# Parametric Study on Mixed Torsional Behavior of U-Shaped Thin-Walled RC Girders

Authors: Jianchao Xu; Bo Diao; Quanquan Guo; Yinghua Ye; Y. L. Mo; Tianmin Zhou
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

Nowadays, U-shaped thin-walled concrete girders have been widely applied in the urban construction of rail viaducts in China as well as worldwide. However, the mixed torsional behaviors of these structures are not well understood. In this paper, the mixed torsional behaviors of the U-shaped thin-walled RC girders are theoretically analyzed, and a method predicting failure modes and ultimate torques is proposed. Nonlinear FE models based on ABAQUS to simulate the mixed torsional behaviors are built and calibrated with the test results. Parametric studies considering three crucial parameters (boundary condition, span length-section height ratio, and ratio of longitudinal bars to stirrups) are conducted based on both the above suggested calculating method and the FE modeling. The calculated and the simulated results agree well with each other and with the test results. It is found that the failure modes of the U-shaped thin-walled RC girders under torsion are influenced by all the three parameters. Three kinds of failure modes are observed: flexural failures dominated by warping moment, shear failures caused by warping torque and circulatory torque, and flexural-shear failures in the cases where flexural failure and shear failure appear almost at the same time.


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## Body

## 1. Introduction

U-shaped thin-walled concrete girders are composed of two webs and a bottom slab, with trains travelling through inside [1–4]. Comparing with traditional girders like box girders, U-shaped thin-walled girders have some advantages such as lower construction elevation (since trains travel inside), easier power system layout (since there is enough space inside for the power system), and attractive appearance. For the U-shaped thin-walled RC girders, torsion is a critical factor considering the transverse wind load and the eccentric traffic load (in multilane cases and curved structure cases).The elastic torsional response of an open thin-walled structure was first formulated by Vlasov in 1961 [5]. Then, some improvements on Vlasov’s theory have been made for some specific cases such as focusing on the shear deformation caused by warping torque [6, 7] or focusing on the secondary warping effect [8]. According to Vlasov’s theory, when an open thin-walled structure is under torsion, it is a mixed torsional case of circulatory torsion and warping torsion. Three kinds of internal forces exist simultaneously, which are circulatory torque Tc (comes from circulatory torsion), warping torque Tω, and warping moment Mω (come from warping torsion).The postcracking circulatory torsional behavior of RC closed-section structures has been well simulated with the space truss model [9–16]. The nonlinear FEA method was also applied to simulate the circulatory torsional behavior of RC closed-section structures [17–19]. As to the postcracking mixed torsional behavior of open thin-walled RC structures, the aforementioned space truss model for circulatory torsion is not accurate anymore due to the considerable warping effect [3, 20]. In 1968, the postcracking mixed torsional behavior of a U-shaped thin-walled RC girder was first addressed [21]. Then in 1981, the mixed torsional behavior of a large U-shaped thin-walled RC girder was systematically tested (with both ends fixed) and analyzed by Krpan and Collins [22, 23]. Although immature anchoring failure occurred, the test results clearly showed the dominating role that warping moment played. In 1983, Hwang and Hsu proposed a nonlinear analysis model based on the Fourier series approach to simulate the mixed torsional behavior of U-shaped thin-walled RC girders [1]. Apart from the U-shaped thin-walled RC girders, the I-shaped RC girders are also open section members applied in Civil Engineering, although their warping effect is not very strong. In 1991, the combined torsional and flexural behaviors of eight I-shaped RC girders were experimentally and analytically studied [20], and a method to calculate the ultimate load was suggested by modifying the skew bending theory [24]. In the following three decades, few research outcomes on the mixed torsional behaviors of U-shaped thin-walled RC girders were reported in the literature. In the 21st century, their torsional performance has again been addressed due to their wide application in rail viaduct engineering. In 2016, three large fixed-fixed U-shaped thin-walled RC girders were tested and analyzed by Chen et al. [3, 25]. Flexural failures dominated by the warping moment were observed for all the girder specimens, based on which a simple method to calculate the ultimate torques was proposed, where only the warping moment was taken into account.As listed above, the current research outcomes on the mixed torsional behavior of U-shaped thin-walled RC girders are derived from a limited number of experimental or analytical studies. The torsional behavior of such girders is still unclear. A comprehensive parametric study is needed to thoroughly understand the mixed torsional behavior of such structures. In this paper, the mixed torsional behavior of these girders is first theoretically analyzed based on Vlasov’s theory, and a method able to calculate both the failure mode and the ultimate torque is derived. Since in modern research the structural engineering finite element analysis (FEA) is essential for supplementing experimental investigations [26, 27], nonlinear FE modeling based on ABAQUS is carried out. Finally, a comprehensive parametric study based on both proposed calculating method and FE modeling is performed considering three crucial parameters, i.e., boundary condition, span-height ratio, and ratio of longitudinal bars to stirrups.


## 2. Theoretical and Ultimate Equilibrium Analyses

### 2.1. Theoretical Analysis

The theoretical analysis is performed based on Vlasov’s theory [5], and two typical distributions of internal forces are shown in Figure 1(a) (fixed-fixed supported case) and Figure 1(b) (simply supported case), respectively. The simply supported boundary condition here means that the rotation of the girder end is constrained but the girder end is free to warp in the longitudinal direction. According to Vlasov’s theory [5], the corresponding formula of the internal forces in the two boundary conditions are given in Equations (1) (fixed-fixed supported) and (2) (simply supported), respectively, where Mω is the warping moment; Tω is the warping torque; Tc is the circulatory torque; k is the characteristic length defined as the square root of the stiffness ratio of warping torsion to circulatory torsion, k=EIω/GKm; E is the elasticity modulus; Iω is the principal sectorial moment of inertia calculated by Iω=∫Aω2dA, in which ω is principal sectorial coordinate; G is the shear modulus taken as 0.4 E; K is St. Venant’s torsional constant; L is the span length; and “sinh” and “cosh” are the hyperbolic sine and the hyperbolic cosine functions, respectively. It should be noted that, in the mixed torsional cases, the characteristic length k is a key cross-sectional property, and for the U-shaped thin-walled RC girder studied in this paper, it has a value of 2.06 m, which is 4.12 times the section height. The characteristic length k is discussed in detail in Section 5.4:(1)Mωz=0.5Tkcoshz/k−cosh0.5L−z/ksinh0.5L/k,0≤z≤0.5L,Tωz=0.5Tsinhz/k+sinh0.5L−z/ksinh0.5L/k,0≤z≤0.5L,Tcz=0.5T−Tωz,0≤z≤0.5L,(2)Mωz=0.5Tksinhz/kcosh0.5L/k,0≤z≤0.5L,Tωz=0.5Tcoshz/kcosh0.5L/k,0≤z≤0.5L,Tcz=0.5T−Tωz,0≤z≤0.5L.Figure 1
Distributions of internal forces and the corresponding stresses: (a) internal forces in a fixed-fixed supported case, (b) internal forces in a simply supported case, and (c) stress distributions. Note:C=0.5Tkcosh0.5L/k−1/sinh0.5L/k; D=0.5Tk⋅sinh0.5L/k/cosh0.5L/k.
(a)
(b)
(c)It can be seen from Figures1(a) and 1(b) and Equations (1) and (2) that the total internal torque 0.5 T is the sum of warping torque Tω and circulatory torque Tc. Moreover, like the relationship between bending moment and shear force, the warping torque Tω is the differential formula of the warping moment Mω. For the fixed-fixed supported case, warping moment Mω and warping torque Tω reach the maximum at support and midspan, while circulatory torque Tc reaches its maximum at quarter span. As to the simply supported case, warping moment Mω and warping torque Tω reach their maximum at midspan, whereas the maximum circulatory torque Tc exists at support. The distributions of warping normal stress, warping torsional shear stress, and circulatory torsional shear stress are shown in Figure 1(c). As shown in Figure 1(c), warping normal stress antisymmetrically distributes around the U-shaped section. Warping torsional shear flow flows along the centerline of the U-shaped section, while the circulatory torsional shear flow forms a U-shaped circle. According to the distributions of internal forces discussed above, at ultimate state, two kinds of potential failure may occur, namely, a flexural failure dominated by warping moment (in the fixed-fixed supported case, it occurs at midspan and support, while in the simply supported case, it occurs at midspan) and a shear failure dominated by the combined action of circulatory torque and warping torque (in the fixed-fixed supported case, it occurs at quarter span, while in the simply supported case, it occurs at support). The potential shear failure occurs at the location where the maximum circulatory torque Tc appears because the U-shaped thin-walled section is weaker at resisting circulatory torque Tc than resisting warping torque Tω due to the small circulatory torsional lever arm.When the ultimate state is the concern, more attention should be paid to the maximum values of internal forces. Referring to Equations (1) and (2), noting γ=L/k (ratio of span length to characteristic length), the maximum internal forces in the fixed-fixed supported case and in the simply supported case can be, respectively, expressed in Equations (3) and (4), where “tanh” is hyperbolic tangent function:(3)Mω,max=0.5Tkcosh0.5γ−1sinh0.5γ,Tω,max=0.5T,Tc,max=0.5T1−2sinh0.25γsinh0.5γ,(4)Mω,max=0.5Tktanh0.5γ,Tω,max=0.5T,Tc,max=0.5T1−1cosh0.5γ.The relationship between the maximum internal forces andγ in Equations (3) and (4) are visualized in Figure 2. It can be seen that, for both fixed-fixed supported and simply supported cases, with the increase of γ, warping moment Mω and circulatory torque Tc have the upper limit values of 0.5 Tk and 0.5 T, respectively. Moreover, they increase much faster in the simply supported case than in the fixed-fixed supported case. Besides, it can be observed from Figure 2 that, in both fixed-fixed and simply supported cases, when the span length-characteristic length ratio γ is small, warping torque Tω dominates, while warping moment Mω and circulatory torque Tc have small values; thus shear failure dominated by warping torque may occur. When γ has a medium value, warping moment Mω dominates and flexural failure dominated by warping moment may occur. When γ has a large value, circulatory torque Tc dominates, and shear failure dominated by circulatory torque may occur.Figure 2
Variation tendency of the maximum internal forces with the increasing of span length. “FS” and “SS” mean “fixed-fixed supported” and “simply supported,” respectively.


