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A softened membrane model for singlebox multicell composite boxgirders with corrugated steel webs under pure torsion
Advances in Bridge Engineering volume 3, Article number: 2 (2022)
Abstract
To analyze the nonlinear torsional behavior of singlebox multicell prestressed concrete (PC) or reinforced concrete (RC) composite boxgirders with corrugated steel webs (BGCSWs), a new theoretical model called unified softened membrane model for torsion (USMMT) is proposed in this study. It is developed from the softened membrane model for torsion which was formerly proposed for the torsional analysis of singlebox singlecell PC/RC composite BGCSWs. This proposed model incorporates the different contributions of inner and outer corrugated steel webs (CSWs) in the multicell BGCSW to the torsional capacity by introducing a rational relationship of shear strains between inner CSWs and outer CSWs. The satisfactory accuracy of the proposed model is obtained when comparing with the experimental and the finite element analysis results, including the torquetwist curves and the smeared shear strains in concrete slabs and CSWs in the fullrange torsional analysis. The comparison indicates that the proposed model is capable of predicting the overall torsional behavior of singlebox multicell BGCSWs.
Introduction
The composite boxgirders with corrugated steel webs (BGCSWs) have been extensively applied in bridge engineering (Zhu 2020; Corrugated SteelWeb Bridge Association 2021) since the first bridge was built in France in1986 (Cheyrezy and Combault 1990), for which they show many remarkable advantages over the concrete boxgirders, such as light weight, high prestress efficiency and few web cracks (Jiang et al. 2015; He et al. 2021). Currently, this kind of steelconcrete composite structures have been used in the design of multicell boxgirders or curved boxgirders (Liu et al. 2021) for satisfying the rapid increasing of traffic demand. But on the other hand, the reduction of torsional stiffness of such structures is about 60–70% of that of the conventional concrete boxgirders (Prestressed Concrete Technology Association 2005), leading to a great attention to the research of the torsional behavior of BGCSWs until now (Li 2017).
In recent years, some researchers have introduced the unified theory of concrete structures (Hsu and Mo 2010) into the nonlinear torsional analysis of BGCSWs. These theoretical models can be divided into two types according to their characteristics (Shen et al. 2018a, b): rotating angle torsional theory and fixed angle torsional theory. The typical development path of the application of unified theory of concrete structures into the nonlinear torsional analysis of BGCSWs is shown in Fig. 1.
Review of rotating angle torsional theory for BGCSWs
In the beginning, the rotatingangle softened truss model for torsion (RASTMT) in prestressed concrete (PC) structures (Hsu and Mo 1985) was extended to the nonlinear torsional analysis of PC composite BGCSWs (Mo et al. 2000) owing to its features of high solving efficiency and simple mechanics concept. It assumed the shear strains in CSWs to be the same as those in concrete slabs to consider the torsional contribution of CSWs. Then a stepbystep procedure was developed to make a torsional design of PC composite BGCSWs (Mo and Fan 2006). Some researchers (Nie and Tang 2007a) believed that the shear flows in CSWs and concrete slabs should be the same during the twisting of BGCSWs. The RASTMT was also employed in the torsional analysis of BGCSWs in the postcracking stage, combining a correction in the precracking stage (Shen et al. 2017). Note that the RASTMT is unable to predict the torsional behavior before the cracking of concrete, a tensionstiffened softened truss model for torsion (TSSTMT) was accordingly developed by considering the tension stiffening effect of concrete (Ko et al. 2013). To better predict the torsional behavior at the cracking stage and ultimate stage, the new average stress coefficients with three stages and a new shear strain relationship between CSW and concrete slab were considered in a modified rotating angle softened truss model (MRASTMT) (Zhu et al. 2020b).
To predict the torsional behavior of singlebox multicell BGCSWs, a unified softened truss model for torsion (USTMT) (Shen et al. 2018c) was proposed on the basis of RASTMT by introducing the new shear strain relationships between the CSWs of multicell BGCSWs (Shen et al. 2018d). Subsequently, the new threestage average stress coefficients including initial stress produced by prestress and a new shear strain relationship between inner and outer CSWs were introduced to predict the torsional behavior of multicell BGCSWs (Zhu et al. 2021). With the rational modifications, the new model, unified RASTMT (URASTMT), can better predict the torsional behavior in the precracking stage.
Review of fixed angle torsional theory for BGCSWs
While the modified USTMT can well predict the overall mechanical performance of BGCSWs under torsion, such as the torque, twist, and smeared shear strain, it is incapable of explaining the contribution of shear stress on the interface of concrete cracks. Hence, the fixed angle softened truss model for torsion (FASTMT) in reinforced concrete (RC) structures (Nie and Tang 2007b) was then extended to the torsional analysis of PC composite BGCSWs (Ding et al. 2013). Furthermore, to describe the biaxial strains which consider the Poison effect (Hsu/Zhu ratios), the fixedangle softened membrane model for torsion (SMMT) (Jeng and Hsu 2009) was introduced to the torsional analysis of singlebox singlecell PC/RC composite BGCSWs (Shen et al. 2018a, b). In the same year, different from assuming that the shear strains in the concrete slabs and CSWs before the yielding of CSWs, the continuous shear flow condition was employed to extend the SMMT to the torsional analysis of PC composite BGCSWs (Zhou et al. 2018). Then the algorithms with a new solution process were developed in the improved SMMT (ISMMT) (Zhou et al. 2019), including a general algorithm for the fullrange twisting and a simplified algorithm for the stage when the girder was all in the elastic state. These research has shown that the SMMT theory can produce better accuracy in describing the real stress state of concrete in the torsional analysis of BGCSWs than those models based on rotatingangle theory.
Research motivation and innovations
From above literature review, it can be obviously found that both rotatingangle STMT theory and SMMT theory can well predict the overall mechanical performance of singlebox multicell BGCSWs under torsion. But the rotatingangle STMT theory can neither reflect the shear stress on the interface of concrete cracks nor describe the biaxial strains in concrete slabs which considers the Poison effect (Hsu/Zhu ratios), whereas, by contrast, the SMMT theory can almost perfectly address the two problems.
