Softened membrane model for forecasting torsional response of reinforced concrete beams strengthened using FRP sheets

Abstract

A new computational method has been developed to predict the full torsional response of reinforced concrete (RC) beams strengthened with FRP (fiber-reinforced polymer) materials. The proposed model was a modification of the softened membrane model for torsion (SMMT) for RC members. It accounts for the confinement effect of FRP strengthening on the softened compressive and tensile behavior of concrete, improving predictions of torsional response (MSMMT-FRP). Then, a trial-and-error algorithm was developed in the MATLAB environment. A total of 25 RC beams, including 20 experimental specimens previously strengthened with FRP material and five new test beam specimens in this study, were used to evaluate the applicability and reliability of the proposed model. In addition, the predictions of torsional strength from the theoretical fib Bulletin 90 were compared with both analytical predictions and experimental results. The experimental and analytical results showed good agreement in predicting the overall torsional behavior of strengthened RC beams. Comparisons show that the MSMMT-FRP model provides better predictions of torsional strength than fib Bulletin 90. Finally, a parametric study was performed, and the results demonstrated that torsional strength could be improved by increasing the concrete compressive strength, modulus of elasticity of FRP, and its ratio.

Keywords

Torsional response reinforced concrete beam fiber-reinforced polymer strengthening softening membrane model for torsion

Introduction

RC members may be subjected to loads exceeding their design capacity, such as bending moments, shear forces, axial forces, torsional forces, or combinations thereof. In the design of a secure structural component, it is essential to account for several types of forces. In most design scenarios, bending moments and shear stresses are considered primary factors, while torsional forces are treated as secondary effects. The torsional response of RC components has not been examined as thoroughly as their responses to bending and shear stresses (Zojaji and Kabir, 2012). Torsion is a critical factor in specific cases, such as spandrel or curved beams, eccentrically loaded bridge girders, and seismically loaded bridge columns (Shen et al., 2017). Thus, it is essential to have a detailed understanding of the behavior of RC components under torsional loading. Furthermore, the effectiveness of FRP strengthening regarding enhancements in torsional strength and response remains inadequately understood. This is due to the complex relationship between FRP confinement effects on concrete softening under compression and its stiffness contributions to tensile behavior. This research aims to address the knowledge gap through analytical and experimental methodologies.

FRP materials are commonly used for externally bonded strengthening and repair of RC components in structures such as buildings and bridges (Meyyada and Alabdulhady, 2019). In the past, numerous researchers have examined the torsional responses of RC beams strengthened with externally bonded FRP sheets. The initial research investigation on the application of FRP in enhancing the torsional strength of RC solid beams was conducted by Ghobarah et al. (2002). The subsequent research studies were conducted by Panchacharam and Belarbi (2002), , Chalioris (2006), Hii and Al-Mahaidi (2006b), Ameli et al. (2007), Mohammadizadeh et al. (2009), Deifalla et al. (2013), Kandekar and Talikoti (2019), and Chandan et al. (2019) regarding RC solid beams strengthened with FRP. Conversely, Hii and Al-Mahaidi (2006a) and Chai et al. (2014) tested torsional response for strengthening box sections of RC beams with FRP sheets. Several key factors have been explored in these studies, including different fiber materials (e.g., carbon-FRP (CFRP), glass-FRP (GFRP), and aramid-FRP (AFRP)) and various wrapping configurations, such as full continuous wrapping, strip wrapping, and U-wrapping with or without mechanical anchoring. The experiments also investigated the influence of FRP ratio, number of FRP layers, and orientation of the fibers. Furthermore, cross-sectional geometry has also been examined in these experimental studies: rectangular, L-shaped, and T-shaped. Recently, there has been a focus on investigating the effects of different externally bonded techniques, as conducted by Chandan et al. (2019) and Mahshid Abdoli et al. (2024). Multiple FRP materials have been used to improve the torsional performance of beams, demonstrating their effectiveness in enhancing torsional strength. However, no experimental studies employing basalt-FRP (BFRP) to assess the torsional behavior of RC beams have been previously undertaken.

Given the high cost and time demands of experimental studies, finite element methods (FEM) using software like ANSYS, ABAQUS, DIANA, and LS-DYNA provide a valuable alternative for analyzing torsional behavior. Numerical investigations regarding cracking and crushing patterns (Ameli and Ronagh, 2007), damage simulation (Ganganagoudar et al., 2016), the influence of CFRP and reinforcing steel bars on torsional behavior, and torque-twist curves (Hii and Al-Mahaidi, 2006a). Muhanad et al. (2021) used the DIANA software program to assess the torsional response and capacity of CFRP-strengthened RC multicell box girders subjected to torsion. This investigation demonstrates that there was good agreement for torque-angle of twist responses, internal reinforcement and CFRP sheet responses, and crack patterns (Muhanad et al., 2021). Parametric research was conducted using FEM, including variables such as the number of FRP layers, concrete compressive strength, and orientations of FRP strips as analyzed by Muhanad et al. (2021). In addition, Allawi et al. (2023) performed experimental and numerical simulations to investigate the influence of CFRP strips on the torsional behavior of both non-damaged and pre-damaged RC multi-cell box girders. This numerical model assessed the impact of concrete compressive strength, transverse reinforcement spacing, and CFRP strip spacing on torsional response.

Furthermore, analytical models have been performed to forecast the torsional response of RC beams enhanced with FRP material through various methodologies. Some standard designs and technical guidelines, such as the fib Bulletin 90 (2019), CNR-DT 200/2013 (2013), and NCHRP Report 655 (2010) based on the truss mechanism, have presented straightforward design formulas to assess the contribution of FRP to the torsional capacity of RC beams enhanced with FRP sheets. Ameli and Ronagh (2007) utilizing the compression field model (CFT) (Mitchell, 1974), also proposed a formula to estimate the torsional capacity of FRP-strengthened RC beams; however, the torsional behavior was not addressed. The modified CFT (MCFT) (Vecchio and Collins, 1986) integrates the hollow tube analogy (Mitchell, 1974) with compatibility conditions at cross-section corners (Onsongo, 1978) to predict the full torsional response of FRP-strengthened RC beams, as shown by Deifalla and Ghobarah (2010). This model takes into account tensile stresses in concrete and various FRP materials, different wrapping configurations, failure mechanisms, and fiber orientations; nevertheless, the confining effect of FRP was not incorporated into the concrete material constitutive law. An analytical approach was proposed by Chalioris (2007a), combining two models: a smear crack model (Karayannis, 2000) and a modified softened truss model for torsion (MSTMT) (Hsu and Mo, 1985; Hsu, 1988), to estimate the torsional response during the pre-cracking and post-cracking phases, respectively. The MSTMT was modified and expanded by Chai et al. (2014), and Muhanad et al. (2020) to predict the nonlinear torsional response for strengthening RC members with FRP laminates, accounting for the impact of FRP strengthening on concrete confinement under compression; however, the tensile stress in concrete was not addressed. This MSTMT, which considered the impact of FRP confinement of concrete on tensile stress and compressive concrete, aimed to evaluate the complete torsional response proposed by Shen et al. (2017). Nonetheless, this model did not account for the impact of the biaxial on the steel reinforcement embedded in the concrete, nor the influence of FRP confinement on softening concrete. (Shen et al., 2017). Recently, Jeng and Hsu (2009) developed the SMMT based on the softened membrane model (SMM) (Zhu, 2002), to forecast the comprehensive torsional behavior of RC members. Several analytical models extend SMMT to predict the full torsional response of FRP-strengthened RC beams, as performed by Kabir (2012) and Ganganagoudar et al. (2016). These models have examined various strengthening configurations, failure modes, and different FRP materials, including solid and hollow sections; nevertheless, the impact of FRP wrapping on concrete softening, as well as FRP stiffness on concrete tensile stress, have not been considered. This SMMT modification by Shen et al. (2017, 2018, 2019) to predict the full torsional behavior for composite concrete box girders with corrugated steel webs with CFRP-strengthened, both with and without prestressing. In this model, the contribution of the corrugated steel webs to the torsional capacity was determined by assumptions about the shear strain relationships of the concrete and corrugated steel webs.

Additionally, a novel material model was recently introduced, derived from research on RC slabs strengthened with externally bonded FRP sheets (Yang et al., 2015, 2017). The findings of this investigation indicated that the application of FRP reduced the softening of concrete and enhanced its tensile strength. Consequently, a constitutive equation for the relationship between stress and strain of concrete was proposed based on these modifications. Moreover, it is noted that when RC beams experience pure torsion, the effective shear flow region encircling the cross-section is affected by softening phenomena (ACI-445.1R-12, 2012). It can be shown that the effective thickness of the cross-sections was significantly decreased by mitigating the softening effects in the shear flow zone (ACI-445.1R-12, 2012).

Motivation and aims of research

The evaluation of the aforementioned study indicates that current analytical methods for forecasting the torsional responses of RC elements augmented with FRP sheets primarily rely on CFT, MCFT, MSTMT, or SMMT. The modified CFT for enhancing RC beams with FRP materials under pure torsion did not account for tensile stress in concrete, both before and after cracking. Meanwhile, the MSTMT for FRP-strengthened RC members subjected to torsion neglected to account for the impact of FRP on the softening of concrete and the biaxial stress effects on steel reinforcement embedded in the concrete. Moreover, the SMMT model for FRP-strengthened RC members does not account for the confinement effect on concrete softening under compression or the influence of FRP stiffness on concrete tensile stress. Furthermore, no investigations have been conducted on the torsional response of RC beams enhanced with BFRP sheets, either through experimental or analytical studies, as previously mentioned.

