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Finite element analysis of curved steel girders with tubular flanges
Engineering Structures 32 (2010) 319–327
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Finite element analysis of curved steel girders with tubular flanges Jun Dong a,∗ , Richard Sause b a b
T.Y.Lin International, San Francisco, CA 94105, USA ATLSS Center, Department of Civil and Environmental Engineering, Lehigh University, Bethlehem, PA 18015, USA
article
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Article history: Received 11 July 2008 Received in revised form 20 July 2009 Accepted 24 September 2009 Available online 23 October 2009 Keywords: Steel girder Curved bridge Finite element Residual stresses Load capacity
abstract A tubular flange girder is an I-shaped steel girder with either rectangular or round tubes as flanges. A tubular flange girder has a much larger torsional stiffness than a conventional I-shaped plate girder and less potential for cross section distortion than a box-girder, and is, therefore, an interesting alternative girder for horizontally curved steel bridges. Finite element (FE) models of curved tubular flange girders are presented in this paper, considering material inelasticity, second-order effects, initial geometric imperfections, and residual stresses. A parametric study is performed using the FE models to study the effects of stiffeners, tube diaphragms, geometric imperfections, residual stresses, and cross section dimensions on the load capacity of curved tubular flange girders. Finally, the FE results for curved tubular flange girders are compared with results for corresponding curved I-shaped plate girders, and the advantages of tubular flange girders are summarized. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Horizontally curved steel girder highway bridges are often used at locations where the roadway alignment is constrained. The horizontal curvature induces significant torsion in the bridge girder system, which is an important design consideration. Currently, I-shaped plate girders (I-girders) or box-shaped girders (box-girders) are used for curved steel bridges. An I-girder is an ‘‘open’’ section and has very little torsional resistance. During bridge erection, curved I-girders are often supported temporarily at intermediate locations between piers before the girders are attached to permanent cross frames (or diaphragms). After the Igirders are attached to the cross frames, the I-girders and cross frames work together to resist the torsion of a curved bridge. Both the I-girders and the cross frames must be designed as primary load-carrying members. A box-girder is a ‘‘closed’’ section which has a relatively large torsional resistance. However, owing to the large cross section width and depth of box-girders, cross section distortion is a concern. Thus, internal diaphragms and stiffeners are needed to maintain the box shape, which makes the design, fabrication, construction, inspection, and maintenance of box-girders complex and expensive. The use of innovative I-shaped girders with hollow tubular flanges in curved steel bridges is studied in this paper. This new
∗
Corresponding author. E-mail address: [email protected] (J. Dong).
0141-0296/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2009.09.018
girder type combines the advantage of curved I-girders, namely, easier design, fabrication, and inspection, with the advantage of curved box-girders, namely, the large torsional resistance. Previous work by Kim and Sause [1–3] addressed the behavior of straight tubular flange girders with a round concrete-filled tube as the compression flange and a flat plate as the tension flange (CFTFGs), as shown in Fig. 1(a). Kim and Sause [1–3] summarized several advantages of CFTFGs relative to conventional Igirders for bridges: (1) the concrete-filled tubular flange provides more strength, stiffness, and stability than a flat plate flange with same amount of steel, and (2) fewer cross frames are needed for CFTFGs to maintain lateral–torsional stability compared to similar I-girders, which reduces the fabrication and erection effort needed to construct a bridge. Dong and Sause [4] investigated the flexural strength of straight hollow tubular flange girders (HTFGs) with rectangular tubes for both the compression and tension flanges, as shown in Fig. 1(b). The effects of stiffeners, geometric imperfections, residual stresses, cross section dimensions, and bending moment distribution on the lateral–torsional buckling flexural strength of HTFGs were studied and formulas for determining the flexural strength of HTFGs were evaluated. Fan and Sause [5] conducted theoretical analyses of individual curved tubular flange girders and systems of multiple curved tubular-flange girders braced by cross frames. The analysis method of Dabrowski [6] was extended and used for the theoretical analyses. Fan and Sause also developed finite element models of curved tubular flange girders and compared the behavior of curved tubular flange girders with curved I-girders. These analyses were firstorder linear elastic analyses.
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(a) Round CFTFG.
(b) HTFG.
(c) I-Girder.
Fig. 1. Cross section shapes.
Curved I-girders have been the subject of substantial previous research. For example, Pi et al. [7,8] studied the nonlinear behavior of curved I-girders. Under vertical loading, a curved I-girder develops both primary bending action and non-uniform torsion action, and vertical deflections are coupled with out-of-plane cross-section rotations. These primary actions and deformations couple together to produce second-order bending actions about the minor axis. The second-order effects are significant for I-girders with large initial curvatures. Pi and Bradford [9] proposed formulas for the design of curved I-girders against the combined bending and torsion actions. The 2004 AASHTO LRFD Bridge Design Specifications [10] have standard provisions for the design of curved I-girders for US highway bridges. This paper focuses on the second-order nonlinear behavior of individual curved hollow tubular flange girders (CHTFGs) with rectangular tubes for both the compression and tension flanges, as shown in Fig. 1(b). The tubes and webs are sufficiently compact so that local buckling does not control the girder strength. The FE models include material inelasticity, second-order effects, geometric imperfections, and residual stresses. FE models are used to study the effects of transverse web stiffeners, tube diaphragms, geometric imperfections, residual stresses, and cross section dimensions on the load capacity of CHTFGs. Finally, the behavior of individual CHTFGs is compared with the behavior of corresponding curved I-girders, and the advantages of CHTFGs are summarized.
Fig. 2. The coordinate system and boundary condition of CHTFG FE models.
2.1. Coordinate system The geometry of a curved girder is easily described within a cylindrical coordinate system whose origin is located at the center of curvature. The global 1, 2 and 3 axes are defined in the radial (lateral), circumferential (longitudinal) and vertical direction, as shown in Fig. 2. U1, U2, U3, UR1, UR2 and UR3 are the displacements and rotations about the global 1, 2 and 3 axes respectively.
2. Finite element models The Finite Element (FE) program ABAQUS (Version 6.7) was used for the FE analyses reported later in the paper. Experimental data for CHTFGs is not currently available to validate the FE models. However, the approach used to develop the FE models is similar to that used previously by the authors and other researchers to develop reliable nonlinear FE (i.e., nonlinear shell element) models for curved and straight steel bridge girders using ABAQUS. For example, Shanmugam et al. [11] performed tests on curved I-girders with different curvature-to-span ratios, and the deformations and ultimate strength from the tests were are found to be in good agreement with the FE results obtained from ABAQUS shell element models. Linzell et al. [12] obtained data from a fullscale curved steel bridge during erection. Comparisons between this experimental data and FE results obtained from an ABAQUS shell element model showed the FE model provided good predictions of erection behavior. Kim and Sause [1–3] validated nonlinear ABAQUS shell element models for straight tubular flange girders using experimental data. For the present study, the FE models were developed for an individual girder with simply supported boundary conditions, and with a uniformly distributed vertical load over the span. The details of the procedure used to develop the models are given by Dong and Sause [13]. The important aspects of the models are summarized below.