The main objective of this study is to extend the SMMT in singlecell BGCSWs to the torsional analysis of multicell BGCSWs. A unified method (USMMT) is proposed and an algorithm is compiled to predict the torsional behavior of both singlecell and multicell PC/RC composite BGCSWs. A rational relationship of shear strains between inner CSWs and outer CSWs should also be introduced in the proposed model. The symbols used in the present study are defined in the notation list of Appendix 1.
Theoretical model for multicell BGCSWs under torsion
Existing equivalent method for singlebox multicell boxgirder
In previous studies (Fu and Yang 1996; Fu and Tang 2001; Allawi et al. 2017), it was generally assumed that the singlebox multicell box girder under pure torsion could be decomposed into multiple cells with the same twist, so that torsional analysis could be performed separately. This assumption is reasonable in the elastic torsional analysis, but it will underestimate the torsional resistance of the boxgirder in the nonlinear torsional analysis. For example, in a singlebox triplecell boxgirder as shown in Fig. 2, the applied torque is carried by the three cells together and can be expressed as \(T={\sum}_{i=1}^32{A}_{0i}{q}_i\). However, when the areas enclosed by the centerline of the shear flow of the three cells are equal (A_{01} = A_{02} = A_{03}), the shear flows are also equal (q_{1} = q_{2} = q_{3}), thus giving \(T={\sum}_{i=1}^32{A}_{0i}{q}_i=2{\sum}_{i=1}^3{A}_{0i}q=2{A}_0q\). This formulas shows that the singlebox triplecell boxgirder can be equivalent to a singlebox singlecell boxgirder with the same outer contour size.
According to the analysis in previous research (Shen et al. 2018d), this simplified calculation method is reasonable for elastic torsional analysis. In the elastic phase, each cell can work together with a coordinated deformation. The shear strains in concrete slabs decrease from the middle of the cross section to both sides, and the shear strains gradually decrease from the outer CSWs to the inner CSWs. At this stage, the shear strains of the inner corrugated steel webs are small and can be ignored. However, in the nonlinear stage after concrete cracking, the shear deformations of the inner webs will also increase greatly in addition to the significant changes in the shear deformation of the outer CSWs. Moreover, the shear deformations of the inner CSWs cannot be ignored in the nonlinear stage, since the outer CSWs will be destroyed first and gradually lose the bearing capacity, and then the inner CSWs will carry a greater load until the failure of the specimens occurs. Therefore, the torsional capacity of the boxgirder will be underestimated if only considering the contribution of the outer CSWs to it.
Rational simplified calculation method for multicell BGCSWs under torsion
In the present study, the calculation method for singlebox multicell BGCSWs under torsion is completely different from the previous simplified method for multicell concrete boxgirders. The rectangular BGCSW with 2n (or 2n1) cells was decomposed into n (n = 1, 2, 3, 4, 5) pieces of singlebox singlecell box girders with the same torsion center as shown in Fig. 3, and then the torsional analysis on each of them were performed. Note that the shear strains in concrete slabs and CSWs in every independent singlecell box can be assumed to be equal before the yielding of CSWs according to the previous research (Mo et al. 2000). Supposing the shear strains (shear stresses) in concrete slab and CSWs of these independent singlecell boxes “Boxi” (i = 1, 2, 3, ..., n) are γ_{lti} (τ_{lti}) and γ_{wi} (τ_{wi}) respectively, the shear strain and shear stress in CSWs of “Boxi” can be obtained as (Shen et al. 2018c):
where R_{γi} is the ratio of the shear strain of CSWs in Boxi (γ_{wi}) to the shear train in outermost webs (γ_{w1}). It can be calculated by a fitting formula relating to R_{di} (Shen et al. 2018d), the ratio of the distance between the torsional center and CSWs in Boxi, d_{i}, to that in the Box1, d_{1}:
It should be noted that Eq. (3) is a fitting formula for simplification, it needs more investigation to better describe the relations among the shear strain of CSWs.
For simplification, the nonuniform shear strain of concrete slab in multicell BGCSW is represented by the smeared shear strain of concrete in the Box1 decomposed from multicell BGCSW. It needs to note that this simplification will underestimate the shear strains in concrete slabs, but it can be offset by the overestimation of shear strains in the outermost webs. Then the applied torque of multicell BGCSW T_{m} can be expressed by a unified formula:
where m is a subscript that represents the number of cells in the multicell BGCSW; T_{f1} is the torque resisted by the concrete slabs in Box1; T_{wi} is the torque resisted by the webs in Boxi. They can be calculated by:
USMMT for multicell BGCSWs under torsion
The advanced SMMT models for singlebox singlecell PC/RC composite BGCSWs under torsion have been proposed by the same authors (Shen et al. 2018a, b). Combining the above calculation formula, the USMMT for torsional analysis of singlebox multicell PC/RC composite BGCSWs can be obtained, and the solution process is shown as Fig. 4. The relevant equations which have been deduced in the authors’ previous researches (Shen et al. 2018a, b) are listed in Appendix 2 to reduce the length of the main text. From the equations listed in the analysis process of the USMMT for the singlebox multicell PC composite BGCSWs, it can be seen that there are a total of 32 unknown parameters. (ε_{2}, ε_{1}, γ_{21}, ε_{l}, ε_{t}, γ_{lt}, (v_{12})_{torsion}, \({\overline{\varepsilon}}_1\), \({\overline{\varepsilon}}_2\), \({\overline{\varepsilon}}_l\), \({\overline{\varepsilon}}_t\), \({\overline{\varepsilon}}_{2s}\), \({\overline{\varepsilon}}_{1s}\), t_{d}, A_{0f}, p_{0f}, k_{c}, β, ζ, \({\sigma}_2^c\), \({\sigma}_1^c\), \({\tau}_{21}^c\), \({f}_s^c\), f_{ps}, θ_{1}, τ_{lt1}, T_{f1}, R_{γi}, γ_{wi}, τ_{wi}, T_{wi}, T_{m}), 31 effective Eqs. ((1) ~ (6), (9) ~ (40)). In order to solve these unknown parameters, a series of values of ε_{2} should be first selected, and then the remaining 31 unknown parameters can be solved with 31 effective equations using the trial and error method. For the nonprestressed singlebox multicell RC composite BGCSWs, the prestress values just need to be set as zero in this algorithm, which means no prestress is applied. The specific solution process is as follows:

(1)
Enter the given parameters, including the cell number of singlebox multicell BGCSW (m = 2n) or (2n1), i = 1, 2, 3, …, n), geometric parameters (b_{1}, h_{1},_{,}R_{di}, A_{l}, A_{t}, A_{c}, A_{ps}, s, t_{h}, t_{w}, a_{w}, b_{w}, c_{w}), material parameters (f_{c}′, ε_{0}, E_{s}, E_{c}, G_{s}, E_{ps}, E′_{ps}, f_{ty}, f_{ly}, τ_{wy}, f_{pu}, f_{pi}).

(2)
Select the initial value of ε_{2}, which can be varied from 0 to –0.0018 monotonically by an increment of − 0.0000001.

(3)
Analyze the torsional behavior of the concrete slabs and CSWs in Box1.

Step 1. Assume γ_{21}.

Step 2. Assume ε_{1}.

Step 3. Calculate ε_{l}, ε_{t}, γ_{lt}, (v_{12})_{torsion} according to Eqs. (10) ~ (13).

Step 4. Calculate \({\overline{\varepsilon}}_1\), \({\overline{\varepsilon}}_2\), \({\overline{\varepsilon}}_l\), \({\overline{\varepsilon}}_t\), \({\overline{\varepsilon}}_{2s}\), \({\overline{\varepsilon}}_{1s}\) according to Eqs. (14) ~ (19).

