Engineering Structures 23 (2001) 779–789 www.elsevier.com/locate/engstruct

Behaviour of unpropped composite girders curved in plan under construction loading M.A. Bradford *, B. Uy, Yong-Lin Pi School of Civil and Environmental Engineering, University of New South Wales, Sydney, N.S.W. 2052, Australia Received 25 October 1999; received in revised form 10 August 2000; accepted 17 August 2000

Abstract Composite steel and concrete curved bridges are often used in highways, particularly for motorway interchanges where high speeds require smooth changes in direction. A composite steel and concrete curved bridge often consists of horizontally curved steel girders and concrete deck slabs. The use of steel I-section curved girders is commonplace because of their economy and ease of construction. During construction, the curved steel girder carries a uniformly distributed load, comprising its self-weight and the wet concrete deck. At this stage, each girder acts separately as an individual simply supported curved beam, with much of its load acting at the top flange level. It is often considered that linear analysis may be used to predict the structural behaviour of composite steel and concrete I-section girders curved in plan during construction. However, when an I-section girder curved in plan is subjected to a vertical uniformly distributed load, it experiences primary bending and non-uniform torsion actions. Because of this, the vertical deflections perpendicular to the plane of the girder are coupled with the twist rotations of the cross-section. The primary bending and torsion actions, vertical deflections and twist rotations couple together to produce second-order bending actions in and out of the plane of the curved girder. The interactions between these actions can grow rapidly, and produce early nonlinear behaviour and even early yielding. Hence, the predictions of a linear theory may be very misleading for the behaviour of composite steel and concrete I-section girders curved in plan during construction. This paper uses an efficient nonlinear inelastic curved beam finite element model developed by the authors to investigate the structural behaviour of composite steel and concrete I-section girders curved in plan during construction. It is found that a nonlinear analysis is needed to predict the structural behaviour of the girders under construction loading. The load carrying capacity of the steel curved girders during construction has to be ascertained and precautions may have to be taken to prevent the failure during construction of the composite curved girders.  2001 Elsevier Science Ltd. All rights reserved. Keywords: Behaviour; Composite; Concrete; Construction; Curved in plan; I-section; Nonlinear; Steel

1. Introduction Composite steel and concrete curved bridges are often used in highways, particularly for motorway interchanges where high speeds require smooth changes in direction. A composite curved bridge often consists of steel girders curved in plan and a concrete slab deck (Fig. 1(a)). Unpropped steel I-sections are used predominantly in composite curved girders because of their economy and ease of construction. During construction, the curved steel girder supports a uniformly distributed load,

* Corresponding author. Tel.: +61-2-93855014; fax: +61-293856139. E-mail address: [email protected] (M.A. Bradford).

provided by its self-weight and that of the concrete deck. Immediately after the concrete deck is poured, it has no stiffness, so that the girders act separately as individual simply supported curved girders when unpropped (Fig. 1(b)), subjected to a load uniformly distributed along the axis of the steel I-section girder, and with much of this load acting at the top flange of the steel I-section curved girder. It is usually considered that a linear analysis can be used to predict the behaviour of unpropped composite steel and concrete girders curved in plan during construction. However, the vertical loads induce primary vertical bending and torsion actions in the curved girder, which produce primary vertical displacements and twist rotations of the cross-sections. Because of this, twist rotation deformations may be coupled with vertical

0141-0296/01/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 1 - 0 2 9 6 ( 0 0 ) 0 0 0 9 7 - 3

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Fig. 1. Composite steel and concrete girder curved in plan. (a) Composite steel and girder curved in plan, and (b) loading during construction.

deformations. The primary bending and torsion actions, vertical deflections and twist rotations couple together to produce second-order bending actions in and out of the plane of the curved girder. The interactions between these actions with increasing load can grow rapidly, and cause early nonlinear behaviour and even early yielding. Hence, the predictions of a linear theory may be very misleading for the structural behaviour of composite steel and concrete girders curved in plan. Efforts have been made to predict the nonlinear behaviour of I-section girders curved in plan by calculating the ‘flexural–torsional buckling load’ of a curved girder. The classical methods of predicting elastic buckling loads consider the bifurcation from a primary prebuckling equilibrium path to an adjacent secondary buckling path. In the case of a curved girder, the torsional deformations are primary, and it is difficult to see the significance of bifurcation-type of flexural–torsional buckling. Although theoretical buckling loads may be calculated for horizontally curved girders, they represent a hypothetical condition of neutral equilibrium when in fact the equilibrium condition is usually unstable. A nonlinear analysis is therefore needed to investigate the flexural– torsional behaviour of I-section girders curved in-plan. Hitherto, most of the research in this problem has been confined to linear elastic behaviour, as identified by the Structural Stability Research Council—Task Group 14 [1]. Research on the grey area of the nonlinear inelastic behaviour of horizontally curved beams and girders is much needed. Tan et al. [2] used the ABAQUS [3] software package to analyse curved steel beams. However, advanced commercial finite element software pack-