The FRP strengthening of RC components creates confinement that increases the stiffness of concrete struts subjected to compression. The current constitutive laws for concrete under compression do not consider the confinement effect on the softening behavior. This work aims to fill this knowledge gap by providing a revised concrete constitutive law incorporating the confinement effect on the softening behavior of concrete under compression. Moreover, the influence of FRP confinement helps control concrete crack formation under tensile stress. This study proposes a new tension stiffening model for concrete to improve the accuracy of torsional response predictions in FRP-strengthened beams. Additionally, an experimental program was conducted, including five RC beam specimens with fully continuous and strip wrap configurations using CFRP and BFRP sheets. A combination of analytical and experimental approaches provides a thorough understanding of this complex issue. The primary aims of the study are listed below:

(1) To improve the compressive constitutive relationships of concrete to account for the confinement effect on the softening behavior of concrete under compression induced by FRP wrapping.

(2) To improve the concrete in tensile constitutive relationships accounting for the impact of FRP stiffness on tensile stress and improve prediction torsional behavior.

(3) To conduct an experimental program of RC beams strengthened with CFRP and BFRP sheets to evaluate the torsional behavior and validate the proposed model.

Modified softened membrane model for RC beams strengthened with FRP (MSMMT-FRP)

Overview and summary of MSMMT-FRP

In this study, an improved analytical model (MSMMT-FRP) based on SMMT was initially proposed by Jeng and Hsu (2009), to predict the complete torsional responses of RC beams strengthened with FRP materials. It is extended to incorporate the impact of FRP strengthening on the softening behavior of concrete under compression by utilizing a more accurate constitutive relationship for concrete in compression, as demonstrated by Yang and Belarbi (2017). Moreover, a new tensile stress model is utilized due to the influence of FRP strengthening on concrete tensile behavior, using the stiffness ratio between stirrups and FRP fibers for enhanced predictions, as proposed by Yang et al. (2015). These two modifications are incorporated into the constitutive laws of concrete through the modified stress-strain curve of concrete in the case of external FRP sheets wrapping around the cross-sectional beam. Finally, the equilibrium equations on Navier’s principle are solved together with the constitutive laws for materials of a purely torsional member. A trial-and-error algorithm was developed in MATLAB (2023) to determine each point of the torque-angle of twist relationship. The calculation steps of the MSMMT-FRP solution are summarized in Figure 1.

Figure 1.

Summary of MSMMT-FRP solution for strengthening RC beams using FRP material.

Equilibrium equations

A rectangular RC beam strengthened with externally bonded FRP sheets and subjected to pure torque (T) is illustrated in Figure 2. Figure 3 shows the actual stress distribution and the theoretical stress distribution according to Saint-Venant’s theory for a rectangular beam subjected to pure torsion. The following characteristics defined the actual stress distribution of a rectangular beam subjected to pure torsion: the maximum stress is located at the center of the wide face “h”; the stress at the corners is zero; the stress distribution at any other point is less than that at the center and greater than zero (Figure 3(a)). In this study, using Saint-Venant’s theory, an internal force was generated in circular shear flow (q) along the periphery of the cross-section to counteract the action of the external force caused by the torque T. The shear flow q exists inside a shear flow zone characterized by an effective thickness t_d. Per equivalent thin-walled tube theory was proposed by Bredt (1896), shear flow is defined as τ_tl = q/t_d, which maintains a constant value across the entire perimeter of the cross-section (Figure 2(a) and Figure 3(b)).

Figure 2.

RC beam strengthened with FRP fibers under pure torsion and plane stress state of tiny element A. (a) RC beam strengthened with FRP strips, (b) In-plane stress state of shear element A, (c) In-plane equilibrium of the (I) part in shear element A, (d) l-t and 1-2 coordinates, (e) In-plane equilibrium of the (II) part in shear element A.

Figure 3.

Stress distribution of rectangular beam under pure torsion (a) Actual, (b) Thin-walled tube theory.

After the concrete cracks, the concrete compression struts are established, and the inclined angle α₂ is created by the direction of the crack and the beam axis, which is also the angle created by the 2-axis and l-axis. Consider a plane element of unit length (A) subjected to pure shear stress, assuming an inclined crack in the concrete passing through element A divides this element into two wedge-shaped parts (I) and (II) (Figure 2(b)). The stress equilibrium equations in the plane for element A may be derived from either part (I) or (II). In part (I), the sum of all stresses in the l-direction gives the equilibrium equation (1), shown in Figure 2(c). Similarly, the equilibrium equations in the t-direction are obtained as equation (2), and the shear stress equilibrium equation is equation (3), shown in Figure 2(e). Table 1 summarizes the equilibrium equations for torsional members for both cross-sectional and in-plane conditions. The three equilibrium equations of element A are expressed by equations (1)–(3).

Table 1.

Equilibrium and Compatibility Equations.

Equations of equilibrium
In-plane equilibrium equations	$σ_{l} = σ_{2}^{c} \cos^{2} α_{2} + σ_{1}^{c} \sin^{2} α_{2} + 2 τ_{21}^{c} \sin α_{2} \cos α_{2} + ρ_{s l} f_{s l} + ρ_{f l} f_{f l}$	(1)
	$σ_{t} = σ_{2}^{c} \sin^{2} α_{2} + σ_{1}^{c} \cos^{2} α_{2} - 2 τ_{21}^{c} \sin α_{2} \cos α_{2} + ρ_{s t} f_{s t} + ρ_{f t} f_{f t}$	(2)
	$τ_{l t} = (σ_{2}^{c} + σ_{1}^{c}) \sin^{2} α_{2} - 2 τ_{21}^{c} \sin α_{2} \cos α_{2} + ρ_{s t} f_{s t} + ρ_{f t} f_{f t}$	(3)
	$ρ_{s l} = \frac{A_{s l}}{p_{0} t_{d}}$ and $ρ_{s t} = \frac{A_{s t}}{s t_{d}}$	(4)
	$ρ_{f l} = \frac{A_{f l}}{p_{0} t_{d}} = \frac{(n_{f l} t_{f l}) p_{f l}}{p_{0} t_{d}}$ and $ρ_{f t} = \frac{A_{f t} p_{c}}{p_{0} s_{f} t_{d}} = \frac{(n_{f t} t_{f t}) w_{f} p_{c}}{p_{0} s_{f} t_{d}}$	(5)
Section equilibrium equations	$T = τ_{l t} (2 A_{0} t_{d})$	(6)
Equations of compatibility
In-plane compatibility equations	$ε_{l} = ε_{2} \cos^{2} α_{2} + ε_{1} \sin^{2} α_{2} + γ_{21} \sin α_{2} \cos α_{2}$	(7)
	$ε_{t} = ε_{2} \sin^{2} α + ε_{1} \cos^{2} α_{2} - γ_{21} \sin α_{2} \cos α_{2}$	(8)
	$γ_{l t} = 2 (- ε_{2} + ε_{1}) \sin α_{2} \cos α_{2} + γ_{21} (\cos α_{2} - \sin α_{2})$	(9)
Hsu/Zhu amplification factor for torsion	${\bar{ε}}_{1} = \frac{1}{1 - υ_{12} υ_{21}} ε_{1} + \frac{υ_{12}}{1 - υ_{12} υ_{21}} ε_{2}$	(10)
	${\bar{ε}}_{2} = \frac{υ_{12}}{1 - υ_{12} υ_{21}} ε_{1} + \frac{1}{1 - υ_{12} υ_{21}} ε_{2}$	(11)
	${\bar{γ}}_{21} = γ_{21}$	(12)
	${(υ_{12})}_{t o r s i o n} = 0.16 + 680 ε_{s f} k h i ε_{s f} \leq ε_{y}$	(13)
	${(υ_{12})}_{t o r s i o n} = 1.52 k h i ε_{s f} > ε_{y}$	(14)
	${\bar{ε}}_{1} = ε_{1} + {(υ_{12})}_{t o r s i o n} ε_{2}$	(15)
	${\bar{ε}}_{2} = ε_{2}$	(16)
	${\bar{ε}}_{l} = {\bar{ε}}_{2} \cos^{2} α_{2} + {\bar{ε}}_{1} \sin^{2} α_{2} + γ_{12} \sin α_{2} \cos α_{2}$	(17)
	${\bar{ε}}_{t} = {\bar{ε}}_{2} \sin^{2} α_{2} + {\bar{ε}}_{1} \cos^{2} α_{2} - γ_{12} \sin α_{2} \cos α_{2}$	(18)
Out-of-plane bending of concrete struts	$w = θ x y$	(19)
Out-of-plane bending of concrete struts	$ψ = θ \sin 2 α_{2}$	(20)
Relationship between curvature and strain of concrete struts	$t_{d} = \frac{{\bar{ε}}_{2 s}}{ψ} o r t_{d} = \frac{{\bar{ε}}_{1 s}}{φ}$	(21)
Relationship between shear strain and angle of twist	$θ = \frac{p_{0}}{2 A_{0}} γ_{l t}$	(22)
	$t_{d} = \frac{1}{2 \times (R + 4)} [p_{c} (1 + \frac{R}{2}) - \sqrt{{(1 + \frac{R}{2})}^{2} {(p_{c})}^{2} - 4 R (R + 4) A_{c}}]$	(23)
	$R = \frac{2 {\bar{ε}}_{2 s}}{γ_{l t} \sin 2 α_{2}} = \frac{4 {\bar{ε}}_{2}}{γ_{l t} \sin 2 α_{2}}$	(24)

The ratio of longitudinal steel, stirrups, and composite fibers in longitudinal and vertical directions in equations (1)–(3) relative to the thickness of the shear flow zone, t_d, are delineated in equations (4)–(5). The torsional moment T can be determined from equation (6), which is based on Bredt’s equation for an equivalent thin-walled cross-section (Bredt, 1896).