Step 5. Calculate t_{d}, A_{0f}, p_{0f}, β according to Eqs. (20), (22), (23), (29).

Step 6. Determine whether the BGCSW is prestressed. If yes, proceed directly to the next step; if not, skip to step 10.

Step 7. Calculate ζ, \({\sigma}_2^c\), \({\sigma}_1^c\), \({\tau}_{21}^c\), \({f}_s^c\), f_{ps} according to Eqs. (24), (26), (27), (30), (32), (34), (35).

Step 8. Determine whether the convergence criteria (37) is satisfied. If yes, proceed directly to the next step, if not, return to step 2.

Step 9. Determine whether the convergence criteria (38) is satisfied. If yes, skip to step 13, if not, return to step 1.

Step 10. Calculate ζ, k_{c}, k_{t},\({\sigma}_2^c\), \({\sigma}_1^c\), \({\tau}_{21}^c\), f_{s} according to Eqs. (25), (26), (28), (31), (33), (36).

Step 11. Determine whether the convergence criteria (37) is satisfied, if yes, proceed directly to the next calculation, if not, return to step 2;

Step 12. Determine whether the convergence criteria (37) is satisfied, if yes, skip to step 13, if not, return to step 1;

Step 13. Calculate θ_{1} according to Eq. (21);

Step 14. Calculate τ_{lt1}, T_{f1}, γ_{w1}, τ_{w1}, T_{w1} according to Eqs. (1), (2), (5), (6), (9).


(4)
Analyze the torsional mechanical behavior of the corrugated steel web in the range of 2 ≤ i ≤ n.

(5)
Calculate the total torque T_{m} according to Eq. (4).

