Improved softened truss model of prestressed concrete box girders subjected to combined bending and torsion

Abstract

Prestressed concrete (PC) box girder is an innovative structure that is widely used in large-span bridges. Under eccentric loads such as vehicles, PC girders are inevitably under combined bending and torsion. To study its bending-torsional behavior, a refined full-range analysis model based on the combined action softening truss model (CA-STM) was proposed, where a simple and effective solution procedure and failure criteria were provided. The equilibrium equation, initial estimation equation for torque, and material constitutive model were modified by considering the prestressing effect to improve the CA-STM. An optimized algorithm was also employed to simplify the solution procedure instead of the traditional trial-and-error method, thus increasing the solution rate and stability. The theoretical curves predicted from the improved CA-STM exhibited great agreement with the available experimental results, and the predicted cracking and ultimate loads were also close to the experimental values. Hence, the improved CA-STM can reasonably predict the full-range mechanical response and failure modes of PC box girders subjected to combined bending and torsion, which provides great support to the design evaluation of such bridges.

Keywords

bending-torsional behavior prestressed concrete torsional full-range analysis combined action softening truss model optimized algorithm failure criteria

Introduction

Prestressed concrete (PC) box girders introduce an efficient structural system that uses prestressing strands to improve the cracking resistance of concrete significantly (Alnuaimi and Bhatt, 2006; Roberts et al., 2012; Shao et al., 2010). It provides several advantages, including high stiffness, low deformation, and good durability (Bernardo et al., 2020), when compared with reinforced concrete (RC) box girders. It also has a great torsional resistance due to the compressive stress generated by the prestressing offsetting the tensile stress of the torque (Jeng and Chao, 2018). Hence, the PC box girder is suitable for modern large-span bridge engineering, as the demand for improved strength and durability performance significantly increases (Bernardo et al., 2013). In practice, PC box girders are commonly subjected to eccentric loads such as vehicles, which may be inevitably subjected to a combination of bending and torsion (Ashour et al., 1999; Cheng et al., 2007). Simple empirical formulas based on codes (ACI Committee, 2019; GB50010 Mohurd, 2010) or finite element method (FEM) are usually adopted to calculate the bearing capacity of box girders. However, the simple formulas can only predict the cracking and ultimate loads, while the FEM involves a complex modeling process and time-consuming calculations for large models. Therefore, it is essential to develop a simple and efficient theoretical model to clearly understand the overall behavior of PC box girders during the bending-torsional coupling loading, which provides a basis for enhancement in bridge design.

Scholars have proposed various analytical models to predict the full-range torsional response of RC box girders since the 1980s, among which the softened truss model (STM) was the most widely used. Hsu and Mo (1985) developed a rotating angle softened truss model for torsion (RA-STMT), based on the space truss analogy. This model took into account the softening effect of concrete under biaxial stresses, which was suitable for predicting the performance of RC box girders under in-plane stress states (Mondal and Prakash, 2015). However, it had not captured the torsional behavior before concrete cracking, as it neglected the tension-stiffening effect of concrete. To solve this problem, Greene (2006) modified the RA-STMT by introducing the tensile constitutive relationship of concrete. The tension-stiffening effect of concrete was successfully considered and a reasonable tension-stiffened-softened truss model for torsion (TS-STMT) was proposed. Nevertheless, the above models can only be used to analyze the mechanical properties of RC box girders subjected to pure torsion.

To predict the mechanical performance of RC box girders under coupled bending-shear-torsion, the TS-STMT was modified to establish a combined action softened truss model (CA-STM) by Greene and Belarbi (2009a). This model idealized the cross-section of the box girder as four cracked RC panels, and assumed that the external loads applied to the box girder acted on these four panels in the form of uniform normal and shear stresses. The truss model of each cracked RC shear panel was established in accordance with RA-STMT. Greene and Belarbi (2009b) found that CA-STM was capable of predicting the mechanical response of RC box girders under combined loadings well, comparing the theoretical results with the collected 28 experimental results. However, the efficiency and stability of the solution algorithm of CA-STM were not satisfactory, as the calculation process of the traditional trial-and-error method used in CA-STM was complicated, and numerous unknown parameters needed to be solved.

To improve the solution rate and stability, Silva et al. (2017) adopted the gradient descent method instead of the traditional trial-and-error method to solve for all unknown parameters. It is an iterative optimization algorithm that decreases the gradient vector along the direction of the objective function to find its minimum value. The solution problem of the nonlinear system of equations is transformed into a least-squares problem by inputting the initial values and geometric parameters, thus establishing a modified combined action softened truss model (MCA-STM). It should be noted that the torsional mechanism of each shear panel in this model is established based on the RA-STMT, and therefore its judgment criterion is to take the principal compressive strain reaching the ultimate strain point as the termination point of analysis. However, the failure mode of different reinforced box girders under combined loading is not only controlled by this single criterion, but there are also several other damage forms. Additionally, since the effect of the initial stress-strain relationship of prestressing on the mechanical performance is not considered, the applicability of existing CA-STM for analyzing the bending-torsional behavior of PC box girders is still questionable.

This study aimed at predicting the bending-torsional behavior of PC beams under combined bending and torsion, based on the MCA-STM by inducing prestressing effect and failure criteria. The constitutive relation of prestressing strands was first introduced to modify the material constitutive and equilibrium equations of improved CA-STM. Subsequently, the initial estimation equation for torque was modified, considering the initial stress-strain relationship of prestressing. Then four new failure criteria for completely over-reinforced, partially over-reinforced, balanced-reinforced and under-reinforced beams was presented. Additionally, a simple equivalent area conversion method for reinforcement and prestressing strands was proposed, and the gradient descent method was employed instead of the traditional trial-and-error method to solve the nonlinear equation system. Finally, data from 20 PC box girders was collected to benchmark the improved CA-STM, by comparing the experimental results with the theoretical results.

Overview of the analysis model

Force mechanism

A box girder is typically idealized as four cracked RC shear panels for CA-STM, with each panel being modeled based on the RA-STMT. Figure 1(a) illustrates the force mechanism of the ideal section, i.e., concrete in a diagonal strut bears the compressive stress, and longitudinal reinforcement, transverse reinforcement, and prestressing strands resist the tensile force. Figure 1(b)–1(d) demonstrate the transformation process from an actual section to an ideal section. For a rectangular solid section, the maximum allowed thickness is represented by $t_{i}$ , with $i$ corresponding to the number of each panel, where the torsional contribution of the concrete core is neglected (Greene and Belarbi, 2009a). For a rectangular hollow section, $t_{i}$ is the real wall’s thickness. According to Bredt’s thin-tube theory, the shear flow generated by torsion is constant in the direction of the thin-wall thickness (Kollbrunner and Basler, 1969). When considering the rectangular PC section as an ideal thin wall, the ideal wall thickness $t_{d, i}$ can be determined by defining the geometry of the shear flow (Greene and Belarbi, 2009a). CA-STM assumes that the centerline of the shear flow coincides with the axis of the ideal thin wall, and then the closed region formed by the centerline of the shear flow can be represented by equations (1a)–(1c).

b_{0} = b - 0.5 t_{d, 1} - 0.5 t_{d, 3}

(1a)

h_{0} = h - 0.5 t_{d, 2} - 0.5 t_{d, 4}

(1b)

A_{0} = b_{0} h_{0}

(1c)

Where

b

and

h

are the width and height of the real section, respectively;

b_{0}

and

h_{0}

are the width and height of the ideal section, and

A_{0}

is the area enclosed by the center line of the shear flow.

Figure 1.

Idealization of the cross-section. (a) Idealized box girder. (b) Real cross-section. (c) Idealized cross-section. (d) Cross-section of shear flow.