Under pure torsion, the cross-sectional shape remains unchanged and perpendicular to the longitudinal axis of the beam at all positions, as per Saint Venant’s theory. This means that the only stress component present under pure torsion is the in-plane shear stress, and normal stresses in the l-directions and t-directions in the tiny element are zero (i.e. σ_l = σ_t = 0, and α₂ = 45^o) (Hsu, 1993), as shown in Figure 2.

Equations of compatibility

In-plane compatibility

Table 1 delineates the compatibility equations for torsional members in both sectional and in-plane configurations. Similarly, the in-plane strain of the shear element, A, as illustrated in Figure 2, must satisfy three compatibility equations calculated using equations (7)–(9) (Zhu, 2002).

Hsu/Zhu amplification factor for torsion

Since the stress-strain relationship of materials is derived from uniaxial strains, it is crucial to determine uniaxial strains from biaxial strains. The relationships between biaxial and uniaxial strains, together with the corresponding algebraic equations, are expressed by equation (10)–(12) (Hsu, 2002).

The Hsu/Zhu amplification factors v₁₂ and v₂₁ in equations (10) and (11) are determined for shear members, denoted by ${(υ_{12})}_{s h e a r}$ and ${(υ_{21})}_{s h e a r}$ (Hsu, 2002). Then, Jeng and Hsu (2009) proposed the first Hsu/Zhu coefficient for torsion ${(υ_{12})}_{t o r s i o n}$ equal to 80% due to the strain reduction by out-of-plane bending. The second Hsu/Zhu coefficient for torsion ${(υ_{21})}_{s h e a r}$ is zero. The coefficient ${(υ_{12})}_{t o r s i o n}$ is defined equations (13) and (14). Substituting equations (13), and (14) into equations (10), and (11), and combining with ${(υ_{21})}_{s h e a r} = 0$ , the biaxial strains are determined in equations (15), and (16). From this, the biaxial strains in the l-direction (ε_l) and t-direction (ε_t) are determined using the following equations (17) and (18).

Out-of-plane bending deformation of concrete struts

The angle of twist (θ) in a torsional element causes warping in the member’s wall, leading to out-of-plane bending of the concrete struts. The concrete struts undergo compression from circular shear and bending due to wall warping. The box member, characterized by four walls of a thickness (t_fe) and subjected to apply torque T, is illustrated in Figure 4(a), and equation (19) describes the hyperbolic paraboloid surface OBED.

Figure 4.

Out-of-plane torsional effects on the concrete struts (Hsu, 1993) (a) Bending of concrete strut, (b) Strains and stresses in concrete strut.

To find the slope of the OE curve, impose an axis 2 through OC in the direction of the diagonal concrete struts. The curvature of the concrete struts, represented as ψ, can be derived as the second derivative of (w) concerning the 2-axis, resulting in equation (20).

Relationship between curvature and strain of concrete struts

The curvature (ψ) and (φ) create a non-uniform distribution of stress and strain in the concrete strut. Analyze a concrete strut of unit width within a hollow section exhibiting an effective wall thickness t_fe (Figure 4 (b)). Per the analogous thin-walled tube theory, the hollow core region within the cross-section is neglected, and the applied torque T is resisted by the compression strut with a shear flow thickness t_d (t_d ≤ t_fe). The strain distribution resulting from torque is presumed to be linear throughout this thickness, t_d, as illustrated in Figure 4(b). Equation (21) expresses the relationship between curvature and the maximum strain on the outer surface of the concrete strut is associated with the effective thickness t_d.

Relationship between shear strain and angle of twist

The shear strain (γ_lt) in the shear flow area of the member, induced by the angle of twist θ, can be ascertained by analogy with an enclosed circumference tube of infinite length (Hsu, 1993). The correlation between the shear strain γ_lt and the angle of twist θ of the torsion component can be established by the compatibility criterion of warpage strain (Jeng and Hsu, 2009). Substituting θ from equation (22) into equation (20), along with the equations for determining (p₀) and (A₀), yields the calculation of t_d by equation (23). For hollow section beams, the effective thickness t_d calculated for the solid beam in equation (23) must be less than the hollow beam wall thickness (t_H) (Chyuan-Hwan Jeng, 2014).

Constitutive laws for materials

Improved constitutive relationship for concrete in compressive

When an RC element undergoes shear stress, its compressive strength significantly decreases due to tensile cracking or the ‘softening’ of the concrete, as suggested by Robinson (1968). Figure 5(a) demonstrates that the constitutive law of softened concrete must incorporate three characteristics: (1) a substantial decrease in the peak levels of stress and strain; (2) the pre-cracking ascending portion of the stress-strain curve is defined as parabolic; (3) the gently sloping curve of the post-peak segment intersects the strain axis at a notable value of 2ε₀ (Chalioris, 2007a).

Figure 5.

Constitutive relationships between stress and strain in concrete. (a) Compressive in concrete, (b) Tensile in concrete.

The stress-strain relationship of concrete in the SMMT model was developed by Belarbi and Hsu (1995) and then updated by Yang et al. (2017) for softened compressive concrete based on experimental studies on large-scale RC slabs strengthened with externally bonded FRP sheets using the Universal Plate Tester (UPT), as given in equation (25) (Yang et al., 2015, 2017).

σ_{(ε)} = {\begin{cases} σ_{p} (2 \frac{{\bar{ε}}_{2}}{ε_{p}} - {(\frac{{\bar{ε}}_{2}}{ε_{p}})}^{2}) f o r \frac{ε}{ε_{p}} \leq 1.0 \\ σ_{p} (1 - {(\frac{{\bar{ε}}_{2} - ε_{p}}{4 ε_{0} - ε_{p}})}^{2}) f o r \frac{ε}{ε_{p}} > 1.0 \end{cases}

(25)

Moreover, the test results indicate that the stiffness of externally bonded FRP sheets significantly increases the concrete softening coefficient (ζ). An additional coefficient, (f_(FRP)), accounts for this effect based on experiments by Yang et al. (2017). This factor (f_(FRP)) basically depends on the strengthening configurations, the elastic modulus and the ratio of FRP. The proposed model integrates an enhanced constitutive model for concrete compression to forecast the torsional behavior of FRP-strengthened RC members subjected to torsional loading. The coefficient ζ considers the softening of concrete struts based on four variables: compressive strength of concrete $f_{c}^{'}$ , principal tensile strain ${\bar{ε}}_{1}$ , deviation angle $\bar{β}$ (Wang, 2006), and FRP stiffness E_fρ_f (Yang et al., 2017), as expressed in equations (26)–(30).

ζ = f_{(f_{c}^{'})} f_{({\bar{ε}}_{1})} f_{(\bar{β})} f_{(F R P)}

(26)

f_{(f_{c}^{'})} = \frac{5.8}{\sqrt{f_{c}^{'}}} \leq 0.9

(27)

f_{({\bar{ε}}_{1})} = \frac{1}{\sqrt{1 + 400 {\bar{ε}}_{1}}}

(28)

f_{(\bar{β})} = (1 - \frac{| \bar{β} |}{24^{0}}) a n d \bar{β} (\deg) = \frac{1}{2} \tan^{- 1} (\frac{γ_{21}}{ε_{2} - ε_{1}}) \frac{180}{π}

(29)

f_{(F R P)} = {\begin{cases} 1 + 0.0076 \sqrt{E_{f} ρ_{f t}} & for strengthened sides \\ 1.0 & for unstrengthened sides \end{cases}

(30)

The maximum cylindrical compressive stress and the associated strain at peak stress of softening concrete are defined by equations (31) and (32) (Chalioris, 2007a).

σ_{p} = ζ f_{c}^{'}

(31)

ε_{p} = ζ ε_{0}

(32)

In the strengthened member with a fully wrapped cross-sectional configuration, the impact of external FRP on concrete confinement is evaluated through the application of multi-playing compressive stress with a confinement coefficient (k_f). Thus, the ultimate strain of the softened concrete under confinement (ε_cu) and the coefficient k_f are defined (Chalioris, 2007a).

ε_{c u} = γ_{F R P} k_{f}^{2} (1, 5 ε_{0})

(33)

k_{f} = 1 + 2.8 \times α_{n} α_{s} ω_{n}

(34)

The three coefficients α_n, α_s and ω_w are given by equations (35)–(37).

