectively. A similar set of load combinations may be found in both the ACI and AISC specifications, though in the case of the ACI code the load factors (developed earlier) are slightly different. These load combinations were developed in order to ensure essentially equal exceedence probabilities for all combinations, U. A discussion of the bases for these load combinations may be found in [12]. A comparison of LRFD with other countries’ codes may be found in [21]. One important tool used in the development of the load combinations is known as Turkstra’s Rule [25, 26], developed as an alternative to more complicated load combination analysis. This rule states that, in effect, the maximum of a combination of two or more load effects will occur when one of the loads is at its maximum value while the other loads take on their instantaneous or arbitrary point-in-time values. Therefore, if n time-varying loads are being considered, there are at least n corresponding load combinations that would need to be considered. This rule may be written generally as n X X (t) + Xj (t) max max {Z} = max i i
T
(26.69)
j =1
j 6 =i
where max {Z} = maximum combined load, Xi (t), i = 1, . . . , n are the time-varying loads being considered in combination, and t = time. In the equation above, the first term in the brackets represents the maximum in the lifetime (T ) of load Xi , while the second term is the sum of all other 1999 by CRC Press LLC
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loads at their point-in-time values. This approximation may be unconservative in some cases where the maximum load effect occurs as a result of the combination of multiple loads at near maximum values. However, in most cases, the probability of this occurring is small, and thus Turkstra’s Rule has been shown to be a good approximation for most structural load combinations [27].
26.6.5
Evaluation of Load and Resistance Factors
Recall that for the generalized case of non-normal random variables, the following expression was developed (see Equation 26.50): N (26.70) Xi∗ = µN i − αi βσi If we further define the design point value Xi∗ in terms of a nominal (design) value Xn : Xi∗ = γi Xn
(26.71)
where γi = partial factor on load Xi (or the inverse of the resistance factor). Therefore, for the popular LRFD format in the U.S. in which the design equation has the form X γi Xn,i (26.72) φRn ≥ i
the load factors may be computed as γi =
ˆ N µN i − αi βσi Xn,i
(26.73)
and the resistance factor is given by φ=
Rn
µN i
ˆ N − αi βσ i
(26.74)
In Equations 26.73 and 26.74, αi = direction cosine from the convergent iterative solution for random variable i, β =convergent reliability index (i.e., the target reliability), and Xn,i and Rn are the nominal load and resistance values, respectively. Additional information on the evaluation of load and resistance factors based on FORM/SORM techniques, as well as comparisons between different code formats, may be found in the literature [3, 12, 21].
26.7
Defining Terms2
Allowable stress design (or working stress design): A method of proportioning structures such that the computed elastic stress does not exceed a specified limiting stress. Calibration: A process of adjusting the parameters in a new standard to achieve approximately the same reliability as exists in a current standard or specification. Factor of safety: A factor by which a designated limit state force or stress is divided to obtain a specified limiting value. Failure: A condition where a limit state is reached.
2 Selected terms taken from [12].
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FORM/SORM (FOSM): First- and Second-order reliability methods (first-order secondmoment reliability methods). Methods that involve (1) a first- or second-order Taylor series expansion of the limit state surface, and (2) computing a notional reliability measure that is a function only of the means and variances (first two moments) of the random variables. (Advanced FOSM includes full distribution information as well as any possible correlations of random variables.) Limit state: A criterion beyond which a structure or structural element is judged to be no longer useful for its intended function (serviceability limit state) or beyond which it is judged to be unsafe (ultimate limit state). Limit states design: A design method that aims at providing safety against a structure or structural element being rendered unfit for use. Load factor: A factor by which a nominal load effect is multiplied to account for the uncertainties inherent in the determination of the load effect. LRFD: Load and resistance factor design. A design method that uses load factors and resistance factors in the design format. Nominal load effect: Calculated using a nominal load; the nominal load frequently is determined with reference to a probability level; e.g., 50-year mean recurrence interval wind speed used in calculating the wind load for design. Nominal resistance: Calculated using nominal material and cross-sectional properties and a rationally developed formula based on an analytical and/or experimental model of limit state behavior. Reliability: A measure of relative safety of a structure or structural element. Reliability-based design (RBD): A design method that uses reliability (probability) theory in the safety checking process. Resistance factor: A factor by which the nominal resistance is multiplied to account for the uncertainties inherent in its determination.
Acknowledgments The author is grateful for the comments and suggestions provided by Professor James T. P. Yao at Texas A&M University and Professor Theodore V. Galambos at the University of Minnesota. In addition, discussions with Professor Bruce Ellingwood at Johns Hopkins University were very helpful in preparing this chapter.
References [1] Abramowitz, M. and Stegun, I.A., Eds. 1966. Handbook of Mathematical Functions, Applied Mathematics Series No. 55, National Bureau of Standards, Washington, D.C. [2] Ang, A.H.-S. and Tang, W.H. 1975. Probability Concepts in Engineering Planning and Design, Volume I: Basic Principles, John Wiley & Sons, New York. [3] Ang, A.H.-S. and Tang, W.H. 1975. Probability Concepts in Engineering Planning and Design, Volume II: Decision, Risk, and Reliability, John Wiley & Sons, New York. [4] American Society of Civil Engineers. 1996. Minimum Design Loads for Buildings and Other Structures, ASCE 7-95, New York. [5] Benjamin, J.R. and Cornell, C.A. 1970. Probability, Statistics, and Decision for Civil Engineers, McGraw-Hill, New York. 1999 by CRC Press LLC
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[6] Bratley, P., Fox, B.L., and Schrage, L.E. 1987. A Guide to Simulation, Second Edition, SpringerVerlag, New York. [7] Chalk, P. and Corotis, R.B. 1980. A Probability Model for Design Live Loads, J. Struct. Div., ASCE, 106(10):2017-2033. [8] Chen, X. and Lind, N.C. 1983. Fast Probability Integration by Three-Parameter Normal Tail Approximation, Structural Safety, 1(4):269-276. [9] Der Kiureghian, A. and Liu, P.L. 1986. Structural Reliability Under Incomplete Probability Information, J. Eng. Mech., ASCE, 112(1):85-104. [10] Ditlevsen, O. 1981. Uncertainty Modelling, McGraw-Hill, New York. [11] Ellingwood, B. and Culver, C.G. 1977. Analysis of Live Loads in Office Buildings, J. Struct. Div., ASCE, 103(8):1551-1560. [12] Ellingwood, B., Galambos, T.V., MacGregor, J.G., and Cornell, C.A. 1980. Development of a Probability Based Load Criterion for American National Standard A58, NBS Special Publication SP577, National Bureau of Standards, Washington, D.C. [13] Ellingwood, B., MacGregor, J.G., Galambos, T.V. and Cornell, C.A. 1982. Probability Based Load Criteria: Load Factors and Load Combinations, J. Struct. Div., ASCE, 108(5):978-997. [14] Ellingwood, B. and Redfield, R. 1982. Ground Snow Loads for Structural Design, J. Struct. Eng., ASCE, 109(4):950-964. [15] Galambos, T.V., Ellingwood, B., MacGregor, J.G., and Cornell, C.A. 1982. Probability Based Load Criteria: Assessment of Current Design Practice, J. Struct. Div., ASCE, 108(5):959-977. [16] Galambos, T.V. and Ravindra, M.K. 1978. Properties of Steel for Use in LRFD, J. Struct. Div., ASCE, 104(9):1459-1468. [17] Grigoriu, M. 1989. Reliability of Daniels Systems Subject to Gaussian Load Processes, Structural Safety, 6(2-4):303-309. [18] Harris, M.E., Corotis, R.B., and Bova, C.J. 1981. Area-Dependent Processes for Structural Live Loads, J. Struct. Div., ASCE, 107(5):857-872. [19] Hohenbichler, M. and Rackwitz, R. 1983. Reliability of Parallel Systems Under Imposed Uniform Strain, J. Eng. Mech. Div., ASCE, 109(3):896-907. [20] MacGregor, J.G., Mirza, S.A., and Ellingwood, B. 1983. Statistical Analysis of Resistance of Reinforced and Prestressed Concrete Members, ACI J., 80(3):167-176. [21] Melchers, R.E. 1987. Structural Reliability: Analysis and Prediction, Ellis Horwood Limited, distributed by John Wiley & Sons, New York. [22] Rubinstein, R.Y. 1981. Simulation and the Monte Carlo Method, John Wiley & Sons, New York. [23] Thoft-Christensen, P. and Baker, M.J. 1982. Structural Reliability Theory and Its Applications, Springer-Verlag, Berlin. [24] Thoft-Christensen, P. and Murotsu, Y. 1986. Application of Structural Systems Reliability Theory, Springer-Verlag, Berlin. [25] Turkstra, C.J. 1972. Theory of Structural Design Decisions, Solid Mech. Study No. 2, University of Waterloo, Ontario, Canada. [26] Turkstra, C.J. and Madsen, H.O. 1980. Load Combinations in Codified Structural Design, J. Struct. Div., ASCE, 106(12):2527-2543. [27] Wen, Y.-K. 1977. Statistical Combinations of Extreme Loads, J. Struct. Div., ASCE, 103(6):10791095. [28] Wen, Y.-K. 1990. Structural Load Modeling and Combination for Performance and Safety Evaluation, Elsevier, Amsterdam.
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Further Reading Melchers [21] provides one of the best overall presentations of structural reliability, both its theory and applications. Ang and Tang [3] also provides a good summary. For a more advanced treatment, refer to Ditlevsen [10], Thoft-Christensen and Baker [23], or Thoft-Christensen and Murotsu [24]. The International Conference on Structural Safety and Reliability (ICOSSAR) and the International Conference on the Application of Statistics and Probability in Civil Engineering (ICASP) are each held every 4 years. The proceedings from these conferences include short papers on a variety of stateof-the-art topics in structural reliability. The conference proceedings may be found in the engineering libraries at most universities. A number of other conferences, including periodic specialty conferences cosponsored by ASCE, also include sessions pertaining to reliability.
Appendix Some Useful Functions for Simulation
1. 8(·) = standard normal cumulative distribution function Approximate algorithm [1]: 8(x) = 1 −
1 2
1 + c1 x + c2 x 2 + c3 x 3 + c4 x 4 |ε(x)|
0.5Py Py Py
(28.3a) (28.3b)
Parabolic Function
The tangent modulus model in Equation 28.3 is suitable for P /Py > 0.5, but it is not sufficient to represent the stiffness degradation for cases with small axial forces and large bending moments. A gradual stiffness degradation of plastic hinge is required to represent the distributed plasticity effects associated with bending actions. We shall introduce the hardening plastic hinge model to represent the gradual transition from elastic stiffness to zero stiffness associated with a fully developed plastic hinge. When the hardening plastic hinges are present at both ends of an element, the incremental force-displacement relationship may be expressed as [24]: 1999 by CRC Press LLC
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FIGURE 28.3: Strength interaction curves for wide-flange sections.
M˙ A M˙ B = Et I L P˙
S22 S1
ηA S1 − (1 − ηB ) ηA ηB S2 0
ηA ηB S2
ηB S1 −
S22 S1
0
(1 − ηA )
0 0 A/I
θ˙A θ˙B e˙ (28.4)
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FIGURE 28.4: Member tangent stiffness degradation derived from the CRC column curve. where M˙ A , M˙ B , P˙ = incremental end moments and axial force, respectively = stability functions S1 , S2 = tangent modulus Et = element stiffness parameters ηA , ηB The parameter η represents a gradual stiffness reduction associated with flexure at sections. The partial plastification at cross-sections in the end of elements is denoted by 0 < η < 1. The η may be assumed to vary according to the parabolic expression (Figure 28.5): η = 4α(1 − α) for α > 0.5
(28.5)
where α is the force state parameter obtained from the limit state surface corresponding to the element end (Figure 28.6): α
=
α
=
8 M P 2 M P + for ≥ Py 9 Mp Py 9 Mp P M P 2 M + for < 2Py Mp Py 9 Mp
(28.6a) (28.6b)
where P , M = second-order axial force and bending moment at the cross-section = plastic moment capacity Mp
28.2.2
Analysis of Semi-Rigid Frames
Practical Connection Modeling
The three-parameter power model contains three parameters: initial connection stiffness, Rki , ultimate connection moment capacity, Mu , and shape parameter, n. The power model may be written as (Figure 28.7): 1999 by CRC Press LLC
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FIGURE 28.5: Parabolic plastic hinge stiffness degradation function with α0 = 0.5 based on the load and resistance factor design sectional strength equation.
FIGURE 28.6: Smooth stiffness degradation for a work-hardening plastic hinge based on the load and resistance factor design sectional strength curve.
m=
θ (1 + θ n )1/n
for θ > 0,
m>0
(28.7)
where m = M/Mu , θ = θr /θo , θo = reference plastic rotation, Mu /Rki , Mu = ultimate moment capacity of the connection, Rki = initial connection stiffness, and n = shape parameter. When the connection is loaded, the connection tangent stiffness, Rkt , at an arbitrary rotation, θr , can be derived by simply differentiating Equation 28.7 as: 1999 by CRC Press LLC
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Rkt =
Mu dM = d |θr | θo (1 + θ n )1+1/n
(28.8)
When the connection is unloaded, the tangent stiffness is equal to the initial stiffness as:
FIGURE 28.7: Moment-rotation behavior of the three-parameter model.