2.2. Boundary and load conditions Simply supported boundary conditions are applied to the end sections, as shown in Fig. 2. At each end section, the vertical displacement (U3) of the nodes of the bottom wall of the bottom tube, the lateral displacements (U1) of all nodes along the line through the web mid-surface, and the twist rotations about 2-axis (UR2) of all nodes on the section are restrained. The longitudinal displacement (U2) of the centroid of the web is restrained at only the left end section, and the rotations about the 3-axis (UR3) are not restrained. Along the length of the FE models (i.e., along the span of the girders), the lateral displacement and twist are unrestrained (i.e., the girders are braced only at the ends). The FE models are loaded with a distributed vertical load that is uniform across the girder cross section and is uniform over the span. This load is similar in distribution to the self weight of the girder. 2.3. Finite element mesh The fully integrated, three-dimensional, four-node shell element S4 is used to model the flanges, webs, and transverse web stiffeners. The S4 element has four integration points across the element and five section points (integration points) through the thickness of the shell element. Dong and Sause [13] conducted a FE
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a
b
c
d
Fig. 3. Components of longitudinal residual stress: (a) membrane (tube); (b) bending (tube); (c) layering (tube); (d) web [7].
mesh study for CHTFGs. The total number of S4 elements in the different meshes for a single CHTFG was varied from 7200 to 243,000. A curved I-shaped girder develops flange lateral bending under vertical loads, and the normal stress in the flanges varies across the width of the flange and along the span. Therefore, a sufficient number of elements are needed across the flange width and also in the longitudinal direction of girder. By comparing displacements, stresses, and ultimate load capacities, it was found that an optimized FE mesh with 40,000 elements produced accurate results, and this mesh was used for all FE models in the present study. 2.4. Material models and properties An elastic–perfectly-plastic material is used for the steel. In the elastic range, Young’s modulus is 200 GPa (29,000 ksi) and Poisson’s ratio is 0.3. In the inelastic range, Von Mises yield criteria is used to define isotropic yielding. The yield strength of the steel is 345 MPa (50 ksi). 2.5. Initial geometric imperfections Dong and Sause [4] showed that the lateral–torsional buckling strength of straight HTFGs depends on the initial geometric imperfections. The effect of initial geometric imperfections on the behavior of CHFTGs is discussed later in the paper. Elastic lateral–torsional buckling shapes, obtained from an elastic buckling analysis, are scaled and added to the perfect geometry of the girders to create an initial geometry with imperfections. Combinations of different buckling shapes and imperfection magnitudes are considered. 2.6. Residual stresses The residual stresses in the FE model vary through the thickness of the shell elements, across the elements, and between different elements on the cross section. The residual stresses are assumed to develop from two stages of the fabrication of the curved hollow tubular flanges: (1) the manufacture (cold forming) of the straight tube, (2) the cold bending of the straight tube
into the desired curved tube. The distribution and magnitude of residual stresses from Stage 1 are based on the measured residual stresses presented by Key and Hancock [14] for a square hollow tube. As shown in Fig. 3, the residual stresses from cold forming of the tube have three components: (a) the membrane residual stresses, which are constant through the tube wall; (b) the bending residual stresses; and (c) the ‘‘layering’’ residual stresses. The bending and layering residual stresses have the same pattern around the tube, and vary linearly (or bi-linearly) through the thickness of the tube wall. The influence of the different components of residual stresses from cold forming the tubes on the lateral–torsional buckling flexural strength of straight tubular flange girders was investigated by Dong and Sause [4]. They showed that the bending residual stress component (shown in Fig. 3(b)) has the most significant effect on the flexural strength, because the magnitude of the bending residual stress component is much larger than the magnitude of the other residual stress components. Thus, only the bending residual stress component from tube cold forming is included in Stage 1. A model for the residual stresses from Stage 2, cold bending, was developed from a theoretical analysis of the cold bending process. In this theoretical cold bending analysis, the steel is assumed to be elastic–perfectly-plastic, and the tube initially has the cold forming residual stresses. The theoretical analysis of the cold bending process was validated using nonlinear FE analysis. The following example illustrates the FE analysis method and compares the results with those from the theoretical analysis. In the example, a 508 mm by 305 mm tube with a thickness of 12.7 mm and a length of 27 m is bent to a final radius of curvature of 60 m. The tube is bent about its strong axis. The analysis is conducted in three steps: (1) the bending residual stress component from cold forming the tube (Stage 1) is included in the model; (2) the tube is loaded under uniform curvature until the target lateral displacement at midspan U1M -max is achieved; (3) the loading is reversed until the bending moment equals zero. For this example, U1M -max equals 2.259 m and the residual lateral displacement U1M equals 1.537 m, which results in the final radius of 60 m for the curvature. U1M max is determined from the final desired curvature, considering
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Table 1 Dimensions of curved tubular flange girder and I-girder. Girders
Flanges (mm)
Web (mm)
Depth (mm)
Area (mm2 )
Ix (mm4 )
Iyf (mm4 )
J (mm4 )
Iw (mm6 )
TG1 TG2 IG1 IG2
508 × 101.6 × 12.7 508 × 304.8 × 12.7 508 × 25.4 508 × 27.9
1066.8 × 12.7 660.4 × 7.9 1257.3 × 14.9 1254.8 × 14.5
1282.7 1282.7 1282.7 1282.7
44,516 46,553 44,516 46,553
1.20E+10 1.05E+10 1.31E+10 1.40E+10
4.45E+08 7.78E+08 2.77E+08 3.05E+08
2.23E+08 1.50E+09 6.94E+06 8.62E+06
3.04E+14 3.62E+14 2.28E+14 2.51E+14
Fig. 4. Distribution of residual stresses at end of step 1.
Fig. 6. Distribution of residual stresses at end of step 3.
Fig. 7. Normalized load capacity versus vertical displacement at mid-span. Fig. 5. Distribution of residual stresses at end of step 2.
the elastic recovery in step 3. The distributions of residual stresses in different layers of the top (and bottom) wall of the tube at the end of each step are shown in Fig. 4 through Fig. 6. As noted above, the residual stresses at the end of step 1 (Fig. 4) include only the bending residual stresses from cold forming the tube. The residual stresses at the end of step 2 and step 3 from the theoretical analysis are compared with results from the FE simulation in Figs. 5 and 6. It is observed that the differences between the FE simulation results and the theoretical results are very small, and the residual stresses from the theoretical analysis were used as the input to the FE models for the remaining studies. The residual stresses in the web are constant through the web thickness and vary bi-linearly over the web height, as shown in Fig. 3(d). 2.7. Analysis method Nonlinear load–displacement analyses with material inelasticity and second-order effects were used to study the behavior of CHTFGs and curved I-girders. The modified Riks method available in ABAQUS was used for these analyses.
3. Distortion of the cross section Dong and Sause [4] showed that distortion of the cross section affects the flexural strength of HTFGs. Thus, the effects of cross section distortion on the load capacity of CHTFGs were studied. 3.1. Sources of cross section distortion To study CHTFG cross section distortion, a modified FE model which did not include cross section distortion was developed. In this modified model (denoted as MWOD, model without distortion), the rotations about the longitudinal axis (UR2) of all nodes on each cross section are constrained to be equal. The model without this constraint is denoted as MWD (model with distortion). Nonlinear analyses of a MWOD and a MWD of a single CHTFG with a cross section denoted TG2 were conducted. The dimensions of TG2 are shown in Table 1. The span L (arc length of the girder) is 27 m, the radius of the curvature is 60 m, and the L/R ratio equals 0.45. A uniformly distributed vertical load was applied, and the analyses were taken to levels of displacement beyond the maximum load. The load–displacement curves are presented in Fig. 7, where the load is normalized by the weight of the girder,
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(a) MWOD.
(b) MWD.
Fig. 8. Deformed shape of cross section near mid-span for different models.
(a) Without distortion.