α_{n} = 1 - \frac{b^{2} + h^{2}}{3 A_{c}}

(35)

α_{s} = {\begin{cases} 1.0 & f o r s t r e n g t h e n e d w i t h f u l l \\ (1 - \frac{s_{f} - w_{f}}{2 b}) (1 - \frac{s_{f} - w_{f}}{2 h}) & f o r s t r e n g t h e n e d w i t h s t r i p s \end{cases}

(36)

ω_{n} = \frac{volume of FRP sheets}{volume of confined concrete} \frac{f_{f u}}{f_{c}^{'}}

(37)

As per RA-STM (Belarbi and Hsu, 1995), the average compression stress of concrete is ascertained by the non-dimensional coefficient k_1c, which is the ratio of average stress to the peak compression stress of concrete. The coefficient k_1c can be obtained from the integration and simplification of equations (38) and (39) for both the ascending and descending branches.

\begin{array}{l} k_{1 c} = \frac{1}{{\bar{ε}}_{2 s} σ_{p}} \int_{0}^{{\bar{ε}}_{2 s}} σ^{c} ({\bar{ε}}_{2 s}) d {\bar{ε}}_{2} \\ = [\frac{{\bar{ε}}_{2 s}}{ε_{p}} - \frac{1}{3} {(\frac{{\bar{ε}}_{2 s}}{ε_{p}})}^{2}] f o r \frac{{\bar{ε}}_{2 s}}{ε_{p}} \leq 1 \end{array}

(38)

\begin{array}{l} k_{1 c} = \frac{1}{{\bar{ε}}_{2 s} σ_{p}} \int_{0}^{{\bar{ε}}_{2 s}} σ^{c} ({\bar{ε}}_{2 s}) d {\bar{ε}}_{2} \\ = 1 - \frac{1}{3} \frac{ε_{p}}{{\bar{ε}}_{2 s}} - \frac{{({\bar{ε}}_{2 s} - ε_{p})}^{3}}{3 {\bar{ε}}_{2 s} {(4 ε_{0} - ε_{p})}^{2}} f o r \frac{{\bar{ε}}_{2 s}}{ε_{p}} > 1 \end{array}

(39)

Furthermore, the compressive strength of concrete under torsion is (η) times greater than that predicted by Yang’s model for shear. This difference is due to the strain gradient effect, as observed in previous studies by Jeng and Hsu (2009). Thus, the mean compressive stress of the concrete struts is determined by:

σ_{2}^{c} = - η k_{1 c} σ_{p}

(40)

Improved constitutive relationship for concrete in tension

Numerous investigations have noted that the cracking pattern and tension stiffening are considerably influenced by the stiffness and wrapping design of FRP (Yang et al., 2015). This phenomenon can be attributed to the reduction in mean crack width resulting from enhanced FRP-to-concrete bond activity, which leads to an increase in the number of cracks and tension stiffening, as well as a decrease in both crack spacing and width. Consequently, the tensile stress in the concrete increases due to the impact of the stiffness of the external FRP wrapping.

Figure 5(b) shows the uniaxial stress-strain relationship of tensile concrete, which is proposed by combining the models by Yang et al. (2015) and Jeng and Hsu (2009). The tensile stress in concrete accounts for the effect of FRP stiffness. A new parameter, exponent C, considers FRP wrapping techniques and the stiffness ratio of FRP to stirrup reinforcement (Yang et al., 2015). Additionally, the cracking strength and the modulus of elasticity of pre-cracked concrete are determined using equations (43) and (44).

σ_{1}^{c} = E_{c} {\bar{ε}}_{1} f o r {\bar{ε}}_{1} \leq ε_{c r}

(41)

σ_{1}^{c} = f_{c r} {(\frac{ε_{c r}}{{\bar{ε}}_{1}})}^{C} f o r {\bar{ε}}_{1} > ε_{c r}

(42)

f_{c r} = = {\begin{cases} 0.31 \sqrt{f_{c}^{'}} & f o r u n s t r e n g t h e n e d \\ λ μ (0.31) \sqrt{f_{c}^{'}} & f o r s t r e n g t h e n e d \end{cases}

(43)

E_{c} = {\begin{cases} 3875 \sqrt{f_{c}^{'}} & f o r u n s t r e n g t h e n e d \\ λ (3875) \sqrt{f_{c}^{'}} & f o r s t r e n g t h e n e d \end{cases}

(44)

For a solid RC beam under pure torsion, the three factors η = μ = 1.45 and λ = 1.0 (Jeng and Hsu, 2009). For hollow section beams, these three factors are determined based on the ratio t_H /t_d,cr,solid and the compressive strength of concrete (f_c^’) proposed by Jeng et al. (2014), (t_d,cr,solid is the thickness of the solid beam calculated from equation (23)). The exponent C is determined based on the strengthening configuration and the stiffness ratio of FRP to steel as follows (Yang et al., 2015):

C = {\begin{cases} 0.4 & f o r u n s t r e n g t h e n e d s i d e s \\ K_{w} / K_{f / s} & f o r s t r e n g t h e n e d s i d e s \end{cases}

(45)

The coefficients K_w and K_f/s given by equation (46) and (47), respectively. In equation (47), the stiffness ratio $ρ_{f} E_{f} / ρ_{t} E_{s t}$ ranges from 0.25 to 1.11 (Yang et al., 2015).

K_{w} = {\begin{cases} 1.0 & f o r f u l l - w a r p p e d s c h e m e \\ 1.6 & f o r s i d e - b o n e d s c h e m e \end{cases}

(46)

K_{f / s} = 0.25 (\frac{ρ_{f} E_{f}}{ρ_{t} E_{s t}}) + 0.15

(47)

Similarly, an average tensile stress factor k_1t can be obtained by integrating and simplifying equations (48)–(50):

σ_{1}^{c} = k_{1 t} f_{c r}

(48)

k_{1 t} = \frac{{\bar{ε}}_{1 s}}{2 ε_{c r}} f o r \frac{{\bar{ε}}_{1 s}}{ε_{c r}} \leq 1

(49)

k_{1 t} = \frac{{\bar{ε}}_{1 s}}{2 ε_{c r}} + \frac{{(ε_{c r})}^{C}}{(1 - C) {\bar{ε}}_{1 s}} [{({\bar{ε}}_{1 s})}^{(1 - C)} - {(ε_{c r})}^{(1 - C)}] f o r \frac{{\bar{ε}}_{1 s}}{ε_{c r}} > 1

(50)

Shear stress relationships in concrete

According to Hsu (2002), concrete shear stress correlates with shear strain as follows:

τ_{21}^{c} = \frac{σ_{1}^{c} - σ_{2}^{c}}{2 (ε_{1} - ε_{2})} γ_{21}

(51)

Tension in steel

Due to concrete cracking, the uniaxial stress-strain relationship of conventional reinforcement (both longitudinal and vertical reinforcement) is replaced with the biaxial reinforcement embedded in concrete for the MSMMT-FRP (Belarbi and Hsu, 1995), as depicted in Figure 6(a), as follows:

f_{s} = E_{s} {\bar{ε}}_{s} f o r {\bar{ε}}_{s} \leq {\bar{ε}}_{n}

(52)

f_{s} = f_{y} [(0.91 - 2 B) + (0.02 + 0.25 B) (\frac{{\bar{ε}}_{s}}{ε_{y}})] f o r {\bar{ε}}_{s} > {\bar{ε}}_{n}

(53)

{\bar{ε}}_{n} = f_{y} (0.93 - 2 B) a n d B = \frac{{(f_{c r} / f_{y})}^{1.5}}{ρ}

(54)

Figure 6.

Constitutive stress-strain laws of steel and FRP fiber. (a) Steel, (b) FRP fiber.

Tension in FRP

Figure 6(b) illustrates that the relationship between stress and strain for the composite fibers is assumed to be elastic linear-elastic behavior till failure (Zojaji and Kabir, 2012). Consequently, the constitutive model is represented by a straightforward Hooke’s law equation as shown in equation (55):

f_{f} = E_{f} ε_{f}, ε_{f} \leq ε_{e f}

(55)

In equation (55), the effective strain is determined by analyzing the failure modes regardless of the FRP material used, which include deboning end, peeling off, and fiber rupture (Teng et al., 2002). As per Zojaji and Kabir (2012), Deifalla and Ghobarah (2010) the effective strain was determined for these three failure modes based on the proposal by Teng et al. (2002). The equations used to calculate the effective strain corresponding to these three failure modes are equations (56), (57), and (58), as shown below:

ε_{e f} = \frac{0.33}{L_{e}} \frac{w_{f}}{s_{f}}

(56)

ε_{e f} = \frac{0.02 α_{f}}{L_{e}}

(57)

ε_{e f} = 0.1 {(E_{f} ρ_{f})}^{- 0.86} ε_{f u}

(58)

The effective bond length (L_e) and a constant parameter (α_f) are determined by equations (Teng et al., 2002):

L_{e} = \sqrt{\frac{E_{f} t_{f}}{\sqrt{f_{c}^{'}}}}

(59)

α_{f} = \sqrt{\frac{2 - \frac{w_{f}}{s_{f} \sin (β_{f})}}{1 + \frac{w_{f}}{s_{f} \sin (β_{f})}}}

(60)

Solution algorithm for MSMMT-FRP

Convergence conditions were considered in the MSMMT-FRP: the first condition that the normal stress in the l-direction is zero, represented as σ_l = 0 in equation (1); the second condition that the normal stress in the t-direction is also zero, denoted as σ_t = 0 in equation (2). From these two conditions, adding equation (1) to equation (2) yields equation (61), which serves as the initial convergence criterion. Similarly, subtracting equation (1) from equation (2) gives equation (62), which serves as the second convergence condition. Figure 7 illustrates the solution algorithm of MSMMT-FRP for strengthening RC beams using FRP fibers in MATLAB (2023).

ρ_{s l} f_{s l} + ρ_{f l} f_{f l} + ρ_{s t} f_{s t} + ρ_{f t} f_{f t} = (σ_{l} + σ_{t}) - (σ_{2}^{c} + σ_{1}^{c})

(61)

ρ_{s l} f_{s l} + ρ_{f l} f_{f l} - ρ_{s t} f_{s t} - ρ_{f t} f_{f t} = (σ_{l} - σ_{t}) - (- σ_{2}^{c} + σ_{1}^{c}) \cos 2 α_{2} - 2 τ_{21}^{c} \sin 2 α_{2}

(62)

Figure 7.

Flowchart illustrates the solution methodology for the MSMMT-FRP.

Validation of the MSMMT-FRP

The MSMMT-FRP was employed to predict the entire torsional response for strengthening RC beams utilizing various FRP materials. The calculated results of the proposed model were compared with those from experimental studies and other previous analytical models in the following subsections.

Collected database

To validate the proposed model, experimental data from 16 solid and four hollow RC beams available in the previous literature were collected. This dataset of RC members was strengthened with FRP material, specifically CFRP and GFRP. These strengthened beams had a full wrap configuration with continuous sheets or strips with fiber orientation at 90° to the beam axis. Note that this study does not include U-shaped wrap configurations or fiber orientations of 45° or 0°. The geometrical, material, steel reinforcement arrangements, and strengthening configuration characteristics of the beams are shown in Table 2.