ages are computationally inefficient in their analysis of beams of open thin-walled cross-section curved in plan, and moreover are unable to model accurately the significant effects of residual stresses in a more than cursory fashion. For example, the integration scheme of ABAQUS beam element in space for the analysis of open thin-walled sections is not consistent with the distribution of shear strains due to uniform torsion, as pointed out by Pi and Bradford [4], so it may fail to predict correctly the three-dimensional (3-D) nonlinear inelastic behaviour of I-beams curved in plan. Pi and Trahair [5] studied the nonlinear elastic behaviour of Isection beams curved in plan. Pi et al. [6] developed an efficient and accurate 3-D finite curved line element model for the nonlinear inelastic analysis of I-section beams and girders curved in plan, and this was validated against test results. The aim of this paper is to use the 3-D finite curved beam element model [6] to investigate the structural behaviour of unpropped composite I-section girders curved in plan during construction, so at that stage, composite action [7,8] between the steel and concrete has not been achieved. The effects of various factors on the load carrying capacity of the steel I-section girder curved in plan during construction are demonstrated.

2. Nonlinear finite element model A nonlinear inelastic finite element model for steel girders curved in plan developed by Pi et al. [6] is used in the investigation, and only the pertinent points are outlined here. The formulation of the finite element model is based on the following assumptions and considerations. 1. Use of the Euler–Bernoulli theory of bending and of Vlasov’s theory of torsion. 2. Nonlinear strain–displacement relationships that allow for large displacements, twists and rotations. 3. Inclusion of the linear and nonlinear geometric effects of the load position on the tangent stiffness relationships. 4. Use of von Mises criterion as yield surface and use of a consistent scheme of integration sampling points over the cross-section as shown in Fig. 2(a). The integration scheme divides the cross-section into 24 tri-

Fig. 2. Integration scheme. (a) Integration scheme; (b) sheer stress distribution, and (c) ABAQUS default integration.

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angles. Each triangle has three, five or seven sampling points distributed over the triangle according to routine Gauss quadrature rules. It is consistent with the distribution of the shear strains due to uniform torsion shown in Fig. 2(b). On the other hand, the sampling points of integration scheme of the ABAQUS beam element in space for the analysis of open thin-walled cross-sections shown in Fig. 2(c) are located in the mid-line of the wall-thickness where the shear strains due to uniform torsion are equal to zero. Therefore, the integration scheme for the ABAQUS beam element used in open cross-sections with thin-walls reduces the von Mises yield criterion in a 2-D hyperspace to a uniaxial yield criterion as pointed out by Pi and Bradford [3]. 5. Use of the elastic–plastic incremental stress–strain relationship derived from the von Mises yield criterion, with the associated flow rule and the isotropic strain hardening rule. 6. Inclusion of different distributions of longitudinal normal residual stresses, described subsequently.

thickness, and tw is the web wall thickness as shown in Fig. 3(a). The elastic–fully plastic strain hardening stress–strain relationship shown in Fig. 3(b) was adopted, with Young’s modulus of elasticity E=206 000 MPa, elastic Poisson’s ratio n=0.3, yield stress sy0=314 MPa, and strain hardening modulus Es=6000 MPa. Strain hardening commences at a strain ⑀s=0.01375. The curved flange plates were cut out from 8-mm-thick plate, and the flat web was bent plastically and welded into position to give a specified curvature and to form the curved beam. The residual stresses shown in Fig. 3(c) were used in the computational model. They satisfy the moment equilibrium conditions

From the principle of virtual work, the nonlinear incremental-iterative equilibrium equations for the total Lagrangian analysis can be obtained as [6]

A

(kT)i⌬rji ⫽(⌬pe)ji ⫹(⌬pr)ij−1

(1)

where i denotes the load step, j denotes the iterative cycle, (kT)i is the tangent stiffness matrix obtained from the last converged load step (i⫺1) and used at the current load step i, ⌬rji is the increment of the nodal displacement vector r in the current iteration j, (⌬pe)ji is the increment of the external load vector pe at the current is the out-of-balance force vector iteration j, and (⌬pr)j−1 i of the previous iteration (j⫺1). A standard modified Newton–Raphson method is used in conjunction with an arc-length control iterative strategy [9] and an automatic incrementation procedure to solve the nonlinear Eq. (1). Full details of the model and of the method of solution of the resulting nonlinear incremental-iterative equilibrium Eq. (1) are given in Pi et al. [6] and Pi and Trahair [10].

