Meso-Scale Numerical Analysis of the Torsional Size Effect of RC Beams Reinforced with CFRP Sheets Under Combined Bending and Torsion

Abstract

In practical engineering, buildings are predominantly subjected to combined forces, and reinforced concrete (RC) beams serve as the primary load-bearing components of buildings. However, there is a paucity of research on the torsional effects of RC beams, particularly concerning the torsional failure mechanisms of large-size beams. To address this gap, this paper establishes a meso-scale numerical analysis model for RC beams reinforced with Carbon Fiber Reinforced Polymer (CFRP) sheets under combined bending and torsion pressures. The research analyzes how the fiber ratio and torsion-bending ratio govern torsion-induced failure characteristics and size effects in CFRP-strengthened RC beams. The results indicate that an increase in the fiber ratio leads to accumulated damage distribution in the RC beam, a gradual decrease in CFRP sheet strain, and an increase in peak load and peak torque, albeit with diminishing amplitudes; as the torsion-bending ratio increases, crack distribution becomes more concentrated, the angle between cracks and the horizontal direction decreases, overall peak load decreases, peak torque increases, and CFRP sheet strain increases; and the nominal torsional capacity of CFRP-strengthened RC beams declines with increasing size, exhibiting a reduction of 24.1% to 35.6%, which distinctly demonstrates the torsional size effect under bending–torsion coupling conditions. A modified Torque Size Effect Law is formulated, characterizing in quantitative terms the dependence of the fiber ratio and the torsion-bending ratio.

1. Introduction

Modern architectural design concepts have evolved, resulting in increased complexity of building structural forms, larger structure sizes, and heightened seismic requirements [1]. Consequently, an increasing number of buildings are now subjected to heightened torque, particularly in structures such as spiral staircases and oblique bridges, making the role of torsion more prominent [2]. However, structural design has traditionally treated torque as a secondary factor compared to shear force, bending moments, or axial force, often employing conservative calculations and structural measures to address it. Concrete members experience physical and chemical changes in the natural environment over their service life, leading to reduced strength and compromised structural function [3,4]. Several methodologies are available for reinforcing existing structures, broadly classified as material enhancement [5], member strengthening [6], and structural system optimization [7]. Within the category of material enhancement, fiber-reinforced polymer (FRP) composites have demonstrated considerable effectiveness in reinforcing RC members. In particular, CFRP offers significant advantages for strengthening RC structural elements, owing to its lightweight properties, resistance to corrosion, ease of installation, and minimal space requirement [5,8]. Moreover, the potential for sustainable recyclability presents further environmental benefits: reprocessed fibers have been shown to retain over 83% of the tensile strength of virgin fibers (2866/3450 MPa), while material reuse processes can reduce global warming potential by 38.47% [9]. The complex stress patterns in CFRP-reinforced components necessitate an in-depth study of their mechanical properties and structural effects [10].

Currently, extensive research has been conducted on the enhancement effects of CFRP sheets on the flexural [11], shear [12], and compressive [13] strengths of RC beams. Nevertheless, research investigating the torsion behavior of CFRP-strengthened RC beams is still scarce. Empirical evidence confirms that CFRP sheets substantially enhance the torsional resistance of RC members [5,14]. Askandar et al. [15] investigated eight RC beams under combined bending and torsion pressures, emphasizing reinforcement methods and fiber ratios. Their findings revealed that CFRP-reinforced RC beams exhibit enhanced torsional resistance and ductility, with optimal reinforcement effects when CFRP sheets are applied in a circular direction. Ban et al. [16] conducted an in-depth study on the impact of CFRP sheets on the torsional performance of RC beams under varying torsion-bending ratios and pure torsion. The results indicated that the presence of bending moments diminishes the torsional capacity of beams, whereas CFRP sheets significantly enhance this capacity. Notably, the reinforcement effect of CFRP sheets increases with higher torque-to-moment ratios. Specifically, at a torsion-bending ratio of 0.75, the ultimate torque increased by 69.6%, at a torsion-bending ratio of 1.38, it increased by 108.5%, and for beams under pure torsion, the ultimate torque increased by 150%. Although Ban et al. considered the influence of different torsion-bending ratios on CFRP reinforcement, the study only selected two sets of combined loading conditions, which does not allow for determining the trend of how the torsion-bending ratio affects the torsional strength of CFRP-reinforced RC beams. The contribution of CFRP toward boosting RC beams’ resistance to torsion is primarily determined by the CFRP quantity used, which is conventionally quantified as the fiber ratio. An increase in the CFRP reinforcement ratio enhances the torsional capacity of RC beams, albeit with a progressive reduction in this enhancement effect [11]. Current theoretical models and design approaches addressing the torsional behavior of CFRP-strengthened RC beams subjected to simultaneous bending and torsion remain insufficiently established and need additional investigation.

The experimental work of Bažant et al. [17] on plain concrete revealed a reduction in nominal torsional strength as beam height increased. Subsequent investigations have corroborated these size effects under combined loading conditions. Lei et al. [18] experimentally confirmed significant size effects in basalt fiber-reinforced polymer (BFRP)-reinforced concrete beams subjected to bending–shear–torsion. They observed a negligible influence of shear-span ratio, but a non-monotonic effect of the torsion-bending ratio, with a peak effect occurring at 0.6. Choi et al. [19], through experimentation on single-crystal copper beams, observed a diminishing size effect as the loading regime transitioned from pure bending to dominant torsion. Within the scope of their study, the size effect decreased from 40% to 15% with increasing torsion-bending ratios. Concurrently, Li et al. [20] demonstrated, via numerical simulations, that CFRP-strengthened RC beams exhibit analogous size effects under combined loads, similarly identifying a peak torsion-bending ratio effect at 0.6. However, a key limitation in the existing body of knowledge is the reliance on small-scale specimens in many torsional studies of RC beams, often dictated by testing facility limitations and funding constraints. These small-scale specimens may not adequately represent the behavior of full-scale structures encountered in real-world engineering applications. Furthermore, research on size effects specifically under combined bending and torsion pressures remains relatively sparse, warranting further investigation.