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(b) Web distortion.
(c) Tube distortion.
Fig. 9. Cross section distortion.
Fig. 11. Variation of RMCD with span and Ts for MWD.
stiffener elements are modeled using three-dimensional shell elements S4 and are connected to the cross section by constraining the nodes in the stiffeners to the corresponding nodes in the flanges and web. The stiffeners are on both sides of the web, as shown in Fig. 10. Diaphragms are commonly used to control distortion of boxlike cross sections. To avoid the need to install diaphragms within the tubes and simplify CHTFG fabrication, diaphragms are introduced only at the ends of the tube (at the end sections) to reduce the tube distortion. In the FE models, the tube diaphragms are modeled using three-dimensional shell elements S4 and are connected to the cross section by constraining the nodes in the diaphragms to the corresponding nodes in the tubes. A parametric study was conducted to investigate the use of transverse web stiffeners and tube diaphragms to control cross section distortion of CHTFGs. The stiffener plate thickness, the number of stiffeners, and the diaphragm plate thickness were selected as variables in the study. 3.3. Effect of stiffener thickness (Ts )
Fig. 10. Typical stiffener configuration.
and U3M is the vertical displacement U3 of the centroid of the midspan cross section (positive downward). By comparing the results of the MWOD and MWD, it is seen that cross section distortion reduces substantially the strength of CHTFGs, which indicates the importance of the distortion. Fig. 8 shows the deformed cross sections of the MWOD and MWD at mid-span. The cross section distortion is clearly seen in the MWD results. Since the tubular flanges are torsionally stiff and the web is comparatively flexible, the web tends to deform by bending out-of-plane. The thin-walled tubular flanges also change shape. Therefore, the cross section distortion includes both web distortion and tube distortion, as shown in Fig. 9. 3.2. Controlling cross section distortion Transverse web stiffeners are usually used to reduce web distortion in I-girders, and for a CHTFG, transverse web stiffeners also reduce the tube distortion where the stiffeners are attached to the wall of the tubes. Therefore, stiffeners are introduced into the CHTFG FE models to reduce the cross section distortion. The
Nonlinear load–displacement analyses were performed on the MWD with seven intermediate stiffeners, which are uniformly distributed along the span between the bearing stiffeners. The span was varied from 15 m to 120 m, and the stiffener plate thickness (Ts ) was either 12.7 m, 25.4 m or 50.8 mm. Results for the ratio of the load capacity of the MWD, including the stiffeners, to that of the MWOD, denoted as RMCD , are plotted in Fig. 11. It is observed that the effect of Ts is not very significant and the differences between the cases with Ts = 25.4 mm and Ts = 50.8 mm are small. Therefore, 25.4 mm thick stiffeners are used for the remaining studies. 3.4. Effect of number of stiffeners (Nis ) Nonlinear load–displacement analyses were performed on the MWD with different numbers of web stiffeners. For all cases, bearing stiffeners were included at the supports, and the number of intermediate transverse web stiffeners between the bearing stiffeners (Nis ) was varied. The intermediate stiffeners are uniformly distributed (i.e., the spacing of the stiffeners is constant) and Ts = 25.4 mm. RMCD is plotted in Fig. 12. It is found that as Nis increases, the load capacity of the MWD with stiffeners approaches the load capacity of the MWOD (i.e., RMCD approaches 1), indicating that cross section distortion is not reducing the load capacity. The differences between the results for Nis = 7 and Nis = 9 are small. Therefore, seven intermediate web stiffeners and two bearing stiffeners with Ts = 25.4 mm are used for the remaining studies.
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Fig. 12. Variation of RMCD with span and Nis for MWD.
Fig. 14. Effect of initial geometric imperfections as span varies.
no residual stresses with residual stresses
Fig. 15. Effect of residual stresses as span varies. Fig. 13. Variation of RMCD with span and Tdia for MWD.
3.5. Effect of tube diaphragms at ends Nonlinear load–displacement analyses were performed on the stiffened MWD (Nis = 7, Ts = 25.4 mm) with a tube diaphragm at both end sections. The diaphragm thickness (Tdia ) was either 12.7 mm, 25.4 mm or 50.8 mm. RMCD is shown in Fig. 13 for the different values of Tdia , and for the case without tube diaphragms. It is observed that including tube diaphragms improves the load capacity, but using a thicker diaphragm (with a larger value of Tdia ) has an effect only for very long spans. Therefore, Tdia = 25.4 mm is used for the remaining studies.
the first two elastic buckling mode shapes, where the mode 1 shape is scaled so the maximum top flange lateral deflection is L/1000 and the mode 2 shape is scaled so the maximum top flange lateral deflection is L/2000, and the two shapes are added together. FE results for models without initial geometric imperfections and for models with these two different initial geometric imperfections are shown in Fig. 14. The span is varied and L/R equals 0.45 for all cases. Since a CHTFG has an initial curvature, it is observed that including initial geometric imperfections does not significantly affect the load capacity. Initial geometric imperfections are not included in the FE models for the remaining studies. 4.2. Effect of residual stresses
4. Load capacity of CHTFGs FE models of the CHTFG with cross section TG2 were used to investigate the load capacity of CHTFGs. Non-linear load–displacement analyses were performed, and the load capacity is expressed as the maximum applied load normalized by the girder weight. 4.1. Effect of initial geometric imperfections As discussed earlier, the initial geometric imperfections used in the present study are derived from elastic buckling analysis results and introduced into the FE models for the nonlinear load–displacement analyses. Two combinations of buckling shapes and imperfection magnitudes are considered. For the initial imperfection denoted as EM1, the elastic buckling shape for mode 1 is scaled so that the maximum top flange lateral deflection is L/1000, where L is the span. Initial imperfection COM1 is a combination of
The effect of residual stresses on the load capacity of TG2 is shown in Fig. 15. The span is varied and only results for L/R = 0.1 and L/R = 0.45 are shown. Fig. 15 shows that the effect of residual stresses is small. The ratio of the load capacity of the model with residual stresses to that of the model without residual stresses is denoted as RMRS . Results for RMRS are plotted in Fig. 16, where L/R is varied from 0.1 to 0.45. It is observed that RMRS is slightly less than one for all cases, which means the load capacity of a CHTFG with residual stresses is slightly smaller than the load capacity without residual stresses. Compared to straight tubular flange girders [4], the effects of residual stresses on the load capacity of a CHTFG are small. A comparison of the behavior of CHTFGs without and with residual stresses is as follows. For girder TG2 without residual stresses and with L = 27 m and L/R of 0.45, the distribution of normal stress across the top and bottom walls of the tubes at the mid-span cross section under the girder self-weight is shown in Fig. 17. Due to the flange
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L/R=0.1 L/R=0.2 L/R=0.3 L/R=0.45
Fig. 16. Variation of RMRS with span and L/R. Fig. 19. Distribution of normal stresses under the ultimate load.
a small region of the flange has large residual stress, and the tubes can carry increasing loads after these small regions yield. Figs. 18 and 19 show the distributions of normal stress across the width of the top wall of the top tube and the bottom wall of the bottom tube at the mid-span cross section for the girder with and without residual stresses under the ultimate load. It is observed that the average value of the normal stress of the inner and outer layers for the model with residual stresses is very close to the normal stress of the model without residual stresses, which illustrates why residual stresses have little effect on the load capacity of a CHTFG. 4.3. Comparison with I-girders
Fig. 17. Distribution of normal stresses for the girder without residual stresses.