(6)
Determine whether it meets ε_{2} > 0.0018. If yes, output the required data and stop the calculation. If not, return to section (2) and select the next value of ε_{2}.
Using above method, the required parameters can be obtained through continuous iterative calculations in an algorithm. Obviously, the USMMT can not only predict the torsional behavior of a singlebox multicell BGCSW, but also the torsional behavior of a singlebox singlecell BGCSW. What’s more, it can also both predict the torsional behavior of the PC/RC composite BGCSWs. In the calculation module with prestress, the effect of initial strains and stresses on the constitutive laws of concrete is considered.
Model verification with experimental results
Experimental specimens
In order to validate the proposed unified model, the experimental data of six pieces of singlebox multicell BGCSWs available from literatures are collected. These specimens were tested under pure torsion, including two singlebox doublecell BGCSWs named T2C (Shen et al. 2018d) and T2CS (Shen et al. 2018a), two singlebox triplecell BGCSWs named T3C (Shen et al. 2018d) and S3 (Zhu et al. 2020c), two singlebox fivecell BGCSWs named S5 and SU5 (Zhu et al. 2020c).
Moreover, to better understand the torsional behavior of singlebox multicell BGCSWs, a specimen named T3CS was also tested by the authors after the torsion experiment of specimen T3C. The test situation of T3CS, including the physical parameters and the experimental procedure, is almost the same to that of T3C except the different yield tensile stress of CSW, thus the relevant information of this specimen will not be repeated in this study.
The geometric and material parameters of all specimens are shown in Table 1. It should be pointed out that the number of tests used to verify the USMMT in this study is relatively small and more torsional test of the specimens with more cells and larger sizes are still needed.
Torquetwist curves
Figure 5 shows the comparisons of the torquetwist curves of all test beams under pure torsion between the experimental results and the calculated results from the USMMT. In these tests, the twists were measured by several displacement meters (DMs) and inclinometers (IMs) respectively. It can be seen that the torquetwist curves calculated by the USMMT are in good agreement with the test results on the overall trend. Furthermore, the calculated and tested values of the torques and twists at the characteristic points of concrete cracking, CSW yielding and ultimate stage of all test beams are listed in Tables 2 and 3. It can be seen that the predicted values of torques and twists at the characteristic points fit the test results well except for the large difference in the values of twists at some characteristic points. Therefore, the comparison shows that the proposed model in this study can accurately predict the test results of singlebox multicell PC/RC composite BGCSWs before the ultimate torque is reached.
As shown in Tables 2 and 3, the proposed model has a better prediction on the torques than that on the twists. This is due to the fact that the torques are accurately calculated by Eqs. (5) and (6), however, the twists are roughly calculated by Eq. (21) for analogy to thinwalled circular rod.
Shear strains in concrete slabs and CSWs
Figures 6 and 7 respectively show the comparisons of the shear strains in concrete slabs and CSWs of the test beam under pure torsion between the experimental results and the calculated results from the USMMT. It should be noted that the shear strains calculated by the USMMT are the average shear strains in concrete slabs and CSWs, whereas the shear strains measured from the test are the shear strains at each measuring point. The shear strains in this study are taken as the absolute values, regardless of the positive and negative values.
In Fig. 6, it can be seen that the calculated results from the USMMT agree well with the test results before the cracking of concrete. Furthermore, the calculated results from the USMMT are basically consistent with the test results after the cracking of concrete in the overall trend, and the predicted values of average shear strains lie between the minimum and maximum tested concrete shear strains at the measuring points.
In Fig. 7, it can be seen that the shear strains of the CSWs are small before the yielding of CSWs, which is similar to the shear strains in concrete slabs. The difference is small between the shear strains measured in the test and the average shear strains in CSWs obtained from Eq. (1). Since the average shear strains in CSWs are assumed to be equal to the average shear strains in concrete slabs in Eq. (1), the curves between the torques and average shear strains in CSWs calculated by the USMMT, however, are broken lines before the yielding of CSWs, which is different from the test results. After the yielding of CSWs, the second assumption in Eq. (1) will cause the average shear strains in CSWs to have a large abrupt change, which is also different from the test results. Therefore, the assumptions in Eq. (1) still needs to be optimized, but its accuracy needs to be improved though the average shear strains in CSWs obtained from Eq. (1) can roughly reflect the change rules of the average shear strains in CSWs measured in the test. Moreover, the acceptable agreements between the measured shear strains in inner/outer CSWs and the predicted ones demonstrate that the shear strains relationship between the inner CSWs and outer CSWs in Eq. (3) is reasonable.
To sum up, it is reasonable to use the two assumptions in Eq. (1) of the USMMT to express the shear strain relationships between the CSWs and concrete slabs. It can roughly reflect the change rules of average shear strains in concrete slabs and CSWs.
Model verification with FEA results
In the previous research (Shen et al. 2018d), a finite element model of the specimen T3C was established, and its accuracy was verified by test results. Based on the finite element model, the effect of the lateral position distribution of the CSWs and the number of cells on the torsional bearing capacity of the BGCSWs was explored by a parameter analysis. In order to verify the accuracy and applicability of the proposed USMMT, this section will perform a corresponding parameter analysis and compare the calculated results from USMMT with the FEA results.
Table 4 shows the comparisons of ultimate torques of the singlebox multicell BGCSWs between the calculated results from the USMMT and the FEA results listed in the literature (Shen et al. 2018d). Furthermore, the ultimate torques from the calculated results of the USMMT and the FEA results are shown in Fig. 8 to display the results more vividly. It can be seen from Table 4 and Fig. 8 that the difference between the predicted values of ultimate torques from USMMT and the ones from FEA is within the allowable range of 10% in engineering for the most of BGCSWs. The mean value of the ratios between the predicted values from USMMT and the ones from FEA in all BGCSWs is 1.029 with the standard deviation of 0.076. Therefore, the unified SMMT model in this study can accurately predict the torsional bearing capacity of the BGCSWs with no more than 10 cells.
Further, the comparisons of the torquetwist curves of BGCSWs among the calculated results from the USMMT, the FEA results and the experimental results are shown in Fig. 9. These BGCSWs have an odd number of cells and an arrangement of CSWs at equal intervals in the transverse direction. It can be seen that the results of the torquetwist curves of the singlebox multicell BGCSWs predicted by the USMMT are in good agreement with the ones from the FEA results and the test results. In addition, the calculated results from the USMMT show that the torquetwist curve of each singlebox multicell BGCSWs has little difference in the elastic stage before the cracking of concrete, and the difference of torques at the cracking of concrete is also very small. However, the total torques increase significantly when the number of cells in the boxgirders increases after the cracking of concrete. This shows that the inner CSWs have little effect on the total torque before the cracking of concrete, whereas they have a significant influence on the total torque after the cracking of concrete. This change rule is basically consistent with the change rule obtained from the FEA. Therefore, the treatment of the inner CSWs by the USMMT model in this study is reasonable. Equation (4) can basically reflect the torsional contribution of the inner CSWs to the total torque of multicell BGCSWs.
From above analysis, it can be seen that the proposed model is rational and can be used to predict the overall mechanical properties of the singlebox multicell PC/RC composite BGCSWs under the action of applied torque, such as the torques, twists, shear strains and so on.
Conclusions
In this study, the analytical model SMMT, formerly developed for the singlebox singlecell PC/RC composite BGCSWs under pure torsion, is extended to the torsional analysis of singlebox multicell PC/ RC composite BGCSWs by incorporating the different contributions of the inner and outer CSWs. The proposed model, USMMT for singlebox multicell PC/RC composite BGCSWs, is verified by the experimental results and the FEA results. The main conclusions can be drawn from the results as follows:

(1)
The predicted results from the USMMT agree well with the experimental results and FEA results available in the literature, indicating that the proposed model is able to predict the fullrange torsional behavior of singlebox multicell BGCSWs, such as the torquetwist curves and the torques at the characteristic points.