Rkt =
Mu dM = = Rki d |θr | θo
(28.9)
It is observed that a small value of the power index, n, makes a smooth transition curve from the initial stiffness, Rkt , to the ultimate moment, Mu . On the contrary, a large value of the index, n, makes the transition more abruptly. In the extreme case, when n is infinity, the curve becomes a bilinear line consisting of the initial stiffness, Rki , and the ultimate moment capacity, Mu . Practical Estimation of Three Parameters Using Computer Program
An important task for practical use of the power model is to determine the three parameters for a given connection configuration. One difficulty in determining the three parameters is the need for numerical iteration, especially to estimate the ultimate moment, Mu . A set of nomographs was proposed by Kishi et al. [22] to overcome the difficulty. Even though the purpose of these nomographs is to allow the engineer to rapidly determine the three parameters for a given connection configuration, the nomographs require other efforts for engineers to know how to use them, and the values of the nomographs are approximate. Herein, one simple way to avoid the difficulties described above is presented. A direct and easy estimation of the three parameters may be achieved by use of a simple computer program 3PARA.f. The operating procedure of the program is shown in Figure 28.8. The input data, CONN.DAT, may be easily generated corresponding to the input format listed in Table 28.2. As for the shape parameter, n, the equations developed by Kishi et al. [22] are implemented here. Using a statistical technique for n values, empirical equations of n are determined as a linear function of log10 θo , shown in Table 28.3. This n value may be calculated using 3PARA.f. 1999 by CRC Press LLC
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FIGURE 28.8: Operating procedure of computer program estimating the three parameters. TABLE 28.2
Input Format
Line
Input data
1 2 3
ITYPE lt tt la ta
ITYPE
=
Fy E lt tt kt gt W d la ta ka ga
= = = = = = = = = = = =
Fy kt ka
E gt ga
Remark W
d
Connection type and material properties Top/ seat-angle data Web-angle data
Connection type (1 = top and seat-angle connection, 2 = with web-angle connection) yield strength of angle Young’s modulus (= 29, 000 ksi) length of top angle thickness of top angle k value of top angle gauge of top angle(= 2.5 in., typical) width of nut (W = 1.25 in. for 3/4D bolt, W = 1.4375 in. for 7/8D bolt) depth of beam length of web angle thickness of web angle k value of web angle gauge of web angle
Note: (1) Top- and seat-angle connections need lines 1 and 2 for input data, and top and seat angle with web-angle connections need lines 1, 2, and 3. (2) All input data are in free format. (3) Top- and seat-angle sizes are assumed to be the same. (4) Bolt sizes of top angle, seat angle, and web angle are assumed to be the same.
TABLE 28.3
Empirical Equations for Shape Parameter, n n
Connection type Single web-angle connection Double web-angle connection Top- and seat-angle connection Top- and seat-angle connection with double web angle
0.520 log10 θo + 2.291 0.695 1.322 log10 θo + 3.952 0.573 2.003 log10 θo + 6.070 0.302 1.398 log10 θo + 4.631 0.827
for log10 θo > −3.073 for log10 θo < −3.073 for log10 θo > −2.582 for log10 θo < −2.582 for log10 θo > −2.880 for log10 θo < −2.880 for log10 θo > −2.721 for log10 θo < −2.721
From Kishi, N., Goto, Y., Chen, W. F., and Matsuoka, K. G. 1993. Eng. J., AISC, pp. 90-107. With permission.
Load-Displacement Relationship Accounting for Semi-Rigid Connection
The connection may be modeled as a rotational spring in the moment-rotation relationship represented by Equation 28.10. Figure 28.9 shows a beam-column element with semi-rigid connections at both ends. If the effect of connection flexibility is incorporated into the member stiffness, the 1999 by CRC Press LLC
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FIGURE 28.9: Beam-column element with semi-rigid connections.
incremental element force-displacement relationship of Equation 28.1 is modified as [24]: ∗ Sii M˙ A I E t S∗ M˙ B = ij L P˙ 0
Sij∗ ∗ Sjj 0
0 0 A/I
θ˙A θ˙B e˙
(28.10)
where =
Et I Sij2 Et I Sii Sjj Sij + − LRktB LRktB
∗ Sjj
=
Et I Sij2 Et I Sii Sjj + − LRktA LRktA
Sij∗
=
R∗
=
Sii∗
Sjj
! /R ∗
(28.11a)
/R ∗
(28.11b)
!
Sij /R ∗ Sij2 Et I Sjj Et I Sii Et I 2 1+ 1+ − LRktA LRktB L RktA RktB
(28.11c) (28.11d)
where RktA , RktB = tangent stiffness of connections A and B, respectively; Sii Sij = generalized ∗ = modified stability functions that account for the presence of stability functions; and Sii∗ , Sjj end connections. The tangent stiffness (RktA , RktB ) accounts for the different types of semi-rigid connections (see Equation 28.8).
28.2.3
Geometric Imperfection Methods
Geometric imperfection modeling combined with the CRC tangent modulus model is discussed in what follows. There are three: the explicit imperfection modeling method, the equivalent notional load method, and the further reduced tangent modulus method. Explicit Imperfection Modeling Method
Braced Frame The refined plastic hinge analysis implicitly accounts for the effects of both residual stresses and spread of yielded zones. To this end, refined plastic hinge analysis may be regarded as equivalent to the plastic-zone analysis. As a result, geometric imperfections are necessary only to consider fabrication error. For braced frames, member out-of-straightness, rather than frame out-of-plumbness, needs to be used for geometric imperfections. This is because the P − 1 effect due to the frame out-ofplumbness is diminished by braces. The ECCS [10, 11], AS [30], and Canadian Standard Association (CSA) [4, 5] specifications recommend an initial crookedness of column equal to 1/1000 times the 1999 by CRC Press LLC
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column length. The AISC code recommends the same maximum fabrication tolerance of Lc /1000 for member out-of-straightness. In this study, a geometric imperfection of Lc /1000 is adopted. The ECCS [10, 11], AS [30], and CSA [4, 5] specifications recommend the out-of-straightness varying parabolically with a maximum in-plane deflection at the midheight. They do not, however, describe how the parabolic imperfection should be modeled in analysis. Ideally, many elements are needed to model the parabolic out-of-straightness of a beam-column member, but it is not practical. In this study, two elements with a maximum initial deflection at the midheight of a member are found adequate for capturing the imperfection. Figure 28.10 shows the out-of-straightness modeling for a braced beam-column member. It may be observed that the out-of-plumbness is equal to 1/500
FIGURE 28.10: Explicit imperfection modeling of a braced member.
when the half segment of the member is considered. This value is identical to that of sway frames as discussed in recent papers by Kim and Chen [16, 17, 18]. Thus, it may be stated that the imperfection values are essentially identical for both sway and braced frames. It is noted that this explicit modeling method in braced frames requires the inconvenient imperfection modeling at the center of columns although the inconvenience is much lighter than that of the conventional LRFD method for frame design. Unbraced Frame The CSA [4, 5] and the AISC codes of standard practice [2] set the limit of erection out-ofplumbness at Lc /500. The maximum erection tolerances in the AISC are limited to 1 in. toward the exterior of buildings and 2 in. toward the interior of buildings less than 20 stories. Considering the maximum permitted average lean of 1.5 in. in the same direction of a story, the geometric imperfection of Lc /500 can be used for buildings up to six stories with each story approximately 10 ft high. For taller buildings, this imperfection value of Lc /500 is conservative since the accumulated geometric imperfection calculated by 1/500 times building height is greater than the maximum permitted erection tolerance. In this study, we shall use Lc /500 for the out-of-plumbness without any modification because the system strength is often governed by a weak story that has an out-of-plumbness equal to Lc /500 [27] 1999 by CRC Press LLC
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and a constant imperfection has the benefit of simplicity in practical design. The explicit geometric imperfection modeling for an unbraced frame is illustrated in Figure 28.11.
FIGURE 28.11: Explicit imperfection modeling of an unbraced frame.
Equivalent Notional Load Method
Braced Frame The ECCS [10, 11] and the CSA [4, 5] introduced the equivalent load concept, which accounted for the geometric imperfections in unbraced frames, but not in braced frames. The notional load approach for braced frames is also necessary to use the proposed methods for braced frames. For braced frames, an equivalent notional load may be applied at midheight of a column since the ends of the column are braced. An equivalent notional load factor equal to 0.004 is proposed here, and it is equivalent to the out-of-straightness of Lc /1000. When the free body of the column shown in Figure 28.12 is considered, the notional load factor, α, results in 0.002 with respect to one-half of the member length. Here, as in explicit imperfection modeling, the equivalent notional load factor is the same in concept for both sway and braced frames. One drawback of this method for braced frames is that it requires tedious input of notional loads at the center of each column. Another is the axial force in the columns must be known in advance to determine the notional loads before analysis, but these are often difficult to calculate for large structures subject to lateral wind loads. To avoid this difficulty, it is recommended that either the explicit imperfection modeling method or the further reduced tangent modulus method be used. Unbraced Frame The geometric imperfections of a frame may be replaced by the equivalent notional lateral loads expressed as a fraction of the gravity loads acting on the story. Herein, the equivalent notional load factor of 0.002 is used. The notional load should be applied laterally at the top of each story. For sway frames subject to combined gravity and lateral loads, the notional loads should be added to the lateral loads. Figure 28.13 shows an illustration of the equivalent notional load for a portal frame. 1999 by CRC Press LLC
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FIGURE 28.12: Equivalent notional load modeling for geometric imperfection of a braced member.
FIGURE 28.13: Equivalent notional load modeling for geometric imperfection of an unbraced frame.
Further Reduced Tangent Modulus Method
Braced Frame The idea of using the reduced tangent modulus concept is to further reduce the tangent modulus, Et , to account for further stiffness degradation due to geometrical imperfections. The degradation of member stiffness due to geometric imperfections may be simulated by an equivalent reduction of member stiffness. This may be achieved by a further reduction of tangent modulus as [15, 16]: 1999 by CRC Press LLC
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Et0
=
Et0
P 4 Py
=
Eξi for P ≤ 0.5Py
P 1− Py
Eξi for P > 0.5Py
(28.12a) (28.12b)
where Et0 = reduced Et ξi = reduction factor for geometric imperfection Herein, the reduction factor of 0.85 is used, and the further reduced tangent modulus curves for the CRC Et with geometric imperfections are shown in Figure 28.14. The further reduced tangent
FIGURE 28.14: Further reduced CRC tangent modulus for members with geometric imperfections.
modulus concept satisfies one of the requirements for advanced analysis recommended by the SSRC task force report [29], that is: “The geometric imperfections should be accommodated implicitly within the element model. This would parallel the philosophy behind the development of most modern column strength expressions. That is, the column strength expressions in specifications such as the AISC-LRFD implicitly include the effects of residual stresses and out-of-straightness.” The advantage of this method over the other two methods is its convenience for design use, because it eliminates the inconvenience of explicit imperfection modeling or equivalent notional loads. Another benefit of this method is that it does not require the determination of the direction of geometric imperfections, often difficult to determine in a large system. On the other hand, in other two methods, the direction of geometric imperfections must be taken correctly in coincidence with the deflection direction caused by bending moments, otherwise the wrong direction of geometric imperfection in braced frames may help the bending stiffness of columns rather than reduce it. 1999 by CRC Press LLC
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Unbraced Frame The idea of the further reduced tangent modulus concept may also be used in the analysis of unbraced frames. Herein, as in the braced frame case, an appropriate reduction factor of 0.85 to Et can be used [18, 19, 20]. The advantage of this approach over the other two methods is its convenience and simplicity because it completely eliminates the inconvenience of explicit imperfection modeling or the notional load input.
28.2.4
Numerical Implementation
The nonlinear global solution methods may be divided into two subgroups: (1) iterative methods and (2) simple incremental method. Iterative methods such as Newton-Raphson, modified Newton-Raphson, and quasi-Newton satisfy equilibrium equations at specific external loads. In these methods, the equilibrium out-of-balance present following the linear load step is eliminated (within tolerance) by taking corrective steps. The iterative methods possess the advantage of providing the exact load-displacement frame; however, they are inefficient, especially for practical purposes, in the trace of the hinge-by-hinge formation due to the requirement of the numerical iteration process. The simple incremental method is a direct nonlinear solution technique. This numerical procedure is straightforward in concept and implementation. The advantage of this method is its computational efficiency. This is especially true when the structure is loaded into the inelastic region since tracing the hinge-by-hinge formation is required in the element stiffness formulation. For a finite increment size, this approach approximates only the nonlinear structural response, and equilibrium between the external applied loads and the internal element forces is not satisfied. To avoid this, an improved incremental method is used in this program. The applied load increment is automatically reduced to minimize the error when the change in the element stiffness parameter (1η) exceeds a defined tolerance. To prevent plastic hinges from forming within a constant-stiffness load increment, load step sizes less than or equal to the specified increment magnitude are internally computed so plastic hinges form only after the load increment. Subsequent element stiffness formations account for the stiffness reduction due to the presence of the plastic hinges. For elements partially yielded at their ends, a limit is placed on the magnitude of the increment in the element end forces. The applied load increment in the above solution procedure may be reduced for any of the following reasons: 1. Formation of new plastic hinge(s) prior to the full application of incremental loads. 2. The increment in the element nodal forces at plastic hinges is excessive. 3. Nonpositive definiteness of the structural stiffness matrix. As the stability limit point is approached in the analysis, large step increments may overstep a limit point. Therefore, a smaller step size is used near the limit point to obtain accurate collapse displacements and second-order forces.
28.3
Verifications
In the previous section, a practical advanced analysis method was presented for a direct twodimensional frame design. The practical approach of geometric imperfections and of semi-rigid connections was also discussed together with the advanced analysis method. The practical advanced analysis method was developed using simple modifications to the conventional elastic-plastic hinge analysis. In this section, the practical advanced analysis method will be verified by the use of several benchmark problems available in the literature. Verification studies are carried out by comparing with the plastic-zone solutions as well as the conventional LRFD solutions. The strength predictions and 1999 by CRC Press LLC
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the load-displacement relationships are checked for a wide range of steel frames including axially loaded columns, portal frame, six-story frame, and semi-rigid frames [15]. The three imperfection modelings, including explicit imperfection modeling, equivalent notional load modeling, and further reduced tangent modulus modeling, are also verified for a wide range of steel frames [15]).
28.3.1
Axially Loaded Columns
The AISC-LRFD column strength curve is used for the calibration since it properly accounts for second-order effects, residual stresses, and geometric imperfections in a practical manner. In this study,hthe column q strength ofiproposed methods is evaluated for columns with slenderness param-
Fy /(π 2 E) , varying from 0 to 2, which is equivalent to slenderness ratios (L/r) eters, λc = KL r from 0 to 180 when the yield stress is equal to 36 ksi. In explicit imperfection modeling, the two-element column is assumed to have an initial geometric imperfection equal to Lc /1000 at column midheight. The predicted column strengths are compared with the LRFD curve in Figure 28.15. The errors are found to be less than 5% for slenderness ratios up to 140 (or λc up to 1.57). This range includes most columns used in engineering practice.
FIGURE 28.15: Comparison of strength curves for an axially loaded pin-ended column (explicit imperfection modeling method).
In the equivalent notional load method, notional loads equal to 0.004 times the gravity loads are applied midheight to the column. The strength predictions are the same as those of the explicit imperfection model (Figure 28.16). In the further reduced tangent modulus method, the reduced tangent modulus factor equal to 0.85 results in an excellent fit to the LRFD column strengths. The errors are less than 5% for columns of all slenderness ratios. These comparisons are shown in Figure 28.17.
