Total tension

L= distance between anchorages Figure 20.10 Suspension-bridge notation Precast or prefabricated elements can be made as transverse rather than longitudinal elements and then joined together on site by prestressing in concrete structures or welding or bolting in steel structures. This approach, sometimes known as segmental construction, was used for the structures of Figure 20.1 l(d), (e), (f), (h), (i), (j) and (k). It was also used for the steel structures of Figure 20.11 (1) and (m). The remaining steel structures shown in Figure 20.11(n) to (r) were constructed by a similar process but with the subdivision taken a stage further. Each transverse slice was built up on the end of the cantilevering structure from several stiffened panels. In situ concrete, reinforced or prestressed, can be used to form complete spans in one operation or else the cantilevering approach can be used. In the latter case, the speed of construction is limited by the time required for the concrete to reach a cube strength adequate for the degree of prestress necessary to support the next section of the cantilever and the erection equipment. Segmental methods of construction8 avoid such delays. In shorter spans, provided that the restrictions on construction depth are not too severe, in situ concrete structures can be built economically using the cross-section of Figure 20.1 l(c). The simple cross-section9 was developed to suit the use of formwork which, after supporting a complete span, could be moved rapidly to the next span. The resulting machine is only economical for multispan structures. The stiffened steel plates (Table 20.10) are used for deck systems of long-span, and movable, bridges in order to reduce the self-weight of the structure.

20.3 Characteristics of bridge structures The following theories have been chosen and developed for their value in demonstrating the principal characteristics of various types of bridge structure. Other methods of calculation, based on finite elements, for example, may be more accurate and more economical in certain circumstances. The theories are, however, linked to the main structural properties of the bridge types considered and are meant to assist the process of synthesis necessary before detailed calculations begin. The concepts described are also useful for idealizing structures when using computer programs and for interpreting and checking the computer output. 20.3.1 Theory of suspension bridges and arch bridges The basic theory of arch and suspension bridges is the same and the equation derived below for suspension bridges is applicable to arches if a change in sign of H and y is made.

20.3.1.1 Suspension bridges with external anchorages The dead load of the cable and stiffening girder is supported by the force per unit length of span produced by the horizontal component of the cable force and the rate of change of slope of the cable: #,y(*) + * = 0

(20.1)

where y, etc. are shown in Figure 20.10. For a parabolic shape of cable corresponding to constant intensity of load across the span /, y"(x) = - 8///2 and: Hg=glW

(20.2)

The cable tension increases under live load p(x) to: H=HK + Hp

(20.3)

The increase in support from the cable is — [Hv" (x) + H^(X)] where v(x) is the vertical deflection of the cable and stiffening girder. The stiffening girder contributes a supporting reaction per unit length of [EIv"(x)]" and adding the cable and stiffening girder contributions and equating them to the intensity of the applied load gives: [EIv"(x)]" - Hv"(x) =p(x) + Hfy"

(20.4)

The term H^y" is added to the live load in order to show that the equation can be represented physically by the substitute structure of Figure 20.12. y" is — 8///2 and therefore represents a force in the opposite direction to the live load. Hp depends on the change in length of the cable and if A&x is the horizontal projection of the change in length of an element ds then for fixed anchorages: JjJdX = O

(20.5)

Integrating along the cable and allowing for a change in temperature of A T gives: fcW*-*Pl^

iMJV^fr-Wfc-O

(20.6)

Approximate values of Lk and L7 are (see Figure 20.13): 2 L^ V°/ 0 ) + -^-+ * (l+8-£ \ /2 + ^tan 2 COS2V1 ~4COS2V2

(20.7) L 1 ^(I + ^ + tan 2 V 0 ) + J j U JL.