3. Verification of numerical model Two examples are used here to demonstrate the accuracy of the finite element model against independent test results. The first example is the six horizontally curved I-beams tested by Fukumoto and Nishida [11], whose dimensions are given in Table 1, where S is the developed length of the curved beam, ⌰ is the included angle of the curved beam, R is the radius of the curved beam as shown in Fig. 1(a), D is the overall depth of the I-section, B is the flange width, tf is the flange wall

冕

xsr dA⫽0

(2a)

A

and

冕

ysr dA⫽0

(2b)

and the axial force equilibrium condition

冕

sr dA⫽0

(3)

A

was used to determine the mid-web residual stress srcw. Each curved beam was simply supported at both ends with lateral displacement and twist rotation prevented ( u=v=w=f=0, My=B=0) and subjected to a vertical concentrated load at the centre of the top flange of its midspan through a sole plate of thickness 25 mm. The variations of the absolute value of the central twist rotation |fc| with the load Q are shown in Fig. 4 for the AR group of curved beams, and in Fig. 5 for the BR group of curved beams. Also shown in Figs. 4 and 5 are the test results of Fukumoto and Nishida [11] and those of the ABAQUS [3] shell element analysis of Tan et al. [2]. To obtain convergent results, 140 ABAQUS shell elements were used for each beam [2] while eight curved beam elements were used for each beam in the present model. The load–central rotation behaviour of the present finite element model is similar to the test results of Fukumoto and Nishida [11] and the results of ABAQUS shell elements. The ultimate loads obtained from the tests, the present model, and the ABAQUS [3] shell elements of Tan et al. [2] are also summarised in Table 2. It can be seen that the agreement of the results of the present finite element model with the test results is very good. The average accuracy of the present model for the ultimate load predictions is better than that of the ABAQUS shell elements of Ref. 2. The second example considered is the two groups of curved beams that were tested by Shanmugam et al. [12].

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Table 1 Dimensions of curved beams tested by Fukumoto and Nishida [11] Beam (1)

D (mm) (2)

B (mm) (3)

tf (mm) (4)

tw (mm) (5)

S/8R (6)

S (m) (7)

⌰ (8)

AR-1 AR-2 AR-3 BR-1 BR-2 BR-3

251.6 251.9 251.9 250.1 251.8 250.4

101.6 101.0 100.9 100.6 100.6 100.9

8.4 8.3 8.3 8.4 8.3 8.3

5.6 5.7 5.8 5.5 5.7 5.6

1/109 1/238 1/798 1/97 1/206 1/1379

1.7 1.7 1.7 2.8 2.8 2.8

4.025° 1.926° 0.574° 4.725° 2.225° 0.332°

Fig. 5. Fig. 3. Residual stresses of Fukumoto’s I-beams curved in plan. (a) Cross section; (b) stress-strain curve, (c) residual stresses.

Fig. 4.

Fukumoto’s AR group of I-beams curved in plan.

The CB group of curved beams was made of a hot-rolled I-section UB305×127×42 kg/m, with the dimensions D=306.6 mm, B=124.3 mm, tf=12.1 mm, and tw=8.0 mm. The hot-rolled beam was then cold bent to the required horizontal curvature using a hydraulic gap

Fukumoto’s BR group of I-beams curved in plan.

press. The WB group of curved beams was fabricated from flat mild steel plates, with each component being cut separately from steel plate. The straight I-section beam was then formed by fillet welding the web to the flanges, and the welded assemblage was then cold bent to the required curved profile. The dimensions of the cross-section of the WB group are: D=306.0 mm, B=124.0 mm, tf=12.0 mm, and tw=8.0 mm. A vertical concentrated load was applied at the profile length S1=3.8 m along the centroidal axis from one end. The lateral displacement of the loading point was prevented (uq=0), and both ends of all the beams were simply supported in plane (v=0). The end lateral restraints for each beam are shown in Table 3 where ‘F’ denotes that the end was fixed laterally (u=u⬘=f=f⬘=0) and ‘SS’ denotes that the end was simply supported laterally ( u=f=0). The developed length, the radius, and the included angle of the curved beams are shown in Table 3. Also shown in Table 3 are the average Young’s modulus E and the average first yield stress sy0 of the beams. The residual stresses recommended by the ECCS [13] that were adopted by Shanmugam et al. [12] were used for CB and WB groups of curved beams, respectively. The present finite element model was used to analyse the nonlinear behaviour of these curved beams, using eight curved elements for each beam. The boundary con-