(2)
The proposed model is a unified model that is suitable for both PC and RC composite BGCSWs. It considers the effect of the initial stresses and strains on the constitutive laws of concrete in the calculation module with prestress. As a unified model, it also can predict the torsional behavior of singlecell BGCSWs.

(3)
The average shear strains in concrete slabs and CSWs calculated by the USMMT reflect the change rules of shear strains from the test, which validates the reasonability of assumption on the shear strain relations between concrete slabs and CSWs, as well as the applicability of shear strains relationship formula between inner CSWs and outer CSWs.
It should be admitted that the proposed model is verified by a small number of test specimens and finite element models of the multicell BGCSWs with no more than 10 cells, it should be validated by more tests. The shear strain relation between concrete slabs and CSWs and the one between inner CSWs and outer CSWs are still need to be optimized and improved in the future work. This work mainly focuses on a softened membrane model using unified theory, it is suggested by the authors to propose simplified explicit equations in the future to predict the torque and twist for singlebox multicell composite BGCSWs under pure torsion for practical design.
Availability of data and materials
The data and materials in current study are available from the corresponding author on reasonable request.
Abbreviations
 PC:

Prestressed concrete
 RC:

Reinforced concrete
 FE:

Finite element
 FEA:

Finite element analysis
 BGCSWs:

Boxgirders with corrugated steel webs
 USMMT:

Unified softened membrane model for torsion
 CSWs:

Corrugated steel webs
 RASTMT:

Rotatingangle softened truss model for torsion
 TSSTMT:

Tensionstiffened softened truss model for torsion
 MRASTMT:

Modified rotating angle softened truss model
 USTMT:

Unified softened truss model for torsion
 URASTMT:

Unified rotatingangle softened truss model for torsion
 FASTMT:

Fixed angle softened truss model for torsion
 SMMT:

Softened membrane model for torsion
 ISMMT:

Improved softened membrane model for torsion
 DM:

Displacement meters
 IM:

Inclinometers
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Acknowledgements
The authors would like to thank Pro. Y.L. Mo and Pro. T.T.C. Hsu for their guidance regarding the SMMT theory. All their supports are greatly appreciated.
Funding
This study is supported by the National Natural Science Foundation of China (No. 52008095).
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Kongjian Shen: Developing the USMMT theory and corresponding algorithm; performing FEA and parametric study, comparative analysis, writing original draft. Shui Wan: Providing guidance in methodology development, financial supports. Yingbo Zhu: Providing the experimental data and substantially revising the draft. Debao Lyu: Revising and editing the draft. Zhiqiang Wu: Revising and editing the draft. The authors read and approved the final manuscript.
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Appendices
Appendix 1
T  applied torque of the BGCSW 
T_{f}, T_{w}  torques contributed by concrete slabs and CSWs, respectively 
θ  twist of the BGCSW (angle per unit length), the twist of the boxgirder is assumed to be represented by the twist of concrete slabs 
b  center distance between CSWs 
h  distance between the outer surfaces of upper and lower concrete slabs 
t_{h}, t_{w}  thickness of concrete slabs and CSWs, respectively 
t _{ d }  thickness of shear flow zone in concrete slabs, named effective thickness 
d _{ i }  the distance between the torsional center and CSWs in Boxi 
q,q_{i}  shear flows in the singlecell and ith cell boxgirder, respectively 
A_{0}, A_{0i}  areas enclosed by the centerline of shear flow, q and q_{i} respectively 
A_{0f}, A_{0w}  areas enclosed by the centerline of shear flow in concrete slabs and CSWs, respectively. For rectangular BGCSWs, A_{0f} = A_{0w} = b(ht_{d})/2 
p _{0f}  perimeter of the centerline of shear flow, for rectangular BGCSWs, p_{0f} = 2b 
a_{w}, b_{w}, c_{w}  lengths of the flat plate, the projection of inclined plate and the inclined plate, respectively, in one half of wavelength of CSW, l_{w} 
A _{ c }  area of reinforced concrete section, A_{c} = 2t_{h}b 
A _{ l }  total crosssection area of longitudinal (l) steel bars 
A _{ t }  crosssection area of one transverse (t) steel bar 
A _{ ps }  total crosssection area of prestressed tendons 
s  spacing of steel bars in the tdirection 
ρ _{ s }  ratios of steel bars in the l and t directions respectively, ρ_{s} can be ρ_{l} or ρ_{t}, ρ_{l} = A_{l}/(p_{0f}t_{d}), ρ_{t} = A_{t}/(t_{d}s) 
ρ_{li}, ρ_{pi}  ratios of steel bars and prestressed steel in the l direction with respect to the net area of concrete, respectively, ρ_{li} = A_{l}/(A_{c} − A_{l} − A_{ps}), ρ_{pi} = A_{ps}/(A_{c} − A_{l} − A_{ps}) 
ρ _{ ps }  ratio of prestressed tendons in the l direction, ρ_{ps} = A_{ps}/(p_{0f}t_{d}) 
η _{ w }  parameter of corrugation, η_{w} = (a_{w} + b_{w})/(a_{w} + c_{w}) 
μ  Poisson’ ratio of steel 
β  deviation angle 
ζ  softened coefficient of concrete in compression 
E_{c}, E_{s}, E_{ps}  Young’s modulus of concrete, steel bar and prestressing steel, respectively 
\({E}_c^{\prime }\), \({E}_c^{{\prime\prime} }\)  tensile modulus of concrete before and after the decompression of concrete, respectively 
E′ _{ ps }  initial tangent modulus of RambergOsgood curve 
G _{ s }  shear modulus of steel plate, G_{s} = E_{s}(1 + μ) 
G _{ e }  effective shear modulus of CSW, G_{e} = G_{s}η_{w} 
f_{c}′  cylinder compressive strength of concrete 
f _{ wy }  yield tensile stress of CSW 
f _{ s }  smeared (average) stresses of steel bars induced only by torsion in the l and t directions, respectively, f_{s} can be f_{l} or f_{t} 
f _{ li }  initial stress of longitudinal steel bars 
\({f}_s^c\)  smeared (average) stresses of steel bars induced by torsion and prestressing in the l and t directions, respectively, \({f}_s^c\) can be \({f}_l^c\) or \({f}_t^c\) 
f _{ sy }  smeared (average) yield stresses of steel bars in l and t directions, respectively, f_{sy} can be f_{ly} or f_{ty} 
f _{ ps }  smeared (average) stress of prestressed tendon 
f _{ pi }  initial stress of longitudinal prestressed tendon 
f _{ pu }  ultimate strength of prestressing steel 
\({\sigma}_1^c,\kern0.4em {\sigma}_2^c\)  smeared (average) normal stresses in the 1 and 2directions, respectively, considering the effect of strain gradient 
σ _{ ci }  initial compressive stress of concrete caused by prestressing, \({\sigma}_{ci}={E}_c{\overline{\varepsilon}}_{li}\) 
\({\tau}_{21}^c\)  smeared (average) shear stress of concrete in 21coordinate 
τ_{lt}, τ_{w}  smeared (average) shear stress in concrete slabs and CSWs, respectively 
τ _{ wy }  yield shear stress of CSW, \({\tau}_{wy}={f}_{wy}/\sqrt{3}\) 
ε_{cr}, f_{cr}  cracking tensile strain of concrete and its corresponding tensile stress 
ε _{0}  compressive strain at peak concrete strength f_{c}′, taken as − 0.002 
ε_{l}, ε_{t}  smeared (average) biaxial strains in the l and t directions, respectively 
ε _{ sy }  smeared (average) biaxial yield strains of steel bars in the l and t directions, respectively, ε_{sy} can be ε_{ly} or ε_{ty} 
ε _{ sf }  smeared (average) biaxial strain of steel bars which yield first, ε_{sf} can be ε_{l} or ε_{t} 
\({\overline{\varepsilon}}_{sn}\)  smeared (average) uniaxial yield strain of the steel bars which yield first, \({\overline{\varepsilon}}_{sn}\) can be \({\overline{\varepsilon}}_{ln}\) or \({\overline{\varepsilon}}_{tn}\), \({\overline{\varepsilon}}_{sn}=\left(0.932{B}_s\right){\varepsilon}_{sy}\kern0.1em\) 
\({\overline{\varepsilon}}_s\)  smeared (average) uniaxial strain induced only by torsion in the l and tdirections, respectively, \({\overline{\varepsilon}}_s\) can be \({\overline{\varepsilon}}_l\) or \(\kern0.2em {\overline{\varepsilon}}_t\) 
\({\overline{\varepsilon}}_s^c\)  smeared (average) uniaxial strain induced by torsion and prestressing in the l and tdirection, respectively, \({\overline{\varepsilon}}_s^c\) can be \({\overline{\varepsilon}}_l^c\) or \({\overline{\varepsilon}}_t^c\) 
ε_{2}, ε_{1}  smeared (average) biaxial strains in the 2 and 1 directions, respectively 
\({\overline{\varepsilon}}_{1s},{\overline{\varepsilon}}_{2s}\)  maximum uniaxial strains in the 1 and 2directions, respectively 
\(\kern0.1em {\overline{\varepsilon}}_{2s}^c\)  maximum uniaxial strains in the 2direction induced by torsion and prestressing 
\({\overline{\varepsilon}}_1,{\overline{\varepsilon}}_2\)  smeared (average) uniaxial strain in the 1 and 2directions, respectively 
\({\overline{\varepsilon}}_1^c\)  smeared (average) uniaxial strain in the 1direction induced by torsion and prestressing 
\({\overline{\varepsilon}}_{ps}\)  uniaxial strain of prestressing steel 
\({\overline{\varepsilon}}_{pi}\)  initial strain of prestressing steel after loss, \({\overline{\varepsilon}}_{pi}={f}_{pi}/{E}_{ps}\) 
\({\overline{\varepsilon}}_{li}\)  initial strain of steel bars in the ldirection, \(\kern0.1em {\overline{\varepsilon}}_{li}={A}_{ps}{f}_{pi}/\left[{A}_l{E}_s+\left({A}_c{A}_l{A}_{ps}\right){E}_c\right]\) 
\({\overline{\varepsilon}}_{2i},\kern0.3em {\overline{\varepsilon}}_{1i}\)  initial uniaxial strain in the 2–1 coordinate 
\({\overline{\varepsilon}}_{cx}\)  extra strain at the end of decompression of concrete 
γ _{ w }  smeared (average) shear strain of CSW 
γ _{ lt }  smeared (average) shear strain of concrete in the lt coordinate 
γ _{21}  smeared (average) shear strain of concrete in the 2–1 coordinate 
B _{ s }  parameter defined in the constitutive law of mild steel embedded in concrete, B_{s} can be B_{l} or B_{t}, B_{s} = (1/ρ_{s}) (f_{cr}/f_{sy})^{1.5} 
(v_{12})_{torsion}  Hsu/Zhu ratio used in the SMMT for torsion 
k _{ c }  ratio of the average compressive stress to the peak compressive stress 
k _{ t }  ratio of the average tensile stress to the peak tensile stress 
R _{ di }  ratio of the distance between the torsional center and CSWs in Boxi, d_{i}, to that in the Box1, d_{1} 
R _{ γi }  ratio of the shear strain of CSWs in Boxi, γ_{wi}, to the shear train in outermost webs, γ_{w1} 
Appendix 2
The following equations can be found in the literature (Shen et al. 2018a, b).