CA-STM assumes that the external loads applied to the box girder result in uniform normal and shear stresses on the four shear panels, as shown in Figure 1(a). Among them, the counterclockwise shear flow caused by the torque ( $T_{X}$ ) acts on the entire ideal section, which is specified to be in the positive direction. The shear flow caused by the shear forces ( $V_{Y}$ and $V_{Z}$ ) only affects the shear panels that are parallel to the direction of loading. Figure 2 shows the superposition of the shear flow due to individual shear forces and torques in the ideal section. The net shear flow acting on each panel can be calculated by equations (2a)–(2e). The CA-STM assumes that the bending moments ( $M_{Z}$ and $M_{Y}$ ) and axial forces ( $N_{X}$ ) applied to the exterior of the box girder can be resisted by uniform normal stresses in each shear panel (see Figure 3), as shown in equations (3a)–(3c). Note that the positive bending moments are defined as moments along the counterclockwise direction of right cross-section and clockwise direction of left cross-section.

q_{1} = \frac{T_{X}}{2 A_{0}} + \frac{V_{Y}}{2 h_{0}}

(2a)

q_{2} = \frac{T_{X}}{2 A_{0}} + \frac{V_{Z}}{2 b_{0}}

(2b)

q_{3} = \frac{T_{X}}{2 A_{0}} - \frac{V_{Y}}{2 h_{0}}

(2c)

q_{4} = \frac{T_{X}}{2 A_{0}} - \frac{V_{Z}}{2 b_{0}}

(2d)

τ_{l t, i} = \frac{q_{i}}{t_{d, i}}

(2e)

M_{Z} = (σ_{l, 4} t_{d, 4} b_{0} - σ_{l, 2} t_{d, 2} b_{0}) \frac{h_{0}}{2}

(3a)

M_{Y} = (σ_{l, 3} t_{d, 3} h_{0} - σ_{l, 1} t_{d, 1} h_{0}) \frac{b_{0}}{2}

(3b)

N_{X} = σ_{l, 2} t_{d, 2} b_{0} + σ_{l, 4} t_{d, 4} b_{0} + σ_{l, 1} t_{d, 1} h_{0} + σ_{l, 3} t_{d, 3} h_{0}

(3c)

Where

q_{i}

and

τ_{l t, i}

are the shear flow and shear stress at each panel;

σ_{l, 1}

and

σ_{l, 3}

are the normal stresses on panels 1 and 3, and

σ_{l, 2}

and

σ_{l, 4}

are the normal stresses on panels 2 and 4.

Figure 2.

Distribution form of shear flow.

Figure 3.

Normal stress distribution of panels 2 and 4.

Equilibrium, compatibility and additional equations

Each shear panel will generate the corresponding shear flow $q_{i}$ to resist the external torque under torsional loading (Silva et al., 2017). A shear element of unit length in an arbitrary shear panel is chosen for analysis, assuming that its internal concrete and reinforcement form a truss model. Based on the in-plane equilibrium relationships, the equilibrium and coordination equations within each shear plane are thus established, as shown in equations (4a)–(4c) and (5a)–(5b).

σ_{l, i} = σ_{d, i} \cos^{2} (α_{d, i}) + σ_{r, i} \sin^{2} (α_{d, i}) + f_{l, i} p_{l, i} + f_{p s, i} p_{p s, i}

(4a)

σ_{t, i} = σ_{r, i} \cos^{2} (α_{d, i}) + σ_{d, i} \sin^{2} (α_{d, i}) + f_{t, i} p_{t}

(4b)

τ_{l t, i} = (σ_{r, i} - σ_{d, i}) \sin (α_{d, i}) \cos (α_{d, i}) s i g n (q_{i})

(4c)

γ_{l t, i} = 2 (ε_{r, i} - ε_{d, i}) \sin (α_{d, i}) \cos (α_{d, i}) s i g n (q_{i})

(5a)

ε_{t, i} = ε_{r, i} + ε_{d, i} - ε_{l, i}

(5b)

Where

f_{l, i}

and

f_{t, i}

are the stresses of longitudinal and transverse reinforcement;

A_{l, i}

and

A_{t}

are the cross-sectional areas of longitudinal and transverse reinforcement;

σ_{l, i}

and

σ_{t, i}

are the stresses in l- and t-directions;

σ_{d, i}

and

σ_{r, i}

are the principal compressive and tensile stresses of concrete;

ε_{l, i}

and

ε_{t, i}

are the strains of longitudinal and transverse reinforcement;

ε_{d, i}

and

ε_{r, i}

are the principal compressive and tensile strains of concrete;

p_{l, i}

is the longitudinal reinforcement rate in each shear panel,

{p_{l, i} = A}_{l, i} / (t_{d, i} w_{0, i})

;

p_{p s, i}

is the reinforcement rate of the prestressing strands for each shear panel,

{p_{p s, i} = A}_{p s, i} / (t_{d, i} w_{0, i})

;

A_{p s, i}

is the cross-sectional area of prestressing strands for each shear panel;

p_{t}

is the transverse reinforcement ratio,

p_{t} = A_{t} / (t_{d, i} s)

;

s

is the distance between transverse reinforcement, and

γ_{l t, i}

is the shear strain;

w_{0, i}

is the width of the panel (equal to

b_{0}

for panels 2 and 4, and equal to

h_{0}

for panels 1 and 3),

α_{d, i}

is the angle of the concrete strut, and

s i g n (q_{i})

is the sign of shear flow corresponding to different panels.

According to the compatibility of equations (5a) and (5b), and the trigonometric transformation relationship, the angle of the concrete struts in thin walls $α_{d, i}$ can be expressed by the equations (6a) and(6b). Equation (6c) can be obtained by multiplying equations (6a), (6b), and (6d) can be obtained by combining equations (6a) and (6b).

\sin^{2} (α_{d, i}) = \frac{ε_{l, i} - ε_{d, i}}{ε_{r, i} - ε_{d, i}} = \frac{ε_{r, i} - ε_{t, i}}{ε_{r, i} - ε_{d, i}}

(6a)

\cos^{2} (α_{d, i}) = \frac{ε_{r, i} - ε_{l, i}}{ε_{r, i} - ε_{d, i}} = \frac{ε_{t, i} - ε_{d, i}}{ε_{r, i} - ε_{d, i}}

(6b)

\cos (α_{d, i}) \sin (α_{d, i}) = \frac{\sqrt{ε_{r, i} - ε_{l, i}} \sqrt{ε_{l, i} - ε_{d, i}}}{ε_{r, i} - ε_{d, i}}

(6c)

α_{d, i} = \arctan (\sqrt{\frac{ε_{l, i} - ε_{d, i}}{ε_{t, i} - ε_{d, i}}}) s i g n (q_{i})

(6d)

The torque causes torsional deformation of the external surface of PC box girders, resulting in a hyperbolic paraboloid shape of the central surface of shear element. As a result, concrete strut experiences both bending and compression. The curvature $ψ_{i}$ and twist $θ$ of the concrete strut on each shear panel can be expressed by equations (7a) and (7b).