### 2.2. Ultimate Equilibrium Analysis

Ultimate equilibrium analyses for the two abovementioned potential failures are conducted here. As stated in the literature review, the flexural failures dominated by warping moment were experimentally observed and theoretically calculated by Chen et al. [3]. In this paper, a calculating method for the potential shear failure is proposed. Moreover, the calculating method for the flexural failure suggested by Chen et al. will be improved by taking the shear effect of warping torque into consideration.


#### 2.2.1. Calculation of the Potential Flexural Failure

As shown in Figures1(a) and 1(b), apart from the maximum warping moment Mω, the warping torque Tω also exists at the flexural critical segments. Thus, the calculation method of flexural failure only considering warping moment [3] is modified here by taking the warping torque into account. As shown in Figure 1(c), the warping moment Mω is a self-balanced internal force; thus at ultimate state, the warping effect is considered on three separated rectangular thin walls. As shown in Figure 3, every thin wall is equivalently subjected to a bending moment, and the equivalent bending moment on the right web can be obtained by Meq=∫Aσωy1dA. As to the treatment of warping torque Tω, as shown in Figure 1(c), the warping torque is totally resisted by the warping torsional shear stresses on the two webs, while the warping torsional shear stress in the bottom slab makes no contribution due to its self-counteraction. Therefore, at the ultimate state, the warping torque is considered equivalent to a couple of shear forces Vω=Tω/dw on the two webs acting in opposite directions, where dw is the distance between the centerlines of two webs. Considering Equation (3) and σω=Mωω/Iω, defining η1=∫Aωy1dA, then for the fixed-fixed supported case, the equivalent shear force and bending moment on the webs can be expressed by(5)Vω=0.5Tdω,Meq=0.5Tkη1cosh0.5γ−1Iωsinh0.5γ.Figure 3
Distributing method for warping moment.Figure 4
Ultimate equilibriums under the combined actions of warping moment and warping torque: (a) the truss model bearing the equivalent shear force and (b) ultimate equilibrium in the normal direction.
(a)
(b)Considering Equation (4) for the simply supported case, they can be expressed by(6)Vω=0.5Tdω,Meq=0.5Tktanh0.5γη1Iω.At the ultimate state as shown in Figure4(a), the shear force is resisted by the tension in reinforcement and the compression in the diagonal concrete struts [15, 24], and the tangential equilibrium can be expressed by Equation (7). The normal equilibrium on the web is shown in Figure 4(b), and it can be expressed by Equation (8) referring to the calculation method of bending-shear capacities of shear walls [28]:(7)Vω=2Fty1h0cotθs,(8)0=fluAs+fluAsdh0−1.5xhd−0.85fcbx−fly′As′−Vωcotθ,Meq=fluAsh0−0.5x+fluAsdh0−1.5xhd0.5h0+0.25x+fly′As′0.5x−a′−Vωcotθ0.5h0−0.5x,where Fty1 is the yielding force of one stirrup leg; s is the stirrup spacing; flu and fly′ are the ultimate tensile stress and yielding compressive stress (absolute value) of the longitudinal bars, respectively; Asd is the area of all distributing bars; and x is the depth of a concrete equivalent rectangular stress block with a uniform compressive stress of 0.85 fc [29]. Since the stress near the neutral axis is low, only the distributing bars located beyond 1.5x (expressed as Asdh0−1.5x/hd) are considered to contribute to the tensile resistance. Then, the ultimate torque of the potential flexural failure can be solved by combining Equations (7), (8), and (5) or (7), (8), and (6) for the fixed-fixed supported case or for the simply supported case, respectively.


#### 2.2.2. Calculation of the Potential Shear Failure

A method to calculate the shear failure in this mixed torsional case is developed here based on the skew bending theory, also called the plastic truss model [15, 24]. At the ultimate state as shown in Figure 5(a), the circulatory torsional shear flow qc can be obtained as qc=Tc/2A0, where A0 is the area enclosed by the shear flow, and it is assumed that the center line of the shear flow coincides with the inner face of the stirrup. The warping torsional shear flow qω on the webs can be expressed by qω=Tω/2dwd1. Under the combined action of warping torque and circulatory torque, the two webs are more critical than the bottom slab due to the interaction of the two kinds of shear flows. The ultimate equilibrium on the right web of the U-shaped girder is shown in Figure 5(b), where the shear flows due to circulatory and warping torques are additive on the right side, while are subtractive on the left side. On the top and bottom sides, only circulatory torsional shear flow exists. The combined shear flows can be obtained as(9)q1=qc+qω,q3=qc−qω,q2=q4=qc.Figure 5
Ultimate equilibrium under the combined actions of warping torque and circulatory torque: (a) simplified distribution of torsional shear stresses and (b) ultimate equilibrium on the right web of the U-shaped girder.
(a)
(b)Considering Equation (3), for the fixed-fixed supported case,(10)q1=0.5T1−2sinh0.25γ/sinh0.5γ2A0+0.5T2sinh0.25γ/sinh0.5γ2dwd1,q2=q4=0.5T1−2sinh0.25γ/sinh0.5γ2A0.Considering Equation (4), for the simply supported case,(11)q1=0.5T1−1/cosh0.5γ2A0+0.5T/cosh0.5γ2dwd1,q2=q4=0.5T1−1/cosh0.5γ2A0.The web fails with the yielding of reinforcement on the right, top, and bottom sides, and at the ultimate state, the equilibrium is expressed as(12)qi=Fty1cotθis,i=1,2,4,0=flyAsrd2−q1d1d2cotθ1−0.5q2d22cotθ2−0.5q4d22cotθ4,where Asr is the area of the longitudinal bars on the right side of the web and fly is the yielding strength of the longitudinal bars. The ultimate torque of the potential shear failure can then be calculated by combining Equations (12) and (10) or (12) and (11) for the fixed-fixed supported case or for the simply supported case, respectively. The angle θ should not be less than 30° [29], in case that the longitudinal bars cannot reach yielding because of overreinforcement. If the solved angle θ < 30°, the ultimate torque can be simply obtained by setting θ = 30°.