Fig. 18. Distribution of normal stresses under the ultimate load.
lateral bending moment introduced by the torsion in the girder, the maximum normal stress is located at the flange tips and side walls. As shown in Fig. 6, the residual stresses at the flange tips and side walls are small after cold bending of the tubes. Thus, the residual stresses are small at the location of maximum stress from loading. Large residual stresses are located near the center of the top and bottom walls of the tubes, and the maximum stress is compressive for the inner layer and tensile for the outer layer of the top and bottom walls of the tubes. The effect of these high residual stresses on the primary bending moment capacity is neutralized by different signs of residual stresses in the different layers. Also, only
To enable the behavior of an individual curved tubular flange girder to be compared with the behavior of an individual curved I-girder, curved I-girders similar to (corresponding to) the CHTFGs were developed, shown in Fig. 2(c). Two curved I-girders, denoted IG1 and IG2, have the same girder weight (cross section area), cross section depth, and flange width as the corresponding CHTFGs, TG1 and TG2, respectively, as shown in Table 1. To illustrate the main differences between the CHTFG and I-girder cross-sections, important cross section properties, such as the cross section moment of inertia about the major axis (Ix ), the flange moment of inertia about the cross section minor axis (Iyf ), the St. Venant torsional constant (J), and the warping moment of inertia (Iw ) are also listed in the table. The corresponding curved I-girders have a slightly larger flexural rigidity but a much smaller St. Venant torsional rigidity than the corresponding CHTFGs. For the studies comparing CHTFGs with curved I-girders, the span, L, is constant and equal to 27 m, and the radius of curvature, R, is varied so that L/R varies from 0.1 to 0.45. Figs. 20 through 24 compare the maximum primary bending normal stress, maximum warping (flange lateral bending) normal stress, maximum total normal stress, mid-span vertical displacement (U3M ), and mid-span cross section rotation (UR2M ) for the CHTFGs and the corresponding curved I-girders under the girder self-weight. The primary bending normal stress is derived from the primary bending moment which is determined by integrating the normal stresses on the cross section. The primary bending normal stress is the primary bending moment divided by the section modulus of the cross section about the strong axis of the cross section. The warping normal stress is derived from the lateral bending moment of the top flange, which is determined by integrating the normal stress on the flange. The warping normal stress is the flange lateral bending moment divided by the section modulus of the flange about the cross section minor axis. The distribution of the top flange lateral bending moment along the span of the girders
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Fig. 20. Variation of maximum bending normal stress with L/R.
Fig. 23. Variation of mid-span vertical displacement with L/R.
Fig. 21. Variation of maximum warping normal stress with L/R. Fig. 24. Variation of mid-span cross section rotation with L/R.
Fig. 22. Variation of maximum total normal stress with L/R.
with L/R of 0.45 is shown in Fig. 25. The load capacity (normalized by the girder weight) is compared in Fig. 26. The results indicate that the bending normal stress is dominant and the warping normal stress is not significant for the CHTFGs, however, the warping normal stress is large for the curved Igirders. Since CHTFGs have a large torsional rigidity, most of the torsion is resisted by St. Venant torsion, and the flange lateral bending moment from warping torsion of the CHTFGs is much smaller than that of the curved I-girders, as shown in Fig. 25. The CHTFGs also have a larger flange moment of inertia and section modulus about the cross section minor axis. Therefore, the maximum warping normal stress for the CHTFGs is much smaller than that of the curved I-girders. The curved I-girders have smaller
Fig. 25. Variation of top flange lateral bending moment along the span (L/R = 0.45).
primary bending normal stresses than the corresponding CHTFGs due to their slightly larger flexural rigidity. In summary, the CHTFGs develop much smaller warping normal stresses, total normal stresses, displacements, and cross section rotations, and have much larger load capacities. For example, for TG2 with L/R = 0.45 under girder self-weight, the mid-span vertical displacement, U3M is 0.019 m which is 1/1500 of the span L, and the cross section rotation at mid-span UR2M is 0.005 rad. The load capacity is 17 times the girder weight. However, for the corresponding curved I-girder IG2 under girder self-weight, U3M is 0.394 m which is 1/70 of L, and UR2M is 0.296 rad. The load
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that a curved I-girder is better at resisting primary bending, but develops much larger warping normal and total normal stresses, and much larger vertical displacements and cross section rotations than a CHTFG. Curved I-girders are generally braced by permanent cross frames within the span in the final constructed condition, and they usually require temporary support or bracing within the span during girder erection before the cross frames are fully installed. The results presented in the paper show that individual CHTFGs have much better structural behavior than individual curved Igirders (e.g., only 1/20th of the midspan vertical displacement and only 1/60th of the midspan cross section rotation under self weight). These results suggest that such temporary support or bracing within the span may not be needed for CHTFGs during girder erection, which could lead to faster and more economical curved bridge construction. Fig. 26. Variation of load capacity with L/R.
capacity is 3.9 times the girder weight. In other words, the midspan vertical deflection of the curved I-girder is about 20 times larger than that of the CHTFG and the mid-span cross section rotation of the curved I-girder is about 60 times larger than that of the CHTFG, while the load capacity of the CHTFG is about 4 times larger than that of the curved I-girder. Under the girder self-weight, the displacements and cross section rotations of the curved I-girder IG2 are quite large, which suggests that temporary support within the span would be needed for this curved I-girder during erection, while the displacements and cross section rotations of CHTFG TG2 are quite reasonable, which suggests that such temporary support within the span would not be needed during erection. As usual, the installation of permanent cross frames between the CHTFGs or the curved I-girders would be used to control the final erected geometry of the girders in a curved bridge. 5. Conclusions Finite element (FE) models of individual curved hollow tubular flange girders (CHTFGs) were presented. The models account for material inelasticity, second-order effects, initial geometric imperfections, and residual stresses. With these FE models, the effect of cross section distortion on the load capacity of CHTFG was studied. An arrangement of transverse web stiffeners and tube end diaphragms was identified to reduce the effect of cross section distortion on the load capacity. Nonlinear load–displacement analyses of the FE models were performed to study the load capacity of CHTFGs. A parametric study showed that the influences of initial geometric imperfections and residual stresses on the load capacity of CHTFGs are small. By comparing the behavior of individual CHTFGs with the behavior of corresponding individual curved I-girders, it is found