Figure 20.11 Elevated roadways, (a) Westway, Section One; (b) Tunnel relief flyover, Liverpool; (c) Vorlandbrucke Obereisesheim; (d) llltal; (e) West Gate approach viaducts; (f) Westway, section five; (g) Bendorf, section at pier; (h) Mancunian Way; (i) Gladesville; (j) London; (k) Narrows; (I) Annacis; (m) Severn; (n) Europa; (o) Duisberg-Neuenkamp; (p) Concordia; (q) Kniebrucke; (r) Sava I; (s) Zoo

Equations (20.4) and (20.6) must be satisfied simultaneously and, although this makes the problem nonlinear, the correct value can be satisfactorily determined by interpolation by solving for two assumed values of H. Each assumed H gives an incorrect solution to Equation (20.6) and, assuming the error varies linearly, the correct value of H can be found. For each assumed value of//, the structure behaves as a simple beam and influence lines can be constructed for bending moments, etc., and for fo(x)dx. Hawranek and Steinhardt10 suggest that for a particular loading case the bending moment and shear forces be found from both sets of influence lines as well as the ftv(x)dx values. H is found by interpolation and then the final bending moments and shears are found by interpolating between the two sets of values already found from the influence lines. Typical results for a continuous stiffening girder are shown in Figure 20.14. The above treatment follows that given by Hawranek and Steinhardt10 who also give a comprehensive set of standard

solutions for the substitute girder. The result quoted below illustrates the form the solutions take. Using: It = HIEI For the load case of Figure 20.15, deflections as a function of x are given by: V(X9 O = PG(X1 &

• p ^ f f (i-jV^^*'"0] H L/ \

//

/i/sinh///

J

for{

(20.8) v(x,Q=PG(x9Q

"A[K-?)-^a?-*] -«-

G(x,
+

V ^ + aJfcJ

(20.15)

«u,= —V a, a k r r+2

r 2 r+l r+]

where the effective distortional bending stiffness of the box in bay r is EIr and an additional /* is a shear stiffness for the warping rotation produced by distortional shears in the box: (20.16)

The various results given above can be found from concepts of virtual work using the mechanism shown in Figure 20.34 consisting of a series of shear webs and booms of axial stiffness. It can be used to obtain more general results such as those for boxes of trapezoidal cross-section." 13 The warping produced by torsion results in stresses only if there is a change in torsion and therefore incompatible warping. Longitudinal stresses act to remove the lack of continuity. An upper bound estimate of these stresses can be made by assuming that the cross-section cannot deform. The warping moments produced by a change in torque T= Pb is: X= ^ ^^expl-V(T1Y/,)*]

where r and r + 1 refer to bays between diaphragms as in Figure 20.30.

Diaphragm r-2

Ff,

Lt>

J

The deflections of the diaphragms or springs induce relative rotations at each hinge. The rotations at the releases are increased by the warping produced by the torque fed into the box at each diaphragm and the effect of the load between diaphragms. Denoting O1 as the relative rotation due to spring deflections and local loads, the total relative rotation is: °'^ -f

l °"->-a,_}a^

EO E

f-G 8c2 E b/t3 + b/t2 + 2c/tl

P _ applied distortional forces v distortional deflection

a-e++ Q±i-&

The influence coefficients for solving the series of compatibility equations are:

(20 17)

where ;c is the distance from the cross-section at which the change occurs.

Springs of stiffness k

Equivalent beam Smaller steel box beams which rely on the stiffness of the sides plus the frame action of web and flange stiffness or larger box beams with special frames which leave the interior of the box unobstructed, have similar characteristics to the concrete boxes. Frames are generally flexible compared with braced or plate diaphragms but stiff frames can be made which will have properties that can only be explored fully by a treatment which is suitable for discrete diaphragms. The validity of the following approach for steel boxes can be determined from the half wavelength which should be greater than twice the spacing of the frames. The effect of twisting loads P applied at the corners of the web on the vertical deflections caused by distortional bending of the box and allowing for the diaphragm action of the cross-section is:14

Diaphragm Transverse frames Diaphragm Figure 20.31 Equivalent beam on elastic supports

Associated twisting forces

•-£H'[i*i(A-7)]-*

Figure 20.32 Distortional force system 4a

The distribution of shear forces associated with warping stresses can be found from the warping stress distribution (Figure 20.35). Starting from the edge of the cantilever and assuming CT is the change in longitudinal stress over a unit length: q- f aids J