Table 2.

Material Characteristics and Geometrical Cross-sections of the test Beams.

Ref.	Beams	B × h × t_H (mm)	Fiber type	f_c^’ (MPa)	A_sl (mm²)	A_st/s (mm² /mm)	f_yl (MPa)	f_yt (MPa)	n _f	t_f (mm)	w_f (mm)	s_f (mm)
Saravanan Panchacharam (2002)	A90W4	279 × 279	GFRP	34	792	0.468	420	460	1	0.154	114.3	228.6
Saravanan Panchacharam (2002)	A90S4	279 × 279	GFRP	34	792	0.468	420	460	1	0.154	114.3	114.3
Ghobarah et al. (2002)	C1	150 × 350	CFRP	37	603	0.542	409	466	1	0.165	100	100
	C2	150 × 350	CFRP	37	603	0.542	409	466	1	0.165	200	350
	C4	150 × 350	CFRP	37	603	0.542	409	466	1	0.165	100	300
	C5	150 × 350	CFRP	37	603	0.542	409	466	1	0.165	100	250
	G1	150 × 350	GFRP	37	603	0.542	409	466	1	0.353	100	100
	G2	150 × 350	GFRP	37	603	0.542	409	466	1	0.353	100	200
Hii and Al-Mahaidi (2006a)	FS050D2	350 × 500	CFRP	52.6	1100	0.226	398	426	2	0.176	50	175
	FH075D1	350 × 500 × 50	CFRP	48.9	1100	0.226	398	426	1	0.176	50	262.5
	FH050D1	350 × 500 × 50	CFRP	56.4	1100	0.226	398	426	1	0.176	50	175
	FH050D2	350 × 500 × 50	CFRP	52.8	1100	0.226	398	426	2	0.176	50	175
Chalioris (2007b)	Ra-SFs150(2)	100 × 200	CFRP	27.5	201	0.183	560	350	2	0.11	100	200
Chalioris (2007b)	Rb-SFs200(1)	150 × 300	CFRP	28.8	201	0.238	560	350	1	0.11	100	200
Ameli and Ronagh (2007)	CFE	150 × 350	CFRP	39	804	0.353	502	251	1	0.165	100	100
	CFE2	150 × 350	CFRP	39	804	0.353	502	251	2	0.165	100	100
	CFS	150 × 350	CFRP	39	804	0.353	502	251	1	0.165	100	200
	GFE	150 × 350	GFRP	36	804	0.353	502	251	1	0.154	100	100
	GFS	150 × 350	GFRP	36	804	0.353	502	251	1	0.154	100	200
Chai et al. (2014)	ST-S	250 × 350 × 50	CFRP	45	452	0.283	605	402	1	0.142	100	200

Note: t_H is the thickness of the hollow beam wall.

Torque - angle of twist response

The torque-angle of twist behavior of all test beams, calculated using the MSMMT-FRP, was compared with experimental results and prior analytical models for beams strengthened with CFRP and GFRP material. These comparisons are presented in Figures 8 and 9 for solid and hollow beams, respectively. In general, the results shown in Figures 8 and 9 demonstrate that the MSMMT-FRP reasonably predicts the complete torsional response during both the pre-cracking and post-cracking phases, consistent with the experimental results. Initially, the beams exhibited linear elastic behavior until the torsional moment reached the cracking torque (T_cr) value and the corresponding cracking angle of twist (θ_cr). Subsequently, there was a sudden decrease in torque and a rapid increase in the angle of twist due to the redistribution of stress from the concrete to the steel reinforcement (longitudinal steel and stirrups) and the externally bonded FRP fibers. This transition marked the beginning of the non-linear stage, during which the torque increased until it reached the ultimate torque (T_u), followed by a decrease in torque and an increase in the angle of twist until the beam eventually failed. Additionally, the proposed model also reasonably agrees with the prediction of the failure modes occurring on the strengthened RC beams. For example, the rupture failure modes of specimens FS050D2 (Hii and Al-Mahaidi, 2006a) and C4 (Ghobarah et al., 2002) are shown in Figure 8(d) and (g), respectively.

Figure 8.

Comparison between the result of the entire torsional behavior of the MSMMT-FRP with experimental and other previous models for RC beams strengthened with FRP material.

Figure 9.

Comparison of the overall torsional behavior of the MSMMT-FRP with experimental results for RC hollow beams strengthened with FRP material.

Table 3 summarizes the values of both experimental and analytical cracking torque T_cr and ultimate torque T_u, as well as the corresponding angle of twist (θ_cr and θ_u) for each beam. The computational results showed that the MSMMT-FRP has a good prediction of the initial stiffness of beams strengthened with FRP composite. The mean ratio T_cr,cal /T_cr,exp for the 20 beams strengthened with CFRP or GFRP is 1.09, with deviation of the standard of 0.22. The mean value and deviation of the standard of θ_cr,cal /θ_cr,exp calculated for all the beams are 0.78 and 0.20, respectively. The MSMMT-FRP also reasonably agrees with the prediction of the torsional capacity T_u with a mean coefficient T_u,cal /T_u,exp of 1.00 and a standard deviation of 0.06. Further, the outcomes of forecasting the ultimate angle of twist with the two coefficients, including the average value and deviation of standard, were 1.17 and 0.39, respectively. Consequently, the forecasting of the angles of twist at the cracking and peak point of the torque-angles of twist relationship obtained by MSMMT-FRP is not as good as the prediction of torques. Based on the analysis of the results, a reasonable prediction with the experimental results of the torque-twist angle relationship of MSMMT-FRP in the entire torsional behavior is found. The proposed analytical model of MSMMT-FRP provides a reliable method for analyzing the overall torsional response for strengthening RC beams with FRP materials.

Table 3.

Comparison of Torques and Angle of Twist From MSMMT-FRP and experimental Results.

Ref.	Beams	T_cr (kN.m)			θ_cr (rad/m)			T_u (kN.m)			θ_u (rad/m)
Ref.	Beams	T _cr,exp	T _cr,cal	T_cr,cal /T_cr,exp	θ _cr,exp	θ _cr,cal	θ_cr,cal /θ_cr,exp	T _u,exp	T _u,cal	T_u,cal /T_u,exp	θ _u,exp	θ _u,cal	θ_u,cal /θ_u,exp
Panchacharam and Belarbi (2002)	A90W4	22	18.31	0.83	0.00250	0.00292	1.17	45	43.07	0.96	0.0580	0.0727	1.25
Panchacharam and Belarbi (2002)	A90S4	21	18.52	0.88	0.00700	0.00292	0.42	34	36.06	1.06	0.1050	0.1031	0.98
Ghobarah et al. (2002)	C1	6.73	6.82	1.01	0.00387	0.00343	0.89	17.92	18.72	1.04	0.1026	0.1194	1.16
	C2	5.53	6.54	1.18	0.00376	0.00326	0.87	13.96	15.42	1.10	0.1107	0.1456	1.32
	C4	6.57	7.64	1.16	0.00370	0.00259	0.70	15.83	15.93	1.01	0.0876	0.0954	1.09
	C5	5.87	6.42	1.09	0.00370	0.00326	0.88	13.42	14.23	1.06	0.0768	0.1289	1.68
	G1	7.17	7.32	1.02	0.00406	0.00382	0.94	18.94	18.34	0.97	0.1119	0.1227	1.10
	G2	6.29	7.41	1.18	0.00388	0.00383	0.99	13.15	13.63	1.04	0.0831	0.1291	1.55
Hii and Al-Mahaidi (2006a)	FS050D2	73.70	61.00	0.83	0.00279	0.00187	0.67	93.80	94.58	1.01	0.1316	0.1098	0.83
	FH075D1	19.60	33.13	1.69	0.00262	0.00263	1.01	67.50	71.01	1.05	0.0803	0.0863	1.07
	FH050D1	21.30	32.98	1.55	0.00279	0.00227	0.81	74.80	75.15	1.00	0.0948	0.1001	1.06
	FH050D2	22.20	31.09	1.40	0.00297	0.00203	0.68	87.70	89.43	1.02	0.0963	0.1203	1.25
Chalioris (2007b)	Ra-SFs150(2)	2.35	2.26	0.96	0.00900	0.00878	0.98	4.33	4.65	1.07	0.0880	0.1447	1.64
Chalioris (2007b)	Rb-SFs200(1)	6.93	5.29	0.76	0.01000	0.00705	0.70	9.80	10.15	1.04	0.0750	0.1686	2.25
Ameli and Ronagh (2007)	CFE	10.40	10.82	1.04	0.00800	0.00356	0.45	28.00	24.93	0.89	0.2269	0.1272	0.56
	CFE2	10.70	10.82	1.01	0.00560	0.00356	0.64	36.50	32.13	0.88	0.3264	0.1142	0.35
	CFS	10.30	10.82	1.05	0.00700	0.00368	0.53	21.70	21.02	0.97	0.1606	0.1390	0.87
	GFE	9.70	10.82	1.12	0.00560	0.00369	0.66	26.30	24.08	0.92	0.0890	0.1129	1.27
	GFS	10.50	10.82	1.03	0.00614	0.00368	0.60	19.90	19.03	0.96	0.1379	0.1376	1.00
Chai et al. (2014)	ST-S	12.20	11.31	0.93	0.00122	0.00131	1.08	21.50	22.62	1.05	0.1257	0.1410	1.12
Average				1.09			0.78			1.00			1.17
Deviation of standard				0.22			0.20			0.06			0.39

Figure 10 shows the MSMMT-FRP results, considering the confinement effect on the softening behavior of concrete under compression (equation (30)) and the impact of FRP stiffness on tensile stress (equation (45)), in comparison to SMMT-FRP, which ignores the confinement effect of FRP on the concrete constitutive law, i.e., C = 0.4 and f_(FRP) = 1.0. The analytical results indicate that SMMT-FRP underestimates both the cracking torque and the post-cracking torque compared to the experimental data. By contrast, MSMMT-FRP provides good predictions. This shows that a more reasonable compressive and tensile constitutive relationship for FRP-strengthening concrete is proposed in this model.