1999 by CRC Press LLC
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FIGURE 28.16: Comparison of strength curves for an axially loaded pin-ended column (equivalent notional load method).
28.3.2
Portal Frame
Kanchanalai [14] performed extensive analyses of portal and leaning column frames, and developed exact interaction curves based on plastic-zone analyses of simple sway frames. Note that the simple frames are more sensitive in their behavior than the highly redundant frames. His studies formed the basis of the interaction equations in the AISC-LRFD design specifications [2, 3]. In his studies, the stress-strain relationship was assumed elastic-perfectly plastic with a 36-ksi yield stress and a 29,000-ksi elastic modulus. The members were assumed to have a maximum compressive residual stress of 0.3Fy . Initial geometric imperfections were not considered, and thus an adjustment of his interaction curves is made to account for this. Kanchanalai further performed experimental work to verify his analyses, which covered a wide range of portal and leaning column frames with slenderness ratios of 20, 30, 40, 50, 60, 70, and 80 and relative stiffness ratios (G) of 0, 3, and 4. The ultimate strength of each frame was presented in the form of interaction curves consisting of the nondimensional first-order moment (H Lc /2Mp in portal frames or H Lc /Mp in leaning column frames in the x axis) and the nondimensional axial load (P /Py in the y axis). In this study, the AISC-LRFD interaction curves are used for strength comparisons. The strength calculations are based on the LeMessurier K factor method [23] since it accounts for story buckling and results in more accurate predictions. The inelastic stiffness reduction factor, τ [2], is used to calculate K in LeMessurier’s procedure. The resistance factors φb and φc in the LRFD equations are taken as 1.0 to obtain the nominal strength. The interaction curves are obtained by the accumulation of a set of moments and axial forces which result in unity on the value of the interaction equation. When a geometric imperfection of Lc /500 is used for unbraced frames, including leaning column frames, most of the strength curves fall within an area bounded by the plastic-zone curves and the LRFD curves. In portal frames, the conservative errors are less than 5%, an improvement on the LRFD error of 11%, and the maximum unconservative error is not more than 1%, shown in Figure 28.18. In leaning column frames, the conservative errors are less than 12%, as opposed to the 17% error of the LRFD, and the maximum unconservative error is not more than 5%, as shown in Figure 28.19. 1999 by CRC Press LLC
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FIGURE 28.17: Comparison of strength curves for an axially loaded pin-ended column (further reduced tangent modulus method).
When a notional load factor of 0.002 is used, the strengths predicted by this method are close to those given by the explicit imperfection modeling method (Figures 28.20 and 28.21). When the reduced tangent modulus factor of 0.85 is used for portal and leaning column frames, the interaction curves generally fall between the plastic-zone and LRFD curves. In portal frames, the conservative error is less than 8% (better than the 11% error of the LRFD) and the maximum unconservative error is not more than 5% (Figure 28.22). In leaning column frames, the conservative error is less than 7% (better than the 17% error of the LRFD) and the maximum unconservative error is not more than 5% (Figure 28.23).
28.3.3
Six-Story Frame
Vogel [32] presented the load-displacement relationships of a six-story frame using plastic-zone analysis. The frame is shown in Figure 28.24. Based on ECCS recommendations, the maximum compressive residual stress is 0.3Fy when the ratio of depth to width (d/b) is greater than 1.2, and is 0.5Fy when the d/b ratio is less than 1.2 (Figure 28.25). The stress-strain relationship is elastic-plastic with strain hardening as shown in Figure 28.26. The geometric imperfections are Lc /450. For comparison, the out-of-plumbness of Lc /450 is used in the explicit modeling method. The notional load factor of 1/450 and the reduced tangent modulus factor of 0.85 are used. The further reduced tangent modulus is equivalent to the geometric imperfection of Lc /500. Thus, the geometric imperfection of Lc /4500 is additionally modeled in the further reduced tangent modulus method, where Lc /4500 is the difference between the Vogel’s geometric imperfection of Lc /450 and the proposed geometric imperfection of Lc /500. The load-displacement curves for the proposed methods together with the Vogel’s plastic-zone analysis are compared in Figure 28.27. The errors in strength prediction by the proposed methods are less than 1%. Explicit imperfection modeling and the equivalent notional load method underpredict lateral displacements by 3%, and the further reduced tangent modulus method shows a 1999 by CRC Press LLC
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FIGURE 28.18: Comparison of strength curves for a portal frame subject to strong-axis bending with Lc /rx = 40, GA = 0 (explicit imperfection modeling method).
FIGURE 28.19: Comparison of strength curves for a leaning column frame subject to strong-axis bending with Lc /rx = 20, GA = 4 (explicit imperfection modeling method).
1999 by CRC Press LLC
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FIGURE 28.20: Comparison of strength curves for a portal frame subject to strong-axis bending with Lc /rx = 60, GA = 0 (equivalent notional load method).
FIGURE 28.21: Comparison of strength curves for a leaning column frame subject to strong-axis bending with Lc /rx = 40, GA = 0 (equivalent notional load method).
1999 by CRC Press LLC
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FIGURE 28.22: Comparison of strength curves for a portal frame subject to strong-axis bending with Lc /rx = 60, GA = 0 (further reduced tangent modulus method).
FIGURE 28.23: Comparison of strength curves for a leaning column frame subject to strong-axis bending with Lc /rx = 40, GA = 0 (further reduced tangent modulus method).
1999 by CRC Press LLC
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FIGURE 28.24: Configuration and load condition of Vogel’s six-story frame for verification study.
good agreement in displacement with Vogel’s exact solution. Vogel’s frame is a good example of how the reduced tangent modulus method predicts lateral displacement well under reasonable load combinations.
28.3.4
Semi-Rigid Frame
In the open literature, no benchmark problems solving semi-rigid frames with geometric imperfections are available for a verification study. An alternative is to separate the effects of semi-rigid connections and geometric imperfections. In previous sections, the geometric imperfections were studied and comparisons between proposed methods, plastic-zone analyses, and conventional LRFD methods were made. Herein, the effect of semi-rigid connections will be verified by comparing analytical and experimental results. Stelmack [31] studied the experimental response of two flexibly connected steel frames. A twostory, one-bay frame in his study is selected as a benchmark for the present study. The frame was fabricated from A36 W5x16 sections, with pinned base supports (Figure 28.28). The connections were bolted top and seat angles (L4x4x1/2) made of A36 steel and A325 3/4-in.-diameter bolts (Figure 28.29). The experimental moment-rotation relationship is shown in Figure 28.30. A gravity load of 2.4 kips was applied at third points along the beam at the first level, followed by a lateral load application. The lateral load-displacement relationship was provided by Stelmack. Herein, the three parameters of the power model are determined by curve-fitting and the program 3PARA.f is presented in Section 28.2.2. The three parameters obtained by the curve-fit are Rki = 40,000 k-in./rad, Mu = 220 k-in., and n = 0.91. We obtain three parameters of Rki = 29,855 kips/rad Mu = 185 k-in. and n = 1.646 with 3PARA.f. 1999 by CRC Press LLC
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FIGURE 28.25: Residual stresses of cross-section for Vogel’s frame.
FIGURE 28.26: Stress-strain relationships for Vogel’s frame.
The moment-rotation curves given by experiment and curve-fitting show good agreement (Figure 28.30). The parameters given by the Kishi-Chen equations and by experiment show some deviation (Figure 28.30). In spite of this difference, the Kishi-Chen equations, using the computer program (3PARA.f), are a more practical alternative in design since experimental moment-rotation curves are not usually available [19]. In the analysis, the gravity load is first applied, then the lateral load. The lateral displacements given by the proposed methods and by the experimental method 1999 by CRC Press LLC
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FIGURE 28.27: Comparison of displacements for Vogel’s six-story frame.
compare well (Figure 28.31). The proposed method adequately predicts the behavior and strength of semi-rigid connections.
28.4
Analysis and Design Principles
In the preceding section, the proposed advanced analysis method was verified using several benchmark problems available in the literature. Verification studies were carried out by comparing it to the plasticzone and conventional LRFD solutions. It was shown that practical advanced analysis predicted the behavior and failure mode of a structural system with reliable accuracy. In this section, analysis and design principles are summarized for the practical application of the advanced analysis method. Step-by-step analysis and design procedures for the method are presented.
28.4.1
Design Format
Advanced analysis follows the format of LRFD. In LRFD, the factored load effect does not exceed the factored nominal resistance of the structure. Two safety factors are used: one is applied to loads, the other to resistances. This approach is an improvement on other models (e.g., ASD and PD) because both the loads and the resistances have unique factors for unique uncertainties. LRFD has the format φRn ≥
m X
γi Qni
i=1
where Rn = nominal resistance of the structural member Qn = nominal load effect (e.g., axial force, shear force, bending moment) 1999 by CRC Press LLC
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(28.13)
FIGURE 28.28: Configuration and load condition of Stelmack’s two-story semi-rigid frame. φ γi i m
= resistance factor (≤ 1.0) (e.g., 0.9 for beams, 0.85 for columns) = load factor (usually > 1.0) corresponding to Qni (e.g., 1.4D and 1.2D + 1.6L + 0.5S) = type of load (e.g., D = dead load, L = live load, S = snow load) = number of load type Note that the LRFD [2] uses separate factors for each load and therefore reflects the uncertainty of different loads and combinations of loads. As a result, a relatively uniform reliability is achieved. The main difference between conventional LRFD methods and advanced analysis methods is that the left side of Equation 28.13 (φRn ) in the LRFD method is the resistance or strength of the component of a structural system, but in the advanced analysis method, it represents the resistance or the load-carrying capacity of the whole structural system.
28.4.2
Loads
Structures are subjected to various loads, including dead, live, impact, snow, rain, wind, and earthquake loads. Structures must be designed to prevent failure and limit excessive deformation; thus, an engineer must anticipate the loads a structure may experience over its service life with reliability. Loads may be classified as static or dynamic. Dead loads are typical of static loads, and wind or earthquake loads are dynamic. Dynamic loads are usually converted to equivalent static loads in conventional design procedures, and it may be adopted in advanced analysis as well.
28.4.3
Load Combinations
The load combinations in advanced analysis methods are based on the LRFD combinations [2]. Six factored combinations are provided by the LRFD specification. The one must be used to determine member sizes. Probability methods were used to determine the load combinations listed in the LRFD 1999 by CRC Press LLC
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FIGURE 28.29: Top and seat angle connection details.
FIGURE 28.30: Comparison of moment-rotation relationships of semi-rigid connection by experiment and the Kishi-Chen equation. 1999 by CRC Press LLC
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FIGURE 28.31: Comparison of displacements of Stelmack’s two-story semi-rigid frame.
specification (LRFD-A4). Each factored load combination is based on the load corresponding to the 50-year recurrence, as follows: (1) 1.4D (2) 1.2D + 1.6L + 0.5(Lr or S or R) (3) 1.2D + 1.6(Lr or S or R) + (0.5L or 0.8W ) (4) 1.2D + 1.3W + 0.5L + 0.5(Lr or S or R) (5) 1.2D ± 1.0E + 0.5L + 0.2S (6) 0.9D ± (1.3W or 1.0E)
(28.14a) (28.14b) (28.14c) (28.14d) (28.14e) (28.14f)
where D = dead load (the weight of the structural elements and the permanent features on the structure) L = live load (occupancy and movable equipment) Lr = roof live load W = wind load S = snow load E = earthquake load R = rainwater or ice load. The LRFD specification specifies an exception that the load factor on live load, L, in combinations (3)–(5) must be 1.0 for garages, areas designated for public assembly, and all areas where the live load is greater than 100 psf. 1999 by CRC Press LLC
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28.4.4
Resistance Factors
The AISC-LRFD cross-section strength equations may be written as 8 M P + φc Py 9 φb Mp M P + 2φc Py φb Mp
= =
P ≥ 0.2 φc Py P 1.0 for < 0.2 φc Py 1.0 for
(28.15a) (28.15b)
where P , M = second-order axial force and bending moment, respectively = squash load Py = plastic moment capacity Mp φc , φb = resistance factors for axial strength and flexural strength, respectively Figure 28.32 shows the cross-section strength including the resistance factors, φc and φb . The
FIGURE 28.32: Stiffness degradation model including reduction factors. reduction factors, φc and φb , are built into the analysis program and are thus automatically included in the calculation of the load-carrying capacity. The reduction factors are 0.85 for axial strength and 0.9 for flexural strength, corresponding to AISC-LRFD specification [2]. For connections, the ultimate moment, Mu , is reduced by the reduction factor 0.9.
28.4.5
Section Application
The AISC-LRFD specification uses only one column curve for rolled and welded sections of W, WT, and HP shapes, pipe, and structural tubing. The specification also uses some interaction equations for 1999 by CRC Press LLC
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doubly and singly symmetric members, including W, WT, and HP shapes, pipe, and structural tubing, even though the interaction equations were developed on the basis of W shapes by Kanchanalai [14]. The present advanced analysis method was developed by calibration with the LRFD column curve and interaction equations described in Section 28.3. To this end, it is concluded that the proposed methods can be used for various rolled and welded sections, including W, WT, and HP shapes, pipe, and structural tubing without further modifications.
28.4.6
Modeling of Structural Members
Different types of advanced analysis are (1) plastic-zone method, (2) quasi-plastic hinge method, (3) elastic-plastic hinge method, and (4) refined plastic hinge method. An important consideration in making these advanced analyses practical is the required number of elements for a member in order to predict realistically the behavior of frames. A sensitivity study of advanced analysis is performed on the required number of elements for a beam member subject to distributed transverse loads. A two-element model adequately predicts the strength of a member. To model parabolic out-of-straightness in a beam-column, a two-element model with a maximum initial deflection at the midheight of a member adequately captures imperfection effects. The required number of elements in modeling each member to provide accurate predictions of the strengths is summarized in Table 28.4. It is concluded that practical advanced analysis is computationally efficient. TABLE 28.4
28.4.7
Necessary Number of Elements Member
Number of elements
Beam member subject to uniform loads Column member of braced frame Column member of unbraced frame
2 2 1
Modeling of Geometric Imperfection
Geometric imperfection modeling is required to account for fabrication and erection tolerances. The imperfection modeling methods used here are the explicit imperfection, the equivalent notional load, and the further reduced tangent modulus models. Users may choose one of these three models in an advanced analysis. The magnitude of geometric imperfections is listed in Table 28.5. TABLE 28.5
Magnitude of Geometric Imperfection
Geometric imperfection method Explicit imperfection modeling method Equivalent notional load method Further reduced tangent modulus method
Magnitude ψ = 2/1000 for unbraced frames ψ = 1/1000 for braced frames α = 2/1000 for unbraced frames α = 4/1000 for braced frames Et0 = 0.85Et
Geometric imperfection modeling is required for a frame but not a truss element, since the program computes the axial strength of a truss member using the LRFD column strength equations, which account for geometric imperfections.