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Table 2 Comparison with test results of Fukumoto and Nishida [11] Beam

Q (kN)

(1)

Test (2)

Present model (3)

AR-1 AR-2 AR-3 BR-1 BR-2 BR-3

117.70 123.19 146.02 43.71 54.10 62.62

116.09 128.07 153.77 44.13 52.00 63.98

(3)/(2)

(4)/(2)

ABAQUS (4)

(5)

(6)

117.70 135.03 156.54 47.34 54.19 65.76

0.986 1.040 1.053 1.010 0.961 1.022

1.000 1.096 1.072 1.083 1.002 1.050

1.012

1.051

Average

Table 3 Dimensions of curved beams tested by Shanmugam et al. [12] Beam (1)

S (m) (2)

R (m) (3)

R/S (4)

⌰ (5)

E (MPa) (6)

sy (MPa) (7)

CB1 CB2 CB3 CB4 CB5 WB1 WB3 WB5

5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

20.0 30.0 50.0 75.0 150.0 20.0 50.0 150.0

4.0 6.0 10.0 15.0 30.0 4.0 10.0 30.0

14.3° 9.5° 5.7° 3.8° 1.9° 14.3° 5.7° 1.9°

213 650 208 3500 216 200 206 700 210 000 208 300 211 650 210 000

366 338.5 348.5 332.5 350.0 362.6 347.1 340.0

ditions are the same as those used by Shanmugam et al. [12] in their ABAQUS shell element model. The ultimate loads obtained from the present model are compared with the test results and the ABAQUS [3] shell element results of Shanmugam et al. [12] in Table 4. Implementation of the current numerical scheme on a contemporary personal computer was very rapid. The variations of the vertical displacement vq of the loading point with the vertical load Q for curved beams

End restraints (8) F/F F/F SS/F SS/F SS/F F/F SS/F SS/F

CB5 and WB1 are shown in Fig. 6. Also shown in this figure are the test results and the ABAQUS shell element results of Shanmugam et al. [12]. It can be seen from Table 4 and Fig. 6 that the results of the present model agree very well with both the test and the ABAQUS shell element results. The average accuracy of the present model for the predictions of the ultimate load is slightly better than that of the ABAQUS shell elements. The load–deflection behaviour of the present finite

Table 4 Comparison with test results of Shanmugam et al. [12] Beam

Q (kN)

(1)

Test (2)

Present model (3)

CB1 CB2 CB3 CB4 CB5 WB1 WB3 WB5

191.50 204.50 207.30 212.20 241.90 189.70 200.3 221.70

184.33 194.56 199.51 203.18 225.87 175.26 189.17 212.85

Average

(3)/(2)

(4)/(2)

ABAQUS (4)

(5)

(6)

176.20 185.00 192.00 202.00 215.009 175.00 182.00 210.00

0.963 0.951 0.962 0.957 0.934 0.924 0.944 0.960

0.920 0.905 0.926 0.952 0.889 0.923 0.909 0.947

0.949

0.921

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the simplified residual stresses of Fig. 3(c) do not satisfy the equilibrium condition [14]

冕

sr(x2⫹y2) dA⫽Wr⫽0

(4)

A

Fig. 6.

Shanmagam’s I-beams curved in plan.

element model is similar to the ABAQUS result, but somewhat different from the test results of Shanmugam et al. [12]. This may be because the actual boundary conditions of the tested curved beams were different from those used in the ABAQUS shell element model [12]. The bottom flanges of the ends of the tested curved beams were bolted at the centre of a lateral beam of UB254×146×31 kg/m I-section with length 1500 mm by four M20 bolts. Both ends of the lateral beam were in turn bolted on other supports by two M25 long bolts. The ends of the curved beams together with the lateral beams may displace vertically when the curved beams are subjected to vertical loading. To obtain correctly the vertical displacements of the curved beams, the central vertical displacements of the lateral beams should be excluded from the measured vertical displacement of the curved beams. However, it is not known if the central vertical displacements of the lateral beams were measured [12].