(1)
Inplane equilibriums in the l and t directions as shown in Fig. 10:

(2)
Compatibility equations for concrete slabs:

(3)
Effective thickness of the concrete slabs t_{d}:
where \(Q=2{\overline{\varepsilon}}_{2s}/{\gamma}_{lt}\)

(4)
Relationship between the twist and shear strain in concrete

(5)
Constitutive laws of the concrete in tension:
For prestressed concrete,
where \({\overline{\varepsilon}}_{cx}={\overline{\varepsilon}}_{1i}{\sigma}_{ci}/2{E}_c^{\prime }\), \({E}_c^{\prime }=2{f}_c^{\prime }/{\varepsilon}_0\), \({E}_c^{{\prime\prime} }={f}_{cr}/\left({\varepsilon}_{cr}{\overline{\varepsilon}}_{cx}\right)\).
For reinforced concrete,

(6)
Constitutive laws of the concrete in compression:
For prestressed concrete,
For reinforced concrete,
For prestressed concrete,
For reinforced concrete,

(7)
Constitutive laws of the concrete in shear:
For prestressed concrete,
For reinforced concrete,

(8)
Constitutive laws of the prestressed steel:
where \({\overline{\varepsilon}}_{ps}=\left({\overline{\varepsilon}}_{pi}{\overline{\varepsilon}}_{li}\right)+{\overline{\varepsilon}}_l^c\).

(9)
Constitutive laws of the steel bars embedded in concrete:
For prestressed concrete,
For reinforced concrete,

(10)
Convergence criteria based on equilibriums:
For prestressed concrete,
For reinforced concrete,
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Shen, K., Wan, S., Zhu, Y. et al. A softened membrane model for singlebox multicell composite boxgirders with corrugated steel webs under pure torsion. ABEN 3, 2 (2022). https://doi.org/10.1186/s43251022000523
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DOI: https://doi.org/10.1186/s43251022000523
Keywords
 Singlebox multicell boxgirders
 Corrugated steel webs
 Softened membrane model
 Nonlinear torsional analysis
 Torquetwist curve
 Shear strain