1999 by CRC Press LLC
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28.4.8
Load Application
It is necessary, in an advanced analysis, to input proportional increment load (not the total loads) to trace nonlinear load-displacement behavior. The incremental loading process can be achieved by scaling down the combined factored loads by a number between 10 and 50. For a highly redundant structure (such as one greater than six stories), dividing by about 10 is recommended, and for a nearly statically determinate structure (such as a portal frame), the incremental load may be factored down by 50. One may choose a number between 10 and 50 to reflect the redundancy of a particular structure. Since a highly redundant structure has the potential to form many plastic hinges and the applied load increment is automatically reduced as new plastic hinges form, the larger incremental load (i.e., the smaller scaling number) may be used.
28.4.9
Analysis
Analysis is important in the proposed design procedures, since the advanced analysis method captures key behaviors including second-order and inelasticity in its analysis program. Advanced analysis does not require separate member capacity checks by the specification equations. On the other hand, the conventional LRFD method accounts for inelastic second-order effects in its design equations (not in analysis). The LRFD method requires tedious separate member capacity checks. Input data used for advanced analysis is easily accessible to users, and the input format is similar to the conventional linear elastic analysis. The format will be described in detail in Section 28.5. Analyses can be simply carried out by executing the program described in Section 28.5. This program continues to analyze with increased loads and stops when a structural system reaches its ultimate state.
28.4.10
Load-Carrying Capacity
Because consideration at moment redistribution may not always be desirable, the two approaches (including and excluding inelastic moment redistribution) are presented. First, the load-carrying capacity, including the effect of inelastic moment redistribution, is obtained from the final loading step (limit state) given by the computer program. Second, the load-carrying capacity without the inelastic moment redistribution is obtained by extracting that force sustained when the first plastic hinge formed. Generally, advanced analysis predicts the same member size as the LRFD method when moment redistribution is not considered. Further illustrations on these two choices will be presented in Section 28.6.
28.4.11
Serviceability Limits
The serviceability conditions specified by the LRFD consist of five limit states: (1) deflection, vibration, and drift; (2) thermal expansion and contraction; (3) connection slip; (4) camber; and (5) corrosion. The most common parameter affecting the design serviceability of steel frames is deflection. Based on the studies by the Ad Hoc Committee [1] and by Ellingwood [12], the deflection limits recommended (Table 28.6) were proposed for general use. At service load levels, no plastic hinges are permitted anywhere in the structure to avoid permanent deformation under service loads.
28.4.12
Ductility Requirements
Adequate inelastic rotation capacity is required for members in order to develop their full plastic moment capacity. The required rotation capacity may be achieved when members are adequately braced and their cross-sections are compact. The limitations of compact sections and lateral unbraced 1999 by CRC Press LLC
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TABLE 28.6
Deflection Limitations of Frame Item
Deflection ratio
Floor girder deflection for service live load Roof girder deflection Lateral drift for service wind load Interstory drift for service wind load
L/360 L/240 H /400 H /300
length in what follows leads to an inelastic rotation capacity of at least three and seven times the elastic rotation corresponding to the onset of the plastic moment for non-seismic and seismic regions, respectively. Compact sections are capable of developing the full plastic moment capacity, Mp , and sustaining large hinge rotation before the onset of local buckling. The compact section in the LRFD specification is defined as: 1. Flange • For non-seismic region bf 65 ≤p 2tf Fy
(28.16)
bf 52 ≤p 2tf Fy
(28.17)
• For seismic region
where bf = width of flange tf = thickness of flange Fy = yield stress in ksi 2. Web • For non-seismic region 640 u ≤√ 1 − 2.75P for φbPPy ≤ 0.125 φb Py Fy 191 253 ≤√ for φbPPy > 0.125 2.33 − φPb Pu y ≥ √ h tw
h tw
Fy
Fy
(28.18a) (28.18b)
• For seismic region 520 u ≤√ 1 − 1.54P for φbPPy ≤ 0.125 φb Py Fy 191 253 ≤√ for φbPPy > 0.125 2.33 − φPb Pu y ≥ √ h tw
h tw
Fy
where h = clear distance between flanges tw = thickness of web Fy = yield strength in ksi 1999 by CRC Press LLC
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Fy
(28.19a) (28.19b)
In addition to the compactness of section, the lateral unbraced length of beam members is also a limiting factor for the development of the full plastic moment capacity of members. The LRFD provisions provide the limit on spacing of braces for beam as: • For non-seismic region Lpd
≤
[3,600 + 2,200(M1 /M2 )] ry Fy
(28.20a)
2,500ry Fy
(28.20b)
• For seismic region Lpd
≤
where = unbraced length Lpd = radius of gyration about y axis ry Fy = yield strength in ksi M1 , M2 = smaller and larger end moment, respectively M1 /M2 = positive in double curvature bending The AISC-LRFD specification explicitly specifies the limitations for beam members as described above, but not for beam-column members. More studies are necessary to determine the reasonable limits leading to adequate rotation capacity of beam-column members. Based on White’s study [33], the limitations for beam members seem to be used for beam-column members until the specification provides the specific values for beam-column members.
28.4.13
Adjustment of Member Sizes
If one of three conditions — strength, serviceability, or ductility — is not satisfied, appropriate adjustments of the member sizes should be made. This can be done by referring to the sequence of plastic hinge formation shown in the P.OUT. For example, if the load-carrying capacity of a structural system is less than the factored load effect, the member with the first plastic hinge should be replaced with a stronger member. On the other hand, if the load-carrying capacity exceeds the factored load effect significantly, members without plastic hinges may be replaced with lighter members. If lateral drift exceeds drift requirements, columns or beams should be sized up, or a braced structural system should be considered instead to meet this serviceability limit. In semi-rigid frames, behavior is influenced by the combined effects of members and connections. As an illustration, if an excessive lateral drift occurs in a structural system, the drift may be reduced by increasing member sizes or using more rigid connections. If the strength of a beam exceeds the required strength, it may be adjusted by reducing the beam size or using more flexible connections. Once the member and connection sizes are adjusted, the iteration leads to an optimum design. Figure 28.33 shows a flow chart of analysis and design procedure in the use of advanced analysis.
28.5
Computer Program
This section describes the Practical Advanced Analysis Program (PAAP) for two-dimensional steel frame design [15, 24]. The program integrates the methods and techniques developed in Sections 28.2 and 28.3. The names of variables and arrays correspond as closely as possible to those used in 1999 by CRC Press LLC
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FIGURE 28.33: Analysis and design procedure.
theoretical derivations. The main objective of this section is to present an educational version of software to enable engineers and graduate students to perform planar frame analysis for a more realistic prediction of the strength and behavior of structural systems. The instructions necessary for user input into PAAP are presented in Section 28.5.4. Except for the requirement to input geometric imperfections and incremental loads, the input data format of the program is basically the same as that of the usual linear elastic analysis program. The user is advised to read all the instructions, paying particular attention to the appended notes, to achieve an overall view of the data required for a specific PAAP analysis. The reader should recognize that no system of units is assumed in the program, and take the responsibility to make all units consistent. Mistaken unit conversion and input are a common source of erroneous results. 1999 by CRC Press LLC
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28.5.1
Program Overview
This FORTRAN program is divided in three: DATAGEN, INPUT, and PAAP. The first program, DATAGEN, reads an input data file, P.DAT, and generates a modified data file, INFILE. The second program, INPUT, rearranges INFILE into three working data files: DATA0, DATA1, and DATA2. The third program, PAAP, reads the working data files and provides two output files named P.OUT1 and P.OUT2. P.OUT1 contains an echo of the information from the input data file, P.DAT. This file may be used to check for numerical and incompatibility errors in input data. P.OUT2 contains the load and displacement information for various joints in the structure as well as the element joint forces for all types of elements at every load step. The load-displacement results are presented at the end of every load increment. The sign conventions for loads and displacements should follow the frame degrees of freedom, as shown in Figures 28.34 and 28.35.
FIGURE 28.34: Degrees of freedom numbering for the frame element.
The element joint forces are obtained by summing the product of the element incremental displacements at every load step. The element joint forces act in the global coordinate system and must be in equilibrium with applied forces. After the output files are generated, the user can view these files on the screen or print them with the MS-DOS PRINT command. The schematic diagram in Figure 28.36 sets out the operation procedure used by PAAP and its supporting programs [15].
28.5.2
Hardware Requirements
This program has been tested in two computer processors. First it was tested on an IBM 486 or equivalent personal computer system using Microsoft’s FORTRAN 77 compiler v1.00 and Lahey’s FORTRAN 77 compiler v5.01. Second, its performance in the workstation environment was tested on a Sun 5 using a Sun FORTRAN 77 compiler. The program sizes of DATAGEN, INPUT, and PAAP are 8 kB, 9 kB, and 94 kB, respectively. The total size of the three programs is small, 111 kB (= 0.111 MB), and so a 3.5-in. high-density diskette (1.44 MB) can accommodate the three programs and several example problems. 1999 by CRC Press LLC
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FIGURE 28.35: Degrees of freedom numbering for the truss element.
FIGURE 28.36: Operating procedures of the proposed program. The memory required to run the program depends on the size of the problem. A computer with a minimum 640 K of memory and a 30-MB hard disk is generally required. For the PC applications, the array sizes are restricted as follows: 1. Maximum total degrees of freedom, MAXDOF = 300 2. Maximum translational degrees of freedom, MAXTOF = 300 1999 by CRC Press LLC
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3. Maximum rotational degrees of freedom, MAXROF = 100 4. Maximum number of truss elements, MAXTRS = 150 5. Maximum number of connections, MAXCNT = 150 It is possible to run bigger jobs in UNIX workstations by modifying the above values in the PARAMETER and COMMON statements in the source code.
28.5.3
Execution of Program
A computer diskette is provided in LRFD Steel Design Using Advanced Analysis, by W. F. Chen and S. E. Kim, [6], containing four directories with the following files, respectively. 1. Directory PSOURCE • DATAGEN.FOR • INPUT.FOR • PAAP.FOR 2. Directory PTEST • DATAGEN.EXE • INPUT.EXE • PAAP.EXE • RUN.BAT (batch file) • P.DAT (input data for a test run) • P.OUT1 (output for a test run) • P.OUT2 (output for a test run) 3. Directory PEXAMPLE • All input data for the example problems presented in Section 28.6 4. Directory CONNECT • 3PARA.FOR (program for semi-rigid connection parameters) • 3PARA.EXE • CONN.DAT (input data) • CONN.OUT (output for three parameters) To execute the programs, one must first copy them onto the hard disk (i.e., copy DATAGEN.EXE, INPUT.EXE, PAAP.EXE, RUN.BAT, and P.DAT from the directory PTEST on the diskette to the hard disk). Before launching the program, the user should test the system by running the sample example provided in the directory. The programs are executed by issuing the command RUN. The batch file RUN.BAT executes DATAGEN, INPUT, and PAAP in sequence. The output files produced are P.OUT1 and P.OUT2. When the compilers are different between the authors’ and the user’s, the program (PAAP.EXE) may not be executed. This problem may be easily solved by recompiling the source programs in the directory PSOURCE. The input data for all the problems in Section 28.6 are provided in the directory PEXAMPLE. The user may use the input data for his or her reference and confirmation of the results presented in Section 28.6. It should be noted that RUN is a batch command to facilitate the execution of PAAP. Entering the command RUN will write the new files, including DATA0, DATA1, DATA2, P.OUT1, 1999 by CRC Press LLC
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and P.OUT2, over the old ones. Therefore, output files should be renamed before running a new problem. The program can generate the output files P.OUT1 and P.OUT2 in a reasonable time period, described in the following. The run time on an IBM 486 PC with memory of 640 K to get the output files for the eight-story frame shown in Figure 28.37 is taken as 4 min 10 s and 2 min 30 s in real time rather than CPU time by using Microsoft FORTRAN and Lahey FORTRAN, respectively. In the Sun 5, the run time varies approximately 2–3 min depending on the degree of occupancy by users.
FIGURE 28.37: Configuration of the unbraced eight-story frame.
The directory CONNECT contains the program that computes the three parameters needed for semi-rigid connections. The operation procedure of the program, the input data format, and two examples were presented in Section 28.2.2.