In real engineering, RC beams are usually in the bending and torsional composite stress state. Jin et al. [21] conducted numerical simulations on the shear-torsional failure of RC columns, showing that torsion-bending ratio affects size effect behavior. However, research on the torsional size effect behavior under compound forces is limited. Most studies have fixed the torsion-bending ratio, leaving its influence on the mechanical behavior and size effect of CFRP-reinforced RC beams underexplored. This paper addresses this gap by establishing a meso-scale numerical model for CFRP-reinforced RC beams. This research investigates how fiber ratio and torsion-bending ratio impact torsion characteristics and size effects, establishing a modified Torque Size Effect Law (SEL) derived from Jin et al. [22], which more accurately quantifies the characteristic torsional capacity of CFRP-reinforced RC beams, accounting for both parameters.

2. Meso-Scale Mechanical Model of RC Beams Reinforced by CFRP Sheets

2.1. Model Building

Figure 1 depicts the establishment process of the meso-scale mechanical model and associated loading scheme for CFRP-sheet-reinforced RC beams.

2.1.1. Meso-Scale Mechanical Model of Plain Concrete Beams

The heterogeneity of concrete materials can be effectively reflected using a meso-scale mechanics analysis model. Referring to the research work of Rios [23] and Jin et al. [24], concrete is considered a three-phase composite material composed of aggregate particles, mortar matrix, and interfacial transition zones (ITZs), as shown in Figure 1a. Although the actual aggregate particle shape is usually irregular, numerical analysis results indicate that the influence of aggregate shape on the concrete stress–strain curve can be ignored [25]. Therefore, coarse aggregate particles are temporarily regarded as standard spheres in this study [26]. Two-gradation concrete is used, with the minimum and maximum equivalent particle sizes of coarse aggregate being 12 mm and 30 mm, respectively [27], and the crude aggregate volume fraction is 40% [28]. Using the Monte Carlo method, coarse aggregate particles were randomly placed into the mortar matrix with Fortran programming [29]. The thin layer enveloping the coarse aggregate is designated as the ITZ. It is found that, when ITZ thickness varies from 0.1 to 2.0 mm, ITZ thickness has little effect on concrete strength and only has a limited impact on softening behavior [30]. Therefore, for computational efficiency, the ITZ thickness is set at 2.0 mm in this study [31]. The finite element grid is discretized across the three-dimensional meso-scale computational framework for plain concrete beams. The element type is determined according to the location of each phase component in the mesh (such as aggregate element, mortar matrix element, and interface element), and the corresponding material characteristics are assigned according to cell type.

2.1.2. Assembly and Meshing

Utilizing the established computational meso-scale three-dimensional model of unreinforced concrete beams, the steel cage is positioned at designated locations, followed by the application of CFRP sheets to the RC beam surfaces. Finally, a three-dimensional meso-scale computational framework for CFRP-reinforced RC beams is formulated. As shown in Figure 1a, each meso-scale concrete component is discretized by an eight-node reduced-integrity solid unit C3D8R, the steel bar is discretized by truss elements, and the CFRP sheet is discretized by shell elements.

2.1.3. Boundary Conditions and Loading Schemes

Figure 1b shows the loading scheme of the model under combined bending and torsion (i.e., both bending moment and torque exist). The static load is applied via displacement control. The established model employs circular arc supports, mirroring the configuration used in Experiment [14]. This design ensures free rotation at the beam ends, effectively precluding any end constraints from influencing the torque distribution within the combined bending-torsion zone of the beam. The torque-bending ratio of RC specimens is modulated by relocating the steel transfer beam (i.e., applied torque). Contact surfaces between the steel torsional arm and the test beam are bound to ensure that the torque applied to the steel torsional arm can be transmitted to the test beam.

2.2. Material Constitutive

The RC beam reinforced by CFRP sheets is composed of CFRP sheets, steel bar, and concrete. The mechanical properties of these materials directly affect the overall mechanical properties of the member.

2.2.1. Concrete Section

Concrete is a three-phase composite material consisting of aggregate particles, mortar matrix, and ITZ. The strength of aggregate particles is usually higher than that of other meso-scale structures, making them difficult to destroy. It is assumed that no damage occurs to aggregate particles under loadings; thus, the aggregate is modeled as an elastic body [32,33]. Concrete is susceptible to irreversible plastic deformation under external loading. The plastic damage constitutive model established by Lubliner et al. [34] and enhanced by Lee and Fenves [35] accurately characterizes this behavior. Therefore, the plastic damage constitutive model is usually used to describe the mechanical properties of concrete materials [36]. The mortar matrix has mechanical properties similar to concrete [37,38], and ITZ is generally considered a weakened mortar matrix [39]. Therefore, the plastic damage constitutive model can also be adopted to describe the mechanical behavior of the mortar matrix and ITZ [30]. The plastic damage constitutive model posits that material failure encompasses both compressive failure and tensile failure. The compressive and tensile stress–strain behavior of concrete is depicted in Figure 2, and the stress–strain relationship formula is presented below.

$$\sigma =(1-D)E_0^{\mathrm{e}\mathrm{l}}:(\epsilon -{\epsilon }^{\mathrm{p}\mathrm{l}})$$

(1)

where $\sigma$ is the stress; D represents a quantifiable variable that reflects the extent of elastic stiffness degradation in a material; $E_0^{\mathrm{e}\mathrm{l}}$ is the linear elastic stiffness of the material without damage; $\epsilon$ is the strain tensor; and ${\epsilon }^{\mathrm{p}\mathrm{l}}$ is the plastic strain tensor.

The damage variable is designated as $D_{\mathrm{t}}$ in tensile and $D_{\mathrm{c}}$ in compression, with 0 signifying undamaged material and 1 denoting complete failure. As described in the Lee and Fenves model [35], the tensile damage variable $D_{\mathrm{t}}$ can be calculated using parameters derived from uniaxial tensile tests. Conversely, the compressive damage variable $D_{\mathrm{c}}$ can be determined using parameters obtained from uniaxial compression tests.