## 2.1. Theoretical Analysis

The theoretical analysis is performed based on Vlasov’s theory [5], and two typical distributions of internal forces are shown in Figure 1(a) (fixed-fixed supported case) and Figure 1(b) (simply supported case), respectively. The simply supported boundary condition here means that the rotation of the girder end is constrained but the girder end is free to warp in the longitudinal direction. According to Vlasov’s theory [5], the corresponding formula of the internal forces in the two boundary conditions are given in Equations (1) (fixed-fixed supported) and (2) (simply supported), respectively, where Mω is the warping moment; Tω is the warping torque; Tc is the circulatory torque; k is the characteristic length defined as the square root of the stiffness ratio of warping torsion to circulatory torsion, k=EIω/GKm; E is the elasticity modulus; Iω is the principal sectorial moment of inertia calculated by Iω=∫Aω2dA, in which ω is principal sectorial coordinate; G is the shear modulus taken as 0.4 E; K is St. Venant’s torsional constant; L is the span length; and “sinh” and “cosh” are the hyperbolic sine and the hyperbolic cosine functions, respectively. It should be noted that, in the mixed torsional cases, the characteristic length k is a key cross-sectional property, and for the U-shaped thin-walled RC girder studied in this paper, it has a value of 2.06 m, which is 4.12 times the section height. The characteristic length k is discussed in detail in Section 5.4:(1)Mωz=0.5Tkcoshz/k−cosh0.5L−z/ksinh0.5L/k,0≤z≤0.5L,Tωz=0.5Tsinhz/k+sinh0.5L−z/ksinh0.5L/k,0≤z≤0.5L,Tcz=0.5T−Tωz,0≤z≤0.5L,(2)Mωz=0.5Tksinhz/kcosh0.5L/k,0≤z≤0.5L,Tωz=0.5Tcoshz/kcosh0.5L/k,0≤z≤0.5L,Tcz=0.5T−Tωz,0≤z≤0.5L.Figure 1
Distributions of internal forces and the corresponding stresses: (a) internal forces in a fixed-fixed supported case, (b) internal forces in a simply supported case, and (c) stress distributions. Note:C=0.5Tkcosh0.5L/k−1/sinh0.5L/k; D=0.5Tk⋅sinh0.5L/k/cosh0.5L/k.
(a)
(b)
(c)It can be seen from Figures1(a) and 1(b) and Equations (1) and (2) that the total internal torque 0.5 T is the sum of warping torque Tω and circulatory torque Tc. Moreover, like the relationship between bending moment and shear force, the warping torque Tω is the differential formula of the warping moment Mω. For the fixed-fixed supported case, warping moment Mω and warping torque Tω reach the maximum at support and midspan, while circulatory torque Tc reaches its maximum at quarter span. As to the simply supported case, warping moment Mω and warping torque Tω reach their maximum at midspan, whereas the maximum circulatory torque Tc exists at support. The distributions of warping normal stress, warping torsional shear stress, and circulatory torsional shear stress are shown in Figure 1(c). As shown in Figure 1(c), warping normal stress antisymmetrically distributes around the U-shaped section. Warping torsional shear flow flows along the centerline of the U-shaped section, while the circulatory torsional shear flow forms a U-shaped circle. According to the distributions of internal forces discussed above, at ultimate state, two kinds of potential failure may occur, namely, a flexural failure dominated by warping moment (in the fixed-fixed supported case, it occurs at midspan and support, while in the simply supported case, it occurs at midspan) and a shear failure dominated by the combined action of circulatory torque and warping torque (in the fixed-fixed supported case, it occurs at quarter span, while in the simply supported case, it occurs at support). The potential shear failure occurs at the location where the maximum circulatory torque Tc appears because the U-shaped thin-walled section is weaker at resisting circulatory torque Tc than resisting warping torque Tω due to the small circulatory torsional lever arm.When the ultimate state is the concern, more attention should be paid to the maximum values of internal forces. Referring to Equations (1) and (2), noting γ=L/k (ratio of span length to characteristic length), the maximum internal forces in the fixed-fixed supported case and in the simply supported case can be, respectively, expressed in Equations (3) and (4), where “tanh” is hyperbolic tangent function:(3)Mω,max=0.5Tkcosh0.5γ−1sinh0.5γ,Tω,max=0.5T,Tc,max=0.5T1−2sinh0.25γsinh0.5γ,(4)Mω,max=0.5Tktanh0.5γ,Tω,max=0.5T,Tc,max=0.5T1−1cosh0.5γ.The relationship between the maximum internal forces andγ in Equations (3) and (4) are visualized in Figure 2. It can be seen that, for both fixed-fixed supported and simply supported cases, with the increase of γ, warping moment Mω and circulatory torque Tc have the upper limit values of 0.5 Tk and 0.5 T, respectively. Moreover, they increase much faster in the simply supported case than in the fixed-fixed supported case. Besides, it can be observed from Figure 2 that, in both fixed-fixed and simply supported cases, when the span length-characteristic length ratio γ is small, warping torque Tω dominates, while warping moment Mω and circulatory torque Tc have small values; thus shear failure dominated by warping torque may occur. When γ has a medium value, warping moment Mω dominates and flexural failure dominated by warping moment may occur. When γ has a large value, circulatory torque Tc dominates, and shear failure dominated by circulatory torque may occur.Figure 2
Variation tendency of the maximum internal forces with the increasing of span length. “FS” and “SS” mean “fixed-fixed supported” and “simply supported,” respectively.


## 2.2. Ultimate Equilibrium Analysis

Ultimate equilibrium analyses for the two abovementioned potential failures are conducted here. As stated in the literature review, the flexural failures dominated by warping moment were experimentally observed and theoretically calculated by Chen et al. [3]. In this paper, a calculating method for the potential shear failure is proposed. Moreover, the calculating method for the flexural failure suggested by Chen et al. will be improved by taking the shear effect of warping torque into consideration.


### 2.2.1. Calculation of the Potential Flexural Failure

As shown in Figures1(a) and 1(b), apart from the maximum warping moment Mω, the warping torque Tω also exists at the flexural critical segments. Thus, the calculation method of flexural failure only considering warping moment [3] is modified here by taking the warping torque into account. As shown in Figure 1(c), the warping moment Mω is a self-balanced internal force; thus at ultimate state, the warping effect is considered on three separated rectangular thin walls. As shown in Figure 3, every thin wall is equivalently subjected to a bending moment, and the equivalent bending moment on the right web can be obtained by Meq=∫Aσωy1dA. As to the treatment of warping torque Tω, as shown in Figure 1(c), the warping torque is totally resisted by the warping torsional shear stresses on the two webs, while the warping torsional shear stress in the bottom slab makes no contribution due to its self-counteraction. Therefore, at the ultimate state, the warping torque is considered equivalent to a couple of shear forces Vω=Tω/dw on the two webs acting in opposite directions, where dw is the distance between the centerlines of two webs. Considering Equation (3) and σω=Mωω/Iω, defining η1=∫Aωy1dA, then for the fixed-fixed supported case, the equivalent shear force and bending moment on the webs can be expressed by(5)Vω=0.5Tdω,Meq=0.5Tkη1cosh0.5γ−1Iωsinh0.5γ.Figure 3
Distributing method for warping moment.Figure 4
Ultimate equilibriums under the combined actions of warping moment and warping torque: (a) the truss model bearing the equivalent shear force and (b) ultimate equilibrium in the normal direction.
(a)
(b)Considering Equation (4) for the simply supported case, they can be expressed by(6)Vω=0.5Tdω,Meq=0.5Tktanh0.5γη1Iω.At the ultimate state as shown in Figure4(a), the shear force is resisted by the tension in reinforcement and the compression in the diagonal concrete struts [15, 24], and the tangential equilibrium can be expressed by Equation (7). The normal equilibrium on the web is shown in Figure 4(b), and it can be expressed by Equation (8) referring to the calculation method of bending-shear capacities of shear walls [28]:(7)Vω=2Fty1h0cotθs,(8)0=fluAs+fluAsdh0−1.5xhd−0.85fcbx−fly′As′−Vωcotθ,Meq=fluAsh0−0.5x+fluAsdh0−1.5xhd0.5h0+0.25x+fly′As′0.5x−a′−Vωcotθ0.5h0−0.5x,where Fty1 is the yielding force of one stirrup leg; s is the stirrup spacing; flu and fly′ are the ultimate tensile stress and yielding compressive stress (absolute value) of the longitudinal bars, respectively; Asd is the area of all distributing bars; and x is the depth of a concrete equivalent rectangular stress block with a uniform compressive stress of 0.85 fc [29]. Since the stress near the neutral axis is low, only the distributing bars located beyond 1.5x (expressed as Asdh0−1.5x/hd) are considered to contribute to the tensile resistance. Then, the ultimate torque of the potential flexural failure can be solved by combining Equations (7), (8), and (5) or (7), (8), and (6) for the fixed-fixed supported case or for the simply supported case, respectively.


### 2.2.2. Calculation of the Potential Shear Failure

A method to calculate the shear failure in this mixed torsional case is developed here based on the skew bending theory, also called the plastic truss model [15, 24]. At the ultimate state as shown in Figure 5(a), the circulatory torsional shear flow qc can be obtained as qc=Tc/2A0, where A0 is the area enclosed by the shear flow, and it is assumed that the center line of the shear flow coincides with the inner face of the stirrup. The warping torsional shear flow qω on the webs can be expressed by qω=Tω/2dwd1. Under the combined action of warping torque and circulatory torque, the two webs are more critical than the bottom slab due to the interaction of the two kinds of shear flows. The ultimate equilibrium on the right web of the U-shaped girder is shown in Figure 5(b), where the shear flows due to circulatory and warping torques are additive on the right side, while are subtractive on the left side. On the top and bottom sides, only circulatory torsional shear flow exists. The combined shear flows can be obtained as(9)q1=qc+qω,q3=qc−qω,q2=q4=qc.Figure 5
Ultimate equilibrium under the combined actions of warping torque and circulatory torque: (a) simplified distribution of torsional shear stresses and (b) ultimate equilibrium on the right web of the U-shaped girder.
(a)
(b)Considering Equation (3), for the fixed-fixed supported case,(10)q1=0.5T1−2sinh0.25γ/sinh0.5γ2A0+0.5T2sinh0.25γ/sinh0.5γ2dwd1,q2=q4=0.5T1−2sinh0.25γ/sinh0.5γ2A0.Considering Equation (4), for the simply supported case,(11)q1=0.5T1−1/cosh0.5γ2A0+0.5T/cosh0.5γ2dwd1,q2=q4=0.5T1−1/cosh0.5γ2A0.The web fails with the yielding of reinforcement on the right, top, and bottom sides, and at the ultimate state, the equilibrium is expressed as(12)qi=Fty1cotθis,i=1,2,4,0=flyAsrd2−q1d1d2cotθ1−0.5q2d22cotθ2−0.5q4d22cotθ4,where Asr is the area of the longitudinal bars on the right side of the web and fly is the yielding strength of the longitudinal bars. The ultimate torque of the potential shear failure can then be calculated by combining Equations (12) and (10) or (12) and (11) for the fixed-fixed supported case or for the simply supported case, respectively. The angle θ should not be less than 30° [29], in case that the longitudinal bars cannot reach yielding because of overreinforcement. If the solved angle θ < 30°, the ultimate torque can be simply obtained by setting θ = 30°.