Figure 10.

Confinement effect in the prediction of G1 tested by Ghobarah et al. (2002).

Comparison of MSMMT-FRP with fib bulletin 90 (2019)

The FRP externally bonded strengthening contributes to the torsional resistance of a structure, as per the fib Bulletin 90 (2019), by applying the concept of the equilibrium truss theory. The standard recommends using a four-sided cross-section wrapping configuration, which could include strip wrapping or continuous wrapping along the beam axis. Additionally, it is essential that the orientation of the fibers is at a right angle to the longitudinal axis of the beam. According to fib Bulletin 90, the torsional resistance of a strengthened member can be expressed as follows:

T_{u} = \min (T_{s t} + T_{f}, T_{c})

(63)

where, T_c and T_st are the torsional resistance of compression struts and torsional strength of internal transverse reinforcement, respectively, in accordance with EN-1992-1-1:2004 (2004); T_f is contribution of FRP to the torsional resistance (fib Bulletin 90, 2019).

Table 4 summarizes the results of the torsional capacity calculations according to fib Bulletin 90 (2019) and comparison with experimental data. The findings indicate that the fib Bulletin 90 (2019) calculation outcomes were lower than the experimental results in terms of the torsional strength of FRP-strengthened beams. The calculation based on this standard yielded a mean value of 0.85 and a standard deviation of 0.37. This outcome may stem from the simplistic mechanical truss model presented in fib Bulletin 90 (2019), which fails to account for the influence of external FRP on the confinement of concrete struts and assumes that the strength of transverse reinforcement achieves full plastic yield at the failure stage, neglecting the hardening phenomenon of steel reinforcement that enhances torsional resistance. Comparison between MSMMT-FRP prediction results and fib Bulletin 90 indicates that the proposed model offers better predictions of torsional strength.

Table 4.

Comparison of Ultimate Torques From fib Bulletin 90 and experimental Results.

Ref.	Beams	T_u,exp (kN.m)	fib bulletin 90 (2019) (kN.m)
Ref.	Beams	T_u,exp (kN.m)	T _st	T _f	T _c	T _u,cal	T_u,cal/ T_u,exp
Panchacharam and Belarbi (2002)	A90W4	45.00	18.85	3.76	53.83	22.62	0.50
Panchacharam and Belarbi (2002)	A90S4	34.00	18.85	7.53	53.83	26.38	0.78
Ghobarah et al. (2002)	C1	17.92	11.88	14.90	28.80	26.78	1.49
	C2	13.96	11.88	8.51	28.80	20.39	1.46
	C4	15.83	11.88	4.97	28.80	16.84	1.06
	C5	13.42	11.88	5.96	28.80	17.84	1.33
	G1	18.94	14.74	9.76	28.80	24.50	1.29
	G2	13.15	11.88	4.88	28.80	16.76	1.27
Hii and Al-Mahaidi (2006a)	FS050D2	93.80	17.40	26.75	168.21	44.16	0.47
	FH075D1	67.50	17.40	8.61	159.31	26.02	0.39
	FH050D1	74.80	17.40	12.47	176.89	29.88	0.40
	FH050D2	87.70	17.40	24.09	168.68	41.49	0.47
Chalioris (2007b)	Ra-SFs150(2)	4.33	1.42	3.43	5.44	4.85	1.12
Chalioris (2007b)	Rb-SFs200(1)	9.80	1.80	1.72	5.49	3.52	0.36
Ameli and Ronagh (2007)	CFE	28.00	5.15	13.66	30.08	18.81	0.67
	CFE2	36.50	5.15	27.32	30.08	30.08	0.82
	CFS	21.70	5.15	6.83	30.08	11.98	0.55
	GFE	26.30	5.15	23.37	30.08	28.52	1.08
	GFS	19.90	5.15	5.84	30.08	10.99	0.55
Chai et al. (2014)	ST-S	21.50	11.24	8.17	66.42	19.41	0.90
Average							0.85
Deviation of standard							0.37

Experimental program

This section presents an experimental program to evaluate the effectiveness of strengthening RC beams with CFRP and BFRP sheets; meanwhile, BFRP has not been previously investigated. The experimental results will be compared and validated with the proposed model.

Details of specimens

The experiment involved testing five solid-RC beams, each with a rectangular shape and identical cross-sectional dimensions (250 × 300 mm). The beams were 2700 mm long and had a concrete cover that was 20 mm thick. The length at each end was 350 mm, and there was additional transverse reinforcement spacing of 50 mm to prevent premature failure when increased loading. The torsional failure occurred in the 2000 mm beam testing zone. This area was reinforced with minimum stirrups torsional reinforcement in accordance with ACI 318-19 (2019). All five testing specimens used the same materials and reinforcement configuration. Four longitudinal reinforcing bars with 18 mm diameter were arranged at the corners of the beam cross-section, while the close stirrups in the test area were 6 mm diameter and spaced 150 mm apart. Figure 11 illustrates the visual and numerical representations of the geometry, reinforcement, and dimensions of all RC beam and details of the strengthening specimens.

Figure 11.

Details of test specimens and strain gauges: (a) details of RC specimen, (b) reference beam (B-0), (c) specimen B-B1-100, (d) specimen B-B1-F, (e) specimen B-C1-100, and (f) specimen B-C1-F.

Measurement schemes of strain and twist

Figure 11 shows the device employed to measure the internal steel reinforcement and exterior FRP strains of the test specimen. Five strain gauges were included in the experiment. Strain gauges G1 and G2 were used to measure the strain in the close stirrups within the test zone, while strain gauges G3 and G4 were employed to assess the strain in the longitudinal reinforcement, as seen in Figure 11(a). Strain gauges (G1, G3) were positioned near the restrained end, whereas (G2, G4) were situated at the midpoint of the beams. In addition, strain gauge G5 was used to assess the strains in the FRP and was positioned adjacent to the restraint end, as depicted in Figure 11(c)–(f).

For measuring vertical displacement, three electronic linear variable differential transformers (LVDT1, LVDT2, and LVDT3) were positioned along the beam (refer to Figure 12(b)). The ratio of the LVDTs vertical displacement parameter to the center distance of the beam gives the twist angle per unit length. Additionally, all test data was gathered using a data acquisition system and a laptop computer. Periodically, the loading procedure was paused to record and plot the cracks on the beam surfaces, capture images, and determine the failure modes that occurred.

Figure 12.

Test of setup and equipment. (a) Test setup, (b) Front view.

Materials

The compressive strengths of the tested specimens on the respective experimental days (ASTM-C39/C39M-16b, 2016), and the yield strength of the stirrup (f_yt) and longitudinal reinforcement (f_yl) (ASTM-A370-16, 2016) are shown in Table 5. The CFRP and BFRP sheets employed in this study were unidirectional and possessed an identical thickness of 0.167 mm. The BFRP sheet material (BJ30 A; Haining Anjie Composite Material CO. LTD; China) had tensile strength, elastic modulus, and ultimate strain values of 1299.78 MPa, 53.71 GPa, and 2.42%, respectively, when subjected to a coated tensile test in accordance with ASTM D3039 (2008). The CFRP sheets (UT70-30G; TORAY INDUSTRIES, INC; Japan) exhibited tensile strength, elastic modulus, and ultimate strain values of 4790.69 MPa, 285.16 GPa, and 1.68%, respectively (ASTM-D3039/D3039M-08, 2008).

Table 5.

Details of Testing Beam Specimens.

Specimens	b×h (mm)	f_c^’ (MPa)	f_yt (MPa)	f_yl (MPa)	FRP types and strengthening technique
B-0	250 × 300	32.56	378.6	364.3	Reference specimen
B-B1-100	250 × 300	32.56	378.6	364.3	BFRP, one layer, strips wrapped with 100 mm width and 200 mm spacing
B-B1-F	250 × 300	32.56	378.6	364.3	BFRP, one layer, fully continuous wrap configuration
B-C1-100	250 × 300	32.27	378.6	364.3	CFRP, one layer, wrap alternately single 100 mm strips with 200 mm spacing
B-C1-F	250 × 300	32.27	378.6	364.3	CFRP, one layer, strips wrapped with 100 mm width and 200 mm spacing

Test setup

Figure 12 illustrates the experimental configuration. The setup consists of a restrained end and a loaded end on a circular arc bearing with the same rotation radius as the cross-sectional beam’s geometric center to create pure torsion. A concentrated load was applied to the beam using a 30-ton hydraulic jack and oil pump, which was measured by a 100-ton load cell. A focused load cell with a lever arm of 500 mm from the centroid of the beam cross-section produced torque. Four 40-mm bolts secure the lever arm to the circular arc bearing, which was installed on a steel support and RC floor system. Moreover, the actual applied torque T is calculated as (T = P×l). T is the applied torque; P is the load measured using the load cell and hydraulic jack; l is the distance of the lever steel arm (l = 500 mm). This study does not account for secondary bending resulting from the beam’s weight and other influences, including lever arm bending.