[£-i(A -j)]si«AJ

(20.I8)

assuming that the distance to a support is infinitely long. EI6 is the effective distortional bending stiffness of the box, and k is the diaphragm stiffness per unit length. A is defined by:

O

V=-±4£/,

where q is the shear flow rt. The complementary shear to q is on the face of the cross-section. The shear at the centre of the top flange can be assumed to be zero at the first stage of the calculation, which enables a simple shear system to be found. A pure torsional shear flow must be added to remove any component of pure torsion acting on the cross-section. 20.3.6 Box beams with continuous diaphragms The frame action of the webs and flanges in concrete boxes provides a continuous resistance to distortion; consequently special diaphragms are not usually necessary except at disturbances such as bearings and other support points.

and:

a = (A2 + k/4Ef*y/2,

The generalized warping stress resultant is:

P X=

We

212)1/2 replaces P and hyperbolic functions are used. The half wavelength is n/2fi. In many cases, the effect of shear is not considerable and a = ff = L The equations then simplify into the standard beam on elastic foundation results. The above equations include, however, the effect of the change in torque due to the twisting loads P and in order to correct the results of beam on elastic foundation theory the warping moment of Equation (20.17) should be added. The correction is likely to be of most significance in boxes which are much wider than their depth. The above equations are valid for boxes of trapezoidal crosssections if the concepts are generalized as was illustrated by Dalton and Richmond.13

box and the magnitude of the stresses is correspondingly great since any loss of torsional strength and stiffness could result in a major increase in the shear stresses and longitudinal stresses with a corresponding reduction in the load factor. The three types of cantilever in Figure 20.36 show:15 (1) Transmission of torque Pe into twisting couple which must be resisted by diaphragm action. (2) Cantilever bracket which, in the position shown, produces horizontal loads which are in the correct ratio to the vertical shear P to give a torsional shear flow without diaphragm action. In a concrete box the transverse moments in walls of the structure would be small. (3) The relative stiffness of the web results in most of the cantilever moment being taken by the web which produces horizontal forces corresponding to a brace at any position of the load. The transverse moments in the walls are therefore reduced rather than increased as the eccentricity increases. If e=6T/2 + 6B/2 they are practically zero.

20.3.7 Box girders with cantilevers

In concrete boxes where the only diaphragm stiffness is due to transverse bending of the walls, except at bearings, the beneficial effects described in (2) and (3) are of considerable importance. The reduction in transverse moments has been described but another important benefit is the reduction in the shear forces in the outer web.

The advantages of box girders over structures of open crosssection are sometimes only marginally linked to structural efficiency. However, where a compact structure is to carry a much wider deck, producing a large cantilevered section, the torsional stiffness and strength of the box is of primary importance. The importance of the interaction of the cantilever and the

The choice of an individual main girder or beam as the member to carry say a concentrated load applied to it with the distribution of that load as the next, correcting, operation is more

20.3.8 Multiple web girders of open cross-section

appropriate to smaller spans where typically there are large numbers of main beams and the width of the bridge is comparable with the span. In most shorter span structures the width is in fact more than the span which lends further weight to the argument. The following section on harmonic analysis adopts this approach.

The results in Figure 20.37 are for a system with zero torsional rigidity; Hendry and Jaeger also give results for a torsionally rigid system.16 Intermediate torsional stiffnesses can be analysed by interpolation. Fixed-ended and continuous beams which they also consider by this method may be more easily solved by an influence coefficient method. Hinge releases at the supports can convert a continuous system of beams into two or more simply supported spans. The behaviour of the released structure and the influence coefficients can be found using the above approach for the loading applied and each influence coefficient. 20.3.8.2 Eigenvalue methods