ψ_{i} = θ \sin (2 α_{d, i})

(7a)

θ = \frac{p_{0}}{2 A_{0}} γ_{l t}

(7b)

As the CA-STM assumes that the torque can be combined with other internal forces, the curvature $ψ_{i}$ of the concrete strut needs to account for the longitudinal and transverse curvature $(ϕ_{l, 13}, ϕ_{l, 24}, ϕ_{t, 13}, ϕ_{t, 24})$ relationships between the thin walls. The twist $θ$ of the concrete strut needs to be considered for the contribution of each panel, and equations (7a) and (7b) are expressed as:

ψ_{i} = θ \sin (2 α_{d, i}) + [\begin{array}{l} - ϕ_{l, 13} \\ - ϕ_{l, 24} \\ ϕ_{l, 13} \\ ϕ_{l, 24} \end{array}] \cos^{2} (α_{d, i}) + [\begin{array}{l} - ϕ_{t, 13} \\ - ϕ_{t, 24} \\ ϕ_{t, 13} \\ ϕ_{t, 24} \end{array}] \sin^{2} (α_{d, i})

(7c)

θ = [(γ_{l t, 2} + γ_{l t, 4}) b_{0} + (γ_{l t, 1} + γ_{l t, 3}) h_{0}] \frac{1}{2 A_{0}}

(7d)

The curvature causes the strain gradient effect inside the concrete strut. The strain of the concrete strut is assumed to be linearly distributed over the range of $t_{d, i}$ in this research. A dimensionless parameter $z_{i}$ $(0 \leq z_{i} \leq 3)$ is introduced to systematically depict the strain distribution along the concrete strut thickness in each situation (Silva et al., 2017). Figure 4 shows the strain distribution in shear flow area. From this, the relationship between average compressive strain $ε_{d, i}$ , curvature $ψ_{i}$ , lower surface compressive strain $ε_{a, i}$ and the ideal thickness of section $t_{d, i}$ with $z_{i}$ of the concrete strut can be obtained, as shown in equations (8a)–(8d). As for the maximum allowed thickness for solid sections, the calculations in this study are based on the torsional design method in the NP EN code (2005), as recommended by Silva et al. (2017) (see equations (9a)–(9d)).

ε_{d, i} = \frac{ε_{d s, i} + ε_{a, i}}{2}

(8a)

ψ_{i} = - \frac{ε_{d s, i} - ε_{a, i}}{t_{d, i}}

(8b)

ε_{a, i} = {\begin{array}{c} 0 for 0 < z_{i} \leq 2 \\ (z_{i} - 2) ε_{d s, i} for 2 < z_{i} \leq 3 \end{array}

(8c)

t_{d, i} = {\begin{cases} \frac{z_{i} t_{i}}{2} for 0 < z_{i} \leq 2 \\ t_{i} for 2 < z_{i} \leq 3 \end{cases}

(8d)

t_{i} = \frac{A_{c p}}{p_{c p}}

(9a)

A_{c p} = b h

(9b)

p_{c p} = 2 b + 2 h

(9c)

A_{g} = (b - t_{1}) t_{4} + (h - t_{2}) t_{1} + (b - t_{3}) t_{2} + (h - t_{4}) t_{3}

(9d)

Figure 4.

Strain distribution in shear flow area. (a)Type 1. (b) Type 2. (c) Type 3. (d) Type 4.

Where $ε_{d s, i}$ and $ε_{a, i}$ are strains at the exterior and interior surfaces of the concrete strut, respectively; $A_{c p}$ is the total area surrounding the outer edge, and $A_{g}$ and $p_{c p}$ are the concrete area and perimeter of the real section.

Compatibility between panels

In addition to the compatibility equations of the RA-STMT, the CA-STM also considers the compatibility relationships between adjacent panels. The difference in longitudinal and transverse strains between panels 1 and 3 causes the formation of longitudinal curvature $ϕ_{l, 13}$ and transverse curvature $ϕ_{t, 13}$ . Similarly, the difference in longitudinal and transverse strains between panels 2 and 4 leads to the formation of longitudinal curvature $ϕ_{l, 24}$ and transverse curvature $ϕ_{t, 24}$ , as described in equations (10a)–(10e). Furthermore, since the CA-STM assumes that the longitudinal strains in the four thin walls are connected by a linear gradient effect, the longitudinal strains between adjacent panels can be represented by equation (10e) in accordance with the optimized approach proposed by Silva et al. (2017).

ϕ_{l, 13} = \frac{ε_{l, 1} - ε_{l, 3}}{b_{0}}

(10a)

ϕ_{l, 24} = \frac{ε_{l, 2} - ε_{l, 4}}{h_{0}}

(10b)

ϕ_{t, 13} = \frac{ε_{t, 1} - ε_{t, 3}}{b_{0}}

(10c)

ϕ_{t, 24} = \frac{ε_{t, 2} - ε_{t, 4}}{h_{0}}

(10d)

ε_{l, 1} + ε_{l, 3} = ε_{l, 2} + ε_{l, 4}

(10e)

Stress-strain relationships

Figure 5 shows the stress-strain relationships. The improved CA-STM incorporates the concrete softening model proposed by Hsu (1993) to consider concrete softening under biaxial tensile and compressive stresses. Where the softening factor $ζ_{i}$ represents the reduction in compressive strength of the concrete, and $k_{1 d, i}$ is the ratio of the average stress to the peak stress of the concrete in an arbitrary shear panel. As a result, the average stress for the compressive softening of concrete is obtained, as expressed in equations (11a)–(11d).

σ_{d, i} = k_{1 d, i} ζ_{i} f_{c}^{'}

(11a)

k_{1 d, i} = \frac{ε_{d s, i}}{ζ_{i} ε_{0}} - \frac{1}{3} {(\frac{ε_{d s, i}}{ζ_{i} ε_{0}})}^{2} if ε_{d s, i} \leq ζ_{i} ε_{0}

(11b)

k_{1 d, i} = 1 - \frac{ζ_{i} ε_{0}}{3 ε_{d s, i}} - \frac{{(ε_{d s, i} - ζ_{i} ε_{0})}^{3}}{3 ε_{d s, i} {(4 ε_{0} - ζ_{i} ε_{0})}^{2}} if ε_{d s, i} > ζ_{i} ε_{0}

(11c)

ζ_{i} = \frac{5.8}{\sqrt{{f_{c}}^{'}}} \frac{1}{\sqrt{1 + 400 ε_{r, i}}} \leq 0.9 and \frac{5.8}{\sqrt{{f_{c}}^{'}}} \leq 0.9

(11d)

Where

{f_{c}}^{'}

is the uniaxial compressive strength of concrete, and

ε_{0}

is the strain value corresponding to the peak stress

{f_{c}}^{'}

, calculated based on NP EN code (2005).

Figure 5.

Stress-strain relationships. (a) Concrete in compression. (b) Concrete in tension. (c) Reinforcement in tension. (d) Prestressing strands in tension.

The tensile stress-strain constitutive relationship of concrete proposed by Jeng and Hsu (2009) was chosen in this study to characterize the stress properties of concrete before and after cracking, as shown in equations (12a)–(12e).

σ_{r, i} = E_{c} ε_{r, i} if ε_{r, i} \leq ε_{c r}

(12a)

σ_{r, i} = f_{c r} {(\frac{ε_{c r}}{ε_{r, i}})}^{0.4} if ε_{r, i} > ε_{c r}

(12b)

f_{c r} = \frac{A_{g}}{2 A_{c p}} \sqrt{f_{c}^{'}}

(12c)

E_{c} = λ (3875) \sqrt{f_{c}^{'}}

(12d)

ε_{c r} = μ (0.00008)

(12e)

Where

λ

and

μ

are the factors proposed by Jeng and Hsu (2009),

λ = μ = 1.45

;

f_{c r}

is the peak tensile stress, and

ε_{c r}

is the peak tensile strain.

To account for the strain gradient effect of concrete strut, the ratio between the average tensile stress and the tensile peak stress $k_{1 r, i}$ is introduced to calculate the average tensile stress of concrete in tension, as shown in equations (13a)–(13c).