The stress–strain relationship under tensile/compressive loading is formulated as

$${\sigma }_{\mathrm{t}}=(1-{{D_{\mathrm{t}})E}_0(\epsilon }_{\mathrm{t}}-{\epsilon }_{\mathrm{t}}^{\mathrm{p}\mathrm{l}})$$

(2)

$${\sigma }_{\mathrm{c}}=(1-{{D_{\mathrm{c}})E}_0(\epsilon }_{\mathrm{c}}-{\epsilon }_{\mathrm{c}}^{\mathrm{p}\mathrm{l}})$$

(3)

The yield function expression of this constitutive model is

$$F={\displaystyle \frac{1}{1-\alpha }}(\overline{q}-3\alpha \overline{p}+\beta ({\tilde{\epsilon }}^{\mathrm{p}\mathrm{l}})〈{\overline{\sigma }}_{\mathrm{m}\mathrm{a}\mathrm{x}}〉-\gamma 〈{\overline{\sigma }}_{\mathrm{m}\mathrm{a}\mathrm{x}}〉)-{\overline{\sigma }}_{\mathrm{c}}({\tilde{\epsilon }}^{\mathrm{p}\mathrm{l}})$$

(4)

where the subscripts t and c represent the tensile and compression conditions, respectively and ${\overline{\sigma }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum principal effective stress; the detailed expressions for parameters $\alpha$, $\beta ({\tilde{\epsilon }}^{\mathrm{p}\mathrm{l}})$, and $\gamma$ can be found in [40]. σ_cu and σ_t0 are uniaxial compression strength and uniaxial tensile strength, respectively; σ_c0 is the elastic limit stress under pressure, $\overline{q}$ is the Mises equivalent of effective stress, $\overline{p}$ is hydrostatic stress, ${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{c}\mathrm{k}}$ and ${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{i}\mathrm{n}}$ represent the inelastic strain, ${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}$, ${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{p}\mathrm{l}}$ and ${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{e}\mathrm{l}}$, ${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{e}\mathrm{l}}$ signify the equivalent plastic strain and equivalent elastic strain during the recovery phase, respectively, and ${\tilde{\epsilon }}_{0\mathrm{c}}^{\mathrm{e}\mathrm{l}}$ and ${\tilde{\epsilon }}_{0\mathrm{t}}^{\mathrm{e}\mathrm{l}}$ denotes the equivalent elastic strain. The plastic damage constitutive model can be found in the literature [40,41].

The mechanical parameters of the mortar matrix can be directly obtained through relevant test data. However, due to the lack of a unified method for measuring the mechanical parameters of the ITZ, obtaining these parameters is challenging. In this study, ITZ-related mechanical parameters were determined using a repeated trial algorithm [37]. Initially, different reduction values (70–85%) based on mortar matrix mechanical parameters were selected as trial values for ITZ parameters. Subsequently, a numerical test of uniaxial compression failure of a concrete cube with a side length of 150 mm was conducted. By continuously refining the ITZ mechanical parameters (including compressive strength, tensile strength, and elastic modulus), the parameters were considered reasonable when the simulated uniaxial compressive strength and curve of concrete matched the test results.

It is important to note that the lack of characteristic length in the plastic damage constitutive model can lead to significant mesh sensitivity in numerical calculations, impacting the simulation results. To mitigate this issue, combined with the fracture energy cracking criterion, the stress–displacement curve replaces the stress–strain curve. This substitution ensures that the energy required for unit failure remains unique.

2.2.2. Steel Bars

Steel, being an isotropic material, can be described by an ideal elastic-plastic constitutive model [42]. The stress–strain relationship is depicted in Figure 3a, and its expression is

$${\sigma }_{\mathrm{s}}=\left\{\begin{array}{c}{E}_{\mathrm{s}}\times {\epsilon }_{\mathrm{s}} {\epsilon }_{\mathrm{s}}<{\epsilon }_{\mathrm{y}}\\ f_{\mathrm{y}} {\epsilon }_{\mathrm{y}}\le {\epsilon }_{\mathrm{s}}\le {\epsilon }_{\mathrm{s}\mathrm{u}}\end{array}\right.$$

(5)

where ε_s, ε_y, and ε_su represent the strain, yield strain, and ultimate strain of the steel bar, respectively. σ_s is the stress of the steel bar, f_y is the yield strength, and E_s is the elastic modulus.

2.2.3. CFRP Sheets

The elastic behavior of CFRP sheets can be simulated using the “Lamina” material model. According to Obaidat et al. [43], the stress–strain relationship of CFRP sheets is nearly ideally elastic until peak strain is reached. Therefore, CFRP sheets can be modeled as ideal linear elastic materials, where the longitudinal elastic modulus E₁ is provided by the manufacturer, and the transverse modulus of elasticity E₂ and shear modulus are small percentages of the elastic behavior in the parallel fiber direction. Both longitudinal and transverse compressive stresses are small percentages of longitudinal tensile stress [44]. After reaching peak strain, CFRP sheets exhibit elastic-brittle failure characteristics, which can be simulated using the Hashin damage criterion [45]; the stress–strain relationship is shown in Figure 3b. The Hashin failure criterion [45] incorporates four distinct damage mechanisms: tensile and compressive fiber failure, along with stretching and crushing at fiber sheet–matrix interfaces. The corresponding equations for these failure modes are

$$F_{\mathrm{f}}^{\mathrm{t}}={({\displaystyle \frac{{\sigma }_{11}}{X^{\mathrm{T}}}})}^2+\alpha {({\displaystyle \frac{{\tau }_{12}}{S^{\mathrm{L}}}})}^2$$

(6)

$${F_{\mathrm{f}}^{\mathrm{c}}=({\displaystyle \frac{{\sigma }_{11}}{X^{\mathrm{C}}}})}^2$$

(7)

$$F_{\mathrm{m}}^{\mathrm{t}}={({\displaystyle \frac{{\sigma }_{22}}{Y^{\mathrm{T}}}})}^2+\alpha {({\displaystyle \frac{{\tau }_{12}}{S^{\mathrm{L}}}})}^2$$

(8)

$$F_{\mathrm{m}}^{\mathrm{c}}={({\displaystyle \frac{{\sigma }_{22}}{2S^{\mathrm{T}}}})}^2+[{({\displaystyle \frac{Y^{\mathrm{C}}}{2S^{\mathrm{T}}}})}^2-1]{\displaystyle \frac{{\sigma }_{22}}{Y^{\mathrm{C}}}}+{({\displaystyle \frac{{\tau }_{12}}{S^{\mathrm{L}}}})}^2$$

(9)

where X^T and Y^T represent the longitudinal and transverse tensile strength, respectively, X^C and Y^C are defined as longitudinal and transverse compressive strength, respectively, and S^L and S^T correspond to longitudinal and transverse shear strength. Additionally, σ signifies effective stress, with the coefficient α characterizing the contribution of shear stress to fiber tensile fracture.

2.3. Interaction Between Materials

The mechanical properties of RC beams reinforced by CFRP sheets are influenced not only by the mechanical properties of their components, but also by the interactions between steel bars and concrete and between CFRP sheets and concrete.

