Abstract
The torsional behaviors of composite box girders with corrugated steel webs are more obvious than traditional concrete girders due to the lower torsional rigidity. In this paper, the torsion and distortion of this kind of composite girder are studied. The formulas for warping normal stress and shear stress were put forward according to the second theory of Umanskii, considering the accordion effect of corrugated steel webs. Then, the influences of different dimensional parameters on the torsional and distortional stress are investigated. Results show that the effect of parameters on stress is different and implicit in composite box girders with corrugated steel webs. Under eccentric loads, the warping torsional and distortional stress in this kind of girder should not be neglected. Compared with girders under corresponding symmetric loads, the total warping stress may be as big as flexural normal stress, and the total shear stress usually reaches 30 to 50 percentage of flexural shear stress. So the warping stress and additional shear stress due to warping torsion and distortion are suggested to be calculated by the proposed equations in structural analysis, which are usually not taken into account in conventional concrete box girders.
1. Introduction
The composite box girders with corrugated steel webs have been used more extensively in bridge engineering since first appearance in France. Under eccentric loads, the box girder may not only produce longitudinal flexural and shear deformation, but also the torsion and distortion. The torsional behaviors in these composite girders were found quite different from traditional concrete box girders [1]. Researchers worldwide have done much work on the torsional and distortional properties of composited box girder with corrugated steel webs. Mo et al. presented an analytical model to predict the torsional behavior of composite girders with corrugated steel webs [2] and developed an analytical model according to the test results [3]. According to experimental results of scaled models and bridge in site, Li et al. [4] and Liu et al. [5] concluded that resistance of this kind of box girder to torsion and distortion is lower than conventional box girders. Yang et al. investigated the warping and shear stress of the box girders with corrugated steel webs under torsion by the second theory of Umanskii [6]. Di et al. [7] and Wang [8] studied the torsional property of composite girders with corrugated steel webs by theoretical analysis and experiment and found its normal stress and shear stress due to torsion are 15% to 50% bigger than flexural normal stress and the distortional warping stress is larger than that due to warping torsion. Nie and Zhang [9] and Chen et al. [10] studied the influencing factors to torsional properties by finite element method, including the diaphragm number, geometry parameters of section, and corrugated web. Ding et al. investigate the behavior under pure torsion and the ultimate torsional strength considering material nonlinearity by finite element models [11] and scaled specimen experiments [12]. According to current studies, the additional normal and shear stress caused by torsion and distortion should be considered properly, which is often neglected for concrete box girders. But little detailed conclusions are found on how the section dimensions influence the torsional and distortional stress.
As to composite box girders with corrugated steel webs, the 3D finite element models are rather complex to build. So, the theoretical analysis, based on the thin-walled beam theory, can be used more conveniently to study the mechanical properties of such composite structures. In the theoretical deduction process, the accordion effect of corrugated steel web should be considered correctly, as discussed by Yang et al. [6]. However, there were still some improper results from current studies. For example, the web shear stress is not influenced by the warping torsion, or the shear stress is not even in web. Additionally, the formula of distortional shear stress has not yet been proposed explicitly. The purpose of this paper is to propose formulas for warping torsional and distortional shear stress and study the influences of different factors to the warping torsional and distortional stress compared to the flexural normal stress.
2. Equivalent Section
2.1. Equivalent Orthotropic Plate of the Corrugated Steel Web
The geometry of the typical corrugated web is shown in Figure 1. The corrugated steel web is recommended to be equivalent to the orthotropic plate [13, 14], which has different elastic modulus in and direction:where is Young’s modulus of steel; , are the equivalent elastic modulus in and direction, respectively; , , , are flat panel width, inclined panel width, wave height, angle of the corrugated web, respectively.
According to (1), the axial stiffness in longitudinal direction can be neglected, for the ratio of and of a corrugated web is always less than 1/100.
Additionally, the effective shear modulus of a corrugated web [15], , is given byin which is the shear modulus of steel.
2.2. Equivalent Section of Composite Box Girders with Corrugated Steel Webs
In order to directly adopt the thin-walled box girder theory to analyze the mechanical properties of the composite box girders with corrugated steel webs, the composite section is transformed to an equivalent cross section of single material, as shown in Figure 2. Considering the corrugated webs bear most of shear force in composite box girder, the cross section is assumed to be composed of steel material alone.