4. Nonlinear behaviour of an unpropped girder 4.1. General In this investigation, North American W36×245 and W10×25 hot-rolled I-sections have been used to demonstrate the nonlinear behaviour. The dimensions of the W36×245 are: D=881.71 mm, B=419.4 mm, tf=34.29 mm, and tw=20.37 mm, and the dimensions of the W10×25 are: D=256 mm, B=146 mm, tf=10.9 mm, and tw=6.4 mm (Fig. 3(a)). The residual stresses consist of two parts, viz. residual stresses induced by manufacturing and by curving. The manufacturing residual stresses shown in Fig. 3(c), which are simplified from those proposed by Bradford and Trahair [14] for hot-rolled beams, were used in the investigation. It should be noted that

and so the Wagner stress resultant Wr will contribute a uniform torque component Wrf⬘ when the curved girder has a twist per unit length f⬘. The residual stress distribution pattern induced in the curving process is related to the method by which the girder is fabricated. There are two methods that can be used to fabricate steel I-section curved girders. The first method is to flame-cut the flanges to the desired curvature from parent steel plate, and fit and weld them to the curved web plate to form the curved girder. The second method is to hot-curve a fabricated straight I-section girder or a hot-rolled I-beam into the required curvature, whilst the third method is to cold-curve a fabricated straight I-section girder or a hot-rolled I-beam into the required curvature. In the cold-curving process, a straight I-section member is bent elastic-plastically to obtain an overbent curved beam, and then released so that the overbent curved beam relaxes elastically into its final shape defined by the permanent set. The depth of the plastic zone after overbending is related to the magnitude of the included angle ⌰, as are the final curving residual stresses. For convenience, the curving residual stresses scrc and scrt at the flange tips are assumed to be given by scrt scrc

冎

冉 冊

⫽⫾sy0

Zpy ⫺1 Zey

(5)

where Zpy and Zey are plastic and elastic moduli of the cross-section about the minor axis, respectively. The cold-curving residual stresses are shown in Fig. 7(a). There appears to be very little information on the residual stresses swrc and swrt developed in flame-cut and welded I-section curved girders. Because the fabrication process for a flame cut and welded curved girder is similar to that for a straight girder formed from welding flame-cut plates, the residual stress pattern for welded Isection straight girders such as suggested by the ECCS

Fig. 7. Residual stresses due to fabrication. (a) Cold-curved girder; (b) flame cut and welded curved girder, (c) hot-curved girder.

M.A. Bradford et al. / Engineering Structures 23 (2001) 779–789

[13] (Fig. 7(b)) can be used for welded I-section curved girders. The magnitude and distribution of the final residual stresses shrc and shrt in a curved steel girder fabricated by hot-curving an I-section straight girder are a function of the dimensions and material properties of the straight girder, the included angle of the curved girder and the hot-curving procedure. Fig. 7(c) shows a typical distribution of the residual stresses suggested by Culver and Nasir [15]. The tensile residual stresses located at the flange tips have a favourable effect on the load carrying capacity of the curved girder. The hot-curving method is the most economical and suitable procedure for curving a large straight girder used typically in a composite bridge (such as the W36×245 used in this paper), so that this method is assumed to be used to fabricate the I-section girder curved in plan that is analysed in the following investigation. Because the residual stresses due to the hot curving procedure have favourable effects on the load carrying capacity of the curved girder, only the unfavourable effects of the manufacturing residual stresses shown in Fig. 3(c) are considered in the following investigation. The tri-linear elastic–plastic strain hardening stress– strain relationship shown in Fig. 3(b) is used, with E=200 000 MPa, G=80 000 MPa, sy0=250 MPa and ⑀y0=0.00125. After the onset of yielding, the stress–strain relationship is assumed to be fully plastic until strain hardening starts at a strain es=0.01375, which is 11 times the initial yield strain and which is often assumed for mild steel [14]. The constant modulus assumed after strain hardening is Es=6000 MPa. During construction, the freshly poured concrete has no stiffness, so that the unpropped supporting steel curved girders act separately as individual simply supported curved girders (u=v=w=f=0, My=B=0), and each curved I-girder carries a load qy uniformly distributed along the girder axis produced by the combination of its own weight at the centroid and the weight of the wet concrete deck acting at the top flange level (Fig. 1(b)). Under the uniformly distributed load, the primary bending moment and torque vary along the curved girder axis. The maximum primary moment Mx occurs at the mid-span, and is given by Mx⫽