Acknowledgments This research was conducted at the Advanced Technology for Large Structural Systems (ATLSS) Research Center of Lehigh University, Bethlehem, PA, USA. Research funding provided by the Federal Highway Administration (FHWA) and the Pennsylvania Infrastructure Technology Alliance (PITA) is gratefully acknowledged. References [1] Kim BG, Sause R. High performance steel girders with tubular flanges. ATLSS report 05-15. Bethlehem (PA, USA): ATLSS Engineering Research Center, Lehigh University; 2005. [2] Kim BG, Sause R. High performance steel girders with tubular flanges. Int J Steel Struct 2005;5(3):253–63. [3] Sause R, Kim BG, Wimer MR. Experimental study of tubular flange girders. J Struct Eng 2008;134(3):384–92. [4] Dong J, Sause R. Flexural strength of tubular flange girders. J Constr Steel Res 2009;65(3):622–30. [5] Fan Z, Sause R. Behavior of horizontally curved steel tubular-flange bridge girders. ATLSS report no. 07-02. Bethlehem (PA): Center for Advanced Technology for Large Structural Systems, Lehigh University; 2007. [6] Dabrowski R. Curved thin walled girders—Theory and analysis [Amerongen CV, Trans.]. Cement and Concrete Association; 1968. [7] Pi Y-L, Bradford MA, Trahair NS. Inelastic analysis and behavior of steel I-beams curved in plan. J Struct Eng 2000;126(7):772–9. [8] Pi Y-L, Trahair NS. Nonlinear elastic behavior of I-beams curved in plan. J Struct Eng 1997;123(9):1201–9. [9] Pi Y-L, Bradford MA. Strength design of steel I-section beams curved in plan. J Struct Eng 2001;127(6):639–46. [10] AASHTO LRFD bridge design specifications. Washington (DC): American Association of State Highway and Transportation Officials; 2004. [11] Shanmugam NE, Thevendran V, Liew JYR, Tan LO. Experimental study on steel beams curved in plan. J Struct Eng 1995;121(2):249–59. [12] Linzell DG, Leon RT, Zureick AH. Experimental and analytical studies of a horizontally curved steel I-girder bridge during erection. J Bridge Eng 2004; 9(6):521–30. [13] Dong J, Sause R. Analytical study of horizontally curved hollow tubular flange girders. ATLSS report 08-15. Bethlehem (PA): ATLSS Engineering Research Center, Lehigh University; 2008. [14] Key PW, Hancock GJ. A theoretical investigation of the column behavior of cold-formed square hollow section. Thin-Walled Struct 1993;17:31–64.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Finite element analysis of curved steel girders with tubular flanges Jun Dong a,∗ , Richard Sause b a b
T.Y.Lin International, San Francisco, CA 94105, USA ATLSS Center, Department of Civil and Environmental Engineering, Lehigh University, Bethlehem, PA 18015, USA
article
info
Article history: Received 11 July 2008 Received in revised form 20 July 2009 Accepted 24 September 2009 Available online 23 October 2009 Keywords: Steel girder Curved bridge Finite element Residual stresses Load capacity
abstract A tubular flange girder is an I-shaped steel girder with either rectangular or round tubes as flanges. A tubular flange girder has a much larger torsional stiffness than a conventional I-shaped plate girder and less potential for cross section distortion than a box-girder, and is, therefore, an interesting alternative girder for horizontally curved steel bridges. Finite element (FE) models of curved tubular flange girders are presented in this paper, considering material inelasticity, second-order effects, initial geometric imperfections, and residual stresses. A parametric study is performed using the FE models to study the effects of stiffeners, tube diaphragms, geometric imperfections, residual stresses, and cross section dimensions on the load capacity of curved tubular flange girders. Finally, the FE results for curved tubular flange girders are compared with results for corresponding curved I-shaped plate girders, and the advantages of tubular flange girders are summarized. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Horizontally curved steel girder highway bridges are often used at locations where the roadway alignment is constrained. The horizontal curvature induces significant torsion in the bridge girder system, which is an important design consideration. Currently, I-shaped plate girders (I-girders) or box-shaped girders (box-girders) are used for curved steel bridges. An I-girder is an ‘‘open’’ section and has very little torsional resistance. During bridge erection, curved I-girders are often supported temporarily at intermediate locations between piers before the girders are attached to permanent cross frames (or diaphragms). After the Igirders are attached to the cross frames, the I-girders and cross frames work together to resist the torsion of a curved bridge. Both the I-girders and the cross frames must be designed as primary load-carrying members. A box-girder is a ‘‘closed’’ section which has a relatively large torsional resistance. However, owing to the large cross section width and depth of box-girders, cross section distortion is a concern. Thus, internal diaphragms and stiffeners are needed to maintain the box shape, which makes the design, fabrication, construction, inspection, and maintenance of box-girders complex and expensive. The use of innovative I-shaped girders with hollow tubular flanges in curved steel bridges is studied in this paper. This new
∗
Corresponding author. E-mail address: [email protected] (J. Dong).
0141-0296/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2009.09.018
girder type combines the advantage of curved I-girders, namely, easier design, fabrication, and inspection, with the advantage of curved box-girders, namely, the large torsional resistance. Previous work by Kim and Sause [1–3] addressed the behavior of straight tubular flange girders with a round concrete-filled tube as the compression flange and a flat plate as the tension flange (CFTFGs), as shown in Fig. 1(a). Kim and Sause [1–3] summarized several advantages of CFTFGs relative to conventional Igirders for bridges: (1) the concrete-filled tubular flange provides more strength, stiffness, and stability than a flat plate flange with same amount of steel, and (2) fewer cross frames are needed for CFTFGs to maintain lateral–torsional stability compared to similar I-girders, which reduces the fabrication and erection effort needed to construct a bridge. Dong and Sause [4] investigated the flexural strength of straight hollow tubular flange girders (HTFGs) with rectangular tubes for both the compression and tension flanges, as shown in Fig. 1(b). The effects of stiffeners, geometric imperfections, residual stresses, cross section dimensions, and bending moment distribution on the lateral–torsional buckling flexural strength of HTFGs were studied and formulas for determining the flexural strength of HTFGs were evaluated. Fan and Sause [5] conducted theoretical analyses of individual curved tubular flange girders and systems of multiple curved tubular-flange girders braced by cross frames. The analysis method of Dabrowski [6] was extended and used for the theoretical analyses. Fan and Sause also developed finite element models of curved tubular flange girders and compared the behavior of curved tubular flange girders with curved I-girders. These analyses were firstorder linear elastic analyses.
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(a) Round CFTFG.
(b) HTFG.
(c) I-Girder.
Fig. 1. Cross section shapes.
Curved I-girders have been the subject of substantial previous research. For example, Pi et al. [7,8] studied the nonlinear behavior of curved I-girders. Under vertical loading, a curved I-girder develops both primary bending action and non-uniform torsion action, and vertical deflections are coupled with out-of-plane cross-section rotations. These primary actions and deformations couple together to produce second-order bending actions about the minor axis. The second-order effects are significant for I-girders with large initial curvatures. Pi and Bradford [9] proposed formulas for the design of curved I-girders against the combined bending and torsion actions. The 2004 AASHTO LRFD Bridge Design Specifications [10] have standard provisions for the design of curved I-girders for US highway bridges. This paper focuses on the second-order nonlinear behavior of individual curved hollow tubular flange girders (CHTFGs) with rectangular tubes for both the compression and tension flanges, as shown in Fig. 1(b). The tubes and webs are sufficiently compact so that local buckling does not control the girder strength. The FE models include material inelasticity, second-order effects, geometric imperfections, and residual stresses. FE models are used to study the effects of transverse web stiffeners, tube diaphragms, geometric imperfections, residual stresses, and cross section dimensions on the load capacity of CHTFGs. Finally, the behavior of individual CHTFGs is compared with the behavior of corresponding curved I-girders, and the advantages of CHTFGs are summarized.
Fig. 2. The coordinate system and boundary condition of CHTFG FE models.
2.1. Coordinate system The geometry of a curved girder is easily described within a cylindrical coordinate system whose origin is located at the center of curvature. The global 1, 2 and 3 axes are defined in the radial (lateral), circumferential (longitudinal) and vertical direction, as shown in Fig. 2. U1, U2, U3, UR1, UR2 and UR3 are the displacements and rotations about the global 1, 2 and 3 axes respectively.