(a) Composite section
(b) Equivalent section
Assuming that the shearing force and stress remain the same, the equivalent thickness of concrete slab, , is given bywhere is the thickness of concrete slab in composite box section; is the shear modulus of concrete; is the ratio of shear modulus of concrete and corrugated steel web.
3. Warping Torsional Stress
3.1. Torsional Warping Stress
According to the second theory of Umanskii, the warping displacement caused by warping torsion is given byin which is the principal unit warping (sectorial coordinate). Its practical formulas may be found in [16–18].
The torsional warping normal stress, , is determined from the bimoment: For concrete slab, For corrugated steel web,where is the warping bimoment caused by warping torsion; is the warping torsional inertia. For composite box girder with corrugated steel webs, it is given by
Considering that the ratio of corrugated steel web is small enough to ignore, the warping torsional inertia is simplified byin which the subscripts and indicate the bottom and top slab, respectively. Equation (9) is used in formulas to calculate warping normal stress for composite girder with corrugated steel webs, which has already been given in [6].
3.2. Total Torsional Shear Stress
The deduction procedure of shear stress for composite box girder with corrugated steel web is similar to that of concrete box girder. However, the accordion effect of corrugated steel web should be considered correctly.
The shear flow in slab can be determined by the warping normal stress considering the equilibrium in the longitudinal direction of a plate element:
The shear flow in different plate is shown as follows. For equivalent cantilever slab, For equivalent top and bottom slab, For corrugated steel web,where
Noting that the external torque is equal to the total action of , we getwhere
In (17), it is noted that the ratio of corrugated steel web is small enough to ignore corrugated steel webs.
From (15),
The warping torsional moment derived from torsional warping bimoment is
Substituting (18) and (19) into (11)~(13), the total shear stress considering the warping torsion can be determined from torsion and warping torsional moment:in which is the generalized area moment of inertia. For equivalent cantilever slab, For equivalent top and bottom slab, For corrugated steel web,
3.3. Differential Equation
Considering the longitudinal displacement closure condition, the differential equation is deduced:in which is load intensity of torque; is the restrained constant of section; is the moment-torsional coefficient, defined byin which is the torsional rigidity; is the warping torsional rigidity. According to specific boundary conditions, the above differential equation can be solved by initial-parameter method in [19, 20].
4. Distortional Stress
4.1. Distortional Warping Stress
According to [20], the distortional warping stress of composite box girder with corrugated steel web has already been given. The most important equations are list as follows.
The distortional differential equation of distortional angle, , isin which is distortional warping rigidity, is distortional rigidity of transverse frame, and is antisymmetric load at intersection point of top slab.
Combined with specific boundary conditions, the analytical solutions of distortional angle, , and distortional bimoment, , can be deduced by initial-parameter method. Then the distortional warping normal stress is given as follows:in which , are the warping stress at the junction between bottom and top slab, respectively. , . The coefficients and can be calculated according to [20].
4.2. Distortional Shear Stress
Little has been studied on the distortional shear stress of composite box girder with corrugated steel webs. So the additional shear stress due to distortion is deduced as follows.
The distortional shear moment is given by
Then, the distortional shear force in different slabs can be calculated as follows. For corrugated steel web, For equivalent bottom slab, For equivalent top slab,
The distortional shear stress is given byin which is the cross-sectional area of equivalent top, bottom, or web slab. Because of the changing of normal stress due to distortional warping, the additional shear stress is also produced. The additional shear stress can be deduced by similar procedures like warping torsional shear stress, and it should be considered that no torsional force in section is generated under distortion. Thus, the distortional additional shear stress can be calculated byin which is the generalized area moment of inertia. For equivalent cantilever slab, For equivalent top and bottom slab, For corrugated steel web,where
Finally, the total distortional shear stress is as follows:
According to the above equation, the distortional shear stress is comprised of two parts, which are the direct shear stress due to distortional shear force and the additional shear stress due to changing normal stress, just like the warping torsional shear stress.