冉

qyS 2 ⌰ ⌰ sin ⫺sin 2⌰ 2 4

冊

and

冉

785

冊

⌰ ⌰ qyS 2 Ms⫽ lim cos ⫺cos ⫽0 4 2 ⌰→0° 2⌰

(9)

are satisfied. 4.2. Effects of included angle Curved W36×245 girders with a developed length S=10 m (group A) and curved W10×25 beams with a developed length S=4.4 m (group B) have been used to investigate the effects of the included angle ⌰ on the inelastic behaviour of composite steel and concrete girders curved in plan before the composite action has developed. The initial curvature of group A of curved girders increases from ␬0=0.001745 m⫺1 for ⌰=1° to ␬0=0.15708 for ⌰=90°, while the initial curvature of group B of curved girders increases from ␬0=0.003967 m⫺1 for ⌰=1° to ␬0=0.356999 for ⌰=90°. The variations of the dimensionless central lateral displacements uc/B, the central twist rotations fc, and the dimensionless central vertical displacements vc/D of these curved girders with the dimensionless central bending moments Mx/Mp are shown in Figs. 8–10 for group A and in Figs. 11–13 for group B, where uc is the central lateral displacement, vc is the central vertical displacement, Mx is the primary central moment given by Eq. (6) and Mp is the full plastic moment of the crosssection given by [16] 1 Mp⫽sy0[Btf(D⫺tf)⫹ tw(D⫺2tf)2] 4

(10)

A curved girder with a small included angle ⌰=1° is closely equivalent to a straight girder with initial crook-

(6)

while the corresponding central torque Ms is given by

冉

qyS 2 ⌰ ⌰ cos ⫺cos Ms⫽ 2⌰ 4 2

冊

(7)

Clearly, for a straight girder, ⌰=0°, so that the familiar expressions

冉

冊

qyS 2 ⌰ ⌰ qyS 2 sin ⫺sin ⫽ Mx⫽ lim 2 4 8 ⌰→0° 2⌰

(8)

Fig. 8. Variations of central lateral displacements with central moments.

786

Fig. 9.

M.A. Bradford et al. / Engineering Structures 23 (2001) 779–789

Variations of central twist rotations with central moments.

Fig. 11. Variations of central lateral displacements with central moments.

Fig. 10. Variations of central vertical displacements with central moments.

Fig. 12. Variations of central twist rotations with central moments.

edness uo=R(1⫺cos 0.5°)⬇S/458, and its inelastic flexural–torsional behaviour is near to the inelastic buckling and postbuckling behaviour of a straight girder because the primary coupling between the vertical deformation and twist rotations is small. The lateral displacements uc and twist rotations fc are small until the inelastic flexural–torsional buckling load is reached, after which the lateral displacement uc and twist rotations fc increase rapidly while the central moment Mx decreases. For the curved girders with an included angle in the range 1°⬍⌰⬍20°, the lateral displacements uc (Figs. 8 and 11) and twist rotation fc (Figs. 9 and 12) become substantial before the member moment capacity is reached. After the moment capacity is attained, the lateral displacements uc and twist rotation fc increase rapidly while the moment capacity of these curved girders decreases slightly. For the curved girders with an included angle

⌰ⱖ20°, however, torsion becomes increasingly important while the significance of the bending action decreases with the increase of the included angle ⌰, so that the vertical displacements vc (Figs. 10 and 13) and the twist rotation fc (Figs. 9 and 12) are large even at small values of the central moment Mx, and no maximum moments are observed. When the central moments reach certain critical values, the lateral displacements uc, twist rotation fc, and the vertical displacements vc increase rapidly while the central moments increase slightly. These values of central moment can be considered sensibly as the member moment capacities of the curved girders under construction loading. As the central moments Mx increase further, the displacements uc and vc, and twist rotation fc stiffen because of the increase of the non-uniform torsional stiffness associated with large twist rotations and Wagner effects.

M.A. Bradford et al. / Engineering Structures 23 (2001) 779–789

Fig. 13. Variations of central vertical displacements with central moments.

The effects of the included angle on the moment capacities of curved girders are significant. For an I-section curved girder with a constant developed length, the moment capacity decreases rapidly as the included angle increases (Figs. 8–13).

Three groups of curved W36×245 girders with an included angle ⌰=1°, 5° or 10° and a group of straight W36×245 girders without initial crookedness and twist, and three groups of curved W10×25 beams with an included angle ⌰=1°, 5° or 10° and a group of straight W10×25 beams without initial crookedness and twist were used to investigate the effects of the modified slenderness on the load-carrying capacity of the composite curved girders during construction. The variations of the dimensionless central moment Mx/Mp with the modified slenderness l of a straight girder of the same length are shown in Fig. 14 where the modified slenderness l is defined by

冪a M Mp m

(11)

o

with the moment modification factor am=1.13 for a uniformly distributed load and Mo is the elastic flexural– torsional buckling moment of a straight girder in uniform bending given by Mo⫽

冪

冉

冊

EIwp2 EIyp2 GJ+ S2 S2

Effects of modified slenderness on strengths.

