A softened membrane model for single-box multi-cell composite box-girders with corrugated steel webs under pure torsion

To analyze the nonlinear torsional behavior of single-box multi-cell prestressed concrete (PC) or reinforced concrete (RC) composite box-girders 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 single-box single-cell PC/RC composite BGCSWs. This proposed model incorporates the different contributions of inner and outer corrugated steel webs (CSWs) in the multi-cell 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 torque-twist curves and the smeared shear strains in concrete slabs and CSWs in the full-range torsional analysis. The comparison indicates that the proposed model is capable of predicting the overall torsional behavior of single-box multi-cell BGCSWs.

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 rotating-angle softened truss model for torsion (RA-STMT) 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 step-by-step 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 RA-STMT was also employed in the torsional analysis of BGCSWs in the post-cracking stage, combining a correction in the pre-cracking stage (Shen et al. 2017).

Fig. 1 Progress of the theoretical models for nonlinear torsional analysis of BGCSWs
Note that the RA-STMT is unable to predict the torsional behavior before the cracking of concrete, a tension-stiffened softened truss model for torsion (TS-STMT) 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 (MRA-STMT) (Zhu et al. 2020b).
To predict the torsional behavior of single-box multi-cell BGCSWs, a unified softened truss model for torsion (USTMT) (Shen et al. 2018c) was proposed on the basis of RA-STMT by introducing the new shear strain relationships between the CSWs of multi-cell BGCSWs (Shen et al. 2018d). Subsequently, the new three-stage 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 multi-cell BGCSWs (Zhu et al. 2021). With the rational modifications, the new model, unified RA-STMT (URA-STMT), 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 (FA-STMT) 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 fixed-angle softened membrane model for torsion (SMMT) (Jeng and Hsu 2009) was introduced to the torsional analysis of single-box single-cell 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 full-range 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 rotating-angle theory.

Research motivation and innovations
From above literature review, it can be obviously found that both rotating-angle STMT theory and SMMT theory can well predict the overall mechanical performance of single-box multi-cell BGCSWs under torsion. But the rotating-angle 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 single-cell BGCSWs to the torsional analysis of multi-cell BGCSWs. A unified method (USMMT) is proposed and an algorithm is compiled to predict the torsional behavior of both single-cell 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.

Existing equivalent method for single-box multi-cell box-girder
In previous studies (Fu and Yang 1996;Fu and Tang 2001;Allawi et al. 2017), it was generally assumed that the single-box multi-cell 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 box-girder in the nonlinear torsional analysis. For example, in a single-box triple-cell box-girder as shown in Fig. 2, the applied torque is carried by the three cells together and can be expressed as T = 3 i=1 2A 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 ), This formulas shows that the singlebox triple-cell box-girder can be equivalent to a single-box single-cell box-girder 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 Fig. 2 The shear flows in a single-box triple-cell box-girder shear strains of the inner corrugated steel webs are small and can be ignored. However, in the non-linear 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 box-girder will be underestimated if only considering the contribution of the outer CSWs to it.

Rational simplified calculation method for multi-cell BGCSWs under torsion
In the present study, the calculation method for single-box multi-cell BGCSWs under torsion is completely different from the previous simplified method for multi-cell concrete box-girders. The rectangular BGCSW with 2n (or 2n-1) cells was decomposed into n (n = 1, 2, 3, 4, 5) pieces of single-box single-cell 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 single-cell 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 single-cell boxes "Box-i" (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 "Box-i" can be obtained as (Shen et al. 2018c): where R γi is the ratio of the shear strain of CSWs in Box-i (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 Box-i, d i , to that in the Box-1, 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 non-uniform shear strain of concrete slab in multi-cell BGCSW is represented by the smeared shear strain of concrete in the Box-1 decomposed from multi-cell 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 multi-cell BGCSW T m can be expressed by a unified formula: Shen et al. Advances in Bridge Engineering (2022) 3:2 Fig. 3 The equivalence of torsion in the BGCSW with 2n (or 2n-1) cells where m is a subscript that represents the number of cells in the multi-cell BGCSW; T f1 is the torque resisted by the concrete slabs in Box-1; T wi is the torque resisted by the webs in Box-i. They can be calculated by:

USMMT for multi-cell BGCSWs under torsion
The advanced SMMT models for single-box single-cell 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 single-box multi-cell 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 (4) ~ (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 non-prestressed single-box multi-cell 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 single-box multi-cell (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 Box-1.
Step 6. Determine whether the BGCSW is prestressed. If yes, proceed directly to the next step; if not, skip to step 10.
Step 7 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.
(4) Analyze the torsional mechanical behavior of the corrugated steel web in the range of 2 ≤ i ≤ n.
(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 single-box multi-cell BGCSW, but also the torsional behavior of a single-box single-cell 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.

Experimental specimens
In order to validate the proposed unified model, the experimental data of six pieces of single-box multi-cell BGCSWs available from literatures are collected. These specimens were tested under pure torsion, including two single-box double-cell BGCSWs Moreover, to better understand the torsional behavior of single-box multi-cell BGC-SWs, a specimen named T-3C-S was also tested by the authors after the torsion experiment of specimen T-3C. The test situation of T-3C-S, including the physical parameters and the experimental procedure, is almost the same to that of T-3C 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. Figure 5 shows the comparisons of the torque-twist 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 torque-twist 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 Table 1 Geometric and material parameters of the specimens   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 thin-walled 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  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 Fig. 7 Comparisons of the shear strains in concrete slabs and CSWs of the specimens under pure torsion between the experimental results and the calculated results from the USMMT 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 T-3C 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 single-box multi-cell BGC-SWs 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 BGC-SWs 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 torque-twist 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 single-box multi-cell 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 torque-twist curve of each single-box multi-cell 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 multi-cell 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 single-box multi-cell 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 single-box singlecell PC/RC composite BGCSWs under pure torsion, is extended to the torsional analysis of single-box multi-cell PC/ RC composite BGCSWs by incorporating the different contributions of the inner and outer CSWs. The proposed model, USMMT for single-box multi-cell 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 full-range torsional behavior of single-box multi-cell BGCSWs, such as the torque-twist 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 single-cell 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, Fig. 9 Comparisons of torque-twist curves between the FEA results and the calculated results from the USMMT 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 multi-cell 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 single-box multi-cell composite BGCSWs under pure torsion for practical design.
(9) Constitutive laws of the steel bars embedded in concrete: For prestressed concrete, For reinforced concrete,