σ_{r, i} = k_{1 r, i} f_{c r}

(13a)

k_{1 r, i} = \frac{ε_{r s, i}}{2 ε_{c r}} if \frac{ε_{r s, i}}{ε_{c r}} \leq 1

(13b)

k_{1 r, i} = \frac{ε_{c r}}{2 ε_{r s, i}} + \frac{{(ε_{c r})}^{0.4}}{0.6 ε_{r s, i}} [{(ε_{r s, i})}^{0.6} - {(ε_{c r})}^{0.6}] if \frac{ε_{r s, i}}{ε_{c r}} > 1

(13c)

As for the tensile property of reinforcement, the CA-STM considers the average tensile stress in concrete (Belarbi and Hsu, 1994), and therefore, the average stress-strain relationship for embedded reinforcement should be utilized, as shown in equations (14a)–(14e).

f_{s, i} = E_{s} ε_{s} if ε_{s, i} \leq ε_{n, i}

(14a)

f_{s, i} = f_{s y} [(0.91 - 2 B_{i}) + (0.02 + 0.25 B_{i}) \frac{ε_{s, i}}{ε_{s y}}] if ε_{s, i} > ε_{n, i}

(14b)

ε_{n, i} = (0.93 - 2 B_{i}) ε_{s y}

(14c)

B_{i} = \frac{1}{ρ_{i}} {(\frac{f_{c r}}{f_{s y}})}^{1.5}

(14d)

ε_{s y} = \frac{f_{s y}}{E_{s}}

(14e)

Where

f_{s, i}

and

f_{s y}

are the average stress and yield stress of reinforcement;

ε_{s, i}

and

ε_{s y}

are the average strain and yield strain of reinforcement;

E_{s}

and

ρ_{i}

are the elastic modulus and reinforcement ratio of reinforcement (

ρ_{i} = ρ_{l, i}

for longitudinal reinforcement and

ρ_{i} = ρ_{t}

for transverse reinforcement).

This model adopts the constitutive relation of prestressing strands proposed by Hsu and Mo (1985) to consider the decompression effect of concrete. Under the longitudinal tensile stress induced by the external torque effect, the compressive stress of concrete caused by the initial prestress will gradually decrease. When it decreases to 0, the prestressing strands are treated as normal reinforcement for analysis, resulting in equations (15a)–(15d).

f_{p s} = E_{p s} ε_{p s} f_{p s} \leq 0.7 f_{p u}

(15a)

f_{p s =} \frac{E_{p s} ε_{p s}}{{[1 + {(E_{p s} ε_{p s} / f_{p u})}^{4.38}]}^{\frac{1}{4.38}}} f_{p s} > 0.7 f_{p u}

(15b)

ε_{p s} = ε_{d e c} + ε_{l, i}

(15c)

ε_{l e} = A_{p s, i} f_{p e} / [A_{l, i} E_{s} + (A_{c, i} - A_{l, i} - A_{p s, i}) E_{c}]

(15d)

Where

ε_{d e c}

is the tensile strain of the prestressing strands at the decompression of concrete,

ε_{d e c} = ε_{p e} + ε_{l e}

;

ε_{p e}

is the initial effective strain of the prestressing strands,

ε_{p e} = f_{p e} / E_{p s}

;

f_{p s}

and

f_{p u}

are the tensile strength and minimum tensile strength of the prestressing strands;

A_{c, i}

is the corresponding area of concrete in each panel,

E_{p s}

and

ε_{p s}

are the elastic modulus and tensile strain of the prestressing strands, and

f_{p e}

is the initial effective stress of the prestressing strands.

Equivalent areas of reinforcement and prestressing strands

When the cross-section is transformed into four shear panels, the distribution position of the reinforcement and prestressing strands will also be adjusted accordingly. In contrast to the CA-STM, the symmetric and asymmetric reinforcement types are no longer distinguished in the improved model (Greene and Belarbi, 2009b). The effective area of the reinforcement and prestressing strands located at the center of each panel is equivalent to their actual cross-sectional area. Half of the area of the longitudinal reinforcement and prestressing strands at the four corners is evenly distributed between the two adjacent panels, as indicated in Figure 6(a). For box girders containing prestressing strands in the center of the section, the total area of the prestressing strands is evenly attributed to each panel, as illustrated in Figure 6(b).

Figure 6.

Equivalent area. (a) Equivalent area of reinforcement. (b) Equivalent area of prestressing strands.

Failure criteria

As mentioned previously, the CA-STM uses the ultimate compressive strain reached by the principal compressive strain of concrete as the convergence criterion to evaluate the full-range stress performance of the box girders, which is not able to reflect the damage forms of the different reinforced box girder under the bending-torsion. To address this problem, this paper establishes four damage mechanisms for completely over-reinforced, partially over-reinforced, balanced-reinforced and under-reinforced beams, corresponding to the criteria 1,2,3 and 4, as shown in equations (16a)–(16d).

ε_{s, i} < ε_{s y} a n d ε_{d s, 1} > ε_{d s, \max} criteria 1

(16a)

ε_{l, i} < ε_{l y}, ε_{t, i} > ε_{t y} a n d ε_{d s, 1} > ε_{d s, \max} or ε_{l, i} > ε_{l y}, ε_{t, i} < ε_{t y} a n d ε_{d s, 1} > ε_{d s, \max} criteria 2

(16b)

ε_{s, i} > ε_{s y} a n d ε_{d s, 1} > ε_{d s, \max} criteria 3

(16c)

ε_{s, i} > ε_{s u} a n d ε_{d s, 1} < ε_{d s, \max} criteria 4

(16d)

Where

ε_{d s, \max}

is the maximum compressive strain of concrete,

ε_{d s, \max} = 0.0035

(Silva et al., 2017);

ε_{s u}

is the ultimate tensile strain of reinforcement,

ε_{s u} = f_{s u} / E_{s}

;

f_{s u}

is the ultimate tensile strength of reinforcement, and

ε_{l y}

and

ε_{t y}

are the yield strains of longitudinal and transverse reinforcement.

Efficient solution procedure

Different from the traditional trial-and-error method of the CA-STM, this paper develops a simple and efficient optimized method to solve the nonlinear system of equations formed by the equilibrium and the coordination equations. The solution problem of the nonlinear system of equations is transformed into a least-squares problem by inputting the initial values and geometric parameters. The gradient descent method is used to predict all primary variables of nonlinear equations with a given tolerance and constraints. The algorithmic steps of the solution procedure are described as follows.

Primary variables

This paper introduces the initial stress-strain relation of prestress to investigate the bending-torsional behavior of PC box girders, based on the optimized algorithm proposed by Silva et al. (2017). To improve the solution rate, the solution process of the traditional trial-and-error method is transformed into a system of nonlinear equations $F_{C A - S T M}$ . The principal strains of concrete ( $ε_{d s}$ and $ε_{r}$ ) and strain of longitudinal reinforcement ( $ε_{l}$ ) are used as unknown parameters to model the behavior of cracked PC panels under in-plane stresses, and $z$ and $T_{X}$ are also adopted as unknown parameters to determine the shear flow thickness and the relationship of bending and torsion, respectively. 16 fundamental primary variables $T_{X}$ , $ε_{d s, j}$ , $ε_{r, i}$ , $ε_{l, i}$ , and $z_{i}$ (i = 1, 2, 3, 4 and j = 2, 3, 4) are finally determined. All primary variables are solved by minimizing the norm $‖ F_{C A - S T M} ‖$ within a given tolerance. The process of iteration is repeated until one of the four failure criteria is reached.

The improved CA-STM assumes that the first solution point (k = 1) is calculated through a linear elastic model of plain concrete under pure shear, and the initial values of primary variables in each panel are equal. Hence, the initial value of the external compressive strain in each panel is equal to the assumed initial value of

ε_{d s, 1}

, and the initial value of the longitudinal strain

ε_{l . i}^{0}

is 0. For the initial value of the principal tensile strain

ε_{r . i}^{0}

in each panel, it is assumed to be equal to half the initial value of the compressive strain

ε_{d s, 1}

at the upper surface of panel 1. The parameter

z_{i}^{0}

is taken to be the middle value of 1 within the allowable range of

0 \leq z_{i} \leq 3

. Table 1 summarizes the calculation of the initial values of all primary variables.

Table 1.

Calculation Formula of Initial Value.