## 2.2.1. Calculation of the Potential Flexural Failure

As shown in Figures1(a) and 1(b), apart from the maximum warping moment Mω, the warping torque Tω also exists at the flexural critical segments. Thus, the calculation method of flexural failure only considering warping moment [3] is modified here by taking the warping torque into account. As shown in Figure 1(c), the warping moment Mω is a self-balanced internal force; thus at ultimate state, the warping effect is considered on three separated rectangular thin walls. As shown in Figure 3, every thin wall is equivalently subjected to a bending moment, and the equivalent bending moment on the right web can be obtained by Meq=∫Aσωy1dA. As to the treatment of warping torque Tω, as shown in Figure 1(c), the warping torque is totally resisted by the warping torsional shear stresses on the two webs, while the warping torsional shear stress in the bottom slab makes no contribution due to its self-counteraction. Therefore, at the ultimate state, the warping torque is considered equivalent to a couple of shear forces Vω=Tω/dw on the two webs acting in opposite directions, where dw is the distance between the centerlines of two webs. Considering Equation (3) and σω=Mωω/Iω, defining η1=∫Aωy1dA, then for the fixed-fixed supported case, the equivalent shear force and bending moment on the webs can be expressed by(5)Vω=0.5Tdω,Meq=0.5Tkη1cosh0.5γ−1Iωsinh0.5γ.Figure 3
Distributing method for warping moment.Figure 4
Ultimate equilibriums under the combined actions of warping moment and warping torque: (a) the truss model bearing the equivalent shear force and (b) ultimate equilibrium in the normal direction.
(a)
(b)Considering Equation (4) for the simply supported case, they can be expressed by(6)Vω=0.5Tdω,Meq=0.5Tktanh0.5γη1Iω.At the ultimate state as shown in Figure4(a), the shear force is resisted by the tension in reinforcement and the compression in the diagonal concrete struts [15, 24], and the tangential equilibrium can be expressed by Equation (7). The normal equilibrium on the web is shown in Figure 4(b), and it can be expressed by Equation (8) referring to the calculation method of bending-shear capacities of shear walls [28]:(7)Vω=2Fty1h0cotθs,(8)0=fluAs+fluAsdh0−1.5xhd−0.85fcbx−fly′As′−Vωcotθ,Meq=fluAsh0−0.5x+fluAsdh0−1.5xhd0.5h0+0.25x+fly′As′0.5x−a′−Vωcotθ0.5h0−0.5x,where Fty1 is the yielding force of one stirrup leg; s is the stirrup spacing; flu and fly′ are the ultimate tensile stress and yielding compressive stress (absolute value) of the longitudinal bars, respectively; Asd is the area of all distributing bars; and x is the depth of a concrete equivalent rectangular stress block with a uniform compressive stress of 0.85 fc [29]. Since the stress near the neutral axis is low, only the distributing bars located beyond 1.5x (expressed as Asdh0−1.5x/hd) are considered to contribute to the tensile resistance. Then, the ultimate torque of the potential flexural failure can be solved by combining Equations (7), (8), and (5) or (7), (8), and (6) for the fixed-fixed supported case or for the simply supported case, respectively.


## 2.2.2. Calculation of the Potential Shear Failure

A method to calculate the shear failure in this mixed torsional case is developed here based on the skew bending theory, also called the plastic truss model [15, 24]. At the ultimate state as shown in Figure 5(a), the circulatory torsional shear flow qc can be obtained as qc=Tc/2A0, where A0 is the area enclosed by the shear flow, and it is assumed that the center line of the shear flow coincides with the inner face of the stirrup. The warping torsional shear flow qω on the webs can be expressed by qω=Tω/2dwd1. Under the combined action of warping torque and circulatory torque, the two webs are more critical than the bottom slab due to the interaction of the two kinds of shear flows. The ultimate equilibrium on the right web of the U-shaped girder is shown in Figure 5(b), where the shear flows due to circulatory and warping torques are additive on the right side, while are subtractive on the left side. On the top and bottom sides, only circulatory torsional shear flow exists. The combined shear flows can be obtained as(9)q1=qc+qω,q3=qc−qω,q2=q4=qc.Figure 5
Ultimate equilibrium under the combined actions of warping torque and circulatory torque: (a) simplified distribution of torsional shear stresses and (b) ultimate equilibrium on the right web of the U-shaped girder.
(a)
(b)Considering Equation (3), for the fixed-fixed supported case,(10)q1=0.5T1−2sinh0.25γ/sinh0.5γ2A0+0.5T2sinh0.25γ/sinh0.5γ2dwd1,q2=q4=0.5T1−2sinh0.25γ/sinh0.5γ2A0.Considering Equation (4), for the simply supported case,(11)q1=0.5T1−1/cosh0.5γ2A0+0.5T/cosh0.5γ2dwd1,q2=q4=0.5T1−1/cosh0.5γ2A0.The web fails with the yielding of reinforcement on the right, top, and bottom sides, and at the ultimate state, the equilibrium is expressed as(12)qi=Fty1cotθis,i=1,2,4,0=flyAsrd2−q1d1d2cotθ1−0.5q2d22cotθ2−0.5q4d22cotθ4,where Asr is the area of the longitudinal bars on the right side of the web and fly is the yielding strength of the longitudinal bars. The ultimate torque of the potential shear failure can then be calculated by combining Equations (12) and (10) or (12) and (11) for the fixed-fixed supported case or for the simply supported case, respectively. The angle θ should not be less than 30° [29], in case that the longitudinal bars cannot reach yielding because of overreinforcement. If the solved angle θ < 30°, the ultimate torque can be simply obtained by setting θ = 30°.


## 3. Mixed Torsional Tests

The mixed torsional tests of U-shaped thin-walled RC girders are introduced here [3]. The detailed dimensions and reinforcing arrangements are shown in Figures 6(a) and 6(b), respectively. The total length, span length, section height, section width, and wall thickness of the tested girders were 7350 mm, 6650 mm, 500 mm, 900 mm, and 70 mm, respectively. The girder specimens were lengthened with solid strengthened blocks at the two ends to restrain warping deformation. A strengthened diaphragm was set at midspan to prevent local failure caused by loading. The material mechanical properties of the girder specimens (numbered as MEM-1, MEM-2, and MEM-3) are summarized in Table 1. In the loading test, the girder specimens were fixed at two ends and the concentrated torque T was introduced at midspan by two jacks acting upward and downward, referring to Figure 7.Figure 6
Detailed information of the U-shaped thin-walled RC girder specimens (unit: mm): (a) dimensions and (b) reinforcing arrangements.
(a)
(b)Table 1
Material properties of the girder specimens.
MEM-1
MEM-2
MEM-3
Longitudinal bars
Yield strength (MPa)
576.2
353.3
353.3
Ultimate strength (MPa)
623.0
573.3
573.3
Diameter (mm)
10
8
8
Stirrups
Yield strength (MPa)
365.0
276.7
276.7
Ultimate strength (MPa)
645.0
446.7
446.7
Diameter (mm)
6
6
6
Spacing (mm)
70
70
70
Concrete
Compressive strength (MPa)
51.0
63.8
51.6
Elastic modules (GPa)
38.5
42.2
34.4Figure 7
FE modeling of the U-shaped thin-walled RC girder under pure torsion.During the test, similar experimental phenomenon was observed for all the specimens. The crack pattern on the external surface of the downward web (the web going down in the loading process) is shown in Figure8. Referring to Figure 8, vertical flexural-type cracks dominated by warping moment occurred first at the support and midspan segments. Then inclined cracks appeared and developed over almost the whole span, with flexural-shear cracks at 1/8 and 3/8 segments, while web shear cracks at the quarter segment. Moreover, diagonal cracks on the external surfaces of the webs appeared earlier and developed more intensively than those on the internal surfaces of the webs, and it is due to that the two kinds of shear flows (Figure 1(c)) flow in the same direction on the external surfaces while in the opposite direction on the internal surfaces. Longitudinal bars yielded first at midspan and support while stirrups yielded afterwards at the quarter span. Finally, concrete crushing was observed at the support and midspan and flexural-type failure occurred.Figure 8
Test photo of the crack pattern on the external surface of the downward web.The ultimate torques of the above girder specimens are calculated with the method proposed in Section2.2. The calculated results together with the test results are compared in Table 2. The tested and calculated ultimate torques of the U-shaped thin-walled RC girder tested by Krpan and Collins [22] are also compared in Table 2 (numbered as MEM-Krpan). The smaller one between the calculated ultimate torques based on the potential flexural failure and potential shear failure is the true ultimate torque. It can be seen from Table 2 that the above proposed calculating method can give good predictions of the ultimate torques (with an average test/calculation value of 1.08) and the failure modes.Table 2
Comparison of the calculated and tested ultimate torques.
Girders
Ultimate torque (kNm)
Test/calculation
Test
Calculation
Flexural failure
Shear failure
MEM-1
199.7
200.7
208.2
0.99
MEM-2
151.0
133.1
135.8
1.13
MEM-3
147.0
131.5
135.8
1.12
MEM-Krpan
266.0
249.5
260.3
1.07
Average value
1.08
Standard deviation
0.07
Coefficient of variation
6.5%


## 4. Nonlinear Finite Element Simulation

The commercial FEA program ABAQUS is applied to build the 3D FEA model to simulate the mixed torsional behavior of the U-shaped thin-walled RC girders, which is shown in Figure7. Elements C3D8R are used for concrete to avoid the shear locking effect [30]. Elements T3D2 are used to model reinforcements. Two concrete elements are used through the thickness of the 70 mm thin walls considering that the circulatory torsional shear flow flows around the thin wall. Perfect bond is assumed between the concrete and reinforcement, using the embedded method. The torsional loading setup in this FE model is designed as the testing setup [3]. The bottom surfaces of the steel support frames are restrained by U1 = U2 = U3 = 0. Surface to surface contacts are introduced between the support frames and the girder ends with the normal behavior of “hard” contact and tangential behavior of penalty friction formulation (the friction coefficient is set as 0.3). It is the same contact definition between the steel loading frame and the girder midspan. The external torque is introduced at the midspan by a couple of concentrated forces acting downward and upward. In this FE model, displacement control is applied with two equal displacements going downward at Reference Point 1 (RP1) and going upward at Reference Point 2 (RP2).