28.5.4
Users’ Manual
Analysis Options
PAAP was developed on the basis of the theory presented in Section 28.2. While the purpose of the program is basically for advanced analysis using the second-order inelastic concept, the program can be used also for first- and second-order elastic analyses. For a first-order elastic analysis, the total factored load should be applied in one load increment to suppress numerical iteration in the nonlinear analysis algorithm. For a second-order elastic analysis, a yield strength of an arbitrarily large value should be assumed for all members to prevent yielding. 1999 by CRC Press LLC
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Coordinate System
A two-dimensional (x, y) global coordinate system is used for the generation of all the input and output data associated with the joints. The following input and output data are prepared with respect to the global coordinate system. 1. Input data • joint coordinates • joint restraints • joint load 2. Output data • joint displacement • member forces Type of Elements
The analysis library consists of three elements: a plane frame, a plane truss, and a connection. The connection is represented by a zero-length rotational spring element with a user-specified nonlinear moment-rotation curve. Loading is allowed only at nodal points. Geometric and material nonlinearities can be accounted for by using an iterative load-increment scheme. Zero-length plastic hinges are lumped at the element ends. Locations of Nodal Points
The geometric dimensions of the structures are established by placing joints (or nodal points) on the structures. Each joint is given an identification number and is located in a plane associated with a global two-dimensional coordinate system. The structural geometry is completed by connecting the predefined joints with structural elements, which may be a frame, a truss, or a connection. Each element also has an identification number. The following are some of the factors that need to be considered in placing joints in a structure: 1. The number of joints should be sufficient to describe the initial geometry and the response behavior of the structures. 2. Joints need to be located at points and lines of discontinuity (e.g., at changes in material properties or section properties). 3. Joints should be located at points on the structure where forces and displacements need to be evaluated. 4. Joints should be located at points where concentrated loads will be applied. The applied loads should be concentrated and act on the joints. 5. Joints should be located at all support points. Support conditions are represented in the structural model by restricting the movement of the specific joints in specific directions. 6. Second-order inelastic behavior can be captured by the use of one or two elements per member, corresponding to the following guidelines: • Beam member subjected to uniform loads: two elements • Column member of braced frame: two elements • Column member of unbraced frame: one element 1999 by CRC Press LLC
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Degrees of Freedom
A two-joint frame element has six displacement components, as shown in Figure 28.34. Each joint can translate in the global x and y directions, and rotate about the global z axis. The directions associated with these displacement components are known as degrees of freedom of the joint. A two-joint truss element has four degrees of freedom, as shown in Figure 28.35. Each join has two translational degrees of freedom and no rotational component. If the displacement of a joint corresponding to any one of its degrees of freedom is known to be zero (such as at a support), then it is labeled an inactive degree of freedom. Degrees of freedom where the displacements are not known are termed active degree of freedoms. In general, the displacement of an inactive degree of freedom is usually known, and the purpose of the analysis is to find the reaction in that direction. For an active degree of freedom, the applied load is known (it could be zero), and the purpose of the analysis is to find the corresponding displacement. Units
There are no built-in units in PAAP. The user must prepare the input in a consistent set of units. The output produced by the program will conform to the same set of units. Therefore, if the user chooses to use kips and inches as the input units, all the dimensions of the structure must be entered in inches and all the loads in kips. The material properties should also conform to these units. The output units will then be in kips and inches, so that the frame member axial force will be in kips, bending moments will be in kip-inches, and displacements will be in inches. Joint rotations, however, are in radians, irrespective of units. Input Instructions
Described here is the input sequence and data structure used to create an input file called P.DAT. The analysis program, PAAP, can analyze any structure with up to 300 degrees of freedom, but it is possible to recompile the source code to accommodate more degrees of freedom by changing the size of the arrays in the PARAMETER and COMMON statements. The limitation of degrees of freedom can be solved by using dynamic storage allocation. This procedure is common in finite element programs [13, 9], and it will be used in the next release of the program. The input data file is prepared in a specific format. The input data consists of 13 data sets, including five control data, three section property data, three element data, one boundary condition, and one load data set as follows: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Title Analysis and design control Job control Total number of element types Total number of elements Connection properties Frame element properties Truss element properties Connection element data Frame element data Truss element data Boundary conditions Incremental loads
Input of all data sets is mandatory, but some of the data associated with elements (data sets 6–11) 1999 by CRC Press LLC
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may be skipped, depending on the use of the element. The order of data sets in the input file must be strictly maintained. Instructions for inputting data are summarized in Table 28.7
28.6
Design Examples
In previous sections, the concept, verifications, and computer program of the practical advanced analysis method for steel frame design have been presented. The present advanced analysis method has been developed and refined to achieve both simplicity in use and, as far as possible, a realistic representation of behavior and strength. The advanced analysis method captures the limit state strength and stability of a structural system and its individual members. As a result, the method can be used for practical frame design without the tedious separate member capacity checks, including the calculation of K factor. The aim of this section is to provide further confirmation of the validity of the LRFD-based advanced analysis methods for practical frame design. The comparative design examples in this section show the detailed design procedure for advanced and LRFD design procedures [15]. The design procedures conform to those described in Section 28.4 and may be grouped into four basic steps: (1) load condition, (2) structural modeling, (3) analysis, and (4) limit state check. The design examples cover simple structures, truss structures, braced frames, unbraced frames, and semi-rigid frames. The three practical models — explicit imperfection, equivalent notional load, and further reduced tangent modulus — are used for the design examples. Member sizes determined by advanced procedures are compared with those determined by the LRFD, and good agreement is generally observed. The design examples are limited to two-dimensional steel frames, so that the spatial behavior is not considered. Lateral torsional buckling is assumed to be prevented by adequate lateral braces. Compact W sections are assumed so that sections can develop their full plastic moment capacity without buckling locally. All loads are statically applied.
28.6.1
Roof Truss
Figure 28.38 shows a hinged-jointed roof truss subject to gravity loads of 20 kips at the joints. A36 steel pipe is used. All member sizes are assumed identical.
FIGURE 28.38: Configuration and load condition of the hinged-jointed roof truss.
1999 by CRC Press LLC
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TABLE 28.7 Data set
Column
Variable
Title
A70
—
Analysis and design control
1–5
IGEOIM
6–10
ILRFD
Job control
1–5 6–10 11–15
NNODE NBOUND NINCRE
Total number of element types Total number of elements
1–5 6–10 11–15
NCTYPE NFTYPE NTTYPE
1–5 6–10 11–15 1-5 6-15∗ 16-25∗ 26-35∗ 1-5 6-15∗ 6-25∗ 26-35∗ 36-45∗ 46-55∗ 55-60
NUMCNT NUMFRM NUMTRS ICTYPE Mu Rki N IFTYPE A I Z E FY IFCOL
1-5 6-15∗ 16-25∗ 26-35∗ 36-45∗ 46-50
ITYPE A I E FY ITCOL
1-5 6-10 11-15
LCNT IFMCNT IEND
16-20 21-25
JDCNT NOSMCN
26-30
NELINC
1-5 6-15∗
LFRM FXO
Connection properties Frame element properties
Truss element properties
Connection element data
Frame element data
1999 by CRC Press LLC
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Input Data Format for the Program PAAP
16-25∗
FYO
26-30 31-35 36-40 41-45
JDFRM IFNODE JFNODE NOSMFE
46-50
NODINC
Description Job title and general comments Geometric imperfection method 0: No geometric imperfection (default) 1: Explicit imperfection modeling 2: Equivalent notional load 3: Further reduced tangent modulus Strength reduction factor, φc = 0.85, φb = 0.9 0: No reduction factors considered (default) 1: Reduction factors considered Total number of nodal points of the structure Total number of supports Allowable number of load increments (default = 100); at least two or three times larger than the scaling number Number of connection types (1–30) Number of frame types (1–30) Number of truss types (1–30) Number of connection elements (1–150) Number of frame elements (1–100) Number of truss elements (1–150) Connection type number Ultimate moment capacity of connection Initial stiffness of connection Shape parameter of connection Frame type number Cross-section area Moment of inertia Plastic section modulus Modulus of elasticity Yield stress Identification of column member, IFCOL = 1 for column (default = 0) Truss type number Cross-section area Moment of inertia Modulus of elasticity Yield stress Identification of column member, ITCOL = 1 for column (default = 0) Connection element number Frame element number containing the connection Identification of element ends containing the connection 1: Connection attached at element end i 2: Connection attached at element end j Connection type number Number of same elements for automatic generation (default = 1) Element number (IFMCNT) increment of automatically generated elements (default = 1) Frame element number Horizontal projected length; positive for i -j direction in global x direction Vertical projected length; positive for i -j direction in global y direction Frame type number Number of node i Number of node j Number of same elements for automatic generation (default = 1) Node number increment of automatically generated elements (default = 1)
TABLE 28.7 Data set Truss element data
Boundary conditions
Incremental loads
Input Data Format for the Program PAAP (continued) Column 1-5 6-15∗
Variable LTRS TXO
16-25∗
TYO
26-30 31-35 36-40 51-55
JDTRS ITNODE JTNODE NOSMTE
56-60
NODINC
1-5 6-10 11-15 16-20 21-25
NODE XFIX YFIX RFIX NOSMBD
26-30
NODINC
1-5 6-15∗ 16-25∗ 26-35∗ 36-40
NODE XLOAD YLOAD RLOAD NOSMLD
41-45
NODINC
Description Truss element number Horizontal projected length; positive for i -j direction in global x direction Vertical projected length; positive for i -j direction in global y direction Truss type number Number of node i Number of node j Number of same elements for automatic generation (default = 1) Node number increment of automatically generated elements (default = 1) Node number of support XFIX = 1 for restrained in global x direction (default = 0) YFIX = 1 for restrained in global y direction (default = 0) RFIX = 1 for restrained in rotation (default = 0) Number of same boundary condition for automatic generation (default = 1) Node number increment of automatically generated supports (default = 1) Node number where a load applied Incremental load in global x direction (default = 0) Incremental load in global y direction (default = 0) Incremental moment in global θ direction (default = 0) Number of same loads for automatic generation (default = 1) Node number increment of automatically generated loads (default = 1)
∗ indicates that the real value (F or E format) should be entered; otherwise input the integer value (I format).
Design by Advanced Analysis
Step 1: Load Condition and Preliminary Member Sizing The critical factored load condition is shown in Figure 28.38. The member forces of the truss may be obtained (Figure 28.39) using equilibrium conditions. The maximum compressive force is 67.1 kips. The effective length is the same as the actual length (22.4 ft) since K is 1.0. The preliminary member size of steel pipe is 6 in. in diameter with 0.28 in. thickness (φPn = 81 kips), obtained using the column design table in the LRFD specification.
FIGURE 28.39: Member forces of the hinged-jointed roof truss. 1999 by CRC Press LLC
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Step 2: Structural Modeling Each member is modeled with one truss element without geometric imperfection since the program computes the axial strength of the truss member with the LRFD column strength equations, which indirectly account for geometric imperfections. An incremental load of 0.5 kips is determined by dividing the factored load of 20 kips by a scaling factor of 40, as shown in Figure 28.40.
FIGURE 28.40: Modeling of the hinged-jointed roof truss.
Step 3: Analysis Referring to the input instructions described in Section 28.5.4, the input data may be easily generated, as listed in Table 28.8. Note that the total number of supports (NBOUND) in the hinged-jointed truss must be equal to the total number of nodal points, since the nodes of a truss element are restrained against rotation. Programs DATAGEN, INPUT, and PAAP are executed in sequence by entering the batch file command RUN on the screen. Step 4: Check of Load-Carrying Capacity Truss elements 10 and 13 fail at load step 48, with loads at nodes 6, 7, and 8 being 24 kips. Since this truss is statically determinant, failure of one member leads to failure of the whole system. Load step 49 shows a sharp increase in displacement and indicates a system failure. The member force of element 10 is 80.4 kips (Fx = 72.0 kips, Fy = 35.7 kips). Since the load-carrying capacity of 24 kips at nodes 6, 7, and 8 is greater than the applied load of 20 kips, the member size is adequate. Step 5: Check of Serviceability Referring to P.OUT2, the deflection at node 3 corresponding to load step 1 is equal to 0.02 in. This deflection may be considered elastic since the behavior of the beam is linear and elastic under small loads. The total deflection of 0.8 in. is obtained by multiplying the deflection of 0.02 in. by the scaling factor of 40. The deflection ratio over the span length is 1/1200, which meets the limitation 1/360. The deflection at the service load will be smaller than that above since the factored load is used for the calculation of deflection above. Comparison of Results
The advanced analysis and LRFD methods predict the same member size of steel pipe with 6 in. diameter and 0.28 in. thickness. The load-carrying capacities of element 10 predicted by these two methods are the same, 80.5 kips. This is because the truss system is statically determinant, rendering inelastic moment redistribution of little or no benefit.
1999 by CRC Press LLC
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TABLE 28.8
28.6.2
Input Data, P.DAT, of the Hinged-Jointed Roof Truss
Unbraced Eight-Story Frame
Figure 28.37 shows an unbraced eight-story one-bay frame with hinged supports. All beams are rigidly connected to the columns. The column and beam sizes are the same. All beams are continuously braced about their weak axis. Bending is primarily about the strong axis at the column. A36 steel is used for all members. Design by Advanced Analysis
Step 1: Load Condition and Preliminary Member Sizing The uniform gravity loads are converted to equivalent concentrated loads in Figure 28.41. The preliminary column and beam sizes are selected as W33x130 and W21x50. Step 2: Structural Modeling Each column is modeled with one element since the frame is unbraced and the maximum moment in the member occurs at the ends. Each beam is modeled with two elements. The explicit imperfection and the further reduced tangent modulus models are used in this example, since they are easier in preparing the input data than equivalent notional load models. Figure 28.42 shows the model for the eight-story frame. The explicit imperfection model uses an out-of-plumbness of 0.2%, and in the further reduced tangent modulus model, 0.85 Et is used. Herein, a scaling factor of 10 is used due to the high indeterminacy. The load increment is automatically reduced if the element stiffness parameter, η, exceeds the predefined value 0.1. The 54 load steps required to converge on the solution are given in P.OUT2. Step 3: Analysis The input data may be easily generated, as listed in Table 28.9a and 28.9b. Programs are executed in sequence by typing the batch file command RUN. Step 4: Check of Load-Carrying Capacity From the output file, P.OUT, the ultimate load-carrying capacity of the structure is obtained as 5.24 and 5.18 kips with respect to the lateral load at roof in load combination 2 by the imperfection 1999 by CRC Press LLC
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Table 28.9a
Input Data, P.DAT, of the Explicit Imperfection Modeling for the Unbraced Eight-Story Frame
Table 28.9b
Input Data, P.DAT, of the Explicit Imperfection Modeling for the Unbraced Eight-Story Frame
1999 by CRC Press LLC
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FIGURE 28.41: Concentrated load condition converted from the distributed load for the two-story frame.
method and the reduced tangent modulus method, respectively. This load-carrying capacity is 3 and 2% greater, respectively, than the applied factored load of 5.12 kips. As a result, the preliminary member sizes are satisfactory. Step 5: Check of Serviceability The lateral drift at the roof level by the wind load (1.0W) is 5.37 in. and the drift ratio is 1/198, which does not satisfy the drift limit of 1/400. When W40x174 and W24x76 are used for column and beam members, the lateral drift is reduced to 2.64 in. and the drift ratio is 1/402, which satisfies the limit 1/400. The design of this frame is thus governed by serviceability rather than strength. Comparison of Results
The sizes predicted by the proposed methods are W33x130 columns and W21x50 beams. They do not, however, meet serviceability conditions and must therefore be increased to W40x174 and W24x76 members. The LRFD method results in the same (W40x174) column but a larger (W27x84) beam (Figure 28.43).