2.3.1. Interaction Between Steel Bars and Concrete

The τ-s relationship specified in GB50010-2010 [46] governs the steel–concrete interaction. Interface behavior is simulated using nonlinear spring elements connecting reinforcement and concrete, with the bond-slip curve defining this mechanism illustrated in Figure 4a. The mathematical formulation follows:

Linear segment:

$$\tau =k_1 s_0\le s\le s_{\mathrm{c}\mathrm{r}}$$

(10)

Split segment:

$$\tau ={\tau }_{\mathrm{c}\mathrm{r}}{+k}_2(s{-s}_{\mathrm{c}\mathrm{r}}) s_{\mathrm{c}\mathrm{r}}\le s\le s_{\mathrm{a}}$$

(11)

Falling segment:

$$\tau ={\tau }_{\mathrm{a}}{+k}_3(s{-s}_{\mathrm{u}}) s_{\mathrm{a}}\le s\le s_{\mathrm{r}}$$

(12)

Remainder:

$$\tau =s s>s_{\mathrm{r}}$$

(13)

where k₁ denotes the linear slope, k₂ characterizes the split-section slope, and k₃ signifies the falling slope; τ corresponds to the bonding force at the concrete–steel bar interface, with s reflecting the relative slip between concrete and steel. The characteristic point parameters are given in

Table 1. Characteristic point parameters of the bond-slip model.

Characteristic Point	Crack (cr)	Peak Value (u)	Remnant (r)
Bond stress τ (MPa)	τcr = 2.5ft	τu = 3ft	τr = ft
Relative slip s (mm)	scr,l = 0.025d	su,l = 0.04d	sr,l = 0.55d

Note: d refers to the diameter of the steel bar; f_t stands for the tensile strength of concrete.

.

2.3.2. Interaction Between CFRP Sheets and Concrete

Numerous studies [47,48] have found that stripping failure of CFRP sheets is one of the most common failure modes of RC beams reinforced with CFRP sheets under load. In this paper, the bilinear bond-slip model proposed by Lu et al. [47] is adopted to reflect the stripping behavior of CFRP sheets. The stress-slip relationship is shown in Figure 4b, and its specific expression is

$$\tau =\left\{\begin{array}{c}{\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}({\displaystyle \frac{s}{s_0}}) 0\le s\le s_0\\ {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}({\displaystyle \frac{s_{\mathrm{u}}-s}{s_{\mathrm{u}}-s_0}}) s_0<s<s_{\mathrm{u}}\\ 0 s>s_{\mathrm{u}}\end{array}\right.$$

(14)

where τ_max is the maximum stress of separation and S₀ and S_u are the initial slip and the maximum bond slip, respectively. The specific parameters of the model are shown in

Table 2. The interface parameters of the bonding-slip model.

K0 (MPa/mm)	τmax (MPa)	S0 (mm)	K (MPa/mm)	Su (mm)	Gf (N/mm)
70	3.500	0.050	23.330	0.200	0.350

. K₀ is the initial stiffness, K is softening stiffness, and G_f is the energy required for stripping to occur.

3. Verification of the Meso-Scale Mechanical Model

3.1. RC Beam Under Combined Bending and Torsion Pressures

To verify the rationality of the RC beam under combined bending and torsion failure model, Ilkhani et al.’s [14] test beam C2 (200 mm × 300 mm × 2300 mm) was selected. A three-dimensional meso-scale numerical analysis model of the RC beam was established, and the combined bending and torsion failure of the RC beam was simulated numerically. The parameter setting and configuration method of the steel bar in the model were consistent with those of test beam C2. An A3 steel bar with a diameter of 20 mm was selected for the longitudinal steel bar, and an A2 steel bar with a diameter of 10 mm was selected for the stirrup.

The concrete mechanical parameters of the steel bar are shown in

Table 3. Mechanical parameters of the meso-scale components of concrete and steel bars.

Mechanical Property	Mortar	ITZ	Aggregate	Longitudinal Steel Bar	Stirrup
Elastic modulus E (GPa)	* 30.3	^ 24.2	# 70	* 174.8	* 165.5
Poisson’s ratio ν	0.2	0.2	0.2	0.3	0.3
Expansion angle ψ (°)	18	15
Fracture energy Gc (J/m2)	50	30
Compressive strength σc (MPa)	* 41.0	^ 30.8
Tensile strength σt (MPa)	* 4.1	^ 3.1
Yield strength fy (MPa)				* 390	* 318

Note: “*” is the data measured in test [14], “^” is the repeated trial parameter, “#” is a direct reference to test data from [35]. Default values were retained for other parameters, and fracture energy parameters are derived from reference [49].

, and other parameters are shown in the literature [14]. The particular mechanical parameters of the three-phase meso-scale concrete components employed in the model are presented in Table 3, with the acquisition methodology described in Section 2.2.1.

The iterative simulation outcomes for standard cubes featuring three mesh sizes (1 mm, 2 mm, and 4 mm) under quasi-static uniaxial compression are illustrated in Figure 5a. Upon implementing the three-phase concrete component mechanical parameters specified in Table 3, the resulting uniaxial compressive strength conforms to the experimental value (41.0 MPa). Additionally, RC beam models incorporating these mesh dimensions are analyzed and validated. Figure 5c demonstrates the comparative failure patterns between numerical simulations and experimental observations for specimen C2. The inclination, morphology, and distribution of the simulated diagonal fracture exhibit fundamental alignment with the test measurements. Correspondingly, Figure 5b presents the load-displacement curve correlation, where both profile characteristics and ultimate load magnitude demonstrate substantial concordance with empirical results.

Consequently, the meso-scale computational approach developed herein proves efficacious for RC beam assessment. Minimal variations emerge among failure mechanisms and stress–strain responses across mesh resolutions, signifying that substituting stress–strain curves with stress–displacement counterparts effectively attenuates mesh sensitivity constraints. To balance calculation accuracy and efficiency, the mesh size for the subsequent model (with a section height of 300 mm) was set at 2.0 mm, and the mesh size was scaled accordingly with the section height.