Figure 20.36 20.3.8.1 Harmonic analysis The interaction between a series of separate simply supported beams of constant cross-section can be investigated for arbitrary loading using harmonic analysis. The sine series is the most suitable approach because any load which varies sinusoidally over the span in a complete number of halfwaves produces a deflected profile for each beam of the same form but of varying magnitudes. This result is justified provided that it is recognized that the interacting forces between the beams will be proportional to the transverse deflected form and will also vary sinusoidally. The interaction between the beams is dependent on the ratio of the transverse to longitudinal stiffness: =

a

12

I

L

V

Dy

n4 \ H J Z)x

(20.20)

where Dy and Dx are stiffnesses of the equivalent orthotropic plate in the transverse and longitudinal directions. For a single half-wave loading on one beam the distribution coefficients giving the fraction of the load taken by each beam have been calculated for a number of different systems by Hendry and Jaeger.16 Figure 20.37 shows the coefficients for a five-beam bridge with beam 2 loaded (Figure 20.38). The coefficients given are for the first harmonic only. Coefficients for subsequent harmonics can be found by varying a as appropriate for the shorter wavelength. Alternatively, if a sufficiently close approximation is given by the first harmonic alone for distribution to the unloaded beams, the behaviour of the loaded beam is given by its 'free deflection' curve less that which has been distributed to the other beams.

The above method of analysing a grillage of beams under outof-plane loading is a particular example of a more general method, i.e. the eigenvalue approach. Whereas a sinusoidal variation of loading of simply supported beams of constant cross-section produces similar deflection forms, there are eigenload systems for beams of all forms which produce deflections with a deflected form similar to the load-intensity curve. Thus, continuous beams of varying cross-section can be investigated by such an eigenvalue approach. Where the transverse member is a continuous concrete slab or steel plate or comprises a large number of transverse beams it may be assumed that a continuous transverse medium is the most appropriate physical model. In some circumstances - where there are a small number of transverse members or other significant variations from a constant transverse medium - the eigenload system becomes a series of discrete loads on the main girder corresponding to the positions of the transverse members. The discrete approach may also be the most appropriate representation of a continuous transverse system in order to facilitate the analysis of more complex systems by numerical techniques. Longer span bridge superstructures are nearer in behaviour to a single beam formed from the aggregate of the individual longitudinal main beams, slabs and plates of the complete bridge cross-section. Consequently, there are both conceptual and numerical advantages in analysing such structures as single aggregate beams with subsequent correcting operations to allow for the deformations of the transverse beams, slabs and diaphragms which contribute to the transverse stiffness of the superstructure. The eigenvalue approach can be used to determine the necessary corrections by considering the characteristics of the transverse structural system of the bridge thereby turning the method described above in section 20.3.8.1 through 90°. The five-beam system for which the distribution coefficients were given in Figure 20.36 for beams of zero torsional stiffness can be used as an example of this approach. Figure 20.39(a) shows a point load A acting at any spanwise position on the central beam of a set of girders which form any system of spans and have any support condition provided that both spans and supports are the same for all girders. Figure 20.39(b) is then the load distribution required to produce the effect of all beams acting as an aggregate beam to carry the point load. The correcting systems are shown in Figure 20.39(c) and (d) and are eigenloads or vectors p, and p2, of the transverse beams or other transverse system. The particular system shown is valid for a transverse beam or slab system that is constant in its flexural properties across the width of the five beams. The eigenload systems are both self-equilibrating and p, and p2 can be calculated readily from statics such that (b), (c) and (d) are equivalent to (a). The eigenloads themselves are found by choosing a pattern of point loads that produces the same ratio of deflection of the transverse system at e.ach beam position relative to the respective component of the eigenload system.