Initial value	Calculation formula
$T_{X}^{0}$	Equation (17c) for the solid section and equation (17d) for the hollow section
$ε_{d s, j}^{0}$	$ε_{d s, j}^{0} = ε_{d s, 1} = 0.0001$
$ε_{l . i}^{0}$	$ε_{l . i}^{0} = 0$
$ε_{r . i}^{0}$	$ε_{r . i}^{0} = ε_{d s, 1} / 2$
$z_{i}^{0}$	$z_{i}^{0} = 1$

The initial value of $T_{x}$ was obtained by Silva et al. (2017), which divided it into the cracking torque $T_{c r}$ of the section according to ACI Committee (2019), as shown in equation (17a). Because the CA-STM assumes that the plain concrete is in pure shear, a relationship between $ε_{d s, 1}$ and $T_{X}^{0}$ can be established based on the stress Mohr circle, as given by equations (17a)–(17d). To allow for the prestressing effects on the initial value, this paper modifies equation (17b) according to ACI Committee (2019) and establishes the equations of $T_{X}^{0}$ for solid and hollow sections in PC box girders, respectively.

T_{c r} = τ_{c r} \frac{A_{c p}^{2}}{p_{c p}} = 0.33 \sqrt{f_{c}^{'}} \frac{A_{c p}^{2}}{p_{c p}}

(17a)

T_{X}^{0} = ε_{d s, 1} \frac{E_{c}}{2} \frac{A_{c p}^{2}}{p_{c p}}

(17b)

T_{X}^{0} = ε_{d s, 1} \frac{E_{c}}{2} \frac{A_{c p}^{2}}{p_{c p}} \sqrt{1 + \frac{{2 f}_{p c}}{{ε_{d s, 1} E}_{c}}}

(17c)

T_{X}^{0} = ε_{d s, 1} \frac{E_{c}}{2} \frac{A_{c p}^{2}}{p_{c p}} (\frac{A_{g}}{A_{c p}}) \sqrt{1 + \frac{{2 f}_{p c}}{{ε_{d s, 1} E}_{c}}}

(17d)

Where

f_{p c}

is the net compressive stress of concrete induced by prestressing force,

f_{p c} = F_{p} / A_{c c}

;

F_{p}

is the prestressing force, and

A_{c c}

is the net cross-sectional area of concrete (for solid box girders

A_{c c} = b h - A_{p}

and hollow box girders

A_{c c} = A_{g}

Nonlinear equations

The 16 non-linear equations are represented by a system of residual equations and their primary variables are calculated by minimizing the residual function. All residual equations are derived from the equilibrium and coordination equations. Among them, $F_{C A - S T M} (i)$ is built by equation (4b), and $F_{C A - S T M} (i + 4)$ is constructed by combining equations (7c) and (8b). $F_{C A - S T M} (i + 8)$ is established by combining equations (2e) and (4c). $F_{C A - S T M} (13)$ to $F_{C A - S T M} (15)$ are derived from equations (3a)–(3c). Finally, $F_{C A - S T M} (16)$ can be obtained based on equation (10e).

F_{C A - S T M} (i) = σ_{r, i} \cos^{2} (α_{d, i}) + σ_{d, i} \sin^{2} (α_{d, i}) + f_{t, i} p_{t}

(18a)

F_{C A - S T M} (i + 4) = θ \sin (2 α_{d, i}) + [\begin{array}{l} - ϕ_{l, 13} \\ - ϕ_{l, 24} \\ ϕ_{l, 13} \\ ϕ_{l, 24} \end{array}] \cos^{2} (α_{d, i}) + [\begin{array}{l} - ϕ_{t, 13} \\ - ϕ_{t, 24} \\ ϕ_{t, 13} \\ ϕ_{t, 24} \end{array}] \sin^{2} (α_{d, i}) - ψ_{i}

(18b)

F_{C A - S T M} (i + 8) = τ_{l t, i} - \frac{q_{i}}{t_{d, i}}

(18c)

F_{C A - S T M} (13) = (σ_{l, 3} t_{d, 3} h_{0} - σ_{l, 1} t_{d, 1} h_{0}) \frac{b_{0}}{2} - M_{Y}

(18d)

F_{C A - S T M} (14) = (σ_{l, 4} t_{d, 4} b_{0} - σ_{l, 2} t_{d, 2} b_{0}) \frac{h_{0}}{2} - M_{Z}

(18e)

F_{C A - S T M} (15) = σ_{l, 1} t_{d, 1} h_{0} + σ_{l, 2} t_{d, 2} b_{0} + σ_{l, 3} t_{d, 3} h_{0} + σ_{l, 4} t_{d, 4} b_{0} - N_{X}

(18f)

F_{C A - S T M} (16) = ε_{l, 1} - ε_{l, 2} + ε_{l, 3} - ε_{l, 4}

(18g)

Solution procedure

Figure 7 shows a simplified flow chart of the optimized algorithm. The solution algorithm is implemented using a MATLAB software development program. Among them, the known parameters include the geometric parameters $b$ and $h$ of cross-section, longitudinal reinforcement area for each panel $A_{l, i}$ , total cross-sectional area of prestressing strands $A_{p s}$ , transverse reinforcement ratio $A_{t} / s$ , initial effective prestressing force $f_{p e}$ and minimum tensile strength $f_{p u}$ of prestressing strands, ultimate tensile strength of reinforcement $f_{u y}$ , net compressive stress $f_{p c}$ produced by the prestressing force, elastic modulus of concrete $E_{c}$ , reinforcement $E_{s}$ and prestressing strands $E_{p}$ , internal force to torque ratios ( $N_{X} / T_{X}$ , $V_{Y} / T_{X}$ , $M_{Y} / T_{X}$ , $M_{Z} / T_{X}$ , $V_{Z} / T_{X}$ ), and the assumed initial compressive strain $ε_{d s, 1}$ . The unknown parameters contain a total of 16 variables, i.e., $T_{X}$ , $ε_{d s, j}$ , $ε_{r, i}$ , $ε_{l, i}$ and $z_{i}$ (i = 1, 2, 3, 4 and j = 2, 3, 4). Initial estimates are also required to start the calculation procedure, including the tolerance $∆ ε_{d s, 1}$ of the residual equation, maximum strain $ε_{d s, \max}$ and initial values of primary variables ( $T_{X}^{0}$ , $ε_{d s, i}^{0}$ , $ε_{r, i}^{0}$ , $ε_{l, i}^{0}$ , $z_{i}^{0})$ . The initial points of each cycle are defined as the solution point from the previous step, with iterations continuing until one of the failure criteria is reached. The torque-twist curves and moment-curvature curves of the RC and PC box girders under the combined bending and torsion can be obtained.

Figure 7.

Flow chart of the solution procedure.

Theoretical model benchmarking

Establishment of database

A total of 20 experimental results of PC girders were collected to compile the database, which was further employed to benchmark the proposed solution algorithm. In the database, 12 PC beams were from McMullen and El-Degwy (1985), two PC beams were from Bernardo et al. (2013), four PC beams were from Tulonen and Laaksonen (2023), and two PC beams were from Mardukhi (1974). Note that the box girders from McMullen and El-Degwy (1985) and Tulonen and Laaksonen (2023) have solid cross-sections, while box girders from Bernardo et al. (2013) and Mardukhi (1974) have hollow cross-sections.

Tables 2 and 3 summarize the geometrical parameters and mechanical properties of the collected box girders, including the length

l

, the width

b

and height

h

of the box girder, wall thickness

t_{i}

of the hollow section, compressive strength of concrete

f_{c}^{'}

, cross-sectional area of reinforcement

A_{l, i}

, total cross-sectional area of the prestressing strands

A_{p s}

, transverse reinforcement ratio

A_{t / s}

, yield strength of the longitudinal

f_{l y}

and transverse reinforcement

f_{t y}

, bending-torsion ratio

{M_{Z} / T}_{X}

, initial effective prestress

f_{p e}

, minimum tensile strength

f_{p u}

, and net compressive stress

f_{p c}

in the concrete resulting from the prestressing force. For box girders with different yield strength longitudinal bars, the average yield strength of all longitudinal bars is taken as

f_{l y}

Table 2.

Summary of Specimen Parameters From the Literature.