5. Comparative Studies
5.1. Case 1
Take the simple supported composite girder with corrugated steel webs in [21] as an illustration (as shown in Figure 3). The span, , is 43.2 m. The height, , is 5.1 m. The top slab width, , is 10 m and thickness, , is 0.30 m. The bottom slab width, , is 6 m and thickness, , is 0.25 m. For the corrugated steel webs, the thickness, , is 0.01 m, the panel width is 0.30 m, is 0.26 m, and is 0.15 m. For concrete, is and is 0.1667. For steel, is and is 0.3. The 1100 kN eccentric load is applied at one conjunction of web on the top slab at the mid-span cross section.
(a) Cross section
(b) Corrugated steel web
Using the finite element program ANSYS, 3D models of composite girders were built to verify the above theoretical analysis expressions. The eight-node Solid element (SOLID45) was used to model the concrete slabs. The four-node Shell element (SHELL63) was chosen to simulate the corrugated steel webs. The concrete and steel materials were linear elastic and no plastic or strain hardening was taken into account. The slip and composite action between the concrete slab and steel web were neglected, so the concrete and steel nodes on their interface were merged to ensure they had the same deformation. As the girder is assumed to be simple supported at both ends, four nodes at the bottom were constrained as shown in Figure 4. Nodes no. 1 to 4 were provided constraints in different directions, but no rotating constraint was given. In finite element analysis (FEA), the torsional loads and the distortional loads were separately applied on nodes of top and bottom concrete slab and steel webs in the mid-span section of the girder.
(a) Girder model
(b) Torsional loads and distortional loads
The warping normal stress and shear stress of this composite girder are shown in Figure 5. As the normal stress in corrugated webs and shear stress in concrete slabs are rather small, only the normal stress in concrete slabs and shear stress in corrugated webs are given. In order to verify the results according to equations in this paper, the stress calculated by FEA is presented separately in bracket .
(a) Warping torsional stress
(b) Distortional stress
As shown in Figure 5, the above equations of warping normal stress and shear stress for composite girder with corrugated steel webs are verified by FEA results. Compared with stress due to distortion, the warping normal stress due to warping torsion is much smaller, while the shear stress in webs is a little bigger. Noticing that the flexural normal stress in bottom slab is 1.45 MPa and the flexural shear stress in webs is 5.39 MPa, the distortional warping stress is even as big as flexural normal stress. This means the influence of both warping torsion and distortion on normal and shear stress must be considered properly. In this girder, the normal stress in bottom slab due to warping torsion has the opposite direction to stress due to distortion.
The results of warping normal stress and web shear stress were also displaced in Figure 6, comparing with the stress according to analytical equations. As shown in Figure 6, the stress from two different methods has little discrepancy, which means the analytical equations in this essay are effective.
(a) Torsional warping stress in bottom slab
(b) Torsional shear stress in web
(c) Distortional warping stress in bottom slab
(d) Distortional shear stress in web
5.2. Case 2
Take the simple supported composite girder with corrugated steel webs in [6] as another illustration (as shown in Figure 7). The span, , is 7.5 m. The height, , is 0.3688 m. The top slab width, , is 1.8 m and thickness, , is 0.1125 m. The bottom slab width, , is 0.4626 m and thickness, , is 0.11 m. For the corrugated steel webs, the thickness, , is 0.003 m, the panel width is 0.063 m, is 0.050 m, and is 0.038 m. For concrete, is MPa amd is 0.1667. For steel, is and is 0.3. The 20 kN eccentric load is applied at one conjunction of web on the top slab at mid-span section.
(a) Cross section
(b) Corrugated steel web
Based on equations for warping torsional and distortional stress, the warping normal stress and shear stress of the composite girder are shown in Figure 8. To verify the results, stress results from [6, 20] and stress calculated by FEA are presented separately in brackets and .
(a) Warping torsional stress
(b) Distortional stress
Figure 8 shows that the above equations of warping normal stress and shear stress are verified by FEA results again. The warping torsional shear stress calculated by equations in this paper is 80% smaller than that in [6, 20], because the accordion effect of corrugated steel web is considered to get the generalized area moment of inertia, . It should be also noted that the additional shear stress due to restrained torsion has the opposite direction to the free torsional stress, which may reduce the total shear stress. The shear stress due to warping torsion is smaller than that by distortion. Additionally, the warping normal stress in bottom slab is much bigger than stress in top slab for trapezoidal sections. Besides, the normal stress in bottom slab caused by warping torsion has the opposite direction to stress due to distortion.