3.2. Rationality of the Simulation Method for CFRP Sheet Reinforcement

While the most standard applications of the Hashin criterion primarily focus on in-plane loading conditions, this section will explore its applicability under combined bending and torsional loads through experimental validation. This investigation will also assess the reliability of the interaction between the concrete substrate and CFRP sheets. The C2-S specimen (200 mm × 300 mm × 2300 mm) with CFRP reinforcement from Ilkhani et al.’s experiments [14] was adopted for simulation verification, with reference to Section 2.1. The specimen was reinforced with a CFRP sheet, with a thickness ($t_{\mathrm{f}}$) of 0.13 mm, a width ($W_{\mathrm{f}}$) of 100 mm, and a spacing (S_f, center to center) of 200 mm. The model utilizes identical mechanical properties for steel reinforcement and concrete meso-scale components as those specified in Section 2.1 (Table 3). The elastic parameters of the CFRP sheet and the damage variable parameters of the Hashin criterion are provided in

Table 4. Elastic mechanical parameters of CFRP sheet.

Parameter	Symbol	Value
Elastic modulus in the fiber direction (GPa)	E1	* 230
Elastic modulus in the transverse direction (GPa)	E2	# 2.227
Longitudinal-transverse Poisson’s ratio	ν12	# 0.3
Shear moduli (GPa)	G12, G13, G23	# 1.127
Tensile strength (MPa)	σt	* 4900
Thickness (mm)	tf	* 0.13

Note: “*” is the data measured in test [14], and “#” is a direct reference to test data from [44].

and

Table 5. Damage-assigned strength parameters in Hashin’s model.

Strength Characteristics	Value	Damage Characteristics	Value
Longitudinal tensile strength (MPa)	# 1188	Longitudinal tensile fracture energy (mJ/mm2)	# 92
Longitudinal compressive strength (MPa)	# 3.96	Longitudinal compression fracture energy (mJ/mm2)	# 1.1
Transverse tensile strength (MPa)	# 3.96	Transverse tensile fracture energy (mJ/mm2)	# 1.1
Transverse compressive strength (MPa)	# 3.96	Transverse compression fracture energy (mJ/mm2)	# 0.2
Longitudinal shear strength (MPa)	# 3.96
Transverse shear strength (MPa)	# 3.96

Note: “#” is a direct reference to test data from [44].

, respectively.

As shown in Figure 5, the angle, shape, and location of inclined cracks simulated by specimen B2U6.5 are in good agreement with the test results (see Figure 5d), and the load-displacement curve (see Figure 5b) is consistent with the experimental data. The correctness and applicability of the simulation method for RC beam reinforcement with CFRP sheets under flexural and torsional failure modes have been verified, making it suitable for subsequent simulation work.

4. Numerical Test Results

Based on the meso-scale mechanical model of RC beams reinforced with CFRP sheets established above, 48 meso-scale numerical analysis models were generated, varying in torsion-bending ratios, fiber ratios, and sizes. Specifically, RC beams were categorized into three groups (S, M, L) based on their structural sizes, with cross-section heights of 300 mm, 600 mm, and 900 mm, respectively. Each group of specimens included four torsion-bending ratios (0, 0.2, 0.4, and 0.8) and four fiber ratios (0%, 0.17%, 0.35%, and 0.69%). The shear span ratio of the beams was 2.0, and the longitudinal reinforcement ratio was 1.0%. The specific parameters of the specimens are detailed in

Table 6. Physical parameters of the RC beams reinforced by CFRP sheets.

Specimen Name	Effective Section Height h0 (mm)	Number of CFRP Layers n	CFRP Strip Width Wf (mm)	CFRP Strip Spacing Sf (mm)	Fiber Ratio ρf
S-0-0	270	0	0	200	0%
S-I-0	270	1	100	200	0.17%
S-II-0	270	2	100	200	0.35%
S-IV-0	270	4	100	200	0.69%
S-0-0.2	270	0	0	200	0%
S-I-0.2	270	1	100	200	0.17%
S-II-0.2	270	2	100	200	0.35%
S-IV-0.2	270	4	100	200	0.69%
S-0-0.4	270	0	0	200	0%
S-I-0.4	270	1	100	200	0.17%
S-II-0.4	270	2	100	200	0.35%
S-IV-0.4	270	4	100	200	0.69%
S-0-0.8	270	0	0	200	0%
S-I-0.8	270	1	100	200	0.17%
S-II-0.8	270	2	100	200	0.35%
S-IV-0.8	270	4	100	200	0.69%
M-0-0	540	0	0	400	0%
M-I-0	540	1	200	400	0.17%
M-II-0	540	2	200	400	0.35%
M-IV-0	540	4	200	400	0.69%
M-0-0.2	540	0	0	400	0%
M-I-0.2	540	1	200	400	0.17%
M-II-0.2	540	2	200	400	0.35%
M-IV-0.2	540	4	200	400	0.69%
M-0-0.4	540	0	0	400	0%
M-I-0.4	540	1	200	400	0.17%
M-II-0.4	540	2	200	400	0.35%
M-IV-0.4	540	4	200	400	0.69%
M-0-0.8	540	0	0	400	0%
M-I-0.8	540	1	200	400	0.17%
M-II-0.8	540	2	200	400	0.35%
M-IV-0.8	540	4	200	400	0.69%
L-0-0	810	0	0	600	0%
L-I-0	810	1	300	600	0.17%
L-II-0	810	2	300	600	0.35%
L-IV-0	810	4	300	600	0.69%
L-0-0.2	810	0	0	600	0%
L-I-0.2	810	1	300	600	0.17%
L-II-0.2	810	2	300	600	0.35%
L-IV-0.2	810	4	300	600	0.69%
L-0-0.4	810	0	0	600	0%
L-I-0.4	810	1	300	600	0.17%
L-II-0.4	810	2	300	600	0.35%
L-IV-0.4	810	4	300	600	0.69%
L-0-0.8	810	0	0	600	0%
L-I-0.8	810	1	300	600	0.17%
L-II-0.8	810	2	300	600	0.35%
L-IV-0.8	810	4	300	600	0.69%

.

The specimen naming convention is as follows: “S, M, L” indicates the size of RC beams, “0, I, II, IV” indicates the number of CFRP sheet layers, and “0, 0.2, 0.4, 0.8” indicates the different torsion-bending ratios. The calculation formulas for the fiber ratio and the torsion-bending ratio are as follows [10,16]:

$${\rho }_{\mathrm{f}}={\displaystyle \frac{n{t}_{\mathrm{f}}{W}_{\mathrm{f}}{P}_{\mathrm{f}}}{A_{\mathrm{c}}{S}_{\mathrm{f}}}}$$

(15)

$$\eta ={\displaystyle \frac{T}{M}}$$

(16)

where the parameters are defined as follows: n denotes the CFRP sheet layer count, t_f indicates the single-layer sheet thickness, W_f represents the CFRP width, P_f is the beam section perimeter, A_c signifies the beam cross-sectional area, S_f expresses the centerline-to-centerline spacing between adjacent CFRP sheets, with T being applied torque and M designating bending moment.