Reference	Beam		$l$ (cm)	$b$ (cm)	$h$ (cm)	$t_{1}$ (cm)	$t_{2}$ (cm)	$t_{3}$ (cm)	$t_{4}$ (cm)	$f_{c}^{'}$ (MPa)	$A_{l, 1}$ (cm²)	$A_{l, 2}$ (cm²)	$A_{l, 3}$ (cm²)	$A_{l, 4}$ (cm²)	$A_{t / s}$ (cm²/m)	$f_{l y}$ (MPa)	$f_{t y}$ (MPa)	${M_{Z} / T}_{X}$
McMullen and El-Degwy (1985)	PA1R	S	360	25.4	25.4	/	/	/	/	43.6	0.71	0.71	0.71	0.71	4.48	435	310	0
McMullen and El-Degwy (1985)	PA2	S	360	25.4	25.4	/	/	/	/	45.6	1.29	1.29	1.29	1.29	7.74	483	310	0
McMullen and El-Degwy (1985)	PA3	S	360	25.4	25.4	/	/	/	/	41.8	2.00	2.00	2.00	2.00	7.93	389	435	0
McMullen and El-Degwy (1985)	PA4	S	360	25.4	25.4	/	/	/	/	42.2	2.84	2.84	2.84	2.84	11	419	435	0
McMullen and El-Degwy (1985)	PB1	S	360	17.8	35.6	/	/	/	/	45.8	0.71	0.71	0.71	0.71	4.48	435	310	0
McMullen and El-Degwy (1985)	PB2	S	360	17.8	35.6	/	/	/	/	45.8	1.29	1.29	1.29	1.29	7.74	483	310	0
McMullen and El-Degwy (1985)	PB3	S	360	17.8	35.6	/	/	/	/	45.5	2.00	2.00	2.00	2.00	7.51	389	435	0
McMullen and El-Degwy (1985)	PB4	S	360	17.8	35.6	/	/	/	/	45.5	2.84	2.84	2.84	2.84	10.2	419	435	0
McMullen and El-Degwy (1985)	PC1	S	360	14.6	43.8	/	/	/	/	42.2	0.71	0.71	0.71	0.71	4.48	435	310	0
McMullen and El-Degwy (1985)	PC2	S	360	14.6	43.8	/	/	/	/	45.1	1.29	1.29	1.29	1.29	6.9	483	310	0
McMullen and El-Degwy (1985)	PC3	S	360	14.6	43.8	/	/	/	/	41.3	2.00	2.00	2.00	2.00	6.79	389	435	0
McMullen and El-Degwy (1985)	PC4	S	360	14.6	43.8	/	/	/	/	42.1	2.84	2.84	2.84	2.84	9.53	419	435	0
Bernardo et al. (2013)	D1	H	590	60	60	11.4	11.4	11.4	11.4	80.8	5.90	5.9	5.90	5.90	11.2	724	715	0
Bernardo et al. (2013)	D2	H	590	60	60	11.5	11.5	11.5	11.5	58.8	5.90	5.9	5.90	5.90	11.2	724	715	0
Tulonen and Laaksonen (2023)	P5-031-LHp	S	290	23.2	23.5	/	/	/	/	44.5	2.76	1.5	2.76	8.04	3.86	554	538	3.23
Tulonen and Laaksonen (2023)	P6-047-LHp	S	290	23.2	23.5	/	/	/	/	44.5	2.76	1.5	2.76	8.04	3.86	554	538	2.13
Tulonen and Laaksonen (2023)	P7-031-HHp	S	290	23.2	23.5	/	/	/	/	44.5	2.76	1.5	2.76	8.04	7.7	554	538	3.23
Tulonen and Laaksonen (2023)	P8-047-HHp	S	290	23.2	23.5	/	/	/	/	44.5	2.76	1.5	2.76	8.04	7.7	554	538	2.13
Mardukhi (1974)	TB2	H	/	30.5	43.2	7.6	8.9	7.6	8.9	45.0	0.71	4.00	0.71	4.00	4.67	370	376	2.96
Mardukhi (1974)	TB3	H	/	30.5	43.2	7.6	8.9	7.6	8.9	33.7	0.71	4.00	0.71	4.00	4.67	370	376	6.95

Table 3.

Mechanical Properties of Prestressing Steel.

Reference	Beam	$f_{p e}$ (MPa)	$f_{p u}$ (MPa)	$f_{p c}$ (MPa)	$A_{p s}$ (cm²)	Type
McMullen and El-Degwy (1985)	PA1R	1207	1654	1.74	0.93	Prestressing strands
McMullen and El-Degwy (1985)	PA2	1207	1680	2.81	1.5
McMullen and El-Degwy (1985)	PA3	1303	1761	4.17	2.06
McMullen and El-Degwy (1985)	PA4	1303	1726	6.03	2.97
McMullen and El-Degwy (1985)	PB1	1207	1654	1.77	0.93
McMullen and El-Degwy (1985)	PB2	1207	1680	2.86	1.5
McMullen and El-Degwy (1985)	PB3	1303	1761	4.25	2.06
McMullen and El-Degwy (1985)	PB4	1303	1726	6.14	2.97
McMullen and El-Degwy (1985)	PC1	1207	1654	1.76	0.93
McMullen and El-Degwy (1985)	PC2	1207	1680	2.84	1.5
McMullen and El-Degwy (1985)	PC3	1303	1761	4.21	2.06
McMullen and El-Degwy (1985)	PC4	1303	1726	6.08	2.97
Bernardo et al. (2013)	D1	715	1687	1.36	4.2
Bernardo et al. (2013)	D2	715	1687	1.81	5.6
Tulonen and Laaksonen (2023)	P5-031-LHp	985	1860	5.42	3
Tulonen and Laaksonen (2023)	P6-047-LHp	1021	1860	5.61	3
Tulonen and Laaksonen (2023)	P7-031-HHp	1035	1860	5.70	3
Tulonen and Laaksonen (2023)	P8-047-HHp	1019	1860	5.61	3
Mardukhi (1974)	TB2	1145	1703	7.63	6.15
Mardukhi (1974)	TB3	1145	1703	7.63	6.15

Model validation

Torque-twist curves

Figures 8 and 9 show the torsion-twist curves of PC box girders. The theoretical curves calculated by the MCA-STM and improved CA-STM are compared. A good agreement between the theoretical curves predicted by the improved CA-STM and measured curves from tests is observed, showing that the proposed model accurately predicts the mechanical behavior of the box girders at the elastic stage, cracking stage, and ultimate load stage. The elastic torsional stiffness of concrete before cracking and ultimate bearing capacity predicted by the MCA-STM do not match the experimental curves due to ignoring the prestressing effect.

Figure 8.

Torque-twist curves of box girders with ${M_{Z} / T}_{X} = 0$ . (a) PA1R. (b) PA2. (c) PA3. (d) PA4. (e) PB1. (f) PB2. (g) PB3. (h) PB4. (i) PC1. (j) PC2. (k) PC3. (l) PC4. (m) D1. (n) D2.

Figure 9.

Bending moment-twist curves of box girders under combined bending and torsion. (a) P5-031-LHp. (b) P6-047-LHp. (c) P7-031-HHp. (d) P8-047-HHp. (e) TB2. (f) TB3.