On the corresponding symmetric load condition, the flexural normal stress in bottom slab is 1.91 MPa and the flexural shear stress in webs is 4.52 MPa. The above results show that the total warping stress in this trapezoidal composite box girder is small, but the total shear stress due to torsion and distortion is 30 percent of flexural shear stress and must be calculated properly.
6. Parametric Analysis
According to the above two examples, the influence of torsion and distortion on stress is different if dimensional parameters of composite box girder change. When calculating the warping torsional and distortional stress, the warping rigidity and transverse form rigidity are not easy to be derived. Besides the fact that the equations for normal and shear stress are complex, it is difficult to find the explicit relationship between girder dimension parameters and warping torsional and distortional rigidity or stress. So, the parametric analysis needs to be done to study how these parameters influence the warping torsion and distortion behaviors for composite girder with corrugated steel webs.
The composite girder dimensions in [21] were used to analyze the torsional and distortional stress under eccentric load at mid-span section under the simple supported boundary condition. The following analysis covers parameters including the span length and section height of girder, the width and thickness of top and bottom flange, and the dimensions of corrugated steel web. Only one parameter is changed in each analysis. The normal stress on the bottom slab at the conjunction of web and web shear stress at mid-span section were studied to compare the effect of warping torsion and distortion.
Six coefficients were defined to analyze the influence of warping torsion and distortion on the normal stress and shear stress, which arewhere , , are normal stress separately caused by bending, torsional warping, and distortional warping; , , , , are shear stress separately caused by bending, free torsion, warping torsion, pure distortion, and distortional addition.
The coefficients , imply the influence of warping torsion and distortion on normal stress, and , imply the influence on shear stress. The coefficients , imply the additional shear stress due to changing of normal stress caused by warping torsion and distortion.
6.1. Effect of Span Length
When the span length is separately 30, 40, 50, 70, 90, 110, 130, and 150 m, the variation of stress coefficients is given in Figure 9. The span length had obvious influence only on both warping normal stress but not shear stress. The effects of torsion and distortion were reduced if the girder span got longer; the coefficient changed from 0.52 to 0.10, and changed from 1.42 to 0.26. On most conditions of girder span within 80 m, the torsional and distortional warping stress are more than 30% of flexural normal stress. So the warping normal stress needs to be considered in normal stress check in structural design.
6.2. Effect of Section Height
The section height is generally 1/40 to 1/25 of the span length. Here, the height is set to be 2.5, 3.8, 5.1, 6.4, 7.7, 9.0, 10.3, 11.6, and 12.9 m. The variation of stress coefficients for different height is given in Figure 10. As the girder height got bigger, the coefficients of normal stress got bigger obviously. The coefficient changed from 0.19 to 0.66 and changed from 0.72 to 1.48. The shear stress coefficients and had small change by less than 0.05. It shows that the influence of section height is much more obvious to warping normal stress than shear stress.
6.3. Effect of Top Flange Width
In following analysis, the girder section is rectangular whose corrugated steel webs are vertical. When the top flange width, measured width between webs, is separately 4, 5, 6, 7, and 8 m, the variation of stress coefficients is given in Figure 11. Only the coefficient nearly remained constant as the chamber width grows, but other coefficients changed. grew from 0.95 to 1.39. The coefficients ~ changed separately from 0.28, 0.39, −0.44, −0.22 to 0.41, 0.46, −0.17, and −0.09. Thus, the influence of top flange width is not ignorable.
6.4. Effect of Cantilever Flange Width
The cantilever flange width (flange width between a web and a free edge) is usually designed within one-half of the top flange width between webs. As the top flange width is 6.0 m, the cantilever flange width is accordingly proposed to be 0, 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 m, respectively. The variation of stress coefficients is given in Figure 12. As the cantilever flange width got bigger, the normal stress coefficients and changed in a limit degree, while the shear stress coefficients changed obviously. The coefficients and decreased separately from 0.67, 0.59 to 0.27, and 0.39, while and changed separately from 0.34, 0.17 to −0.45, −0.22. It should be noted that the direction of shear stress due to warping torsion and distortion may change as cantilever flange width varied.