Also shown in Fig. 14 is the dimensionless nominal member moment capacity Mb/Ms of the Australian Standard AS4100 [17] for a uniformly distributed load acting at the top flange where Mb⫽amasMs

(13)

where as is the slenderness reduction factor given by

4.3. Effects of slenderness

l⫽

Fig. 14.

787

(12)

where Iy is the second moment of the area of the crosssection about the minor principal axis oy, J is the Saint Venant torsion section constant, and Iw is the warping section constant.

冋冪冉 冊

as⫽0.6

册

Ms Ms 2 +3⫺ Moa Moa

(14)

in which Ms is the nominal section moment capacity (for W36×245 I-section, Ms=Mp), and Moa⫽

Mob am

(15)

in which Mob is the elastic buckling bending moment at the mid-span of a straight girder subjected to a uniformly distributed load at the top flange determined using the results of an elastic flexural–torsional buckling analysis. As the modified slenderness l increases, the moment capacity of the curved girders decreases. When the included angle ⌰=1°, the torsion action in these curved girders is very small and can be ignored, so the curved girders can be assumed to be subjected to bending action only. The member moment capacities of these curved girders can be predicted using the nominal member moment capacities Mb for straight girders given in AS4100 [17]. However, girders with ⌰=1° are rarely used in composite bridges, and as the included angle increases, the member moment capacity of the curved girders decreases as the significance of the torsion action increases. When the included angle ⌰=5° or 10° (which is more realistic), then the effects of torsion actions on the load-carrying capacities of the curved girders must be considered so that their load carrying capacities can-

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not directly be predicted by the nominal member moment capacities Mb for straight girders given in AS4100 [17]. 4.4. Effects of residual stresses Two groups of curved W36×245 girders and two groups of curved W10×25 beams with an included angle 1° or 10° were used to investigate the effects of the longitudinal normal residual stresses on the moment capacities of composite curved girders during construction. Variations of the dimensionless central moment Mx/Mp with the modified slenderness l of the corresponding straight girders of the same length are shown in Fig. 15. The effects of the longitudinal normal residual stresses on the moment capacities of curved girders decrease as the included angle ⌰ increases from 1° to 10°. For the curved girder with an included angle ⌰=1°, bending is the major action and the longitudinal normal stress distribution is affected by the longitudinal normal residual stresses, as are the moment capacities of these curved girders. The significance of torsion actions increases with the increase of the included angle ⌰ from 1° to 10°. However, the uniform torsion shear stress distribution is not affected by the longitudinal normal residual stresses, so that the moment capacities of the curved girder with an included angle ⌰=10° are less affected by the longitudinal normal residual stresses. 4.5. Effects of lateral braces Three curved girders with a W36×245 section, a developed length S=15 m and an included angle ⌰=10° were used to investigate the effects of the lateral braces on the moment capacities of composite curved girders under construction loading. The girders have no lateral brace, a central lateral brace, or continuous lateral

Fig. 15. Effects of residual stresses on strengths.

braces, respectively. The variations of dimensionless central moment Mx/Mp with central twist rotation fc are shown in Fig. 16. The lateral braces increase the moment capacities of curved girders during construction. Because of this, proper use of lateral braces can prevent composite steel and concrete curved girders from premature failure during construction.

5. Conclusions A composite steel and concrete girder curved in plan most often consists of horizontally curved steel girders and concrete deck slab cast unpropped. Steel I-section curved girders are mostly used because of their economy and convenience in construction. During construction, the unpropped curved steel girder carries a uniformly distributed load, due to both its self-weight and that of the wet concrete deck. Immediately after the concrete deck is poured, the deck has no stiffness, so that the steel girders act separately as individual simply supported members, with much of the load acting above the shear centre. The structural behaviour of composite steel and concrete I-section girders curved in plan under construction loading was investigated in this paper using a nonlinear inelastic curved beam finite element model developed by the authors elsewhere. The following conclusions may be drawn from this study. 1. It was found that the predictions of a linear theory may be very misleading for the structural behaviour of steel I-section girders curved in plan under construction loading because early nonlinear behaviour, and even early yielding, may be induced by the primary coupling between bending and torsion actions, and vertical deflections and twist rotations. A nonlinear analysis is needed to predict correctly the struc-

Fig. 16.

Effects of lateral bracing on inelastic behaviour.

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2.

3.

4.