### 4.1. Material Modeling

The behavior of concrete can be modeled in ABAQUS with the concrete damage plasticity (CDP) model [18, 26, 27, 30, 31]. The CDP model was first proposed by Lubliner et al. [32] and then modified by Lee and Fenves [33, 34]. To define the CDP model, the uniaxial constitutive laws of concrete under compression and tension together with five constitutive parameters are required. Since only compressive strength and Young’s modulus of concrete were reported in the test [3], as shown in Figure 9(a), the unconfined concrete uniaxial compressive stress-strain curve suggested by Mander et al. [35] is adopted, where ε0=2fc/E0. In tension, the behavior of concrete is assumed to be linearly elastic up to the onset of cracking and then followed by tension softening. In this paper as shown in Figure 9(b), the stress-displacement curve suggested by the CEB code [36] is applied. The fracture energy is calculated by Gf=73fcm0.18, where fcm is the mean compressive strength. The tensile strength of concrete is determined by ftm=0.3fck2/3 [36], where fck is the characteristic value of compressive strength (fck=fcm−8in MPa). The five constitutive parameters are given in Table 3. Since no relative information is available from the experimental tests, default values of the five constitutive parameters [30] are used except for the viscosity parameter. As suggested by Lee and Fenves [34], the value of the viscosity parameter can be set at about 15% of one time increment in order for the solution to be improved without changing the result. In the standard static analysis of the current model, about 3,000 increments are needed in a time period of one second; thus, as shown in Table 3, the viscosity parameter is set at 5·10−5. Damage is introduced into the CDP model in compression and tension according to Figures 9(c) and 9(d). The shape function of the unloading and reloading of compressive concrete suggested by Birtel and Mark [37] is applied to determine the evolution of the compressive damage dc. A tensile damage-displacement curve is used to be coherent with the tension softening curve of concrete [27].Figure 9
Constitutive laws of concrete: (a) compressive stress-strain curve, (b) tensile stress-crack width curve, (c) compressive damage-inelastic strain curve, and (d) tensile damage-crack width curve.
(a)
(b)
(c)
(d)Table 3
Additional parameters for the CDP model.
Dilation angle
Eccentricity
f
b0/fc0
K
Viscosity parameter
35°
0.1
1.16
0.67
5 · 10−5As shown in Figure10, linear hardening is introduced to simulate the behavior of reinforcement after its yielding [26]. The material mechanical parameters used in the FE models are the same as the tested ones, which have been shown in Table 1. The elastic modules of reinforcement are set as 200 (GPa). ABAQUS/standard static analyses are applied for all the FE models.Figure 10
Stress-strain curve for the reinforcement.


### 4.2. Calibration of the FE Model

The test result of MEM-2 is used to calibrate the FE model in terms of torque-rotation curve, crack pattern, and failure mode. Referring to Figure7, the rotation at the midspan is obtained by α = arctan (U22−U21/D12+U11−U12), where the first subscript of U means the direction of displacement and the second subscript means RP1 or RP2. D12 is the original distance between RP1 and RP2. The applied torque is calculated by T=FrD12+U11−U12, where Fr is the average value of the two reaction forces at RP1 and RP2. At the outset, it is essential to discuss the chosen material parameters, the mesh size, and the element type. Since it is known that a larger viscosity parameter will improve the solution but gives a stiffer response [26, 30, 34], the chosen viscosity parameter should be calibrated. As shown in Figure 11(a), the simulated torque-rotation curves with a series of viscosity parameters are compared with each other as well as with the test result. It can be seen that the viscosity parameters of 2 · 10−5, 5 · 10−5(used in this paper), and 10 · 10−5 give almost the same results, which are accurate compared with the test results. With the increase of the viscosity parameter to 50 · 10−5 and 100 · 10−5, stiffer responses are as expected. The chosen viscosity parameter of 5 · 10−5 is proved to be accurate. Due to the localization of the concrete tensile cracks, the FE model is mesh size dependent as happens in most plasticity-based FE models with strain softening [26, 27, 30]. Hence, a mesh convergence study is performed. As shown in Figure 11(b), two mesh sizes of 35 mm and 23 mm were compared with each other to investigate the mesh sensitivity (35 mm mesh size giving two elements through the thickness of the thin wall while 23 mm mesh size giving three). It can be seen from Figure 11(b) that similar responses are obtained for the two mesh sizes, and they are accurate compared with the test result. Considering the computing cost, larger mesh size of 35 mm is selected. Comparison between simulated results with different element types is shown in Figure 11(c). Elements C3D20R are used to obtain the correct solution. It can be seen from Figure 11(c) that elements C3D8R and C3D20R give similar simulation results.Figure 11
Torque-rotation curves with (a) different viscosity parameters, (b) different mesh sizes, and (c) different element types.
(a)
(b)
(c)The simulated crack pattern and failure mode are shown in Figures12(a) and 12(b), respectively, where the simulated crack pattern is approximately represented by the distribution of concrete tensile damage (DAMAGET) and the simulated failure mode is represented by the distribution of concrete compressive damage (DAMAGEC). Comparing Figure 12(a) with Figure 8, it can be seen that the propagation and distribution of cracks are well simulated by the built FE model (detail features are stated in Section 3). Comparing Figure 12(b) with Figure 8, it can be observed that the distribution of DAMAGEC agrees well with the test result. The maximum DAMAGEC occurs at midspan on the top of the downward web and at the support on the top of the upward web, and those were the locations where concrete crushing occurred in the test. Flexural failure dominated by the warping moment occurred both in the simulation and in the test. From the above comparisons between the simulated results and the test results, it can be concluded that the FE model in this paper can accurately simulate the mixed torsional behavior of the tested specimen. The simulated torque-rotation curves of all the three girder specimens are shown in Figure 13.Figure 12
The simulated crack patterns and failure modes: (a) distribution of tensile damage at a load level of 80% ultimate torque and (b) distribution of compressive damage at ultimate load level. “FC” means flexural cracking, “YL” means yielding of longitudinal bars, and “YS” means yielding of stirrups.
(a)
(b)Figure 13
Simulated and tested torque-rotation curves: (a) MEM-1, (b) MEM-2, and (c) MEM-3.
(a)
(b)
(c)


## 4.1. Material Modeling

The behavior of concrete can be modeled in ABAQUS with the concrete damage plasticity (CDP) model [18, 26, 27, 30, 31]. The CDP model was first proposed by Lubliner et al. [32] and then modified by Lee and Fenves [33, 34]. To define the CDP model, the uniaxial constitutive laws of concrete under compression and tension together with five constitutive parameters are required. Since only compressive strength and Young’s modulus of concrete were reported in the test [3], as shown in Figure 9(a), the unconfined concrete uniaxial compressive stress-strain curve suggested by Mander et al. [35] is adopted, where ε0=2fc/E0. In tension, the behavior of concrete is assumed to be linearly elastic up to the onset of cracking and then followed by tension softening. In this paper as shown in Figure 9(b), the stress-displacement curve suggested by the CEB code [36] is applied. The fracture energy is calculated by Gf=73fcm0.18, where fcm is the mean compressive strength. The tensile strength of concrete is determined by ftm=0.3fck2/3 [36], where fck is the characteristic value of compressive strength (fck=fcm−8in MPa). The five constitutive parameters are given in Table 3. Since no relative information is available from the experimental tests, default values of the five constitutive parameters [30] are used except for the viscosity parameter. As suggested by Lee and Fenves [34], the value of the viscosity parameter can be set at about 15% of one time increment in order for the solution to be improved without changing the result. In the standard static analysis of the current model, about 3,000 increments are needed in a time period of one second; thus, as shown in Table 3, the viscosity parameter is set at 5·10−5. Damage is introduced into the CDP model in compression and tension according to Figures 9(c) and 9(d). The shape function of the unloading and reloading of compressive concrete suggested by Birtel and Mark [37] is applied to determine the evolution of the compressive damage dc. A tensile damage-displacement curve is used to be coherent with the tension softening curve of concrete [27].Figure 9
Constitutive laws of concrete: (a) compressive stress-strain curve, (b) tensile stress-crack width curve, (c) compressive damage-inelastic strain curve, and (d) tensile damage-crack width curve.
(a)
(b)
(c)
(d)Table 3
Additional parameters for the CDP model.
Dilation angle
Eccentricity
f
b0/fc0
K
Viscosity parameter
35°
0.1
1.16
0.67
5 · 10−5As shown in Figure10, linear hardening is introduced to simulate the behavior of reinforcement after its yielding [26]. The material mechanical parameters used in the FE models are the same as the tested ones, which have been shown in Table 1. The elastic modules of reinforcement are set as 200 (GPa). ABAQUS/standard static analyses are applied for all the FE models.Figure 10
Stress-strain curve for the reinforcement.