Tables 4 and 5 present the experimental and theoretical values of torque and bending moment in the cracking and ultimate states. The improved CA-STM has better prediction accuracy of ultimate torques and bending moments of the PC-composite box girders, in comparison to the MCA-STM. The ratios of ultimate torque of experimental value to theoretical value (

T_{X, u}^{Exp} / T_{M, u}^{T h}

and

T_{X, u}^{Exp} / T_{I, u}^{T h}

) predicted by the two theoretical models show the average values (Avg.) of 1.234 and 1.009, with standard deviations (St.D.) of 11.6% and 6.3%. For the box girders with

{M_{Z} / T}_{X} \neq 0

(i.e., combined bending and torsion), the average values of ultimate bending moment ratios (

M_{Z, u}^{Exp} / M_{M, u}^{T h}

and

M_{Z, u}^{Exp} / M_{I, u}^{T h}

) are 1.246 and 0.963, respectively, with standard deviations of 11.4% and 4.5%. For the cracking torque and bending moment, the MCA-STM predicted cracking values differed significantly from the experimental results due to the neglect of the prestressing effect on the cracking properties of the concrete. The improved CA-STM can predict the cracking torque of the PC box girders. The average values of cracking torque ratios (

T_{X, c r}^{Exp} / T_{M, c r}^{T h}

T_{X, c r}^{Exp} / T_{I, c r}^{T h}

M_{Z, c r}^{Exp} / M_{M, c r}^{T h}

and

M_{Z, c r}^{Exp} / M_{I, c r}^{T h}

) are 2.638, 1.024, 2.162 and 0.934, with standard deviations of 58.1%, 8.3%, 16.4% and 5.9%. Additionally, it was worth mentioning that the average processing time for calculating the ultimate torques of PC box girders is 14s by a normal computer (intel(R) Core(TM) i7-11,700 CPU @ 2.50 GHz).

Table 4.

Comparison of Torques From the Proposed Model, MCA-STM, and Experimental Results.

Reference	Beam	$\frac{M_{Z}}{T_{X}}$	Cracking state					Ultimate state					$Criterion$
Reference	Beam	$\frac{M_{Z}}{T_{X}}$	$T_{X, c r}^{Exp}$ (kN·m)	$T_{I, c r}^{T h}$ (kN·m)	$T_{M, c r}^{T h}$ (kN·m)	$T_{X, c r}^{Exp} / T_{I, c r}^{T h}$	$T_{X, c r}^{Exp} / T_{M, c r}^{T h}$	$T_{X, u}^{Exp}$ (kN·m)	$T_{I, u}^{T h}$ (kN·m)	$T_{M, u}^{T h}$ (kN·m)	$T_{X, u}^{Exp} / T_{I, u}^{T h}$	$T_{X, u}^{Exp} / T_{M, u}^{T h}$	$Criterion$
McMullen and El-Degwy (1985)	PA1R	0	18.50	17.20	5.75	1.076	3.217	21.80	20.81	17.08	1.048	1.276	3
McMullen and El-Degwy (1985)	PA2	0	21.70	18.60	7.16	1.167	3.031	29.30	27.14	25.40	1.080	1.154	3
McMullen and El-Degwy (1985)	PA3	0	24.40	20.56	7.62	1.187	3.202	34.00	33.53	31.91	1.014	1.065	3
McMullen and El-Degwy (1985)	PA4	0	26.30	22.89	7.88	1.149	3.338	37.40	37.67	35.22	0.993	1.062	3
McMullen and El-Degwy (1985)	PB1	0	15.40	14.99	6.21	1.027	2.480	22.20	23.55	15.82	0.943	1.403	3
McMullen and El-Degwy (1985)	PB2	0	17.80	17.69	6.24	1.006	2.853	27.50	25.67	21.59	1.071	1.274	3
McMullen and El-Degwy (1985)	PB3	0	20.70	18.91	6.56	1.095	3.155	32.60	30.65	26.65	1.064	1.223	3
McMullen and El-Degwy (1985)	PB4	0	21.90	21.85	6.58	1.002	3.328	37.60	36.21	31.23	1.038	1.204	3
McMullen and El-Degwy (1985)	PC1	0	13.30	12.50	5.95	1.064	2.235	19.70	18.76	13.47	1.050	1.463	3
McMullen and El-Degwy (1985)	PC2	0	16.60	17.55	6.77	0.946	2.452	28.60	26.48	21.75	1.080	1.315	3
McMullen and El-Degwy (1985)	PC3	0	17.00	15.62	6.92	1.088	2.457	32.80	31.73	26.70	1.034	1.228	3
McMullen and El-Degwy (1985)	PC4	0	21.60	19.85	7.15	1.088	3.021	38.50	36.95	32.88	1.042	1.171	3
Bernardo et al. (2013)	D1	0	172.90	179.85	58.97	0.961	2.932	396.00	454.71	358.68	0.871	1.104	3
Bernardo et al. (2013)	D2	0	184.70	191.90	54.10	0.962	3.414	447.70	420.48	346.00	1.065	1.294	3
Tulonen and Laaksonen (2023)	P5-031-LHp	3.23	6.90	7.01	3.00	0.984	2.300	19.20	19.92	15.94	0.964	1.205	3
Tulonen and Laaksonen (2023)	P6-047-LHp	2.13	6.84	7.33	3.40	0.933	2.012	18.80	19.94	17.53	0.943	1.072	2
Tulonen and Laaksonen (2023)	P7-031-HHp	3.23	6.90	7.65	3.14	0.902	2.197	25.90	26.07	18.04	0.993	1.436	3
Tulonen and Laaksonen (2023)	P8-047-HHp	2.13	7.40	7.88	3.54	0.939	2.090	24.50	27.72	21.66	0.884	1.131	3
Mardukhi (1974)	TB2	2.96	23.25	25.03	17.61	0.929	1.320	53.20	52.46	41.79	1.014	1.273	3
Mardukhi (1974)	TB3	6.95	15.40	15.76	8.92	0.977	1.726	28.10	28.39	21.23	0.990	1.324	3
	Avg					1.024	2.638				1.009	1.234
	St.D					0.083	0.581				0.063	0.116

Note: $c r$ represents the cracking state, $I$ represents improved CA-STM, $M$ represents MCA-STM, $Exp$ represents the experimental results, $T h$ represents the theoretical result, $u$ represents the ultimate state.

Table 5.

Comparison of bending moments from the proposed model, mca-stm, and experimental results.

Reference	Beam	${M_{Z} / T}_{X}$	Cracking state					Ultimate state					Criterion
Reference	Beam	${M_{Z} / T}_{X}$	$M_{Z, c r}^{Exp}$ (kN·m)	$M_{I, c r}^{T h}$ (kN·m)	$M_{M, c r}^{T h}$ (kN·m)	$M_{Z, c r}^{Exp} / M_{I, c r}^{T h}$	$M_{Z, c r}^{Exp} / M_{M, c r}^{T h}$	$M_{Z, u}^{Exp}$ (kN·m)	$M_{I, u}^{T h}$ (kN·m)	$M_{M, u}^{T h}$ (kN·m)	$M_{Z, u}^{Exp} / M_{I, u}^{T h}$	$M_{Z, u}^{Exp} / M_{M, u}^{T h}$	Criterion
Tulonen and Laaksonen (2023)	P5-031-LHp	3.23	23.00	22.63	9.60	1.016	2.396	61.60	64.34	51.01	0.957	1.208	3
Tulonen and Laaksonen (2023)	P6-047-LHp	2.13	13.80	15.62	7.14	0.883	1.933	40.10	42.48	36.82	0.944	1.089	2
Tulonen and Laaksonen (2023)	P7-031-HHp	3.23	21.56	24.71	10.06	0.873	2.143	82.80	84.21	57.73	0.983	1.434	3
Tulonen and Laaksonen (2023)	P8-047-HHp	2.13	16.20	16.79	7.44	0.965	2.177	52.30	59.04	45.49	0.886	1.150	3
Mardukhi (1974)	TB2	2.96	/	74.09	52.13	/	/	157.30	155.28	123.70	1.013	1.272	3
Mardukhi (1974)	TB3	6.95	/	109.53	61.99	/	/	195.40	197.31	147.55	0.990	1.324	3
	Avg					0.934	2.162				0.963	1.246
	St.D					0.059	0.164				0.045	0.114

Bending moment-curvature curves

Figure 10 presents the bending moment-curvature curves from Tulonen and Laaksonen (2023). It shows that the improved CA-STM can predict the bending moment-curvature curves of the box girders under different bending-torsional effects. Compared with the experimental results, the theoretical calculations typically show slightly larger values. It may be because in the actual tests, ${M_{Z} / T}_{X}$ cannot be loaded exactly according to the test design, resulting in smaller ultimate bending moment values.