2. Finite element models The Finite Element (FE) program ABAQUS (Version 6.7) was used for the FE analyses reported later in the paper. Experimental data for CHTFGs is not currently available to validate the FE models. However, the approach used to develop the FE models is similar to that used previously by the authors and other researchers to develop reliable nonlinear FE (i.e., nonlinear shell element) models for curved and straight steel bridge girders using ABAQUS. For example, Shanmugam et al. [11] performed tests on curved I-girders with different curvature-to-span ratios, and the deformations and ultimate strength from the tests were are found to be in good agreement with the FE results obtained from ABAQUS shell element models. Linzell et al. [12] obtained data from a fullscale curved steel bridge during erection. Comparisons between this experimental data and FE results obtained from an ABAQUS shell element model showed the FE model provided good predictions of erection behavior. Kim and Sause [1–3] validated nonlinear ABAQUS shell element models for straight tubular flange girders using experimental data. For the present study, the FE models were developed for an individual girder with simply supported boundary conditions, and with a uniformly distributed vertical load over the span. The details of the procedure used to develop the models are given by Dong and Sause [13]. The important aspects of the models are summarized below.
2.2. Boundary and load conditions Simply supported boundary conditions are applied to the end sections, as shown in Fig. 2. At each end section, the vertical displacement (U3) of the nodes of the bottom wall of the bottom tube, the lateral displacements (U1) of all nodes along the line through the web mid-surface, and the twist rotations about 2-axis (UR2) of all nodes on the section are restrained. The longitudinal displacement (U2) of the centroid of the web is restrained at only the left end section, and the rotations about the 3-axis (UR3) are not restrained. Along the length of the FE models (i.e., along the span of the girders), the lateral displacement and twist are unrestrained (i.e., the girders are braced only at the ends). The FE models are loaded with a distributed vertical load that is uniform across the girder cross section and is uniform over the span. This load is similar in distribution to the self weight of the girder. 2.3. Finite element mesh The fully integrated, three-dimensional, four-node shell element S4 is used to model the flanges, webs, and transverse web stiffeners. The S4 element has four integration points across the element and five section points (integration points) through the thickness of the shell element. Dong and Sause [13] conducted a FE
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a
b
c
d
Fig. 3. Components of longitudinal residual stress: (a) membrane (tube); (b) bending (tube); (c) layering (tube); (d) web [7].
mesh study for CHTFGs. The total number of S4 elements in the different meshes for a single CHTFG was varied from 7200 to 243,000. A curved I-shaped girder develops flange lateral bending under vertical loads, and the normal stress in the flanges varies across the width of the flange and along the span. Therefore, a sufficient number of elements are needed across the flange width and also in the longitudinal direction of girder. By comparing displacements, stresses, and ultimate load capacities, it was found that an optimized FE mesh with 40,000 elements produced accurate results, and this mesh was used for all FE models in the present study. 2.4. Material models and properties An elastic–perfectly-plastic material is used for the steel. In the elastic range, Young’s modulus is 200 GPa (29,000 ksi) and Poisson’s ratio is 0.3. In the inelastic range, Von Mises yield criteria is used to define isotropic yielding. The yield strength of the steel is 345 MPa (50 ksi). 2.5. Initial geometric imperfections Dong and Sause [4] showed that the lateral–torsional buckling strength of straight HTFGs depends on the initial geometric imperfections. The effect of initial geometric imperfections on the behavior of CHFTGs is discussed later in the paper. Elastic lateral–torsional buckling shapes, obtained from an elastic buckling analysis, are scaled and added to the perfect geometry of the girders to create an initial geometry with imperfections. Combinations of different buckling shapes and imperfection magnitudes are considered. 2.6. Residual stresses The residual stresses in the FE model vary through the thickness of the shell elements, across the elements, and between different elements on the cross section. The residual stresses are assumed to develop from two stages of the fabrication of the curved hollow tubular flanges: (1) the manufacture (cold forming) of the straight tube, (2) the cold bending of the straight tube
into the desired curved tube. The distribution and magnitude of residual stresses from Stage 1 are based on the measured residual stresses presented by Key and Hancock [14] for a square hollow tube. As shown in Fig. 3, the residual stresses from cold forming of the tube have three components: (a) the membrane residual stresses, which are constant through the tube wall; (b) the bending residual stresses; and (c) the ‘‘layering’’ residual stresses. The bending and layering residual stresses have the same pattern around the tube, and vary linearly (or bi-linearly) through the thickness of the tube wall. The influence of the different components of residual stresses from cold forming the tubes on the lateral–torsional buckling flexural strength of straight tubular flange girders was investigated by Dong and Sause [4]. They showed that the bending residual stress component (shown in Fig. 3(b)) has the most significant effect on the flexural strength, because the magnitude of the bending residual stress component is much larger than the magnitude of the other residual stress components. Thus, only the bending residual stress component from tube cold forming is included in Stage 1. A model for the residual stresses from Stage 2, cold bending, was developed from a theoretical analysis of the cold bending process. In this theoretical cold bending analysis, the steel is assumed to be elastic–perfectly-plastic, and the tube initially has the cold forming residual stresses. The theoretical analysis of the cold bending process was validated using nonlinear FE analysis. The following example illustrates the FE analysis method and compares the results with those from the theoretical analysis. In the example, a 508 mm by 305 mm tube with a thickness of 12.7 mm and a length of 27 m is bent to a final radius of curvature of 60 m. The tube is bent about its strong axis. The analysis is conducted in three steps: (1) the bending residual stress component from cold forming the tube (Stage 1) is included in the model; (2) the tube is loaded under uniform curvature until the target lateral displacement at midspan U1M -max is achieved; (3) the loading is reversed until the bending moment equals zero. For this example, U1M -max equals 2.259 m and the residual lateral displacement U1M equals 1.537 m, which results in the final radius of 60 m for the curvature. U1M max is determined from the final desired curvature, considering
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Table 1 Dimensions of curved tubular flange girder and I-girder. Girders
Flanges (mm)
Web (mm)
Depth (mm)
Area (mm2 )
Ix (mm4 )
Iyf (mm4 )
J (mm4 )
Iw (mm6 )
TG1 TG2 IG1 IG2
508 × 101.6 × 12.7 508 × 304.8 × 12.7 508 × 25.4 508 × 27.9
1066.8 × 12.7 660.4 × 7.9 1257.3 × 14.9 1254.8 × 14.5
1282.7 1282.7 1282.7 1282.7
44,516 46,553 44,516 46,553
1.20E+10 1.05E+10 1.31E+10 1.40E+10
4.45E+08 7.78E+08 2.77E+08 3.05E+08
2.23E+08 1.50E+09 6.94E+06 8.62E+06
3.04E+14 3.62E+14 2.28E+14 2.51E+14
Fig. 4. Distribution of residual stresses at end of step 1.
Fig. 6. Distribution of residual stresses at end of step 3.
Fig. 7. Normalized load capacity versus vertical displacement at mid-span. Fig. 5. Distribution of residual stresses at end of step 2.
the elastic recovery in step 3. The distributions of residual stresses in different layers of the top (and bottom) wall of the tube at the end of each step are shown in Fig. 4 through Fig. 6. As noted above, the residual stresses at the end of step 1 (Fig. 4) include only the bending residual stresses from cold forming the tube. The residual stresses at the end of step 2 and step 3 from the theoretical analysis are compared with results from the FE simulation in Figs. 5 and 6. It is observed that the differences between the FE simulation results and the theoretical results are very small, and the residual stresses from the theoretical analysis were used as the input to the FE models for the remaining studies. The residual stresses in the web are constant through the web thickness and vary bi-linearly over the web height, as shown in Fig. 3(d). 2.7. Analysis method Nonlinear load–displacement analyses with material inelasticity and second-order effects were used to study the behavior of CHTFGs and curved I-girders. The modified Riks method available in ABAQUS was used for these analyses.