28.6.3
Two-Story Four-Bay Semi-Rigid Frame
Figure 28.44 shows a two-story four-bay semi-rigid frame. The height of each story is 12 ft and it is 25 ft wide. The spacing of the frames is 25 ft. The frame is subjected to a distributed gravity and 1999 by CRC Press LLC
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FIGURE 28.42: Structural modeling of the eight-story frame.
concentrated lateral loads. The roof beam connections are the top and seat angles of L6x4.0x3/8x7 with double web angles of L4x3.5x1/4x5.5 made of A36 steel. The floor beam connections are the top and seat angles of L6x4x9/16x7 with double web angles of L4x3.5x5/16x8.5. All fasteners are A325 3/4-in.-diameter bolts. All members are assumed to be continuously braced laterally. Design by Advanced Analysis
Step 1: Load Condition and Preliminary Member Size The load conditions are shown in Figure 28.44. The initial member sizes are selected as W8x21, W12x22, and W16x40 for the columns, the roof beams, and the floor beams, respectively. Step 2: Structural Modeling Each column is modeled with one element and beam with two elements. The distributed gravity loads are converted to equivalent concentrated loads on the beam, shown in Figure 28.45. In explicit imperfection modeling, the geometric imperfection is obtained by multiplying the column height by 0.002. In the equivalent notional load method, the notional load is 0.002 times the total gravity load plus the lateral load. In the further reduced tangent modulus method, the program automatically accounts for geometric imperfection effects. Although users can choose any of these three models, the further reduced tangent modulus model is the only one presented herein. The incremental loads are computed by dividing the concentrated load by the scaling factor of 20. Step 3: Analysis 1999 by CRC Press LLC
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FIGURE 28.43: Comparison of member sizes of the eight-story frame.
FIGURE 28.44: Configuration and load condition of two-story four-bay semi-rigid frame.
The three parameters of the connections can be computed by use of the computer program 3PARA. Corresponding to the input format in Table 28.2, the input data, CONN.DAT, may be generated, as shown in Tables 28.10a and 28.10b. Referring to the input instructions (Section 28.5.4), the input data is written in the form shown in Tables 28.11a and 28.11b. Programs DATAGEN, INPUT, and PAAP are executed sequentially by typing “RUN.” The program will continue to analyze with increasing load steps up to the ultimate 1999 by CRC Press LLC
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FIGURE 28.45: Concentrated load condition converted from the distributed load for the two-story one-bay semi-rigid frame. Table 28.10a Input Data, CONN.DAT, of Connection for Roof Beam 2 7 5.5
36.0 0.375 0.25
29000 0.875 0.6875
2.5 2.5
1.25
12
Table 28.10b Input Data, CONN.DAT, of Connection for Floor Beam 2 7 8.5
36.0 0.5625 0.3125
29000 1.0625 0.75
2.5 2.5
1.25
16
state. Step 4: Check of Load-Carrying Capacity As shown in output file, P.OUT2, the ultimate load-carrying capacities of the load combinations 1 and 2 are 46.2 and 42.9 kips, respectively, at nodes 7–13 (Figure 28.45). Compared to the applied loads, 45.5 and 31.75 kips, the initial member sizes are adequate. Step 5: Check of Serviceability The lateral displacement at roof level corresponding to 1.0W is computed as 0.51 in. from the computer output, P.OUT2. The drift ratio is 1/565, which satisfies the limitation 1/400. The preliminary member sizes are satisfactory. Comparison of Results
The member sizes by the advanced analysis and LRFD methods are compared in Figure 28.46. The beam members are one size larger in the advanced analysis method, and the interior columns are one size smaller.
28.7
Defining Terms
Advanced analysis: Analysis predicting directly the stability of a structural system and its component members and not needing separate member capacity checks. ASD: Acronym for Allowable Stress Design. Beam-columns: Structural members whose primary function is to carry axial force and bending 1999 by CRC Press LLC
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Table 28.11a
Input Data, P.DAT, of the Four-Bay Two-Story Semi-Rigid Frame
1999 by CRC Press LLC
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Table 28.11b
Input Data, P.DAT, of the Four-Bay Two-Story Semi-Rigid Frame
1999 by CRC Press LLC
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FIGURE 28.46: Comparison of member sizes of the two-story four-bay semi-rigid frame.
moment. Braced frame: Frame in which lateral deflection is prevented by braces or a shear walls. CRC: Acronym for Column Research Council. Column: Structural member whose primary function is to carry axial force. Drift: Lateral deflection of a building. Ductility: Ability of a material to undergo a large deformation without a significant loss in strength. Factored load: The product of the nominal load and a load factor. Flexural member: Structural member whose primary function is to carry bending moment. Geometric imperfection: Unavoidable geometric error during fabrication and erection. Limit state: A condition in which a structural or structural component becomes unsafe (strength limit state) or unfit for its intended function (serviceability limit state). Load factor: A factor to account for the unavoidable deviations of the actual load from its nominal value and uncertainties in structural analysis. LRFD: Acronym for Load Resistance Factor Design. Notional load: Load equivalent to geometric imperfection. PD: Acronym for Plastic Design. Plastic hinge: A yield section of a structural member in which the internal moment is equal to the plastic moment of the cross-section. Plastic zone: A yield zone of a structural member in which the stress of a fiber is equal to the yield stress. Refined plastic hinge analysis: Modified plastic hinge analysis accounting for gradual yielding of a structural member. Resistance factors: A factor to account for the unavoidable deviations of the actual resistance of a member or a structural system from its nominal value. Second-order analysis: Analysis to use equilibrium equations based on the deformed geometry of a structure under load. Semi-rigid connection: Beam-to-column connection whose behavior lies between fully rigid and ideally pinned connection. Service load: Nominal load under normal usage. Stability function: Function to account for the bending stiffness reduction due to axial force. 1999 by CRC Press LLC
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Stiffness: Force required to produce unit displacement. Unbraced frame: Frame in which lateral deflections are not prevented by braces or shear walls.
References [1] Ad Hoc Committee on Serviceability. 1986. Structural Serviceability: a critical appraisal and research needs, J. Struct. Eng., ASCE, 112(12), 2646-2664. [2] American Institute of Steel Construction. 1994. Load and Resistance Factor Design Specification, 2nd ed., Chicago. [3] American Institute of Steel Construction. 1986. Load and Resistance Factor Design Specification for Structural Steel Buildings, Chicago. [4] Canadian Standard Association. 1994. Limit States Design of Steel Structures, CAN/CSA-S16.1M94. [5] Canadian Standard Association. 1989. Limit States Design of Steel Structures, CAN/CSA-S16.1M89. [6] Chen, W. F. and Kim, S. E. 1997. LRFD Steel Design Using Advanced Analysis, CRC Press, Boca Raton, FL. [7] Chen, W. F. and Lui, E. M. 1992. Stability Design of Steel Frames, CRC Press, Boca Raton, FL. [8] Chen, W.F. and Lui, E.M. 1986. Structural Stability—Theory and Implementation, Elsevier, New York. [9] Cook, R.D., Malkus, D.S., and Plesha, M.E. 1989. Concepts and Applications of Finite Element Analysis, 3rd ed., John Wiley & Sons, New York. [10] ECCS. 1991. Essentials of Eurocode 3 Design Manual for Steel Structures in Building, ECCSAdvisory Committee 5, No. 65. [11] ECCS. 1984. Ultimate Limit State Calculation of Sway Frames with Rigid Joints, Technical Committee 8—Structural Stability Technical Working Group 8.2— System, Publication No. 33. [12] Ellingwood. 1989. Serviceability Guidelines for Steel Structures, Eng. J., AISC, 26, 1st Quarter, pp. 1-8. [13] Hughes, T. J. R. 1987. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ. [14] Kanchanalai, T. 1977. The Design and Behavior of Beam-Columns in Unbraced Steel Frames, AISI Project No. 189, Report No. 2, Civil Engineering/Structures Research Lab., University of Texas at Austin. [15] Kim, S.E. 1996. Practical Advanced Analysis for Steel Frame Design, Ph.D. thesis, School of Civil Engineering, Purdue University, West Lafayette, IN. [16] Kim, S. E. and Chen, W. F. 1996. Practical advanced analysis for steel frame design, ASCE Structural Congress XIV, Chicago, Special Proceeding Volume on Analysis and Computation, April, pp. 19-30. [17] Kim, S. E. and Chen, W. F. 1996. Practical advanced analysis for braced steel frame design, J. Struct. Eng., ASCE, 122(11), 1266-1274. [18] Kim, S. E. and Chen, W. F. 1996. Practical advanced analysis for unbraced steel frame design, J. Struct. Eng., ASCE, 122(11), 1259-1265. [19] Kim, S. E. and Chen, W. F. 1996. Practical advanced analysis for semi-rigid frame design, AISC Eng. J., 33(4), 129-141. [20] Kim, S. E. and Chen, W. F. 1996. Practical advanced analysis for frame design—Case study, SSSS J., 6(1), 61-73. 1999 by CRC Press LLC
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[21] Kish, N. and Chen, W. F. 1990. Moment-rotation relations of semi-rigid connections with angles, J. Struct. Eng., ASCE, 116(7), 1813-1834. [22] Kishi, N., Goto, Y., Chen, W. F., and Matsuoka, K. G. 1993. Design aid of semi-rigid connections for frame analysis, Eng. J., AISC, 4th quarter, pp. 90-107. [23] LeMessurier, W. J. 1977. A practical method of second order analysis, Part 2—Rigid frames. AISC Eng. J., 14(2), 49-67. [24] Liew, J. Y. R. 1992. Advanced Analysis for Frame Design, Ph.D. thesis, School of Civil Engineering, Purdue University, West Lafayette, IN. [25] Liew, J. Y. R., White, D. W., and Chen, W. F. 1993. Second-order refined plastic hinge analysis of frame design: Part I, J. Struct. Eng., ASCE, 119(11), 3196-3216. [26] Liew, J. Y. R., White, D. W., and Chen, W. F. 1993. Second-order refined plastic-hinge analysis for frame design: Part 2, J. Struct. Eng., ASCE, 119(11), 3217-3237. [27] Maleck, A. E., White, D. W., and Chen, W. F. 1995. Practical application of advanced analysis in steel design, Proc. 4th Pacific Structural Steel Conf., Vol. 1, Steel Structures, pp. 119-126. [28] SSRC 1981. General principles for the stability design of metal structures, Technical Memorandum No. 5, Civil Engineering, ASCE, February, 53-54. [29] White, D. W. and Chen, W. F., Eds., 1993. Plastic Hinge Based Methods for Advanced Analysis and Design of Steel Frames: An Assessment of the State-of-the-art, SSRC, Lehigh University, Bethlehem, PA. [30] Standards Australia. 1990. AS4100-1990, Steel Structures, Sydney, Australia. [31] Stelmack T. W. 1982. Analytical and Experimental Response of Flexibly-Connected Steel Frames, M.S. dissertation, Department of Civil, Environmental, and Architectural Engineering, University of Colorado. [32] Vogel, U. 1985. Calibrating frames, Stahlbau, 10, 1-7. [33] White, D. W. 1993. Plastic hinge methods for advanced analysis of steel frames, J. Constr. Steel Res., 24(2), 121-152.
Further Reading [1] Chen, W. F. and Kim, S. E. 1997. LRFD Steel Design Using Advanced Analysis, CRC Press, Boca Raton, FL. [2] Chen, W. F. and Lui, E. M. 1992. Stability Design of Steel Frames, CRC Press, Boca Raton, FL. [3] Chen, W. F. and Toma, S. 1994. Advanced Analysis of Steel Frames, CRC Press, Boca Raton, FL. [4] Chen, W. F. and Sohal, I. 1995. Plastic Design and Second-Order Analysis of Steel Frames, Springer-Verlag, New York. [5] Chen, W. F., Goto, Y., and Liew, J. Y. R. 1996. Stability Design of Semi-Rigid Frames, John Wiley & Sons, New York.
1999 by CRC Press LLC
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Marshall, P.W. “Welded Tubular Connections---CHS Trusses” Structural Engineering Handbook Ed. Chen Wai-Fah Boca Raton: CRC Press LLC, 1999
Welded Tubular Connections—CHS Trusses 29.1 29.2 29.3 29.4 29.5
Introduction Architecture Characteristics of Tubular Connections Nomenclature Failure Modes
Local Failure • General Collapse • Unzipping or Progressive Failure • Materials Problems • Fatigue
29.6 Reserve Strength 29.7 Empirical Formulations 29.8 Design Charts
Joint Efficiency • Derating Factor
Peter W. Marshall MHP Systems Engineering, Houston, Texas
29.1
29.9 Application 29.10Summary and Conclusions References
Introduction
Truss connections in circular hollow sections (CHS) present unique design challenges. This chapter discusses the following elements of the subject: Architecture, Characteristics of Tubular Connections, Nomenclature, Failure Modes, Reserve Strength, Empirical Formulations, Design Charts, Application, and Summary and Conclusions.
29.2
Architecture
“Architecture” is defined as the art and science of designing and successfully executing structures in accordance with aesthetic considerations and the laws of physics, as well as practical and material considerations. Where tubular structures are exposed for dramatic effect, it is often disappointing to see grand concepts fail in execution due to problems in the structural connections of tubes. Such “failures” range from awkward ugly detailing, to learning curve problems during fabrication, to excessive deflections or even collapse. Such failures are unnecessary, as the art and science of welded tubular connections has been codified in the AWS Structural Welding Code [1]. A well-engineered structure requires that a number of factors be in reasonable balance. Factors to be considered in relation to economics and risk in the design of welded tubular structures and their connections include: (1) static strength, (2) fatigue resistance, (3) fracture control, and (4) weldability. 1999 by CRC Press LLC
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Static strength considerations are so important that they often dictate the very architecture and layout of the structure; certainly they dominate the design process and are the focus of this chapter. Many of the other factors also require early attention in design, and arise again in setting up QC/QA programs during construction; these are discussed further in sections of the Code dealing with materials, welding technique, qualification, and inspection.
29.3
Characteristics of Tubular Connections
Tubular members benefit from an efficient distribution of their material, particularly in regard to beam bending or column buckling about multiple axes. However, their resistance to concentrated radial loads are more problematic. For architecturally exposed applications, the clean lines of a closed section are esthetically pleasing and they minimize the amount of surface area for dirt, corrosion, or other fouling. Simple welded tubular joints can extend these clean lines to include the structural connections. Although many different schemes for stiffening tubular connections have been devised [3], the most practical connection is made by simply welding the branch member to the outside surface of the main member (or chord). Where the main member is relatively compact (D/T less than 15 or 20), the branch member thickness is limited to 50 or 60% of the main member thickness, and a prequalified weld detail is used, the connection can develop the full static capacity of the members joined. Where the foregoing conditions are not met, e.g., with large diameter tubes, a short length of heavier material (or joint can) is inserted into the chord to locally reinforce the connection area. Here, the design problem reduces to one of selecting the right combination of thickness, yield strength, and notch toughness for the chord or joint can. The detailed considerations involved in this design process are the subject of this chapter.