Figure 20.37 Distribution coefficients for a five-beam bridge, beam two loaded. (After Hendry and Jaeger (1958) The analysis of grid frameworks and related structures. Chatto and Windus)

Figure 20.38 Five-beam bridge. (After Hendry and Jaeger (1958) The analysis of grid frameworks and related structures. Chatto and Windus) Each beam is thereby effectively supported by a spring of equal stiffness when the complete system is loaded by one of the eigensystems. The spring stiffnesses for a transverse medium of flexural stiffness D per unit length for each beam is: \2o}}D ^3

. , , for eigenload system p,

f

(20 21)

and:

U(O2D —£-

, . . , for eigenload system p2

where Co1 = 0.0759016 and ^2 = 2.35267 Thus, the beam systems under eigenload systems P1 and p2 can be analysed by considering for each load system the behaviour of the appropriate elastically supported beam. The essential

feature of the method is, of course, that only one beam has to be investigated for each load system. Where the transverse medium of stiffness D is appropriate, that single beam can be analysed using beam-on-elastic foundation theory. The supports and continuity can be allowed for as appropriate. Consequently, the two correcting solutions are readily produced either as exact solutions or as a means of assessing the characteristics of the structural system. Grillage programs enable solutions to be obtained by computer with the advantage that complex geometries can be simulated without difficulty. In both cases, it is necessary to estimate the effective top flange unless the more complex form of beam and slab program using finite elements is used. 20.3.9 Multiple single-cell box beams A series of box beams connected by a top deck and, in some cases, cross-beams and stiffening diaphragms, can be analysed by various approaches. The grillage approach using a computer is not necessarily suited to all problems of this type but it is discussed first because, in determining the properties of the members, the essential mode of behaviour of this form of structure emerges. Figure 20.40 shows part of a typical cross-section which could represent a series of concrete main longitudinal beams spanning 20 and 60m or the trough stiffeners on a steel deck system spanning 4 m. In such systems, the interaction between the beams is through the deck slab or deck plate only if no special cross-beams, etc. are provided. The magnitude of the interacting forces will be mainly dependent on the overall deflections of the beams and so the distortional stiffness of the individual boxes will almost equal the frame stiffness of the sides. The distortional bending or warping stiffnesses of the boxes may be assumed to be nil except for local wheel loads, which will be mentioned later. Through isolating one of the boxes and its share of the deck slab, its behaviour can be considered further (Figure 20.41). A unit value of the antisymmetrical component of the vertical shearing forces will act as shown. The twisting effect will be resisted by pure torsion if there is no significant distortional resistance of the box beam except for diaphragm action. The cross-section, acting as a frame, is loaded by the distortional component of the twisting load. The net effect can be obtained by the device illustrated. The pin-jointed bars are placed so as to apply a pure torsional shear flow to the cross-section. They also prevent only a pure rotation of the cross-section since distor-

Transverse slab five equal beams of low torsional stiffness

Load distribution for aggregate beam action

Eigen load system 1

Eigen load system 2

The three load systems (b)+(c)+(d) are equivalent tothesingle load p in (a) if P1 = 0-3247 and P2 =0 4752 Figure 20.39 (a) Point load A acting on central beam; (b) load distribution acting as aggregate for point load p; (c) eigenloads for The effect of local wheel loads on the transverse moments system 1; (d) eigenloads for system 2 must be added to the above transverse moments by assuming

Figure 20.40 Interconnected multiple box girders tional deflections do not have components in the direction of the restraints. A simple plane-frame analysis of the system gives the deflections and, hence, effective stiffness of a transverse member cantilevering out a distance (a + b)/2 from a grillage element with the torsional and flexural stiffnesses of the box. The antisymmetrical moments and symmetrical shears and moments can be applied to find the appropriate stiffnesses and, if the simplest form of grillage is used, a single compromise value must be chosen. The one based on antisymmetrical shears alone has been found to give results which compared well with a threedimensional finite element simulation of a series of concrete boxes at 2-m centres.