3. Distortion of the cross section Dong and Sause [4] showed that distortion of the cross section affects the flexural strength of HTFGs. Thus, the effects of cross section distortion on the load capacity of CHTFGs were studied. 3.1. Sources of cross section distortion To study CHTFG cross section distortion, a modified FE model which did not include cross section distortion was developed. In this modified model (denoted as MWOD, model without distortion), the rotations about the longitudinal axis (UR2) of all nodes on each cross section are constrained to be equal. The model without this constraint is denoted as MWD (model with distortion). Nonlinear analyses of a MWOD and a MWD of a single CHTFG with a cross section denoted TG2 were conducted. The dimensions of TG2 are shown in Table 1. The span L (arc length of the girder) is 27 m, the radius of the curvature is 60 m, and the L/R ratio equals 0.45. A uniformly distributed vertical load was applied, and the analyses were taken to levels of displacement beyond the maximum load. The load–displacement curves are presented in Fig. 7, where the load is normalized by the weight of the girder,
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(a) MWOD.
(b) MWD.
Fig. 8. Deformed shape of cross section near mid-span for different models.
(a) Without distortion.
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(b) Web distortion.
(c) Tube distortion.
Fig. 9. Cross section distortion.
Fig. 11. Variation of RMCD with span and Ts for MWD.
stiffener elements are modeled using three-dimensional shell elements S4 and are connected to the cross section by constraining the nodes in the stiffeners to the corresponding nodes in the flanges and web. The stiffeners are on both sides of the web, as shown in Fig. 10. Diaphragms are commonly used to control distortion of boxlike cross sections. To avoid the need to install diaphragms within the tubes and simplify CHTFG fabrication, diaphragms are introduced only at the ends of the tube (at the end sections) to reduce the tube distortion. In the FE models, the tube diaphragms are modeled using three-dimensional shell elements S4 and are connected to the cross section by constraining the nodes in the diaphragms to the corresponding nodes in the tubes. A parametric study was conducted to investigate the use of transverse web stiffeners and tube diaphragms to control cross section distortion of CHTFGs. The stiffener plate thickness, the number of stiffeners, and the diaphragm plate thickness were selected as variables in the study. 3.3. Effect of stiffener thickness (Ts )
Fig. 10. Typical stiffener configuration.
and U3M is the vertical displacement U3 of the centroid of the midspan cross section (positive downward). By comparing the results of the MWOD and MWD, it is seen that cross section distortion reduces substantially the strength of CHTFGs, which indicates the importance of the distortion. Fig. 8 shows the deformed cross sections of the MWOD and MWD at mid-span. The cross section distortion is clearly seen in the MWD results. Since the tubular flanges are torsionally stiff and the web is comparatively flexible, the web tends to deform by bending out-of-plane. The thin-walled tubular flanges also change shape. Therefore, the cross section distortion includes both web distortion and tube distortion, as shown in Fig. 9. 3.2. Controlling cross section distortion Transverse web stiffeners are usually used to reduce web distortion in I-girders, and for a CHTFG, transverse web stiffeners also reduce the tube distortion where the stiffeners are attached to the wall of the tubes. Therefore, stiffeners are introduced into the CHTFG FE models to reduce the cross section distortion. The
Nonlinear load–displacement analyses were performed on the MWD with seven intermediate stiffeners, which are uniformly distributed along the span between the bearing stiffeners. The span was varied from 15 m to 120 m, and the stiffener plate thickness (Ts ) was either 12.7 m, 25.4 m or 50.8 mm. Results for the ratio of the load capacity of the MWD, including the stiffeners, to that of the MWOD, denoted as RMCD , are plotted in Fig. 11. It is observed that the effect of Ts is not very significant and the differences between the cases with Ts = 25.4 mm and Ts = 50.8 mm are small. Therefore, 25.4 mm thick stiffeners are used for the remaining studies. 3.4. Effect of number of stiffeners (Nis ) Nonlinear load–displacement analyses were performed on the MWD with different numbers of web stiffeners. For all cases, bearing stiffeners were included at the supports, and the number of intermediate transverse web stiffeners between the bearing stiffeners (Nis ) was varied. The intermediate stiffeners are uniformly distributed (i.e., the spacing of the stiffeners is constant) and Ts = 25.4 mm. RMCD is plotted in Fig. 12. It is found that as Nis increases, the load capacity of the MWD with stiffeners approaches the load capacity of the MWOD (i.e., RMCD approaches 1), indicating that cross section distortion is not reducing the load capacity. The differences between the results for Nis = 7 and Nis = 9 are small. Therefore, seven intermediate web stiffeners and two bearing stiffeners with Ts = 25.4 mm are used for the remaining studies.
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Fig. 12. Variation of RMCD with span and Nis for MWD.
Fig. 14. Effect of initial geometric imperfections as span varies.
no residual stresses with residual stresses
Fig. 15. Effect of residual stresses as span varies. Fig. 13. Variation of RMCD with span and Tdia for MWD.
3.5. Effect of tube diaphragms at ends Nonlinear load–displacement analyses were performed on the stiffened MWD (Nis = 7, Ts = 25.4 mm) with a tube diaphragm at both end sections. The diaphragm thickness (Tdia ) was either 12.7 mm, 25.4 mm or 50.8 mm. RMCD is shown in Fig. 13 for the different values of Tdia , and for the case without tube diaphragms. It is observed that including tube diaphragms improves the load capacity, but using a thicker diaphragm (with a larger value of Tdia ) has an effect only for very long spans. Therefore, Tdia = 25.4 mm is used for the remaining studies.
the first two elastic buckling mode shapes, where the mode 1 shape is scaled so the maximum top flange lateral deflection is L/1000 and the mode 2 shape is scaled so the maximum top flange lateral deflection is L/2000, and the two shapes are added together. FE results for models without initial geometric imperfections and for models with these two different initial geometric imperfections are shown in Fig. 14. The span is varied and L/R equals 0.45 for all cases. Since a CHTFG has an initial curvature, it is observed that including initial geometric imperfections does not significantly affect the load capacity. Initial geometric imperfections are not included in the FE models for the remaining studies. 4.2. Effect of residual stresses
4. Load capacity of CHTFGs FE models of the CHTFG with cross section TG2 were used to investigate the load capacity of CHTFGs. Non-linear load–displacement analyses were performed, and the load capacity is expressed as the maximum applied load normalized by the girder weight. 4.1. Effect of initial geometric imperfections As discussed earlier, the initial geometric imperfections used in the present study are derived from elastic buckling analysis results and introduced into the FE models for the nonlinear load–displacement analyses. Two combinations of buckling shapes and imperfection magnitudes are considered. For the initial imperfection denoted as EM1, the elastic buckling shape for mode 1 is scaled so that the maximum top flange lateral deflection is L/1000, where L is the span. Initial imperfection COM1 is a combination of
The effect of residual stresses on the load capacity of TG2 is shown in Fig. 15. The span is varied and only results for L/R = 0.1 and L/R = 0.45 are shown. Fig. 15 shows that the effect of residual stresses is small. The ratio of the load capacity of the model with residual stresses to that of the model without residual stresses is denoted as RMRS . Results for RMRS are plotted in Fig. 16, where L/R is varied from 0.1 to 0.45. It is observed that RMRS is slightly less than one for all cases, which means the load capacity of a CHTFG with residual stresses is slightly smaller than the load capacity without residual stresses. Compared to straight tubular flange girders [4], the effects of residual stresses on the load capacity of a CHTFG are small. A comparison of the behavior of CHTFGs without and with residual stresses is as follows. For girder TG2 without residual stresses and with L = 27 m and L/R of 0.45, the distribution of normal stress across the top and bottom walls of the tubes at the mid-span cross section under the girder self-weight is shown in Fig. 17. Due to the flange
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L/R=0.1 L/R=0.2 L/R=0.3 L/R=0.45
Fig. 16. Variation of RMRS with span and L/R. Fig. 19. Distribution of normal stresses under the ultimate load.