29.4
Nomenclature
Non-dimensional parameters for describing the geometry of a tubular connection are given in the following list. Beta, eta, theta, and zeta describe the surface topology. Gamma and tau are two very important thickness parameters. Alpha (not shown) is an ovalizing parameter, depending on load pattern (it was formerly used for span length in beams loaded via tee connections). β (beta) η (eta) θ (theta) ζ (zeta) γ (gamma) τ (tau)
d/D , branch diameter/main diameter branch footprint length/main diameter angle between branch and main member axes g/D , gap/diameter (between balancing branches of a K-connection) R/T , main member radius/thickness ratio t/T , branch thickness/main thickness
In AWS D1.1 [1], the term “T-, Y-, and K-connection” is used generically to describe simple structural connections or nodes, as opposed to co-axial butt and lap joints. A letter of the alphabet (T, Y, K, X) is used to evoke a picture of what the node subassemblage looks like.
29.5
Failure Modes
A number of unique failure modes are possible in tubular connections. In addition to the usual checks on weld stress, provided for in most design codes, the designer must check for the following failure modes, listed together with the relevant AWS D1.1-96 [1] code sections: 1999 by CRC Press LLC
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Local failure (punching shear) General collapse Unzipping (progressive weld failure) Materials problems (fracture and delamination) Fatigue
29.5.1
2.40.1.1 2.40.1.2 2.40.1.3 2.42, C4.12.4.4, and 2.1.3 2.36.6
Local Failure
AWS design criteria for this failure mode have traditionally been formulated in terms of punching shear. The main member acts as a cylindrical shell in resisting the concentrated radial line loads (kips/in.) delivered to it at the branch member footprint. Although the resulting localized shell stresses in the main member are quite complex, a simplified but still quite useful representation can be given in terms of punching shear stress, vp : acting vp = fn τ sin θ
(29.1)
where fn is the nominal stress at the end of the branch member, either axial or bending, which are treated separately. Punching shear is the notional stress on the potential failure surface, as illustrated in Figure 29.1. The overriding importance of chord thickness is reflected in tau, while sin θ indicates that it is the radial component of load that causes all the mischief. The allowable punching shear stress is given in the Code as: allowable vp =
Fyo 0.6γ
·
Qq ·
Qf
(29.2)
We see that the allowable punching shear stress is primarily a function of main member yield strength (Fyo ) and gamma ratio (main member radius/thickness), with some trailing terms that tend towards unity. The term Qq reflects the considerable influence of connection type, geometry, and load pattern, while interactions between branch and chord loads are covered by the reduction factor Qf . Interactions between brace axial load and bending moments are treated analogous to those for a fully plastic section. Since 1992, the AWS code also includes tubular connection design criteria in total load ultimate strength format, compatible with an LRFD design code formulation. This was derived from, and intended to be comparable to, the original punching shear criteria.
29.5.2
General Collapse
In addition to local failure of the main member in the vicinity of the branch member, a more widespread mode of collapse may occur, e.g., general ovalizing plastic failure in the cylindrical shell of the main member. To a large extent, this is now covered by strength criteria that are specialized by connection type and load pattern, as reflected in the Qq factor. For balanced K-connections, the inward radial loads from one branch member is compensated by outward loads on the other, ovalizing is minimized, and capacity approaches the local punching shear limit. For T and Y connections, the radial load from the single branch member is reacted by beam shear in the main member or chord, and the resulting ovalizing leads to lower capacity. For cross or X connections, the load from one branch is reacted by the opposite branch, and the resulting double dose of ovalizing in the main member leads to still further reductions in capacity. The Qq term also reflects reduced ovalizing and increased capacity, as the branch member diameter approaches that of the main member. Thus, for design purposes, tubular connections are classified according to their configuration (T, Y, K, X, etc.). For these “alphabet” connections, different design strength formulae are often applied 1999 by CRC Press LLC
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FIGURE 29.1: Local failure mode and punching shear Vp . (From Marshall, P., Design of Welded Tubular Connections (1992), Dev. Civ. Eng., Vol. 37. With kind permission from Elsevier Science, Amsterdam, The Netherlands.)
to each different type. Until recently, the research, testing, and analysis leading to these criteria dealt only with connections having their members in a single plane, as in a roof truss or girder. Many tubular space frames have bracing in multiple planes. For some loading conditions, these different planes interact. When they do, criteria for the “alphabet” joints are no longer satisfactory. In AWS, an “ovalizing parameter” (alpha, Appendix L) may be used to estimate the beneficial or deleterious effect of various branch member loading combinations on main member ovalizing. This reproduces the trend of increasingly severe ovalizing in going from K to T/Y to X-connections, and has been shown to provide useful guidance in a number of more adverse planar (e.g., all-tension double-K [9]) and multi-planar (e.g., hub [11]) situations. However, for similarly loaded members in adjacent planes, e.g., paired KK connections in delta trusses, Japanese data indicate that no increase in capacity over the corresponding uniplanar connections should be taken [2]. The effect of a short joint can (less than 2.5 diameters) in reducing the ovalizing or crushing capacity of cross connections is addressed in AWS section 2.40.1.2(2) [1]. Since ovalizing is less severe in K-connections, the rule of thumb is that the joint can need only extend 0.25 to 0.4 diameters beyond the branch member footprints to avoid a short-can penalty. Intermediate behavior would apply to T/Y connections. A more exhaustive discussion would also consider the following modes of general collapse in addition to ovalizing: beam bending of the chord (in T-connection tests), beam shear (in the gap of K-connections), transverse crippling of the main member sidewall, and local buckling due to uneven load transfer (either brace or chord). These are illustrated in Figure 29.2. 1999 by CRC Press LLC
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FIGURE 29.2: Failure modes — general colapse. (a) Ovalizing, (b) beam bending, (c) beam shear in the gap, (d) sidewall (web) crippling, and (e) local buckling due to uneven distribution of axial load. (From Marshall, P., Design of Welded Tubular Connections (1992), Dev. Civ. Eng., Vol. 37. With kind permission from Elsevier Science, Amsterdam, The Netherlands.)
29.5.3
Unzipping or Progressive Failure
The initial elastic distribution of load transfer across the weld in a tubular connection is highly nonuniform, as illustrated in Figure 29.3, with the peak line load often being a factor of two higher than that indicated on the basis of nominal sections, geometry, and statics. Some local yielding is required for tubular connections to redistribute this and reach their design capacity. If the weld is a weak link in the system, it may “unzip” before this redistribution can happen. Criteria given in the AWS code are intended to prevent this unzipping, taking advantage of the higher reserve strength in weld 1999 by CRC Press LLC
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allowable stresses than is the norm elsewhere. For mild steel tubes and overmatched E70 weld metal, weld effective throats as small as 70% of the branch member thickness are permitted.
FIGURE 29.3: Uneven distribution of load across the weld. (From Marshall, P., Design of Welded Tubular Connections (1992), Dev. Civ. Eng., Vol. 37. With kind permission from Elsevier Science, Amsterdam, The Netherlands.)
29.5.4
Materials Problems
Most fracture control problems in tubular structures occur in the welded tubular connections, or nodes. These require plastic deformation in order to reach their design capacity. Fatigue and fracture problems for many different node geometries are brought into a common focus by use of the “hot spot” stress, as would be measured by a strain gauge, adjacent to and perpendicular to the toe of the weld joining branch to main member, in the worst region of localized plastic deformation (usually in the chord). Hot spot stress has the advantage of placing many different connection geometries on a common basis with regard to fatigue and fracture. Charpy impact testing is a method for qualitative assessment of material toughness. The method has been, and continues to be, a reasonable measure of fracture safety when employed with a definitive program of nondestructive testing to eliminate weld area flaws. The AWS recommendations for material selection (C2.42.2.2) and weld metal impact testing (C4.12.4.4) are based on practices that have provided satisfactory fracture experience in offshore structures located in moderate temperature environments, i.e., 40◦ F (+5◦ C) water and 14◦ F (−10◦ C) air exposure. For environments that are either more or less hostile, impact testing temperatures should be reconsidered based on LAST (lowest anticipated service temperature). In addition to weld metal toughness, consideration should be given to controlling the properties of the heat affected zone (HAZ). Although the heat cycle of welding sometimes improves hot rolled base metals of low toughness, this region will more often have degraded toughness properties. A number of early failures in welded tubular connections involved fractures that either initiated in or propagated through the HAZ, often obscuring the identification of other design deficiencies, e.g., inadequate static strength. A more rigorous approach to fatigue and fracture problems in welded tubular connections has been taken by using fracture mechanics [5]. The CTOD (crack tip opening displacement) test is used 1999 by CRC Press LLC
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to characterize materials that are tough enough to undergo some plasticity before fracture. Underneath the branch member footprint, the main member is subjected to stresses in the thruthickness or short transverse direction. Where these stresses are tensile, due to weld shrinkage or applied loading, delamination may occur — either by opening up pre-existing laminations or by laminar tearing in which microscopic inclusions link up to give a fracture having a woody appearance, usually in or near the HAZ. These problems are addressed in API joint can steel specifications 2H, 2W, and 2Y. Pre-existing laminations are detected with ultrasonic testing. Microscopic inclusions are prevented by restricting sulfur to very low levels (< 60 ppm) and by inclusion shape control metallurgy in the steel-making ladle. As a practical matter, weldments that survive the weld shrinkage phase usually perform satisfactorily in ordinary service. Joint can steel specifications also seek to enhance weldability with limitations on carbon and other alloying elements, as expressed by carbon equivalent or Pcm formulae. Such controls are increasingly important as residual elements accumulate in steel made from scrap. In AWS Appendix XI [1], the preheat required to avoid HAZ cracking is related to carbon equivalent, base metal thickness, hydrogen level (from welding consumables), and degree of restraint.
29.5.5
Fatigue
This failure mode has been observed in tubular joints in offshore platforms, dragline booms, drilling derricks, radio masts, crane runways, and bridges. The nominal stress, or detail classification approach, used for non-tubular structures fails to recognize the wide range of connection efficiencies and stress concentration factors that can occur in tubular structures. Thus, fatigue design criteria based on either punching shear or hot spot stress appear in the AWS Code. The subject is also summarized in recent papers on tubular offshore structures [7, 8].
29.6
Reserve Strength
While the elastic behavior of tubular joints is well predicted by shell theory and finite element analysis, there is considerable reserve strength beyond theoretical yielding due to triaxiality, plasticity, large deflection effects, and load redistribution. Practical design criteria make use of this reserve strength, placing considerable demands on the notch toughness of joint-can materials. Through joint classification (API) or an ovalizing parameter (AWS), they incorporate elements of general collapse as well as local failure. The resulting criteria may be compared against the supporting data base of test results to ferret out bias and uncertainty as measures of structural reliability. Data for K, T/Y, and X joints in compression show a bias on the safe side of 1.35, beyond the nominal safety factor of 1.8, as shown in Figure 29.4. Tension joints appear to show a larger bias of 2.85; however, this reduces to 2.05 for joints over 0.12 in. thick, and 1.22 over 0.5 in., suggesting a thickness effect for tests that end in fracture. For overload analysis of tubular structures (e.g., earthquake), we need not only ultimate strength, but also the load-deflection behavior. Early tests showed ultimate deflections of 0.03 to 0.07 chord diameters, giving a typical ductility of 0.10 diameters for a brace with weak joints at both ends. As more different types of joints were tested, a wider variety of load-deflection behaviors emerged, making such generalizations tenuous. Cyclic overload raises additional considerations. One issue is whether the joint will experience a ratcheting or progressive collapse failure, or will achieve stable behavior with plasticity contained at local hotshots, a process called “shakedown” (as in shakedown cruise). While tubular connections have withstood 60 to several hundred repetitions of load in excess of their nominal capacity, a conservative analytical treatment is to consider that the cumulative plastic deformation or energy absorption to failure remains constant. 1999 by CRC Press LLC
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FIGURE 29.4: Comparison of AWS design criteria with the WRC database. (From Marshall, P., Design of Welded Tubular Connections (1992), Dev. Civ. Eng., Vol. 37. With kind permission from Elsevier Science, Amsterdam, The Netherlands.)
When tubular joints and members are incorporated into a space frame, the question arises as to whether computed bending moments are primary (i.e., necessary for structural stability, as in a sidesway portal situation, and must be designed for) or secondary (i.e., an unwanted side effect of deflection which may be safely ignored or reduced). When proportional loading is imposed, with both axial load and bending moment being maintained regardless of deflection, the joint simply fails when it reaches its failure envelope. However, when moments are due to imposed lateral deflection, and then axial load is imposed, the load path skirts along the failure envelope, shedding the moment and sustaining further increases in axial load. Another area of interaction between joint behavior and frame action is the influence of brace bending/rotation on the strength of gap K-connections. If rotation is prevented, bending moments develop which permit the gap region to transfer additional load. If the loads remain strictly axial, brace end rotation occurs in the absence of restraining moments, and a lower joint capacity is found. These problems arise for circular tubes as well as box connections, and a recent trend has been to conduct joint-in-frame tests to achieve a realistic balance between the two limiting conditions. Loads 1999 by CRC Press LLC
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that maintain their original direction (as in an inelastic finite element analysis) or, worse yet, follow the deflection (as in testing arrangements with a two-hinge jack), result in a plastic instability of the compression brace stub which grossly understates the actual joint strength. Existing data bases may need to be screened for this problem.
29.7
Empirical Formulations
Because of the foregoing reserve strength issues, AWS design criteria have been derived from a database of ultimate strength tubular joint tests. Comparison with the database (Figure 29.4) indicates a safety index of 3.6 against known static loads for the AWS punching shear criteria. Safety index is the safety margin, including hidden bias, expressed in standard deviations of total uncertainty. Since these criteria are used to select the main member chord or joint can, the choice of safety index is similar to that used for sizing other structural members, rather than the higher safety margins used for workmanship-sensitive connection items such as welds or bolts. When the ultimate axial load is used in the context of AISC-LRFD, with a resistance factor of 0.8, AWS ultimate strength is nominally equivalent to punching shear allowable stress design (ASD) for structures having 40% dead load and 60% live load. LRFD falls on the safe side of ASD for structures having a lower proportion of dead load. AISC criteria for tension and compression members appear to have made the equivalency trade-off at 25% dead load; thus, the LRFD criteria given by AWS would appear to be conservative for a larger part of the population of structures. In Canada, using these resistance factors with slightly different load factors, a 4.2% difference in overall safety factor results — within calibration accuracy [10].