6.5. Effect of Bottom Flange Width
When the width of top flange remain unchanged, only the bottom flange width is set to be 3, 3.5, 4, 4.5, 5, 5.5, and 6 m. The variation of stress coefficients is given in Figure 13. The coefficients and nearly remained constant as the bottom flange width grew, but other coefficients changed obviously. The normal stress coefficients and grew separately from 0.69, 0.10 to 1.26, 0.43. The coefficients and decreased separately from −0.46, −0.70 to −0.28, −0.13. So, the influence of bottom flange width is not ignorable.
6.6. Effect of Flange Thickness
When the top and bottom flange thickness is separately 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50 m, the variation of stress coefficients is given in Figures 14 and 15. As shown below, the coefficients and in Figure 15 and and in Figure 14 changed less than 15%. Other coefficients nearly remained constant as flange thickness varied. It is meant that the stress is almost independent of flange thickness, except that the stress of the slab whose thickness is changed.
6.7. Effect of Web Thickness
Assuming that the general construction of corrugated steel web is unchanged, the web thickness is set to be 0.006, 0.010, 0.014, 0.018, 0.022, 0.026, 0.030, 0.034, and 0.038 m. The variation of stress coefficients is given in Figure 16. The coefficients and decreased separately from 0.59, 1.34 to 0.02, 1.01, while and increase separately from 0.32, 0.41 to 0.49, 0.47. and changed separately from −0.37, −0.18 to −0.02, −0.05. Results show that web thickness has obvious influence on warping normal stress.
6.8. Effect of Web Construction
As different constructions of corrugated steel web may be adopted in the composite girder, the out-plane bending moment of inertia, , may be different. So the influence of was studied, while the web thickness remained unchanged. The common used constructions of corrugated steel web are listed in Table 1. The variation of stress coefficients is given in Figure 17. It shows the coefficients nearly remained constant, which means the stress due to warping torsion and distortion is independent of the web construction.
7. Conclusions
Assuming that the longitudinal axial stiffness of corrugated steel web can be neglected, the formulas for warping torsional and distortional stress were deduced according to the second theory of Umanskii. Based on the revised formulas, this paper has presented a theoretical study on stress of the simple supported composite box girder under eccentric loads. And the influence of varied parameters were studied in detail, including the span length and section height of girder, the width and thickness of top and bottom flange, and the dimensions of corrugated steel web. The following conclusions are obtained.
The accordion of corrugated steel webs should be considered properly in torsion and distortion analysis of composite box girder. The formulas of warping normal stress and shear stress were revised for analyzing the warping torsion and distortion of the composite box girder of corrugated steel webs. Comparing with the FEA results, the revised formulas are proved to be more effective.
The results show that the influences of warping torsion and distortion are more significant if the girder is shorter, which is easily acceptable. However, even though the whole girder rigidity may increase by enlarging the section dimensions, the warping normal stress and shear stress may not decrease at all. According to studies in this paper, in order to reduce the warping stress ratio, it is suggested to increase the girder span length and the thickness of corrugated steel web height or reduce the girder height and the top, bottom, and cantilever slab width. To reduce the web shear stress ratio, it is suggested to increase the cantilever flange width flange width or reduce the bottom slab width. The thickness of concrete flange and the web construction have a limited impact on both kinds of stress.
As for common composite box girders with corrugated steel webs under eccentric loads, the warping torsion and distortion may cause considerable warping normal stress of shear stress. The total warping normal stress may be as big as flexural normal stress, and the total shear stress is usually 30 to 50 percent of flexural shear stress. Accordingly, it is recommended that both the warping normal stress of concrete flanges and shear stress of corrugated steel webs should be included in stress calculating, when eccentric loads are imposed on composited girders.
The FEA results showed a special feature that the shear stress were different in flat panel and inclined panel of corrugated steel webs, as illustrated in Figure 6. So the detailed mechanical mechanism of corrugated steel webs under different shearing conditions should be studied in future.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The work described herein is part of a research project funded by the National Natural Science Foundation of China (no. 51308053 and no. 51608069) and Chinese Universities Scientific Fund of Chang’an University (no. 300102288110).