The numerical model established above was loaded with combined bending and torsion loads (as shown in Figure 1b). The failure mode, load-displacement curve, torque-twist curve, and CFRP sheet strain evolution with different fiber ratios and torsion-bending ratios were analyzed.

4.1. Failure Mode

Figure 6 illustrates the failure modes of RC beams reinforced with CFRP sheets under varying fiber ratios, torsion-bending ratios, and structural sizes. Figure 6 illustrates that variations in the torsion-bending ratio can substantially influence the failure mode of reinforced concrete beams. CFRP sheet reinforcement effectively constrains crack redistribution and postpones crack penetration. This observed mechanical behavior is consistent with findings documented in prior research [50]. With increasing fiber ratio, crack propagation in the RC beam distributes more widely, and damage severity within the reinforced region reduces. This is due to the increased constraint on the RC beam with a higher fiber ratio, enhancing the inhibition of crack development in the reinforced area. When final failure occurs, the RC beam shows a more concentrated crack tendency. With increasing torsion-bending ratio, cracks in RC beams exhibit amplified concentration, accompanied by a steepening reduction in the inclination angle relative to the horizontal direction. This indicates a shift in the failure mode from shear failure to pure torsional failure due to the increased torsion-bending ratio, consistent with existing test results [51].

4.2. Load-Displacement Curve

Figure 7 presents the load-displacement curves of RC beams reinforced with CFRP sheets under different fiber ratios and torsion-bending ratios.

Elevating the fiber ratio progressively raises the peak load of RC beams, yet with a diminishing enhancement rate, signifying a reduced efficacy in the CFRP sheet’s strengthening mechanism. This is because the reinforcement effect of the CFRP sheet on the RC beam becomes more pronounced with an increasing fiber ratio. Conversely, as the torsion-bending ratio (torque arm) increases, RC beams can sustain a smaller load before cracking, resulting in a decreased peak load, although the rate of this decrease diminishes.

4.3. Torque-Twist Curve

Figure 8 illustrates the torque-twist curves of RC beams strengthened with CFRP sheets under varying fiber ratios and torsion-bending ratios.

An increase in the torsion-bending ratio leads to higher peak torque in the RC beam. This phenomenon occurs because a higher torsion-bending ratio elevates the proportion of torsional failure in the RC beam, thereby increasing peak torque. These findings are consistent with previous experimental results [14]. Additionally, as the fiber ratio increases, the peak torque rises. This tendency aligns with the progression manifested in the load-displacement profile, wherein the ascent rate of peak torque decreases incrementally. Figure 8 illustrates that increasing the fiber ratio from 0.35% to 0.69% results in negligible peak torque enhancement for S-size beams, while L-size beams demonstrate a slight, yet still limited, increase. This suggests that larger beam dimensions provide marginally improved CFRP utilization. However, beyond a certain threshold of CFRP utilization, further increases in fiber thickness lead to material costs that outweigh the associated gains in load-bearing capacity. Therefore, at this inflection point, the exploration of alternative reinforcement methodologies or the implementation of innovative bonding techniques is warranted.

4.4. Strain on CFRP Sheets

Due to consistent simulation tendencies across varying scales, Figure 9 illustrates the strain profile of CFRP sheets along the beam height for the S-group RC specimens. The data reveal that the strain distribution along the beam height is non-uniform. When analyzed in conjunction with Figure 6, it becomes evident that the strain is closely related to the crack distribution in the RC beam. Specifically, the closer the CFRP sheet is to a crack, the greater the strain it experiences. This arises from the mechanism whereby crack depth progression induces amplified loading on CFRP sheets, contributing to elevated strain localization.

Analysis of CFRP strain profiles along the beam height at varying fiber ratios reveals a consistent reduction in strain magnitude corresponding to higher fiber content. For example, when $\eta$ = 0.4 and the fiber ratio increases from 0.17% to 0.69%, the peak strain decreases by about 35%. This reduction occurs because a higher fiber ratio, which corresponds to a thicker CFRP sheet, enhances the protection of RC beams at the reinforcement site, thereby reducing the average strain on the CFRP sheet. Furthermore, when comparing strain distribution patterns of CFRP sheet strain along the beam height for different torsion-bending ratios, it is found that strain levels intensify with a higher torsion-bending ratio. When ${\rho }_{\mathrm{f}}$ = 0.35%, increasing the torsion-bending ratio from 0 to 0.8 results in a 200% rise in peak strain and an upward shift in the position of maximum strain. This increase is due to the higher relative torque associated with a greater torsion-bending ratio, which exacerbates damage in the torsional section and consequently elevates strain on the CFRP sheet.

5. Size Effect Analysis

Figure 10 illustrates the correlation of nominal torsional strength with structural dimensions in RC beams. The nominal torsional strength v_u is defined through the following equation [52]:

$${\upsilon }_{\mathrm{u}}=T_{\mathrm{u}}/W_{\mathrm{t}}$$

(17)

where T_u is the peak torque; W_t is the torsional plastic resistance moment of a rectangular section, and the calculation formula [53] is $W_{\mathrm{t}}=1/6\times b^2(3h-b)$; b corresponds to the cross-sectional width of the beam; and h designates the effective height at the section.

As illustrated in Figure 10, the nominal torsional strength of the RC beam decreases with increasing structural size, indicating a discernible size effect under combined bending and torsional loads for CFRP-strengthened RC beams. Furthermore, the results demonstrate that increasing the fiber ratio enhances the nominal torsional strength and progressively mitigates the observed size effect. As the torsion-bending ratio increases, the reduction in nominal torsional strength of L-beams relative to S-beams initially becomes more pronounced before subsequently diminishing. This trend suggests that the size effect in reinforced concrete beams intensifies and then weakens with increasing torsion-bending ratio. The underlying mechanism for this phenomenon is rooted in the fundamental similarity of the size effect under pure shear and pure torsion. At low torsion-bending ratios, shear forces dominate the behavior, causing the beam’s size effect to more closely resemble that observed under pure shear conditions. As the torsion-bending ratio increases, the contributions of shear and torsion become more balanced, leading to amplification of the size effect. However, beyond a certain threshold, torsion becomes the predominant factor, and the size effect increasingly mirrors that characteristic of pure torsion. However, simply observing the declining trend of nominal torsional strength does not provide accurate rules. Therefore, the theoretical size effect law formula will be elaborated upon below.