5.

tural behaviour of composite curved girders during construction. The load-carrying capacity of the composite curved girders during construction has to be checked and some measures may have to be taken in order to prevent the curved girder from failure before composite action is achieved. The effects of the included angle on the inelastic behaviour of composite curved girders during construction are significant. For curved girders with a small unrealistic included angle (⌰ⱕ1°), the bending is the major action and the inelastic behaviour of these curved girders is similar to that of straight I-girders with initial crookedness. Near inelastic flexural–torsional buckling may occur for these beams. The postbuckling displacement response is associated with a load decrease. The nominal member moment capacity for straight girders given in AS4100 [17] can be used to predict the moment capacities of these composite curved girders during construction. The load carrying capacities of composite curved girders during construction decrease rapidly as the included angle increases. For curved beams with an included angle 1°⬍⌰⬍20°, both bending and torsion become important and the significance of bending decreases while the significance of torsion increases with the increase of the included angle. There is no bifurcation type of inelastic flexural–torsional buckling because the primary twist rotations are increasingly significant. However, near inelastic lateral buckling phenomena can be observed. For curved beams with a large included angle (⌰ⱖ20°), torsion is the major action and the nonlinear inelastic behaviour develops very early. Before the load-carrying capacity is reached, the vertical displacements and twist rotations become very large. Both the bending and the torsion actions should be considered in determining the load-carrying capacity of a composite steel and concrete curved girder with an included angle ⌰⬎ 1°. The load carrying capacities of an unpropped composite steel and concrete curved girder during construction decrease as the modified slenderness of the corresponding straight girder of the same length increases. The effects of residual stresses on the loadcarrying capacities of composite curved girders during construction decrease as the included angle increases. Lateral braces increase the load carrying capacities of composite steel and concrete curved girders during

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construction, and may therefore be used as a measure to prevent the steel curved girders from failure during construction.

Acknowledgements This work has been supported by research grants provided by the Australian Research Council under the ARC Large Grants scheme. References [1] SSRC. Horizontally curved girders—a look to the future. Annual Report, Structural Stability Research Council—Task Group 14, Chicago, 1991. [2] Tan LO, Thevendran V, Liew JYR, Shanmugam NE. Analysis and design of I-beams curved in plan. J Singapore Struct Steel Soc Steel Struct 1992;3(1):39–45. [3] ABAQUS user’s manual. Version 5.8. Pawtucket (RI): Hibbitt, Karlsson and Sorensen, 1998. [4] Pi Y-L, Bradford MA. Effects of approximation in analysis of beams of open thin-walled cross-section: 2. 3-D nonlinear behaviour. Int J Numer Methods Eng 2000, in press. [5] Pi Y-L, Trahair NS. Nonlinear elastic behaviour of I-beams curved in plan. J Struct Eng ASCE 1997;123(9):1201–9. [6] Pi Y-L, Bradford MA, Trahair NS. Nonlinear inelastic behaviour and analysis of steel I-beams curved in plan. J Struct Eng ASCE 2000;126(7):772–9. [7] Oehlers DJ, Bradford MA. Composite steel and concrete structural members: fundamental behaviour. Oxford: Pergamon, 1995. [8] Oehlers DJ, Bradford MA. Elementary behaviour of composite steel and concrete structural members. Oxford: Butterworth Heinemann, 1999. [9] Crisfield MA. A fast incremental/iterative solution procedure that handles snap-through. Comput Struct 1981;13(1):55–62. [10] Pi Y-L, Trahair NS. Nonlinear inelastic analysis of steel beamcolumns. I: theory. J Struct Eng ASCE 1994;120(7):2041–61. [11] Fukumoto Y, Nishida S. Ultimate load behavior of curved Ibeams. J Eng Mech Div ASCE 1981;107(EM2):367–85. [12] Shanmugam NE, Thevendran V, Liew JYR, Tan LO. Experimental study on steel beams curved in plan. J Struct Eng ASCE 1995;121(2):249–59. [13] European Recommendation for Steel Construction (ECCS). New York: Construction Press, 1984. [14] Bradford MA, Trahair NS. Inelastic buckling of beam-columns with unequal end moments. J Construct Steel Res 1985;5(2):195–212. [15] Culver CG, Nasir G. Inelastic flange buckling of curved plate girders. J Struct Div ASCE 1971;97(ST4):1239–55. [16] Trahair NS, Bradford MA. The behaviour and design of steel structures to AS4100. 3rd ed. London: E & FN Spon, 1998. [17] AS4100—1998. Steel structures. Sydney: Standards Australia, 1998.