29.8
Design Charts
Research, testing, and applications have progressed to the point where tubular connections are about as reliable as the other structural elements with which designers deal. One of the principal barriers to more widespread use seems to be unfamiliarity. To alleviate this problem, design charts have been presented in “Designing Tubular Connections with AWS D1.1”, by P. W. Marshall [4]. The capacity of simple, direct, welded, tubular connections is given in terms of punching shear efficiency, Ev , where allowable punching shear stress (29.3) Ev = main member allowable tension stress Charts for punching shear efficiency for axial load, in-plane bending, and out-of-plane bending appear as Figures 29.5 through 29.9. Note that for axial load, separate charts are given for Kconnections, T/Y connections, and X connections, reflecting their different load patterns and different values of the ovalizing parameter (alpha). Within each connection or load type, punching shear efficiency is a function of the geometry parameters, diameter ratio (beta) and chord radius/thickness (gamma), as defined earlier. For K-connections, the gap, g, between braces (of diameter d) is also significant, with the behavior reverting to that of T/Y connections for very large gap. Punching shear efficiency cannot exceed a value of 0.67, the material limit for shear.
29.8.1
Joint Efficiency
The importance of branch/chord thickness ratio tau (t/T ) and of angle (sin θ ) becomes apparent in the expression for joint efficiency, Ej , given by: Ej = 1999 by CRC Press LLC
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Ev · Qf (t/T ) sin θ
·
Fyo (chord) Fy (branch)
(29.4)
FIGURE 29.5: Values of Qq for axial load in K-connections. (From Marshall, P.W., Designing Tubular Connections with AWS D1.1, Welding J., March, 1989. With permission from the American Welding Society.)
FIGURE 29.6: Values of Qq for axial load in T- and Y-connections. (From Marshall, P.W., Designing Tubular Connections with AWS D1.1, Welding J., March, 1989. With permission from the American Welding Society.) where Qf is the derating factor to account for chord utilization (described in the next section), and the ratio of specified minimum yield strengths Fyo /Fy drops out if chord and branch are of the same material. In LRFD, joint efficiency is the characteristic ultimate capacity of the tubular connection, as a fraction of the branch member yield capacity. In ASD, joint efficiency is the branch member nominal stress (as a fraction of tension allowable) at which the tubular connection reaches 1999 by CRC Press LLC
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FIGURE 29.7: Values of Qq for axial load in X-connections and other configurations subject to crushing. (From Marshall, P.W., Designing Tubular Connections with AWS D1.1, Welding J., March, 1989. With permission from the American Welding Society.)
FIGURE 29.8: Values of Qq for in-plane bending. (From Marshall, P.W., Designing Tubular Connections with AWS D1.1, Welding J., March, 1989. With permission from the American Welding Society.) its allowable punching shear. Connections with 100% joint efficiency develop the full yield capacity of the attached branch member, in either design format.
1999 by CRC Press LLC
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FIGURE 29.9: Values of Qq for out-of-plane bending. (From Marshall, P.W., Designing Tubular Connections with AWS D1.1, Welding J., March, 1989. With permission from the American Welding Society.)
29.8.2
Derating Factor
In most structures, the main member (chord) at tubular connections must do double duty, carrying loads of its own (axial stress fa and bending fb ) in addition to the localized loadings (punching shear) imposed by the branch members. Interaction between these two causes a reduction in the punching shear capacity, as reflected in the Qf derating factor, shown in Figure 29.10. In-plane bending experiences the most severe interaction, as localized shell bending stresses at the tubular intersection are in the same direction and directly additive to the chord’s own nominal stresses over a large part of the cross-section. For chords with very high R/T (gamma) and high nominal compressive stresses, buckling tendencies further reduce the capacity for localized shell stresses. Out-of-plane bending is less vulnerable to both these sources of interaction, as high shell stresses only occupy a localized part of the cross-section, and are transverse to P-1 effects. Axially loaded connections of the types tested thus far exhibit intermediate behavior (although the gap region in K-connections might be expected to behave more like in-plane bending).
29.9
Application
What follows is a step-by-step design procedure for simple tubular trusses, applying the charts presented in the foregoing section. Step 1. Lay out the truss and calculate member forces using statically determinate pin-end assumptions. Flexibility of the connections results in secondary bending moments being lower than given by typical rigid-joint computer frame analyses. Step 2. Select members to carry these axial loads, using the appropriate governing design code, e.g., AISC. While doing this, consider the architecture of the connections along the following guidelines: 1. Keep compact members, especially low D/T for the main member (chord). 2. Keep branch/main thickness ratio (tau) less than unity, preferably about 0.5. 1999 by CRC Press LLC
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FIGURE 29.10: Derating factor Qf for (a) axial loads in branch, (b) in-plane bending, and (c) outof-plane bending. (From Marshall, P.W., Designing Tubular Connections with AWS D1.1, Welding J., March, 1989. With permission from the American Welding Society.)
3. Select branch members to aim for large beta (branch/main diameter ratio), subject to avoidance of large eccentricity moments. 4. In K-connections, use a minimum gap of 2 in. between the braces for welding access. For small tubes, this may be reduced to 20% of the branch member diameter. Connection eccentricities up to 25% of the chord diameter may be used to accomplish this. Reconsider truss layout if this gets awkward. Step 3. Calculate and distribute eccentricity moments and moments due to loads applied inbetween panel points. These are not secondary moments, and must be provided for. They may be allocated entirely to the chord, for connection eccentricities less than 25% of the chord diameter, but should be distributed to both chord and braces for larger eccentricities, portal frames, or Vierendeel type trusses. Recheck members for these moments and resize as necessary. 1999 by CRC Press LLC
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Step 4. For each branch member, calculate Ay , utilization against member-end yield at the joint. For allowable stress design, |fa | |fb | or (29.5) Ay = 0.6Fy 0.6Fy where fa = nominal axial stress fb = bending in the branch Where used, the 1/3 increase is applicable to the denominator. Step 5. Also calculate chord utilization, using the formula in Figure 29.10 with chord nominal stresses and specified minimum yield strength. Use the appropriate chart in the figure to determine the derating factor Qf . At heavily sheared gap K-connections and at eccentric bearing shoes, it may (rarely) also be necessary to check beam shear in the main member, and its interaction with other chord stresses, e.g., using AISC criteria. For circular sections, the effective area for beam shear is half the gross area. Step 6. For each end of each branch member, calculate the joint efficiency Ej using Equation 29.4 and the appropriate charts for punching shear efficiency Ev . Joint efficiencies less than 0.5 are sometimes considered poor practice, rendering the structure vulnerable to incidental loads which the members could resist, but not the weaker joints. Step 7. For axial loading alone, or bending alone, the connection is satisfactory if member-end utilization is less than joint efficiency, i.e., Ay /Ej ≤ 1.0. For the general case, with combinations of axial load and bending, the connection must satisfy the following interaction formula: (Ay /Ej )1.75 axial + (Ay /Ej )bending ≤ 1.0
(29.6)
Step 8. To redesign unsatisfactory connections, go back to Step 2 and 1. increase the chord thickness, or 2. increase the branch diameter, or 3. both of the above. Consider overlapped connections (AWS section 2.40.1.6) or stiffened connections only as a last resort. Overlapped connections increase the complexity of fabrication, but can result in substantial reductions in the required chord wall thickness. Step 9. When the designer thinks he is done, he should talk to potential fabricators and erectors. Their feedback could be valuable for avoiding unnecessary, difficult, and expensive construction headaches. Also make sure they are familiar with, and prepared to follow, AWS Code requirements for special welder qualifications, and that they are capable of coping the brace ends with sufficient accuracy to apply AWS prequalified procedures. Considerable savings can be realized by specifying partial joint penetration welds for tubular T-, Y-, and K-connections with no root access, where these are appropriate to service requirements. Fabrication and inspection practices for welded tubular connections have been addressed by Post [12].
29.10
Summary and Conclusions
This chapter has served as a brief introduction to the subject of designing welded tubular connections for circular hollow sections. More detail on the background and use of AWS D1.1 in this area can be found in [6]. 1999 by CRC Press LLC
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References [1] AWS D1.1-96. 1996. Structural Welding Code — Steel, American Welding Society, Miami, FL. [2] Kurobane, Y. 1995. Comparison of AWS vs. International Criteria, ASCE Structures Congress, Atlanta, GA. [3] Marshall, P.W. 1986. Design of Internally Stiffened Tubular Joints, Proc. IIW/AIJ Intl. Conf. on Safety Criteria in the Design of Tubular Structures, Tokyo. [4] Marshall, P.W. 1989. Designing Tubular Connections, Welding J. [5] Marshall, P.W. 1990. Advanced Fracture Control Procedures for Deepwater Offshore Towers,
Welding J. [6] Marshall, P.W. 1992. Design of Welded Tubular Connections: Basis and Use of AWS D.1.1, Elsevier Science Publishers, Amsterdam. [7] Marshall, P.W. 1993. API Provisions for SCF, S-N and Size-Profile Effects, Proc Offshore Tech. Conf., OTC 7155, Houston, TX. [8] Marshall, P.W. 1996. Offshore Tubular Structures, Proc. AWS Intl. Conf. on Tubular Structures, Vancouver. [9] Marshall, P.W. and Luyties, W.H. 1982. Allowable Stresses for Fatigue Design, Proc. Intl. Conf. on Behavior of Off-Shore Structures, BOSS-82 at MIT, McGraw-Hill, New York. [10] Packer, J.A. et al. 1984. Canadian Implementation of CIDECT Monograph 6, IIW Doc. XV-E84-072. [11] Paul, J.C. 1992. The Ultimate Behavior of Multiplanar TT- and KK-Joints Made of Circular Hollow Sections, Ph.D. thesis, Kumamoto University, Japan. [12] Post, J.W. 1996. Fabrication and Inspection Practices for Welded Tubular Connections, Proc. AWS Intl. Conf. on Tubular Structures, Vancouver. [13] Wardenier, J. 1987. Design and Calculation of Predominantly Statically Loaded Joints Between Round (Circular) Hollow Sections, van Leeuwen Technical Information No. 7.
1999 by CRC Press LLC
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Yashinsky, M. “Earthquake Damage to Structures” Structural Engineering Handbook. Ed. Lian Duan Boca Raton: CRC Press LLC, 2001
30 Earthquake Damage to Structures Mark Yashinsky
30.1
Introduction
Caltrans Office of Earthquake Engineering
30.2
Damage as a Result of Problem Soils
30.3
Damage as a Result of Structural Problems
Earthquakes • Structural Damage Liquefaction • Landslides • Weak Clay Foundation Failure • Foundation Connections • Soft Story • Torsional Moments • Shear • Flexural Failure • Connection Problems • Problem Structures
30.4
Secondary Causes of Structural Damage Surface Faulting • Damage Caused by Nearby Structures and Lifelines
30.5
Recent Improvements in Earthquake Performance Soil Remediation Procedures • Improving Slope Stability and Preventing Landslides • Soil-Structure Interaction to Improve Earthquake Response • Structural Elements that Prevent Damage and Improve Dynamic Response
30.1 Introduction Earthquakes Most earthquakes occur due to the movement of faults. Faults slowly build up stresses that are suddenly released during an earthquake. We measure the size of earthquakes using moment magnitude as defined in Equation 30.1.
M = (2/3)[log(Mo) – 16.05]
(30.1)
where Mo is the seismic moment, as defined in Equation 30.2:
Mo = GAD (in dyne-cm)
(30.2)
where G is the shear modulus of the rock (dyne/cm2), A is the area of the fault (cm2), and D is the amount of slip or movement of the fault (cm). The largest magnitude earthquake that can occur on a particular fault is the product of the fault length times its depth (A), the average slip rate times the recurrence interval of the earthquake (D), and the hardness of the rock (G). For instance, the northern half of the Hayward Fault (in the San Francisco Bay Area) has an annual slip rate of 9 mm/yr (Figure 30.1). It has an earthquake recurrence interval of 200 years. It is 50 km long and 14 km deep. G is taken as 3 × 1011 dyne/cm2:
© 2001 by CRC Press LLC
FIGURE 30.1 Map of Hayward Fault. (Courtesy of EERI [1].)
Mo = (.9 × 200) (5 × 106) (1.4 × 106) (3 × 1011) = 3.78 × 1026 M = (2/3)[log 3.78 × 1026 – 16.05] = 7.01 © 2001 by CRC Press LLC
FIGURE 30.2 Attenuation curve developed by Mualchin and Jones [7].
Therefore, an earthquake of a magnitude about 7.0 is the maximum event that can occur on the northern section of the Hayward Fault. Because G is a constant, the average slip is usually a few meters, and the depth of the crust is fairly constant, the size of the earthquake is usually controlled by the length of the fault. Magnitude is not particularly revealing to the structural engineer. Engineers design structures for the peak accelerations and displacements at the site. After every earthquake, seismologists assemble the recordings of acceleration vs. distance to create attenuation curves that relate the peak ground acceleration (PGA) to the magnitude of earthquakes based on distance from the fault rupture (Figure 30.2). All of the data available on active faults are assembled to create a seismic hazard map. The map has contour lines that provide the peak acceleration based on attenuation curves that indicate the reduction in acceleration due to the distance from a fault. The map is based on deterministic-derived earthquakes or on earthquakes with the same return period.
Structural Damage Every day, regions of high seismicity experience many small earthquakes; however, structural damage does not usually occur until the magnitude approaches 5.0. Most structural damage during earthquakes is caused by the failure of the surrounding soil or from strong shaking. Damage also results from surface ruptures, from the failure of nearby lifelines, or from the collapse of more vulnerable structures. We consider these effects as secondary, because they are not always present during an earthquake; however, when there is a long surface rupture (such as that which accompanied the 1999 Ji Ji, Taiwan earthquake), secondary effects can dominate. Because damage can mean anything from minor cracks to total collapse, categories of damage have been developed, as shown in Table 30.1. These levels of damage give engineers a choice for the performance of their structure during earthquakes. Most engineered structures are designed only to prevent TABLE 30.1 Categories of Structural Damage Damage State
Functionality
Repairs Required
Expected Outage
(1) None (pre-yield) (2) Minor/slight (3) Moderate (4) Major/extensive (5) Complete/collapse
No loss Slight loss Some loss Considerable loss Total loss
None Inspect, adjust, patch Repair components Rebuild components Rebuild structure
None