Summary of experimental results

Table 6 summarizes the values of torque T_cr and cracking angle of twist θ_cr, ultimate torque T_u, and ultimate angle of twist θ_u. The reference-tested beam specimen exhibited no enhancement in its post-cracking response. Consequently, the maximum torsion resistance of beam specimens can be ascertained as follows: if T_u ≤T_cr, then T_max = T_cr; otherwise, T_max = T_u. Generally, the specimens strengthened with CFRP or BFRP sheets exhibited improved stiffness, rotational flexibility, and torsional strength. Table 6 indicates that specimens augmented with carbon sheets exhibited superior torsional strength and angle of twist relative to those strengthened with basalt, and the fully wrapped test specimens exhibited more significant torsional strength than the strip-wrapped specimens. The low strength and elastic modulus of the basalt fiber layer may lead to premature fiber rupture, and impact the confinement provided by the wrapping configurations associated with external strengthening using FRP sheets.

Table 6.

Test Results Summary.

Test beams	T _cr (kN.m)	θ _cr (rad/m)	T _max = T _u (kN.m)	θ _u (rad/m)	Increases T_u (%)
B-0	15.18	0.00223	15.18	—	—
B-B1-100	15.28	0.00192	18.18	0.0442	19.8
B-B1-F	15.80	0.00200	22.00	0.0788	44.9
B-C1-100	15.57	0.00199	23.29	0.1028	68.1
B-C1-F	17.18	0.00198	27.47	0.1219	81.0

Additionally, beam specimens strengthened with BFRP demonstrated failure from fiber rupture before concrete crushing, as illustrated in Figure 13 (a)–(b). The primary causes of these failure types are the low tensile strength and elastic modulus of basalt fiber. Figure 13 (c)–(d) demonstrates that the failure modes observed experimentally in CFRP-strengthened specimens included debonding end, followed by concrete crushing.

Figure 13.

Failure mode of test specimens. (a) B-B1-100, (b) B-B1-F, (c) B-C1-100, (d) B-C1-F.

Comparison of experimental results with MSMMT-FRP

The full torsional response of the reference specimen (unstrengthened beam B-0) was obtained using SMMT developed by Jeng and Hsu (2009), and the MSMMT-FRP was employed to predict the overall torsional response for the test beam strengthened with CFRP and BFRP. In addition, the effective strain of the FRP fiber was determined based on the failure mode observed in the test specimens.

Figure 14 illustrates the comparison between the result of the full torque-angle of twist response, torsional capacity at the pre- and post-crack stages of the MSMMT-FRP and the experimental in this study for all test specimens. The torsional behavior of the control specimen was represented by a relationship curve between torque and angle of twist, exhibiting linear and elastic characteristics with elastic torsional stiffness up to the cracking torque T_cr (15.18 kN.m) and the cracking angle of torsion θ_cr (0.00223 rad/m). Subsequently, the torque experiences a sudden decrease while the twist escalates rapidly until specimen failure (Figure 14 (a)). The experimental results of this control specimen conformed to the behavior theory of RC beams with minimal stirrups when exposed to torsion and brittle failure (ACI-318-19, 2019). By contrast, the FRP-strengthened specimens showed behavior through the non-linear phase; the torque and twist increased progressively until the specimen reached its torsional strength, after which the torque decreased. At the same time, the twist continued to increase until failure. Figure 14 also shows that the proposed model gives good predictions of torsional behavior through the loading stages and the failure mode for all test specimens.

Figure 14.

Comparison between the result of the full torsional response of the proposed model with tested beams.

Moreover, it reasonably predicts the torsional capacity at the pre- and post-crack stages. The results of the proposed model in Table 7 indicate for each tested specimen in this study that the MSMMT-FRP successfully predicted the values of T_cr and θ_cr with errors ranging from −11% to +1% and +3% to +30%, respectively. The anticipated values of T_u and θ_u exhibited errors ranging from +1% to +7% and −15% to +64%, respectively. Based on the analytical and experimental results of this study, it is demonstrated that MSMMT-FRP is a dependable approach for evaluating the overall torsional behavior of RC beams strengthened with FRP laminates.

Table 7.

Comparison of Torques and Angle of Twist From Proposed Model and experimental Results.

Beams	T_cr (kN.m)			θ_cr (rad/m)			T_u (kN.m)			θ_u (rad/m)
Beams	T _cr,exp	T _cr,cal	T_cr,cal /T_cr,exp	θ _cr,exp	θ _cr,cal	θ_cr,cal /θ_cr,exp	T _u,exp	T _u,cal	T_u,cal /T_u,exp	θ _u,exp	θ _u,cal	θ_u,cal /θ_u,exp
B-0	15.18	15.31	1.01	0.00223	0.00230	1.03	15.18	15.31	1.01	—	—	—
B-B1-100	15.28	15.31	1.00	0.00192	0.00250	1.30	18.18	19.44	1.07	0.0442	0.0725	1.64
B-C1-100	15.80	15.31	0.97	0.00200	0.00230	1.15	22.00	23.24	1.06	0.0788	0.1126	1.43
B-B1-F	15.57	15.32	0.98	0.00199	0.00250	1.26	23.29	24.77	1.06	0.1028	0.1135	1.10
B-C1-F	17.18	15.33	0.89	0.00198	0.00250	1.26	27.47	29.00	1.06	0.1219	0.1037	0.85

Figure 15(a) and (b) depict the torque-strain relationship curves (strain in stirrups, longitudinal steel, and FRP fibers) of representative beam (B-C1-100), and the strains at the points where cracking torque occurs and at the ultimate torque points in this beam. In addition, the yield strain value of stirrup reinforcement (ε_ty = 1.80 × 10⁻³) and longitudinal reinforcement (ε_ly = 1.85 × 10⁻³) determined from the steel tensile test in Subsection 5.3 are depicted in Figure 15(a), (b), respectively. Figure 15(c) shows the relationship between torque and strain of the external bonded FRP sheet (T-ε_f), and the limit effective strain value of 0.004 to prevent the loss of aggregate interlock or detachment of the FRP composite from the concrete. (ACI-440.2R-17, 2017). Overall, the strain values were very small in the pre-cracking stage and increased slowly. After cracking, these strains increased rapidly and reached their peak at the ultimate torque. The results presented in Figure 15 show that the proposed model reasonably predicts the strain behavior trends of the internal steel reinforcement and external FRP of this specimen through all torsional stages.

Figure 15.

Experimental and analytical torque-strain curves of specimen B-C1-100.

Parametric study

Numerous prior experimental investigations have established factors such as: compressive strength of concrete, strength of FRP material, and FRP ratio substantially influence torsional strength and response (Chalioris, 2006; Deifalla et al., 2013; Ghobarah et al., 2002; Kandekar and Talikoti, 2019; Salom et al., 2004). Additionally, numerical simulation studies have been conducted using various parameters on the torsional capability and response of RC beams strengthened with FRP composites (Majed et al., 2021). However, the limitations of experimental research were both costly and time-consuming, while FEM had difficulty in simulating the boundary conditions as in actual experimental conditions (Ganganagoudar et al., 2016).

In this study, MSMMT-FRP has demonstrated its effectiveness in predicting the torsional strength and behavior of RC beams strengthened with externally bonded FRP sheets. Therefore, certain impacts of parameters significantly influencing torsion will be examined using MSMMT-FRP. The evaluation of parameters on concrete compressive strength, elastic modulus and ratio of FRP with respect to torsional capabilities and responses, based on the G1 specimen analyzed by Ghobarah et al. (2002).

The concrete strength (f_c^’) varied from 20 MPa to 50 MPa. The value of f_c^’ in the baseline model was 37 MPa for the G1 specimen (Ghobarah et al., 2002). The strength of concrete significantly influences the behavior and torsional strength of the test specimen throughout both pre- and post-cracking phases, evidenced by the rise in cracking torque (T_cr) and ultimate torque (T_u) with increasing concrete strength (Figure 16(a)). Prior research has shown that, during the pre-cracking phase, torsional capacity was considerably influenced by concrete strength (Panchacharam and Belarbi, 2002). Meanwhile, increasing the FRP elastic modulus and ratio $(ρ_{f} = \frac{n_{f} t_{f} p_{f}}{A_{c}} \frac{w_{f}}{s_{f}})$ significantly affected the post-cracking stage of concrete and enhanced the torsional resistance T_u, with minimal impact on pre-cracking behavior and cracking torque (T_cr), as depicted in Figure 16(b) and (c).

Figure 16.

Effect of concrete strength, elastic modulus of FRP (E_f), and ratio of FRP (ρ_f) on torsional response of G1 tested by Ghobarah et al. (2002). (a) Effect of concrete strength, (b) Effect of E_f, (c) Effect of ρ_f.

Conclusion

An improved analytical model (MSMMT-FRP) is proposed to predict the complete torsional responses of RC beams strengthened with FRP materials. A trial-and-error algorithm was built in the MATLAB, utilizing equilibrium conditions and new material constitutive relationships to predict the full torque angle of twist curve. Experimental results from 20 RC beams strengthened with CFRP and GFRP sheets from previous studies were considered in this paper to validate the applicability and reliability of the model. In addition, an experimental program was employed, including four specimens that were strengthened using various wrap configurations of CFRP and BFRP laminates, as well as an unstrengthened reference to investigate the torsional effectiveness of the strengthening technique, and the results were compared with those of the analytical model. The primary conclusions are as follows:

(1) The proposed model more reasonably accounts for the confinement effect of FRP-strengthened concrete on its compressive and tensile constitutive relationships under torsional loading.

(2) The MSMMT-FRP reasonably predicts the full torsional response of the experimentally tested 20 beams strengthened with CFRP or GFRP laminates collected from the available literature through loading stages. Specifically, the proposed analytical model provides reliable predictions of cracking and ultimate torsional strength. The mean ratios between the analytical model and experimental results for T_cr and T_u are 1.09 and 1.00, respectively. The results validate the MSMMT-FRP capability to forecast the entire torsional behavior of RC beams strengthened with FRP materials.