Nowadays, the size effect law proposed by Bažant [54] based on fracture mechanics is widely accepted. This approach effectively characterizes size effects governing compressive collapse, shear fracture [55], and torsional rupture [52]. The specific formula is as follows:

$$v_{\mathrm{u}}={\displaystyle \frac{v_0}{\sqrt{1+h/d_0}}}$$

(18)

where v_u denotes the nominal torsional strength obtained from simulation results, h represents the beam height, and v₀ and d₀ are the empirical parameters.

Jin et al. [22] modified Bažant’s size effect formula, establishing a torsional size effect law (Torque SEL) targeted at rectangular RC columns. The modified formula is as follows:

$$v_{\mathrm{u}}={\displaystyle \frac{v_0-v_{\mathrm{\infty }}}{\sqrt{1+h/d_0}}}{+v}_{\mathrm{\infty }}$$

(19)

where $v_{\infty }$ denotes the torsional strength of infinitely-sized beams. Given scarce experimental data concerning torsion in large-scale RC beams, $v_{\infty }$ is set at 0.47$f_{\mathrm{t}}$, where $f_{\mathrm{t}}$ represents concrete’s tensile strength, based on the correction work of Jin et al. [22] and Kim et al. [56].

Figure 11 compares the simulation outcomes against the Torque SEL. To enhance comparative clarity, the strength criterion (horizontal line) and linear elastic fracture mechanics (LEFM) line (−1/2 slope) are added to the figure. The strength criterion corresponds to plastic materials (i.e., without size effects), while LEFM governs brittle failure modes. Figure 11 confirms Torque SEL’s superior accuracy in predicting torsional capacity of dimensionally varied RC beams, evidenced by close congruence between simulations and the predictive model.

Table 7. Parameters v₀ and d₀ for varying fiber ratios and torsion-bending ratios.

Torsion-Bending Ratio η	Fiber Ratios ρf	v0	d0
0.2	0%	12.5	12.9
	0.17%	13.0	13.2
	0.35%	13.5	13.6
	0.69%	13.9	15.2
0.4	0%	13.1	17.8
	0.17%	14.8	15.8
	0.35%	15.4	22.9
	0.69%	17.5	20.5
0.8	0%	10.7	13.0
	0.17%	13.1	13.3
	0.35%	16.0	11.8
	0.69%	18.3	7.5

lists v₀ and d₀ parameters for CFRP-reinforced concrete beams with varying fiber ratios and torsion-bending combinations. Analysis of Table 7 reveals Torque SEL’s inability to incorporate fiber ratio and torsion-bending interaction effects, as the quantitative relationship between v₀ and d₀ is irregular. Therefore, it is necessary to supplement the formula and establish an extended torque size effect law (extended Torque SEL), reflecting fiber ratio and torsion-bending ratio effects on the nominal torsional strength of RC beams reinforced with CFRP sheets.

Figure 11 highlights that fiber ratio and torsion-bending ratio profoundly govern the torsional size effect in RC beams. Based on simulated data analysis, the influences of fiber ratio and torsion-bending ratio on the mechanical properties of RC beams can be categorized into two aspects: strength and size effect.

(1)

Influence on strength:

(a) Fiber Ratio: Increased fiber ratios augment the nominal torsional strength. The enhancement coefficient is represented by ${\phi }_{{\rho }_{\mathrm{f}}}$. (b) Torsion-Bending Ratio: As the torsion-bending ratio increases, the ultimate torsional strength also increases. The enhancement coefficient is expressed by ${\phi }_{\eta }$.

(2)

Influence on size effect:

(a) Fiber Ratio: Increased fiber ratios attenuate size effects. The weakening coefficient is represented by ${\beta }_{{\rho }_{\mathrm{f}}}$. (b) Torsion-Bending Ratio: The size effect undergoes progressive intensification followed by attenuation as torsion-bending ratios escalate. The influence coefficient is expressed by ${\beta }_{\eta }$.

Therefore, by combining the Torque SEL and the above analysis of influence coefficients, an extended Torque SEL can be established. This involves multiplying the corresponding influence coefficients based on the original formula, with parameters affecting the strength of RC beams determined by fiber ratio and torsion-bending ratio, respectively. The formula is as follows:

$$v_{\mathrm{u}}=({\displaystyle \frac{v_0-v_{\mathrm{\infty }}}{\sqrt{1+h/d_0}}}{+v}_{\mathrm{\infty }}){·\phi }_{{\rho }_{\mathrm{f}}}·{{\phi }_{\eta }·\beta }_{{\rho }_{\mathrm{f}}}·{\beta }_{\eta }$$

(20)

5.1. Determination of Strength Improvement Factor ${\phi }_{{\rho }_f}$

To quantify the fiber ratio’s impact on RC member torsional performance, a fixed torsion-bending ratio of 0.2 is maintained. Additionally, S beams are selected as the research object, with the initial torsional strength derived from RC beams without CFRP sheets. Recognizing the continuous increase in nominal torsional strength with increasing fiber ratio, coupled with a diminishing rate of increase, we employed a natural logarithmic function to model the influence coefficient of the fiber ratio on nominal torsional strength. Figure 12a illustrates the results of the fitting process.

$${\phi }_{{\rho }_{\mathrm{f}}}=\left\{\begin{array}{c} 1 {\rho }_{\mathrm{f}}=0\\ Aln({\rho }_{\mathrm{f}})+B {\rho }_{\mathrm{f}}>0\end{array}\right.$$

(21)

where the parameters A and B can be obtained by fitting. The simulation data fitting results show that A is 0.045 and B is 1.4.

5.2. Determination of Strength Enhancement Coefficient ${\phi }_{\eta }$

To elucidate the enhancement degree of the torsion-bending ratio, RC beams from group S, which are not reinforced with CFRP sheets, were selected as the research object. The nominal torsional strength of RC beams with a torsion-bending ratio of 0.2 was taken as the initial reference value. Considering the continuous increase in nominal torsional strength with increasing torsion-bending ratio, but with a rapidly diminishing rate of increase, we selected an exponential function to represent the influence coefficient of the torsion-bending ratio on nominal torsional strength. Figure 12b illustrates the results of the fitting process.

$${\phi }_{\eta }=\mathrm{C}+\mathrm{D}{\mathrm{e}}^{-\mathrm{E}\eta }$$

(22)

The parameters C, D, and E, obtained by fitting, were found to be 1.2, −1.2, and 6.1, respectively.