Virtual displacement

that the boxes provide rigid supports. The maximum moments in the slab were, in the case mentioned, unaffected by the local slab loading. Another effect already mentioned is the distortion of the box due to wheel loads applied on one side only. In fact, for boxes that are spaced at up to 2-m centres, the wheel loads are spread over a width sufficient to make the highest loads fairly symmetrically disposed about an individual box. An allowance can be made, however, by using the single cell theory to calculate stresses which are superimposed on those described above. A similar approach has been used for steel deck systems, assuming points of contraflexure halfway between stringers but, instead of treating the structure as a series of discrete beams, it is transformed into an orthotropic plate. The transverse flexural stiffness is included in the torsional stiffness of the plate and is, therefore, taken as zero in the plate. The longitudinal flexural stiffness is determined in the usual way. Transversely, the deflections of the plate are represented as a sine series in order to solve the plate equation for wheel loading. A large number of terms in the series are required because of poor convergence which, together with the difficulties in obtaining detailed stress values other than longitudinal ones, from the solution, make the method of limited value. Graphs have, however, been produced17 for the wheel loading used in the US which are useful for preliminary estimates. Where a small number of large box girders are used, it may be necessary to allow for the various components of the interacting forces more exactly. An example of this is shown in Figure 20.42 in which the nonuniform component of load on two boxes is split into three load systems with the properties of either symmetry or antisymmetry. Releasing the forces at the centre of

Virtual displacement 6

Figure 20.41 Torsional support system for slice of beam to give frame stiffness

the connecting slab or cross-girder produces a lack of compatibility in each case. The influence coefficients are found from the unit forces of Figure 20.43(a) which relate to the compatibility equations including wa and uc and Figure 20.43(b) for ub. S represents the overall bending deflection of the box beam, ad is the deflection produced by torsional rotation and Sc is the deflection of the cross-girder or deflection of the slab. Sc will be found from Figure 20.43(c) if it is a concrete box of the type already discussed, and Oc similarly from Figure 20.43(d). Sinusoidally varying forces can be used for boxes without discrete diaphragms except at the supports, otherwise the influence coefficients can be related to individual cross-members.

20.3.10 Multicellular bridge decks Bridge structures similar to the top-hat beam deck6 which has 115 mm thick webs and no diaphragms between supports, have cross-sections which are relatively flexible in transverse shear. The usual grillage or orthotropic plate approach in which shear deformations are neglected is consequently invalid. The following treatment6 is also relevant to cellular steel decks which may be even more flexible owing to higher web depth: thickness ratios. Transverse shears are carried by the Vierendeel frame action (Figure 20.44) and the flexibility of the frame can be simulated by an equivalent web area of the transverse beams: m/d A» -< wc/; Fir\ W2/,) + (/*2// )

A

2

(20.21)

The grillage program used must include the effects of deflections due to shear strains. It is necessary to differentiate between rotations of initially horizontal and initially vertical lines when shear deformations are considered. In the grillage program used for Table 18.9 structure the rotation of vertical lines was the variable used. The flexural parameters are derived in the usual way but the torsional stiffnesses of the grillage members require further consideration. 20.3.11 Symmetrical loading Loads disposed symmetrically in the transverse sense produce only relatively small transverse movements. True torsion is absent but Figure 20.45 shows that transverse members of grillage will be subjected to twisting which in the actual structure is simply a set of shear strains leading to shear transfer between the beams in a horizontal plane. This is sometimes referred to as shear lag. If the shear lag is small the shear transfer is high and the whole flange will be stressed uniformly. If the transverse members are assigned a stiffness per unit longitudinal distance of /*2//2, this effect will be simulated. The torsional stiffness of the longitudinals is largely immaterial since they do not rotate significantly. 20.3.12 Antisymmetrical loading Figure 20.42 Interconnected boxes

Figure 20.43 Unit loads and couples applied at releases

Loads which cause twisting produce rotations of the crosssection which are the mean of a rotation of a vertical and horizontal line. The vertical line component is given directly by the grillage. The rate of change of rotation of a horizontal line is

almost equal to the rate of change of rotation of a vertical line considered in the longitudinal and transverse direction respectively, since the true webs only undergo small shear strains compared with the frame of the cross-section. Therefore, by allocating half the torsional stiffness GK of the cross-section to longitudinals and the remainder to transverse members, the mean rotation will be used as required. GK for the whole structure is: 4GA2/§(ds/t). Therefore, the torsional stiffness of members divided by the spacing should be (4GA2/$(l/t)ds)/2b for both sets of members. Comparisons have shown that solving the structure in two stages using the different torsional properties described above does give good agreement, but it is clearly preferable to have one set of properties for any load case. It has been found that adopting H2t/2 as the torsional stiffness for transverse members and dividing the remainder, 4GA2/$(l/t)ds-(h2t/2)b amongst the longitudinals is a satisfactory compromise for all loading cases.