## 4.2. Calibration of the FE Model

The test result of MEM-2 is used to calibrate the FE model in terms of torque-rotation curve, crack pattern, and failure mode. Referring to Figure7, the rotation at the midspan is obtained by α = arctan (U22−U21/D12+U11−U12), where the first subscript of U means the direction of displacement and the second subscript means RP1 or RP2. D12 is the original distance between RP1 and RP2. The applied torque is calculated by T=FrD12+U11−U12, where Fr is the average value of the two reaction forces at RP1 and RP2. At the outset, it is essential to discuss the chosen material parameters, the mesh size, and the element type. Since it is known that a larger viscosity parameter will improve the solution but gives a stiffer response [26, 30, 34], the chosen viscosity parameter should be calibrated. As shown in Figure 11(a), the simulated torque-rotation curves with a series of viscosity parameters are compared with each other as well as with the test result. It can be seen that the viscosity parameters of 2 · 10−5, 5 · 10−5(used in this paper), and 10 · 10−5 give almost the same results, which are accurate compared with the test results. With the increase of the viscosity parameter to 50 · 10−5 and 100 · 10−5, stiffer responses are as expected. The chosen viscosity parameter of 5 · 10−5 is proved to be accurate. Due to the localization of the concrete tensile cracks, the FE model is mesh size dependent as happens in most plasticity-based FE models with strain softening [26, 27, 30]. Hence, a mesh convergence study is performed. As shown in Figure 11(b), two mesh sizes of 35 mm and 23 mm were compared with each other to investigate the mesh sensitivity (35 mm mesh size giving two elements through the thickness of the thin wall while 23 mm mesh size giving three). It can be seen from Figure 11(b) that similar responses are obtained for the two mesh sizes, and they are accurate compared with the test result. Considering the computing cost, larger mesh size of 35 mm is selected. Comparison between simulated results with different element types is shown in Figure 11(c). Elements C3D20R are used to obtain the correct solution. It can be seen from Figure 11(c) that elements C3D8R and C3D20R give similar simulation results.Figure 11
Torque-rotation curves with (a) different viscosity parameters, (b) different mesh sizes, and (c) different element types.
(a)
(b)
(c)The simulated crack pattern and failure mode are shown in Figures12(a) and 12(b), respectively, where the simulated crack pattern is approximately represented by the distribution of concrete tensile damage (DAMAGET) and the simulated failure mode is represented by the distribution of concrete compressive damage (DAMAGEC). Comparing Figure 12(a) with Figure 8, it can be seen that the propagation and distribution of cracks are well simulated by the built FE model (detail features are stated in Section 3). Comparing Figure 12(b) with Figure 8, it can be observed that the distribution of DAMAGEC agrees well with the test result. The maximum DAMAGEC occurs at midspan on the top of the downward web and at the support on the top of the upward web, and those were the locations where concrete crushing occurred in the test. Flexural failure dominated by the warping moment occurred both in the simulation and in the test. From the above comparisons between the simulated results and the test results, it can be concluded that the FE model in this paper can accurately simulate the mixed torsional behavior of the tested specimen. The simulated torque-rotation curves of all the three girder specimens are shown in Figure 13.Figure 12
The simulated crack patterns and failure modes: (a) distribution of tensile damage at a load level of 80% ultimate torque and (b) distribution of compressive damage at ultimate load level. “FC” means flexural cracking, “YL” means yielding of longitudinal bars, and “YS” means yielding of stirrups.
(a)
(b)Figure 13
Simulated and tested torque-rotation curves: (a) MEM-1, (b) MEM-2, and (c) MEM-3.
(a)
(b)
(c)


## 5. Parametric Study

A parametric study considering three crucial parameters, boundary condition (fixed-fixed or simply supported), span length-characteristic length ratio, and the ratio of longitudinal bars to stirrups, is conducted. The design of the simple supporting is shown in Figure14, where the coupling constraint with the continuum distributing and linear weighting method is used between a reference point and its slave surface. Then, the displacements in the Y direction (U2) are restrained for the reference points on the top and at the bottom, while the displacements in the X direction (U1) are restrained for the reference points on the two sides. In this supporting design, the displacements of the girder end in the X and Y directions (U1 and U2) are restrained while the displacement in the Z direction (U3) is free; as to the rotational freedoms, the rotation of the girder end around the Z direction (UR3) is restrained while the other two rotations, UR1 and UR2, are free.Figure 14
Design of the simple supporting.


### 5.1. Effect of Span Length-Characteristic Length Ratio

As discussed in Section2.1 with the changing of span length-characteristic length (SL-CL) ratio, different failure types may occur. The mixed torsional behaviors of these girders with varying span length-characteristic length (SL-CL) ratios are investigated here. For the fixed-fixed supported cases, the SL-CL ratio is designed to have a range of from 2.0 to 5.0 (the corresponding span length-section height ratio is from 8.2 to 20.6), and for the simply supported cases, it is from 2.0 to 4.0 (the corresponding span length-section height ratio is from 8.2 to 16.5). The above specified ranges for the SL-CL ratio should have covered all the engineering practices [2]. The numerical package of MEM-2 is applied here. The simulated torque versus midspan rotation curves for the fixed-fixed supported cases and for the simply supported cases are shown in Figures 15(a) and 15(b), respectively. The simulated results with the largest concrete compressive damage (DAMAGEC) at flexural critical segments (for fixed-fixed supported cases at midspan and support; for simply supported cases at midspan) are plotted with solid lines, while those with largest DAMAGEC at shear critical segments (for fixed-fixed supported cases at quarter segment; for simply supported cases at support) are plotted with dashed lines. Two typical simulated results with largest DAMAGEC at shear critical segments are shown in Figures 16(a) and 16(b). It can be seen from Figures 15(a) and 15(b) that, with the increase of span length, the ultimate torque drops while the ultimate rotation goes up.Figure 15
Simulated and calculated results of girders with various span lengths under different boundary conditions: (a) torque-rotation curves under fixed-fixed supporting, (b) torque-rotation curves under simple supporting, and (c) simulated and calculated failure modes and ultimate torques. In (a) and (b), “SL-CL ratio” means span length-characteristic length ratio; “YL” and “YS” mean yielding of longitudinal bars and yielding of stirrups, respectively. In (c), the first abbreviation “FS” and “SS” means fixed-fixed supported and simply supported, respectively; the second abbreviation “C” and “S” means calculated and simulated, respectively; and the third abbreviation “FF,” “SF,” and “FSF” means flexural failure, shear failure, and flexural-shear failure, respectively.
(a)
(b)
(c)Figure 16
Two typical simulated results with the largest DAMAGEC at shear critical segment: (a) fixed-fixed supported SL-CL ratio 2.0 case and (b) simply supported SL-CL ratio 3.5 case.
(a)
(b)Theoretically the simulated failure type can be determined by the distribution of DAMAGEC, namely, the place where the concrete is first crushed gets the maximum DAMAGEC. However, considering that the shear failure is brittle [3], in this paper, the shear failure is also determined by the yielding of stirrups and longitudinal bars at shear critical segments. Therefore, to determine which kind of failure occurs, the following criterion is applied. When the DAMAGEC at shear critical segment is larger than or as large as the DAMAGEC at bending critical segment, it is classified as a shear failure; if not, check the stress condition of reinforcement at shear critical segment: if both the stirrups and the longitudinal bars yield, it is defined as a flexural-shear failure (which means flexural failure and shear failure occur simultaneously); or if either the stirrups or the longitudinal bars at shear critical segment do not yield, it is classified as a flexural failure. According to the criterion, the simulated failure modes are summarized in Figure 15(c). The calculated results based on the calculating method developed in Section 2.2 are also shown in Figure 15(c). It can be seen from Figure 15(c) that the calculating method can give a good prediction for the failure modes as well as the ultimate torques. As shown in Figure 15(c) for the fixed-fixed supported cases, shear failure dominated by the warping torque occurs when the span length is small (SL-CL ratio 2.0). With the increase of span length to medium values (SL-CL ratios from 3.0 to 4.0), flexural failure dominated by the warping moment occurs, and when the span length further increases to 4.5 times the characteristic length, flexural-shear failures are observed. As for the simply supported cases, with the increasing of span length from 2.0 times to 4.0 times the characteristic length, the failure changes from flexural-shear type to shear type. The evolutions of failure type agree well with the expectations in Section 2.1.It should be mentioned that ductile behaviors are observed even in flexural-shear failure cases and in shear failure cases. It can be explained by that, as shown in Figures15(a) and 15(b), the longitudinal bars at midspan (flexure critical segment) always yield first even in flexural-shear failure cases and shear failure cases. Besides, it should be explained that the negative stiffness behaviors in the simply supported SL-CL 3.5 and SL-CL 4.0 cases (Figure 15(b)) are due to the intensive shear cracking caused by circulatory torque Tc. This negative stiffness behavior due to intensive shear cracking has been often observed in the circulatory torsional tests [10, 14, 15].