Figure 10.

Bending moment-curvature curves. (a) P5-031-LHp. (b) P6-047-LHp. (c) P7-031-HHp. (d) P8-047-HHp. (e) Bending-torsional coupling interaction

For the bending-torsional interaction relationship, the comparison of experimental and theoretical curves is also established using Tulonen’s box girders (Tulonen and Laaksonen, 2023). Since only the experimental curves for four members at ${M_{Z} / T}_{X} = 2.13$ or $3.23$ are provided in the reference (Tulonen and Laaksonen, 2023), this paper predicts the ultimate torques and bending moments of all box girders based on the improved CA-STM for ${M_{Z} / T}_{X} = 0, 1, 2.13, 3.23$ to supplement the bending-torsional interaction curves, as depicted in Figure 10(e). The predicted bending-torsional coupling curves exhibit positive growth for ${M_{Z} / T}_{X} < 1$ , whereas the torque decreases with increasing bending moment for ${M_{Z} / T}_{X} > 1$ . When ${M_{Z} / T}_{X}$ is equal to 2.13, there is only a slight deviation between the predicted results of the improved CA-STM and experimental results. Therefore, the improved CA-STM is able to capture the interaction curves of the box girders under bending moment and torque well.

Torque-strain curve

From Table 4, all box girders are damaged by balanced-reinforced, except for P6, which is damaged by partially over-reinforced. To elaborate the damage form of P6, the reinforcement strains predictions of the improved CA-STM at left end (side A) and right end (side B) of Tulonen’s box girder bottom slab (Tulonen and Laaksonen, 2023) are shown in Figure 11 (Transverse strain curves are not given in the reference (Tulonen and Laaksonen, 2023)). Obviously, the longitudinal strain curves predicted by the improved CA-STM are in good agreement with the test curves due to the consideration of the damage forms caused by reinforcement. For P6, when the concrete is crushed, the theoretical and experimental curves of longitudinal strain did not reach the yield strain, while the transverse reinforcement predicted by the improved CA-STM has reached the yield strength. Hence, the P6 corresponds to the partially over-reinforced damage.

Figure 11.

Torque-bottom reinforcement strain curves. (a) P5-031-LHp. (b) P6-047-LHp. (c) P7-031-HHp. (d) P8-047-HHp.

Based on the above analysis, the improved CA-STM can reasonably predict the overall force behavior and damage modes of PC box girders under combined bending and torsion. However, it should be noted that further research is needed to validate the proposed model for predicting the performance of large-span bridge structures.

Conclusions

To accurately predict the full-range structural behavior of PC box girders subjected to combined bending and torsion, this paper proposes a bending-torsional theoretical model based on the CA-STM. The model successfully considers the decompression effect of concrete due to prestressing force, tensile properties of concrete, failure criteria for different reinforced beams, and the effect of prestressing steel on the initial torque of solid and hollow sections. According to the study results, the following conclusions can be drawn:

1. The torque-twist, bending moment-curvature, and torque-strain curves of the PC box girders, as predicted by the improved model, exhibit good agreement with the experimental results. The improved CA-STM is reasonable for incorporating the initial torque estimation method for both solid and hollow sections and the stress-strain constitutive laws of prestress and concrete.

2. The use of the gradient descent method can significantly improve both the speed and stability of the solution, which is recommended to replace the traditional trial and error method.

3. The improved CA-STM takes into account the deformation coordination relationship between adjacent shear panels, which can accurately predict the cracking and ultimate load points of PC box girders under combined bending and torsion.

4. The improved model utilizes the different damage forms of box girder caused by the steel as failure criteria for predicting the mechanical behavior of the box girder, which is able to accurately predict the overall stress performance and failure modes of the PC box girders. It can be used to quickly evaluate the accuracy and reliability of existing design methods, thus optimizing the design of PC bridges.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (51478120).

ORCID iDs

Hao Zhang

Xiaoqiang Yang

Appendix

References

ACI Committee (2019) Building Code Requirements for Structural Concrete. Farmington Hills, MI: American Concrete Institute.

Alnuaimi

Bhatt

(2006) Direct design of partially prestressed concrete hollow beams. Advances in Structural Engineering 9(4): 459–476.

Ashour

Shihata

Akhtaruraman

, et al. (1999) Prestressed high-strength concrete beams under torsion and bending. Canadian Journal of Civil Engineering 26: 197–207.

Belarbi

Hsu

(1994) Constitutive laws of concrete in tension and reinforcing bars stiffened by concrete. Structural Journal 91: 465–474.

Bernardo

Andrade

Pereira-De-Oliveira

(2013) Reinforced and prestressed concrete hollow beams under torsion. Journal of Civil Engineering and Management 19:S141–S152.

Bernardo

Lopes

Teixeira

(2020) Experimental study on the torsional behaviour of prestressed hsc hollow beams. Applied Sciences 10(2): 642.

Cheng

Cai

Xiao

R-C

(2007) Probabilistic response analysis of cracked prestressed concrete beams. Advances in Structural Engineering 10(1): 1–10.

NP EN 1992-1-1 (2005) Eurocode 2: Design of Concrete Structures-Part 1–1: General Rules and Rules for Buildings. United Kingdom: British Standard Institution.

Greene

(2006) Behavior of reinforced concrete girders under cyclic torsion and torsion combined with shear: experimental investigation and analytical models. PhD Thesis, University of Missouri-Rolla. Rolla, MO.

10.

Greene

Belarbi

(2009a) Model for reinforced concrete members under torsion, bending, and shear. I: theory. Journal of Engineering Mechanics 135: 961–969.

11.

Greene

Belarbi

(2009b) Model for reinforced concrete members under torsion, bending, and shear. II: model application and validation. Journal of Engineering Mechanics 135: 970–977.

12.

Hsu

(1993) Unified Theory of Reinforced Concrete. Florida: CRC Press Inc.

13.

Hsu

(1985) Softening of concrete in torsional members- prestressed concrete. ACI Structural Journal 2: 603–615.

14.

Jeng

Chao

(2018) Torsion experiment and cracking-torque formulas of prestressed concrete beams. ACI Structural Journal 115: 1267–1278.

15.

Jeng

Hsu

(2009) A softened membrane model for torsion in reinforced concrete members. Engineering Structures 31: 1944–1954.

16.

Kollbrunner

Basler

(1969) Torsion in Structures: An Engineering Approach SSBM. Seattle, WA: Semantic Scholar.

17.

Mardukhi (1974) The behaviour of uniformly prestressed concrete box beams in combined torsion and bending. PhD Thesis, University. Toronto.

18.

McMullen

El-Degwy

(1985) Prestressed concrete tests compared with torsion theories. PCI Journal 30: 96–127.

19.

GB50010 Mohurd (2010). Code for Design of Concrete Structures. Beijing, China: China Architecture & Building Press.

20.

Mondal

Prakash

(2015) Effect of tension stiffening on the behaviour of reinforced concrete circular columns under torsion. Engineering Structures 92: 186–195.

21.

Roberts

Lees

Hoult

(2012) Flexural fatigue performance of CFRP prestressed concrete Poles. Advances in Structural Engineering 15(4): 575–587

22.

Shao

Yang

Zhao

, et al. (2010) Experimental research on and application of twice-prestressed concrete beam. Advances in Structural Engineering 13(2): 321–329

23.

Silva

Horowitz

Bernardo

(2017) Efficient analysis of beam sections using softened truss model. ACI Structural Journal 114: 765–774.

24.

Tulonen

Laaksonen

(2023) Variable compression panel thickness for concrete members under bending and torsion: experimental study. Engineering Structures 279(3):115606.