a small region of the flange has large residual stress, and the tubes can carry increasing loads after these small regions yield. Figs. 18 and 19 show the distributions of normal stress across the width of the top wall of the top tube and the bottom wall of the bottom tube at the mid-span cross section for the girder with and without residual stresses under the ultimate load. It is observed that the average value of the normal stress of the inner and outer layers for the model with residual stresses is very close to the normal stress of the model without residual stresses, which illustrates why residual stresses have little effect on the load capacity of a CHTFG. 4.3. Comparison with I-girders
Fig. 17. Distribution of normal stresses for the girder without residual stresses.
Fig. 18. Distribution of normal stresses under the ultimate load.
lateral bending moment introduced by the torsion in the girder, the maximum normal stress is located at the flange tips and side walls. As shown in Fig. 6, the residual stresses at the flange tips and side walls are small after cold bending of the tubes. Thus, the residual stresses are small at the location of maximum stress from loading. Large residual stresses are located near the center of the top and bottom walls of the tubes, and the maximum stress is compressive for the inner layer and tensile for the outer layer of the top and bottom walls of the tubes. The effect of these high residual stresses on the primary bending moment capacity is neutralized by different signs of residual stresses in the different layers. Also, only
To enable the behavior of an individual curved tubular flange girder to be compared with the behavior of an individual curved I-girder, curved I-girders similar to (corresponding to) the CHTFGs were developed, shown in Fig. 2(c). Two curved I-girders, denoted IG1 and IG2, have the same girder weight (cross section area), cross section depth, and flange width as the corresponding CHTFGs, TG1 and TG2, respectively, as shown in Table 1. To illustrate the main differences between the CHTFG and I-girder cross-sections, important cross section properties, such as the cross section moment of inertia about the major axis (Ix ), the flange moment of inertia about the cross section minor axis (Iyf ), the St. Venant torsional constant (J), and the warping moment of inertia (Iw ) are also listed in the table. The corresponding curved I-girders have a slightly larger flexural rigidity but a much smaller St. Venant torsional rigidity than the corresponding CHTFGs. For the studies comparing CHTFGs with curved I-girders, the span, L, is constant and equal to 27 m, and the radius of curvature, R, is varied so that L/R varies from 0.1 to 0.45. Figs. 20 through 24 compare the maximum primary bending normal stress, maximum warping (flange lateral bending) normal stress, maximum total normal stress, mid-span vertical displacement (U3M ), and mid-span cross section rotation (UR2M ) for the CHTFGs and the corresponding curved I-girders under the girder self-weight. The primary bending normal stress is derived from the primary bending moment which is determined by integrating the normal stresses on the cross section. The primary bending normal stress is the primary bending moment divided by the section modulus of the cross section about the strong axis of the cross section. The warping normal stress is derived from the lateral bending moment of the top flange, which is determined by integrating the normal stress on the flange. The warping normal stress is the flange lateral bending moment divided by the section modulus of the flange about the cross section minor axis. The distribution of the top flange lateral bending moment along the span of the girders
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Fig. 20. Variation of maximum bending normal stress with L/R.
Fig. 23. Variation of mid-span vertical displacement with L/R.
Fig. 21. Variation of maximum warping normal stress with L/R. Fig. 24. Variation of mid-span cross section rotation with L/R.
Fig. 22. Variation of maximum total normal stress with L/R.
with L/R of 0.45 is shown in Fig. 25. The load capacity (normalized by the girder weight) is compared in Fig. 26. The results indicate that the bending normal stress is dominant and the warping normal stress is not significant for the CHTFGs, however, the warping normal stress is large for the curved Igirders. Since CHTFGs have a large torsional rigidity, most of the torsion is resisted by St. Venant torsion, and the flange lateral bending moment from warping torsion of the CHTFGs is much smaller than that of the curved I-girders, as shown in Fig. 25. The CHTFGs also have a larger flange moment of inertia and section modulus about the cross section minor axis. Therefore, the maximum warping normal stress for the CHTFGs is much smaller than that of the curved I-girders. The curved I-girders have smaller
Fig. 25. Variation of top flange lateral bending moment along the span (L/R = 0.45).
primary bending normal stresses than the corresponding CHTFGs due to their slightly larger flexural rigidity. In summary, the CHTFGs develop much smaller warping normal stresses, total normal stresses, displacements, and cross section rotations, and have much larger load capacities. For example, for TG2 with L/R = 0.45 under girder self-weight, the mid-span vertical displacement, U3M is 0.019 m which is 1/1500 of the span L, and the cross section rotation at mid-span UR2M is 0.005 rad. The load capacity is 17 times the girder weight. However, for the corresponding curved I-girder IG2 under girder self-weight, U3M is 0.394 m which is 1/70 of L, and UR2M is 0.296 rad. The load
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that a curved I-girder is better at resisting primary bending, but develops much larger warping normal and total normal stresses, and much larger vertical displacements and cross section rotations than a CHTFG. Curved I-girders are generally braced by permanent cross frames within the span in the final constructed condition, and they usually require temporary support or bracing within the span during girder erection before the cross frames are fully installed. The results presented in the paper show that individual CHTFGs have much better structural behavior than individual curved Igirders (e.g., only 1/20th of the midspan vertical displacement and only 1/60th of the midspan cross section rotation under self weight). These results suggest that such temporary support or bracing within the span may not be needed for CHTFGs during girder erection, which could lead to faster and more economical curved bridge construction. Fig. 26. Variation of load capacity with L/R.
capacity is 3.9 times the girder weight. In other words, the midspan vertical deflection of the curved I-girder is about 20 times larger than that of the CHTFG and the mid-span cross section rotation of the curved I-girder is about 60 times larger than that of the CHTFG, while the load capacity of the CHTFG is about 4 times larger than that of the curved I-girder. Under the girder self-weight, the displacements and cross section rotations of the curved I-girder IG2 are quite large, which suggests that temporary support within the span would be needed for this curved I-girder during erection, while the displacements and cross section rotations of CHTFG TG2 are quite reasonable, which suggests that such temporary support within the span would not be needed during erection. As usual, the installation of permanent cross frames between the CHTFGs or the curved I-girders would be used to control the final erected geometry of the girders in a curved bridge. 5. Conclusions Finite element (FE) models of individual curved hollow tubular flange girders (CHTFGs) were presented. The models account for material inelasticity, second-order effects, initial geometric imperfections, and residual stresses. With these FE models, the effect of cross section distortion on the load capacity of CHTFG was studied. An arrangement of transverse web stiffeners and tube end diaphragms was identified to reduce the effect of cross section distortion on the load capacity. Nonlinear load–displacement analyses of the FE models were performed to study the load capacity of CHTFGs. A parametric study showed that the influences of initial geometric imperfections and residual stresses on the load capacity of CHTFGs are small. By comparing the behavior of individual CHTFGs with the behavior of corresponding individual curved I-girders, it is found
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