### 5.2. Effect of Boundary Condition

The comparison of torque-rotation curves under different boundary conditions are plotted in Figures17(a)–17(c), respectively. It can be seen from Figures 17(a)–17(c) that when the boundary condition is changed from fixed-fixed supporting to simple supporting, the ultimate torques decreased dramatically while the ultimate rotations increased significantly. Besides, as shown in Figure 15(c), when the span length is 3.5 times the characteristic length, for the fixed-fixed supported case, flexural failure dominated by the warping moment occurs, while for the simply supported case, shear failure dominated by circulatory torque occurs. It can be concluded from both calculated results and simulated results shown in Figure 15(c) that the evolution of the failure type with the increasing of the span length is faster in the simply supported case than in the fixed-fixed supported case.Figure 17
Simulated torque-rotation curves under different boundary conditions with (a) SL-CL ratio of 2.0, (b) SL-CL ratio of 3.0, and (c) SL-CL ratio of 4.0. “SL-CL ratio” means span length-characteristic length ratio; “YL” means yielding of longitudinal bars; “YS” means yielding of stirrups.
(a)
(b)
(c)


### 5.3. Effect of Ratio of Longitudinal Bars to Stirrups

As indicated by the calculating method in Section2.2, the ultimate torque in a flexural failure depends mainly on the longitudinal bars while the ultimate torque in a shear failure depends on both longitudinal bars and stirrups. Thus, it can be expected that the mixed torsional behavior of the U-shaped thin-walled RC girders varies with different reinforcement ratios. The numerical package of MEM-2 is applied here to do the parametric study. For MEM-2, the original reinforcement ratio is 2.0 (ρlfly/ρtfty = 2.0), and flexural failure occurred. In this parametric study, the section area of stirrups is kept constant and the section area of longitudinal bars is increased to 120%, 140%, 160%, and 180% (corresponding reinforcement ratios: 2.4, 2.8, 3.2, and 3.6) as well as decreased to 80% (the corresponding reinforcement ratio of 1.6). The corresponding simulated torque-rotation curves are shown in Figure 18(a), while the simulated failure modes and ultimate torques together with the calculation results are shown in Figure 18(b). As shown in Figure 18(b), both calculation and simulation give the same results that when the reinforcement ratio is not larger than 2.0, flexural failure occurs. With the increase of the section area of longitudinal bars, flexural-shear failure and then shear failure occur.Figure 18
Simulated and calculated results considering different reinforcement ratios: (a) simulated torque-rotation curves and (b) simulated and calculated ultimate torques. “L-S ratio” means the ratio of longitudinal bars to stirrups; “YL” and “YS” mean yielding of longitudinal bars and yielding of stirrups, respectively.
(a)
(b)


### 5.4. Comparison of the Calculated and the Simulated Torque Capacities

The calculated and simulated ultimate torques for all the above parametrically studied cases are compared in Table4. Considering that the characteristic length (k = 2.06 m) in the above parametrically studied cases is constant, as shown in Figure 19, another two characteristic lengths of 1.8 m and 2.5 m are designed. The designed values of the characteristic length should have covered the practical engineering range [2]. Then the numerical package of MEM-2 is applied to simulate the mixed torsional behaviors of girders with various characteristic values. A fixed-fixed supported case and a simply supported case were studied for each characteristic length, with setting the span length at 3.5 times the characteristic length. The simulated results are shown in Figure 20. The simulated and calculated ultimate torques of these 4 cases are also compared in Table 4. As shown in Table 4, an average value of 1.08 for the simulation/calculation with a coefficient of variation of 5.6% proves that the calculating method can give good predictions for the torque capacities.Table 4
Comparison of the simulated and calculated ultimate torques.
Parametrically studied cases
Ultimate torque (kNm)
Simulation/calculation
Simulation
Calculation
Fixed-fixed supported cases with varying span length (Figure15)
SL-CL ratio 2.0
176.4
174.3
1.01
SL-CL ratio 2.5
169.2
152.2
1.11
SL-CL ratio 3.0
157.6
137.5
1.15
SL-CL ratio 3.5
141.7
126.8
1.12
SL-CL ratio 4.0
126.8
114.7
1.11
SL-CL ratio 4.5
115.6
104.2
1.11
SL-CL ratio 5.0
106.4
95.7
1.11
Simply supported cases with varying span length (Figure15)
SL-CL ratio 2.0
116.2
114.7
1.01
SL-CL ratio 2.5
99.8
95.7
1.04
SL-CL ratio 3.0
89.7
83.5
1.07
SL-CL ratio 3.5
83.2
75.4
1.10
SL-CL ratio 4.0
78.6
69.9
1.12
Cases with varying ratios of longitudinal bars to stirrups (Figure18)
L-S ratio 1.6
127.8
109.5
1.17
L-S ratio 2.0
149.5
133.1
1.12
L-S ratio 2.4
157.7
148.8
1.06
L-S ratio 2.8
159.5
157.5
1.01
L-S ratio 3.2
159.7
157.5
1.01
L-S ratio 3.6
160.2
157.5
1.02
Cases with the characteristic length of 1.8 m (Figure20)
Fixed-fixed supported
130.3
120.2
1.08
Simply supported
79.4
71.1
1.12
Cases with the characteristic length of 2.5 m (Figure20)
Fixed-fixed supported
149.3
134.8
1.11
Simply supported
98.7
90.5
1.09
Average value
1.08
Standard deviation
0.06
Coefficient of variation
5.6%Figure 19
Girders with various characteristic lengths of (a) 1.8 m and (b) 2.5 m.
(a)
(b)Figure 20
Simulated results of girders with various characteristic lengths. “FS” and “SS” mean “fixed-fixed supported” and “simply supported,” respectively; “YL” and “YS” mean “yielding of longitudinal bars” and “yielding of stirrups,” respectively.


## 5.1. Effect of Span Length-Characteristic Length Ratio

As discussed in Section2.1 with the changing of span length-characteristic length (SL-CL) ratio, different failure types may occur. The mixed torsional behaviors of these girders with varying span length-characteristic length (SL-CL) ratios are investigated here. For the fixed-fixed supported cases, the SL-CL ratio is designed to have a range of from 2.0 to 5.0 (the corresponding span length-section height ratio is from 8.2 to 20.6), and for the simply supported cases, it is from 2.0 to 4.0 (the corresponding span length-section height ratio is from 8.2 to 16.5). The above specified ranges for the SL-CL ratio should have covered all the engineering practices [2]. The numerical package of MEM-2 is applied here. The simulated torque versus midspan rotation curves for the fixed-fixed supported cases and for the simply supported cases are shown in Figures 15(a) and 15(b), respectively. The simulated results with the largest concrete compressive damage (DAMAGEC) at flexural critical segments (for fixed-fixed supported cases at midspan and support; for simply supported cases at midspan) are plotted with solid lines, while those with largest DAMAGEC at shear critical segments (for fixed-fixed supported cases at quarter segment; for simply supported cases at support) are plotted with dashed lines. Two typical simulated results with largest DAMAGEC at shear critical segments are shown in Figures 16(a) and 16(b). It can be seen from Figures 15(a) and 15(b) that, with the increase of span length, the ultimate torque drops while the ultimate rotation goes up.Figure 15
Simulated and calculated results of girders with various span lengths under different boundary conditions: (a) torque-rotation curves under fixed-fixed supporting, (b) torque-rotation curves under simple supporting, and (c) simulated and calculated failure modes and ultimate torques. In (a) and (b), “SL-CL ratio” means span length-characteristic length ratio; “YL” and “YS” mean yielding of longitudinal bars and yielding of stirrups, respectively. In (c), the first abbreviation “FS” and “SS” means fixed-fixed supported and simply supported, respectively; the second abbreviation “C” and “S” means calculated and simulated, respectively; and the third abbreviation “FF,” “SF,” and “FSF” means flexural failure, shear failure, and flexural-shear failure, respectively.
(a)
(b)
(c)Figure 16
Two typical simulated results with the largest DAMAGEC at shear critical segment: (a) fixed-fixed supported SL-CL ratio 2.0 case and (b) simply supported SL-CL ratio 3.5 case.
(a)
(b)Theoretically the simulated failure type can be determined by the distribution of DAMAGEC, namely, the place where the concrete is first crushed gets the maximum DAMAGEC. However, considering that the shear failure is brittle [3], in this paper, the shear failure is also determined by the yielding of stirrups and longitudinal bars at shear critical segments. Therefore, to determine which kind of failure occurs, the following criterion is applied. When the DAMAGEC at shear critical segment is larger than or as large as the DAMAGEC at bending critical segment, it is classified as a shear failure; if not, check the stress condition of reinforcement at shear critical segment: if both the stirrups and the longitudinal bars yield, it is defined as a flexural-shear failure (which means flexural failure and shear failure occur simultaneously); or if either the stirrups or the longitudinal bars at shear critical segment do not yield, it is classified as a flexural failure. According to the criterion, the simulated failure modes are summarized in Figure 15(c). The calculated results based on the calculating method developed in Section 2.2 are also shown in Figure 15(c). It can be seen from Figure 15(c) that the calculating method can give a good prediction for the failure modes as well as the ultimate torques. As shown in Figure 15(c) for the fixed-fixed supported cases, shear failure dominated by the warping torque occurs when the span length is small (SL-CL ratio 2.0). With the increase of span length to medium values (SL-CL ratios from 3.0 to 4.0), flexural failure dominated by the warping moment occurs, and when the span length further increases to 4.5 times the characteristic length, flexural-shear failures are observed. As for the simply supported cases, with the increasing of span length from 2.0 times to 4.0 times the characteristic length, the failure changes from flexural-shear type to shear type. The evolutions of failure type agree well with the expectations in Section 2.1.It should be mentioned that ductile behaviors are observed even in flexural-shear failure cases and in shear failure cases. It can be explained by that, as shown in Figures15(a) and 15(b), the longitudinal bars at midspan (flexure critical segment) always yield first even in flexural-shear failure cases and shear failure cases. Besides, it should be explained that the negative stiffness behaviors in the simply supported SL-CL 3.5 and SL-CL 4.0 cases (Figure 15(b)) are due to the intensive shear cracking caused by circulatory torque Tc. This negative stiffness behavior due to intensive shear cracking has been often observed in the circulatory torsional tests [10, 14, 15].