(3) Comparison of MSMMT-FRP prediction results with theoretical fib Bulletin 90 confirms that the proposed model gives better predictions of torsional strength.

(4) The results of the experiment showed that using CFRP and BFRP sheets effectively increased the torsional strength of RC beams. The failure mechanisms obtained in all strengthened specimens, CFRP and BFRP, are attributed to debonding end and rupture failure, respectively, regardless of the wrapping techniques employed.

(5) The good prediction of the analytical model for the torsional behavior and failure modes of the tested beams in this study. The maximum errors in predicting cracking and ultimate torque are 11% and 7%, respectively.

(6) Reasonable agreement was achieved between the experimental and analytical model in strain response of the internal steel reinforcement and external FRP.

(7) The parametric study indicates that beams with enhanced compressive strength display greater cracking and ultimate torque. Simultaneously, increasing the FRP elastic modulus and ratio leads to an elevation in ultimate torque, whereas the rise in cracking torque is negligible.

(8) This study considers fully strengthened beams, including both continuous and strip schemes, and fiber orientation 90° relative to the longitudinal beam axis. In future research, further modifications are needed to extend the model to include U-warpping configurations and other fiber orientations such as 0 and 45°, as well as different cross-sectional geometries (L-shaped, and T-shaped).

Footnotes

Acknowledgments

Vinh Sang Nguyen was funded by the Master, PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), code: VINIF2024.TS.091.

CRediT authorship contribution statement

Vinh Sang Nguyen: Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization. Xuan Huy Nguyen: Writing – review & editing, Resources, Supervision, Methodology, Conceptualization. Dung Le Dang: Writing – review & editing, Methodology. Cao Thanh Tran Ngoc: Writing – review & editing, Methodology. Anh Dung Nguyen: Writing – review & editing, Supervision, Methodology, Conceptualization.

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: Vinh Sang Nguyen was funded by the Master, PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF) (VINIF2024.TS.091).

ORCID iDs

Anh Dung Nguyen

Xuan Huy Nguyen

Appendix

Notation
$A_{0}$	Area enclosed by the centerline of shear flow; $A_{0} = A_{c} - \frac{1}{2} p_{c} t_{d} + t_{d}^{2}$	$α$	Rotating angle, angle of applied principle compressive stress (two-axis) with respect to longitudinal steel bars (l-axis)
$A_{c}$	Cross-sectional area bounded by the outer perimeter of the concrete; $A_{c} = b h$	$α_{2}$	Fixed angle, angle of applied principle compressive stress (two-axis) with respect to the longitudinal steel reinforcing bars (l-axis)
$A_{f l}$	Fiber area in the longitudinal direction	$α_{f}$	Constant parameter taking into account the difference in stress distribution between continuous composite sheets and strips
$A_{f t}$	Fiber area in the transverse direction	$α_{n}$	In-section coefficient of effectiveness of the confinement
$A_{s l}$	Total cross-sectional area of the longitudinal steel bars	$\bar{β}$	Deviation angle
$A_{s t}$	Cross-sectional area of one transverse steel (stirrup) bar	$γ_{21}$	Smeared (average) biaxial shear strain of concrete in the 2-1 direction
$b$	Width of the beam section	${\bar{γ}}_{21}$	Smeared (average) uniaxial shear strain of concrete in the 2-1 direction
B	Variable as defined in the constitutive relationship of embedded mild steel	$γ_{l t}$	Smeared (average) shear strain of steel reinforcing bars in the l-t direction
$E_{c}$	Elastic modulus of the concrete	$ε_{0}$	Concrete strain at the peak compressive stress f_c^’ taken as −0.00235
$E_{f}$	Elastic modulus of the fibers	$ε_{2}, ε_{1}$	Smeared (average) biaxial strain of concrete in the 2-direction and the 1-direction, respectively
$E_{s}$	Elastic modulus of steel reinforcing bars	${\bar{ε}}_{2}, {\bar{ε}}_{1}$	Smeared (average) uniaxial strain of concrete in the 2-direction and the 1-direction, respectively
$f_{c}^{'}$	Cylinder compressive strength of concrete	${\bar{ε}}_{2 s}, {\bar{ε}}_{1 s}$	Maximum uniaxial strain at the surface in the 2-Direction and the 1-direction, respectively ${\bar{ε}}_{2 s} = 2 {\bar{ε}}_{2} a n d {\bar{ε}}_{1 s} = 2 {\bar{ε}}_{1}$
$f_{c r}$	Cracking stress of the concrete	$ε_{c u}$	Maximum strain of concrete
$f_{f}$	Tensile stress in the direction of the fiber	$ε_{f}$	Fiber tensile strain
$f_{f l}, f_{f t}$	Fiber stresses in the longitudinal and transverse directions, respectively	$ε_{f e}$	Effective fiber tensile strain
$f_{f e}$	Effective tensile strength of the fibers	$ε_{f u}$	Ultimate fiber tensile strain
$f_{f u}$	Ultimate tensile strength of the fibers	$ε_{l}, ε_{t}$	Smeared (average) biaxial strain of steel bars in the l-direction and the t-direction, respectively
$f_{s}$	Smeared (average) stress of the steel reinforcing bars	${\bar{ε}}_{l}, {\bar{ε}}_{t}$	Smeared (average) uniaxial strain of steel bars in the l-direction and the t-direction, respectively
$f_{s l}, f_{s t}$	Smeared (average) steel stresses in the longitudinal and transverse directions, respectively	${\bar{ε}}_{n}$	Smeared (average) uniaxial yield strain of the steel reinforcing bars
$f_{s u}$	Maximum stress of the steel reinforcing bars	$ε_{s}$	Smeared (average) strain of steel reinforcing bars
$f_{s y}$	Yield stress of the steel reinforcing bars	$ε_{s f}$	Smeared (average) strain of steel reinforcing bars that yield first
$h$	Height of the beam section	$ε_{s u}$	Maximum strain of steel reinforcing bar
$k_{f}$	Composite confinement parameter	$ε_{s y}$	Yield strain of steel reinforcing bar
$k_{1 c}$	Ratio of the average compressive stress to the peak compressive stress in the concrete struts, taking into account the tensile stress of concrete	${\bar{ε}}_{s}$	Smeared (average) uniaxial strain of steel reinforcing bars
$k_{1 t}$	Ratio of the average tensile stress to the peak tensile stress in the concrete struts	$ζ$	Softened coefficient of concrete in compression
$K_{f / s}$	FRP/steel stiffness ratio coefficient	$τ_{21}^{c}$	Smeared (average) shear stress of concrete in 2-1 coordinate
$K_{w}$	Strengthening scheme coefficient	$τ_{l t}$	Applied shear stress in the l-t coordinate of the steel bars
$L_{e}$	Effective bond length	$ρ$	Steel reinforcement ratio
$n_{f l}$	Number of composite layers in the longitudinal direction	$ρ_{f l}, ρ_{f t}$	Longitudinal and transverse fiber ratios, respectively
$n_{f t}$	Number of composite layers in the transverse direction	$ρ_{s l}, ρ_{s t}$	Longitudinal and transverse steel ratios, respectively
$p_{0}$	Perimeter of centerline of shear flow zone $p_{0} = p_{c} - 4 t_{d}$	$v_{12}, v_{21}$	Hsu/Zhu ratios used in the SMM
$p_{c}$	Perimeter of outer concrete cross section $p_{c} = 2 (b + h)$	${(v_{12})}_{S h e a r}$	Same as $v_{12}$
$p_{f l}$	Perimeter of the strengthened beam cross section enclosed by the composite in the longitudinal direction	${(v_{12})}_{T o r s i o n}$	Modified Hsu/Zhu ratio used in the SMMT for torsion
$p_{s t}$	Perimeter of the area enclosed by the stirrup	$v$	Strength reduction factor for concrete cracked in shear
$p_{f t}$	Perimeter of the strengthened beam cross section enclosed by the composite in the transverse direction	$ψ$	Curvature of the concrete struts along the 2-direction
$q$	Shear flow	$φ$	Curvature of the concrete struts along the 1-direction
$s$	Center-to-center spacing of the transverse reinforcing bars (stirrups)	$θ$	Angle of twist per unit length
$s_{f}$	Center-to-center spacing between the centerline of the composite strips	$θ_{c r}$	Cracking angle of twist per unit length
$T$	Torque	$θ_{u}$	Ultimate angle of twist per unit length
$T_{c r}$	Cracking torque	$ω_{n}$	Volumetric mechanical ratio of external confinement
$t_{d}$	Effective thickness of shear flow zone	$α_{c w}$	A coefficient taking into account the state of the stress in the concrete struts
$t_{H}$	Thickness of the hollow beam wall	$γ_{F R P}$	Coefficient taking into account the type of FRP
$T_{c}$	Torsional resistance of compression struts	$σ_{2}^{c}, σ_{1}^{c}$	Smeared (average) normal stresses of concrete in the 2-direction and the 1-direction, respectively
$T_{u}$	Ultimate torque
$T_{s t}$	Torsional strength of internal transverse reinforcement	$σ_{l}, σ_{t}$	Applied normal stresses of steel reinforcing bars in the l-direction and the t-direction, respectively
$T_{f}$	Design resistance of externally applied FRP	$η$	Multiplier factors for average tensile and compressive stresses of concrete
$t_{f l}$	Fiber thickness in the longitudinal direction of the beam	$μ$	Amplification factor for f_cr and ε_cr
$t_{f t}$	Fiber thickness in the transverse direction of the beam	$λ$	Amplification factor for E_c
$w$	Out-of-plane displacement in the direction normal to the membrane element
$w_{f}$	Width of the composite strip

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