Figure 20.45 Shear strain due to differential longitudinal deflections

20.3.13 Design curves Design curves have been obtained using the above approach for cellular decks constructed using precast 'top-hat' beams. They are of value for other cellular decks of similar proportions for preliminary design studies. Figure 20.46 gives curves for HB coefficients (see Chapter 19). The range of decks covered is for spans from 60 ft to 120 ft (18.3 to 36.6 m) and deck widths from 30 ft to 90 ft (9.1 to 27.4 m), but the curves can be extrapolated to include values outside these figures and approximate solutions can also be derived for skew decks. It is important to note that the curves have been derived by means of a grillage representation. Consequently, the application of the curves and the results obtained from them relate to the grillage solution and are subject to its conditions and limitations. The curves provide a coefficient per foot width of beam of the total moment ML. The coefficients are plotted against the breadth:span ratio B:L and are dependent upon the span L in conjunction with the edge stiffness ratio / and also upon the distance D from the centre of the outer wheels to the edge of the deck. The values of kL and k, are derived from the intersection of B: L and L, and kp from the intersection of B: L and D. The design live load moment for a composite top-hat beam is then given by:

M=kLk9[\ -kfj- l)]bML

(20.22)

where b is the beam width of either the edge beam, which may be asymmetrical, or the adjacent inner beam.

Figure 20.46 Design curves, HB coefficients

Figure 20.44 Transverse frame

20.4 Stress concentrations Sudden changes in loads or in the shape of a structure produce stresses that cannot be calculated by normal beam theory. Concentrated loads such as the reactions at bearings and holes cut out of flanges are obvious examples of such changes but alterations in the direction of flanges, variations in thickness or width may produce significant effects. The introduction of strengthening members such as diaphragms or stiffening around holes are examples of situations in which the stress concentrations may weaken the structure instead of adding strength, if the full implications of the addition are not considered. 20.4.1 Shear lag due to concentrated loads Beams with wide flanges are subject to shear lag effects, since the longitudinal stresses at points remote from the webs must be generated by the shear stress field across the flange. At sudden changes in shear, due to concentrated loads, the necessary changes in the longitudinal stresses in the flange require differential longitudinal movements in the transverse direction which produces the shear stress field in the flange. In other words, a longitudinal strain variation and therefore a stress variation across the flange is produced which is called the shear lag effect. It is important to note that transverse stresses are associated with shear lag as can be demonstrated by a consideration of the statics of the shear stress field alone. Shear lag is most pronounced in girders of rectangular crosssection. The effect of cambering the flanges in the convex sense is to reduce shear lag. In the extreme case of a circular crosssection, shear lag does not occur, provided that all crosssections remain circular. This is because the web of a rectangular box can deform in shear without shear deformations of the flange being necessary, whereas the circular beam can only deform in shear if all parts of the beam are subjected to shear strains. The distribution of shear strain in the latter case, which depends on purely geometrical considerations, agrees with the shear stress distribution produced by the longitudinal bending stresses of normal beam theory. Differential longitudinal movements are not required and no shear lag occurs. Figure 20.47 shows a substitute structure which enables shear lag effects to be found from standard formulae or relatively simple calculation. It is also useful for evaluating more precise results. The area AF is the area of the edge member plus one-third of the web area between the flange and neutral axis. The area AL is equal to half the area of the plate plus longitudinal stiffeners and bs is given by Kuhn18 as: b& = [0.55 + (0.45//I2)]

the interface between the different flanges. In steel it is usual to taper the thicker plate to reduce the effects of fatigue and the possibility of brittle fracture. In some cases only part of the flange is thickened in areas of concentrated load such as forces from supporting cables or prestressing cables. The effect of such thickening is to tend to concentrate all flange forces in that part of the flange which must be allowed for by either gradually tapering out the increased area or by carrying the greater thickness through to a more lowly stressed region. A premature end to the reinforcement may overload the connecting unreinforced section.