## 5.2. Effect of Boundary Condition

The comparison of torque-rotation curves under different boundary conditions are plotted in Figures17(a)–17(c), respectively. It can be seen from Figures 17(a)–17(c) that when the boundary condition is changed from fixed-fixed supporting to simple supporting, the ultimate torques decreased dramatically while the ultimate rotations increased significantly. Besides, as shown in Figure 15(c), when the span length is 3.5 times the characteristic length, for the fixed-fixed supported case, flexural failure dominated by the warping moment occurs, while for the simply supported case, shear failure dominated by circulatory torque occurs. It can be concluded from both calculated results and simulated results shown in Figure 15(c) that the evolution of the failure type with the increasing of the span length is faster in the simply supported case than in the fixed-fixed supported case.Figure 17
Simulated torque-rotation curves under different boundary conditions with (a) SL-CL ratio of 2.0, (b) SL-CL ratio of 3.0, and (c) SL-CL ratio of 4.0. “SL-CL ratio” means span length-characteristic length ratio; “YL” means yielding of longitudinal bars; “YS” means yielding of stirrups.
(a)
(b)
(c)


## 5.3. Effect of Ratio of Longitudinal Bars to Stirrups

As indicated by the calculating method in Section2.2, the ultimate torque in a flexural failure depends mainly on the longitudinal bars while the ultimate torque in a shear failure depends on both longitudinal bars and stirrups. Thus, it can be expected that the mixed torsional behavior of the U-shaped thin-walled RC girders varies with different reinforcement ratios. The numerical package of MEM-2 is applied here to do the parametric study. For MEM-2, the original reinforcement ratio is 2.0 (ρlfly/ρtfty = 2.0), and flexural failure occurred. In this parametric study, the section area of stirrups is kept constant and the section area of longitudinal bars is increased to 120%, 140%, 160%, and 180% (corresponding reinforcement ratios: 2.4, 2.8, 3.2, and 3.6) as well as decreased to 80% (the corresponding reinforcement ratio of 1.6). The corresponding simulated torque-rotation curves are shown in Figure 18(a), while the simulated failure modes and ultimate torques together with the calculation results are shown in Figure 18(b). As shown in Figure 18(b), both calculation and simulation give the same results that when the reinforcement ratio is not larger than 2.0, flexural failure occurs. With the increase of the section area of longitudinal bars, flexural-shear failure and then shear failure occur.Figure 18
Simulated and calculated results considering different reinforcement ratios: (a) simulated torque-rotation curves and (b) simulated and calculated ultimate torques. “L-S ratio” means the ratio of longitudinal bars to stirrups; “YL” and “YS” mean yielding of longitudinal bars and yielding of stirrups, respectively.
(a)
(b)


## 5.4. Comparison of the Calculated and the Simulated Torque Capacities

The calculated and simulated ultimate torques for all the above parametrically studied cases are compared in Table4. Considering that the characteristic length (k = 2.06 m) in the above parametrically studied cases is constant, as shown in Figure 19, another two characteristic lengths of 1.8 m and 2.5 m are designed. The designed values of the characteristic length should have covered the practical engineering range [2]. Then the numerical package of MEM-2 is applied to simulate the mixed torsional behaviors of girders with various characteristic values. A fixed-fixed supported case and a simply supported case were studied for each characteristic length, with setting the span length at 3.5 times the characteristic length. The simulated results are shown in Figure 20. The simulated and calculated ultimate torques of these 4 cases are also compared in Table 4. As shown in Table 4, an average value of 1.08 for the simulation/calculation with a coefficient of variation of 5.6% proves that the calculating method can give good predictions for the torque capacities.Table 4
Comparison of the simulated and calculated ultimate torques.
Parametrically studied cases
Ultimate torque (kNm)
Simulation/calculation
Simulation
Calculation
Fixed-fixed supported cases with varying span length (Figure15)
SL-CL ratio 2.0
176.4
174.3
1.01
SL-CL ratio 2.5
169.2
152.2
1.11
SL-CL ratio 3.0
157.6
137.5
1.15
SL-CL ratio 3.5
141.7
126.8
1.12
SL-CL ratio 4.0
126.8
114.7
1.11
SL-CL ratio 4.5
115.6
104.2
1.11
SL-CL ratio 5.0
106.4
95.7
1.11
Simply supported cases with varying span length (Figure15)
SL-CL ratio 2.0
116.2
114.7
1.01
SL-CL ratio 2.5
99.8
95.7
1.04
SL-CL ratio 3.0
89.7
83.5
1.07
SL-CL ratio 3.5
83.2
75.4
1.10
SL-CL ratio 4.0
78.6
69.9
1.12
Cases with varying ratios of longitudinal bars to stirrups (Figure18)
L-S ratio 1.6
127.8
109.5
1.17
L-S ratio 2.0
149.5
133.1
1.12
L-S ratio 2.4
157.7
148.8
1.06
L-S ratio 2.8
159.5
157.5
1.01
L-S ratio 3.2
159.7
157.5
1.01
L-S ratio 3.6
160.2
157.5
1.02
Cases with the characteristic length of 1.8 m (Figure20)
Fixed-fixed supported
130.3
120.2
1.08
Simply supported
79.4
71.1
1.12
Cases with the characteristic length of 2.5 m (Figure20)
Fixed-fixed supported
149.3
134.8
1.11
Simply supported
98.7
90.5
1.09
Average value
1.08
Standard deviation
0.06
Coefficient of variation
5.6%Figure 19
Girders with various characteristic lengths of (a) 1.8 m and (b) 2.5 m.
(a)
(b)Figure 20
Simulated results of girders with various characteristic lengths. “FS” and “SS” mean “fixed-fixed supported” and “simply supported,” respectively; “YL” and “YS” mean “yielding of longitudinal bars” and “yielding of stirrups,” respectively.


## 6. Distributions of Shear Stress Resultant

The distributions of shear stress resultant around the shear critical section at different loading levels are investigated. The shear stress distributions of the fixed-fixed supported SL-CL ratio 3.5 case and the simply supported SL-CL ratio 3.5 case are shown in Figures21 and 22, respectively. The direction of circulatory torsional shear stress is taken as positive. It is in the elastic range when the load level is at 10% ultimate torque. The load level of 30% ultimate torque is in the postcracking range, while the load level of 80% ultimate torque is in the postyielding range. It can be seen from Figures 21 and 22 that due to the interactions of the two kinds of shear stresses, the shear stress resultant on the external sides of webs is larger than those on the internal sides of webs, which justifies the assumed shear stress distribution in a shear failure.Figure 21
Shear stress distributions around quarter section at different loading levels of (a) 10% ultimate torque, (b) 30% ultimate torque, and (c) 80% ultimate torque.
(a)
(b)
(c)Figure 22
Shear stress distributions around support section at different loading levels of (a) 10% ultimate torque, (b) 30% ultimate torque, and (c) 80% ultimate torque.
(a)
(b)
(c)As shown in Figures21 and 22, the shear stress resultants on the external sides of webs are the summation of the two kinds of shear stresses, while the shear stress resultants on the internal sides of webs are circulatory torsional shear stress minus warping torsional shear stress. Thus, the shear stresses caused by circulatory torque and by warping torque can be separated from each other. Then, the ratios of circulatory torsional shear stress to the summation of shear stresses can be obtained at every loading step, which is shown in Figure 23. As shown in Figure 23, this ratio fluctuated due to variation of the respective stiffness of circulatory torsion and warping torsion. For both cases in Figure 23, the ratio at the ultimate state is relatively smaller than that in the elastic range. This proves the proposed ultimate equilibrium for shear failure (the original ratio in elastic range of circulatory torque to warping torque is applied) to be conservative.Figure 23
Ratio of circulatory torsional shear stress to the summation of shear stresses.


## 7. Conclusions

In this paper, the mixed torsional behaviors of the U-shaped thin-walled RC girders are parametrically investigated based on the theoretical analysis, the ultimate equilibrium analysis, and the nonlinear FE simulation. Three crucial parameters are considered, which are boundary condition (fixed-fixed supported or simply supported), span length-characteristic length ratio, and ratio of longitudinal bars to stirrups. This study gives a thorough understanding on the mixed torsional behavior of the U-shaped thin-walled RC girders. The conclusions are drawn in the following:(1)
The built nonlinear FE models can precisely simulate the mixed torsional behaviors of the tested U-shaped thin-walled RC girders in terms of not only the overall torque-rotation curves but also the cracking patterns and the failure modes.(2)
A method able to calculate both failure mode and ultimate torque is derived from the ultimate equilibriums at the flexural critical and shear critical segments. The calculated results are in good agreement with the test results and with the simulated results.(3)
The evolution of failure type with the increasing of span length is theoretically discovered and numerically verified: when the span length-characteristic length ratioγ is small, shear failure dominated by warping torque may occur; when γ has a medium value, flexural failure dominated by warping moment may occur; when γ has a large value, shear failure dominated by circulatory torque may occur. Besides, the evolution of failure type with the increasing of span length is faster in the simply supported case than that in the fixed-fixed supported case.(4)
The failure mode is also influenced by the reinforcement ratio between longitudinal bars and stirrups. For the specific fixed-fixed supported case with a span length of 3.23 times the characteristic length, when the reinforcement ratio is not larger than 2.0, flexural failure occurs at midspan, and as the reinforcement ratio increases, flexural-shear failure and then shear failure occur.



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*Source: 3497390-2018-11-15.xml*