5.3. Calculation of Size Effect Reduction Factor ${\beta }_{{\rho }_f}$

Figure 10 confirms that higher fiber ratios reduce torsional size effects in RC beams. When ${\rho }_{\mathrm{f}}=0,$ the influence of the CFRP sheet on the torsional size effect of RC beams becomes negligible. At this point, the CFRP sheet does not inhibit the torsional size effect. At this point, ${\beta }_{{\rho }_{\mathrm{f}}}=$ 1. When ${\rho }_{\mathrm{f}}>0$, as fiber content increases, the brittleness of the RC beam decreases, the size effect decreases, and the parameter ${\beta }_{{\rho }_{\mathrm{f}}}$ converges from 1 to ${\beta }_0={\displaystyle \frac{v_1}{\frac{v_0-v_{\infty }}{\sqrt{1+h/d_0}}{+v}_{\infty }}}$. The relationship between the fiber ratio and the size effect weakening coefficient is described using a hyperbolic tangent function, as shown in Figure 12c.

$${\beta }_{{\rho }_{\mathrm{f}}}=\left\{\begin{array}{c} 1 {\rho }_{\mathrm{f}}=0\\ ({\beta }_0-1)\tanh (\alpha {\rho }_{\mathrm{f}})+1 {\rho }_{\mathrm{f}}>0\end{array}\right.$$

(23)

where v₁ represents the nominal torsional strength of the RC beam with the minimum size and no CFRP sheet at a torsion-bending ratio of 0.2, with a simulation result of 4.29 MPa.

The adjustment parameter α represents the fiber ratio’s reduction of the torsional size effect. When α = 1, the theoretical nominal torsional strength aligns well with the simulation results, as depicted in Figure 13.

5.4. Calculation of Size Effect Influence Factor ${\beta }_{\eta }$

As illustrated in Figure 10, the torsional size effect observed in the RC beam exhibits a non-monotonic relationship with the torsion-bending ratio, initially increasing before subsequently decreasing. This trend aligns with the findings reported in previous studies [18]. By examining the S group beams without CFRP sheets under different torsion-bending ratios and combining Equations (21)–(24), the influence coefficient of the torsional size effect at each torsion-bending ratio can be calculated. The relationship between the torsion-bending ratio and the size effect influence coefficient can be fitted by a quadratic function, as shown in Figure 12d, which is specifically expressed as

$${\beta }_{\eta }=\mathrm{F}{\eta }^2+\mathrm{G}\eta +\mathrm{H}$$

(24)

where the parameters F, G, and H, obtained by fitting, are 0.74, −0.67, and 1.0, respectively.

5.5. Preliminary Verification Based on Existing Test Data

To verify the proposed formula, experimental data from Ilkhani [14], Askandar [15], Sahib [16], and Thomas [57] were summarized, as shown in

Table 8. Details of the test database.

Test	Section Size	η	ρf
Ilkhani [14]	200 mm × 300 mm	1.0;	0%; 1.08%
Askandar [15]	150 mm × 250 mm	1.25	0.18%; 0.24%; 0.35%; 0.47%
Sahib [16]	120 mm × 200 mm	0.75; 1.38	0%
Thomas [57]	230 mm × 300 mm	1.0	0%

. Figure 14 compares the theoretical and test values of the nominal torsional strength of RC beams. While the discrepancy between theoretical predictions derived from the extended torque size effect law formula and experimental values remains below 20%, suggesting a reasonable predictive capability for the torsional size effect in reinforced concrete RC beams with varying fiber and torsion-bending ratios, it is crucial to acknowledge the limitations of this study. This concordance indirectly supports the rationality of the proposed model and the generated data. It must be noted that the experimental investigations primarily employed relatively small specimens. Consequently, the reliability of Equation (20) has been verified solely within the investigated parameter ranges: fiber ratio of 0–1.08%, torsion-to-bending ratio of 0–1.38, and beam height of 200–900 mm. Therefore, further experimental validation is necessary to ascertain its applicability to larger structural dimensions and potentially higher fiber ratios, thus expanding the generalizability of the findings.

6. Conclusions

This paper analyzed how the fiber ratio and torsion-bending ratio affect torsional failure and size effect in CFRP-reinforced RC beams, using a meso-scale numerical model under combined bending and torsion pressures. Based on the simulation results, the Torque SEL proposed by Jin et al. [22] was improved. The main conclusions are as follows:

(1)

The application of CFRP sheets demonstrably restricts crack propagation in reinforced concrete RC beams subjected to combined bending and torsion. Specifically, for small-scale RC beams under a torsion-bending ratio of 0.4, increasing the fiber reinforcement ratio results in a 16.5% enhancement in peak torque capacity and a 17.2% increase in peak load, accompanied by a 36.1% reduction in peak strain experienced by the CFRP sheets.

(2)

In S-size RC beams reinforced with a 0.17% fiber ratio, elevating the torsion-bending ratio from 0.2 to 0.8 induces a transition in the failure mode from predominantly bending-controlled to a combined bending–torsion failure mechanism. This shift leads to more dispersed crack distribution, a 16.4% increase in peak torque, a 75.7% decrease in peak load, and a 42.8% elevation in CFRP sheet strain.

(3)

A significant size effect is present in CFRP-strengthened RC beams under combined bending–torsion loading. For beams with a torsion-bending ratio of 0.4 and no fiber reinforcement, the nominal torsional strength reduction amounts to 35.6% when the beam height increases from 300 mm to 900 mm. Increasing the fiber ratio to 0.69% partially alleviates this size effect, diminishing the nominal torsional strength degradation to 28.9%.

(4)

The size effect in RC beams exhibits a non-monotonic relationship with the torsion-bending ratio, initially intensifying and subsequently weakening as the ratio increases. For beams with a 0% fiber ratio, nominal torsional strength reductions are 32.3%, 35.6%, and 30.2% at torsion-bending ratios of 0.2, 0.4, and 0.8, respectively. With a 0.69% fiber ratio, these reductions decrease to 24.9%, 28.9%, and 24.1% under identical torsion-bending conditions.

(5)

To quantitatively investigate how the fiber ratio and torsion-bending ratio affect the torsional strength and size effect of RC beams, an extended Torque SEL, quantifying fiber ratio and torsion-bending ratio effects, was established based on the Torque SEL proposed by Jin et al.