Figure 20.47 Transformation of actual into substitute beam cross-section. (After Kuhn (1956) Stresses in aircraft and shell structures. McGraw-Hill)

(20.23)

where n is the number of stringers in one-half of the flange. The actual distance of the centroid of the half-flange to the web is bc whereas bs is the distance which, together with the shear stiffness of the actual plate per unit width, simulates the shear lag property of the flange relative to the webs. The above calculation gives the increase in stress at the web flange junction, Aa^ above normal beam theory. Figure 20.48 shows how the stress across the flange may be found assuming a cubic law variation. The effects of shear lag are usually more important in steel box girders as the webs are more widely spaced than in concrete structures. Also, the thick diaphragms at bearings in concrete structures reduce the longitudinal stresses. 20.4.2 Changes in thickness and cut-outs The change in thickness of a whole flange causes changes in the shear stresses between web and flange as well as local effects at

Figure 20.48 (After Kuhn (1956) Stresses in aircraft and shell structures. McGraw-Hill)

The reinforcement required for cut-outs must be continued or tapered for similar reasons but the need to do so is more obvious. The transverse and shear stresses associated with cutouts are, however, also of considerable importance. In steel structures they are likely to cause buckling, fatigue and brittle fracture problems, whereas in concrete structures they cause cracking. It is, therefore, necessary to connect diaphragms, etc. to the structure by much more than nominal amounts of reinforcing steel. It is instructive to note with reference to steel structures that a number of large tankers have experienced local failures at cut-outs owing to the effects described above. It seems likely that the extrapolation of design knowledge from smaller structures was not backed-up with sufficient research into the complex stress systems produced and the associated buckling phenomenon. Similarly, the causes of failure of several steel box girder bridges have mainly been due to stress concentrations due to cut-outs in stiffeners, support reactions, and a cut-out produced by partly unbolting a main compression flange splice. An earlier failure of a plate girder bridge due to the stress concentrations produced by a flange cover plate completes an unanswerable case for the importance of allowing for stress concentrations in structural design. The basic engineering solution to this problem is to avoid severe stress concentrations and all those mentioned could have been avoided without significant cost or difficulty. Some degree 6f stress concentration is, however, unavoidable and only by using test data can the distinction be drawn between the acceptable and unsafe forms of structures, structural details, and associated stress levels. The stress levels themselves due to known loads can be found with considerable accuracy using, for example, two- and threedimensional finite element methods. Kuhn18 describes ingenious methods for the approximate analysis of shell structures with cut-outs as well as the shear lag approach described above. They give a valuable insight into the structural behaviour of such systems but are more expensive to use than computer-based techniques using finite elements.

(1) Local flexure due to the transfer of wheel loads to the adjacent beam members. (2) Flexure due to relative movements of various parts of structure. (3) In-plane stresses due to beam action of main and secondary members of structure. Local stresses can be found by assuming that all supporting members are rigid when evaluating the slab moments and shears due to wheel loading. The remaining effects can then be found by applying loads equal to the reactions of the supporting members to those members. It is important that the latter loads should be statically equivalent to the vehicle loading, but the exact span wise distribution is not usually required. There are several publications giving influence surfaces for local slab bending moments for several types of support, i.e. simply supported on four sides, cantilever slabs, fixed on four sides.19-20 A well-known treatment is by Westergaard21 which uses Nadai's equations to obtain the bending moment under a wheel load:

Mx \

•/ f M 1. J

=

(1 +p)P Fln1 /4s A

4n

\_

nv \

1 "I _,_ (1 +p)P

\ncl COS—s J} + =2 J

±

o

-„ Sn ,-(20.24)

The meaning of the symbols is given in Figure 20.49 except for c, which is the equivalent diameter of the loaded area. According to Westergaard, the equivalent diameter C1 is expressed with satisfactory approximation by the following formula, applicable when c