Effects of Concrete Strength and CFRP Cloth Ratio on the Shear Performance of CFRP Cloth Strengthened RC Beams

Abstract

A three-dimensional meso-scale numerical model, which considers (1) the concrete meso-structures, (2) the bond slip between concrete internal structures and steel bars, and (3) the stripping behavior of carbon fiber reinforced polymer (CFRP) cloth from the surface of the concrete component, is established to investigate the effects of concrete strength and the CFRP cloth ratio on the shear performance of reinforced concrete (RC) beams. On the basis of verifying the rationality of the shear failure model and the feasibility of the CFRP reinforcement simulation method, 30 orthogonally designed numerical models of six kinds of concrete strength and five kinds of CFRP cloth ratios were designed and established. Based on the numerical simulation results, this paper analyzes the damage evolution of the CFRP–concrete interface, the variation trend of CFRP strain along the height direction of beam, the failure mode, and the load-displacement curve. Results show the following: (1) With the increase of concrete strength grade and CFRP cloth ratio, the shear strength of RC beams strengthened with increases in CFRP cloth ratio, the influence range of concrete strength grade is 24–50%, and the influence range of the CFRP cloth ratio is 10–25%; (2) The improvement range decreases, and the improvement range of the concrete strength grade decreases, and the reduction range is about 20%; (3) Based on the simulation results, influences of concrete strength and CFRP cloth ratio on the shear strength of CFRP cloth-strengthened RC beams are quantitatively considered.

1. Introduction

Reinforced concrete (RC) beams are widely used in structural engineering due to their high strength and durability. However, one of the most critical failure mechanisms that RC beams often encounter is shear failure. Shear failure occurs when the applied load exceeds the shear capacity of the beam, leading to cracks and potential structural collapse. Therefore, understanding and enhancing the shear strength of RC beams have been the focus of extensive research in recent years. In practical applications, improving the strength of concrete has been identified as the primary direct means to enhance the shear strength of RC beams [1,2,3]. By increasing the concrete strength grade, the ability of the beam to resist shear forces is enhanced, thereby providing a higher level of structural safety. Numerous experimental studies and simulations have been conducted to investigate the influence of concrete strength grade on the shear failure behavior of RC beams [4,5,6,7,8,9,10]. For instance, Mphonde AG et al. [8] and Smith and Vantsiotis [5] conducted separate research, both concluding that an increase in the concrete strength contributes to a corresponding improvement in the shear capacity of beams under similar conditions. Furthermore, Oh and Shin [6] investigated deep beams with varying compressive strengths of concrete, ranging from 23 MPa to 74 MPa, and obtained consistent results, illustrating the correlation between concrete strength grade and shear strength.

Besides the conventional method of reinforcing RC beams with transverse steel bars, a relatively recent method involving the application of carbon fiber reinforced polymer (CFRP) cloth on the beam’s surface has gained attention. Fiber reinforced polymer (FRP) composites have the advantages of light weight, high strength, good corrosion resistance, and excellent fatigue resistance [11,12,13]. This technique has shown promising results in enhancing the shear strength of RC beams [14,15]. Specifically, research by Carolin and Taljsten [14] and Tamer and Yousef [15] demonstrated the potential of shear reinforcement using CFRP to increase the shear capacity of RC beams. The amount of CFRP material applied, often quantitatively indicated by the CFRP cloth ratio, serves as a crucial factor directly influencing the shear capacity of the reinforced beam. Higher CFRP cloth ratios have been found to result in greater increases in the beam’s shear bearing capacity.

In the current design codes, the shear capacity V of CFRP-strengthened RC beams is mainly composed of two parts, one is the shear bearing capacity V_c provided by the concrete beam, and the other is the shear bearing capacity V_f provided by the CFRP cloth. Thus, the total shear bearing capacity V of RC beams that have been strengthened by CFRP may be written as:

$$V=V_{\mathrm{c}}+V_{\mathrm{f}}$$

(1)

Table 1. Shear design equations for CFRP-strengthened RC beams.

Specification Name	Calculation Formula	Characteristics and Comments
Chinese codesꞏGB50367-2013 [25]	$V_{\mathrm{c}}=0.7{\beta }_{\mathrm{h}}{f}_{\mathrm{t}}b{h}_0$ ${\beta }_{\mathrm{h}}={\left(\frac{800}{h_0}\right)}^{1/4}$, influence coefficient of section height: when h0 is less than 800 mm, 800 mm is taken; When h0 is greater than 2000 mm, use 2000 mm; ft is the design value of concrete tensile strength; b and h0 are the width and effective height of the calculated section respectively.	The effect of section size on shear strength is considered; the effect of concrete strength increases linearly with ft
	$V_{\mathrm{f}}=\phi \frac{f_{\mathrm{t}\mathrm{f}}{A}_{\mathrm{f}}{h}_{\mathrm{f}}}{s_{\mathrm{f}}}$ φ is the reduction coefficient of shear strength related to the anchoring method and stress condition of the strip; $f_{\mathrm{t}\mathrm{f}}$ is the design value of tensile strength of fiber composite material for shear reinforcement; Af is the cross-sectional area of CFRP plate, that is, Af = 2nfwf tf, nf is the number of layers, wf is the width of CFRP plate, and tf is the thickness of CFRP plate. sf represents the distance between CFRP plates, here is the distance between two adjacent CFRP plates. hf is the bonding height of CFRP cloth.	Vf increases linearly with the increase of the fiber ratio; $f_{\mathrm{t}\mathrm{f}}$ is multiplied by the adjustment coefficient of 0.56, multiply by the reduction coefficient φ according to the anchoring mode
American codesꞏACI440.2R [26]	$V_{\mathrm{c}}=8{\lambda }_{\mathrm{s}}\lambda {\left({\rho }_{\mathrm{w}}\right)}^{\frac{1}{3}}\sqrt{f_{\mathrm{c}}^{\prime }}{b}_{\mathrm{w}}d$ Λ is the density correction factor of concrete and 1.0 is taken for ordinary concrete; ρw is the ratio of longitudinal tensile reinforcement, which is 1% in this study; λs is the size effect correction factor, which is 1 in this study; $f_{\mathrm{c}}^{\prime }$ is the compressive strength of concrete cylinder; bw and d are the width and effective height of the calculated section, respectively.	V is calculated using the arithmetic square of the compressive strength of a concrete cylinder
	$V_{\mathrm{f}}={\Psi }_{\mathrm{f}}{A}_{\mathrm{f}}{\epsilon }_{\mathrm{f}\mathrm{e}}{E}_{\mathrm{f}}\left(\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\alpha }+\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\alpha }\right)d_{\mathrm{f}}/s_{\mathrm{f}}$ Ψf is the additional reduction factor determined by the reinforcement scheme, 0.85 for the side reinforced and U-shaped wrapped beams, 0.95 for the fully wrapped beams; Af is the cross-sectional area of the CFRP plate, that is, Af = 2nfwf tf, nf is the number of layers, wf is the width of CFRP plate, and tf is the thickness of CFRP plate. sf represents the distance between CFRP plates; εfe is the effective strain of carbon fiber cloth; Ef is the elastic modulus of the CFRP cloth; α is the angle between the direction of the fiber force and the longitudinal axis of the beam; df is the width of CFRP plate; sf represents the distance between CFRP plates.	Compared with GB50367-2013, the tensile strength of CFRP is not reduced, but the effective strain of CFRP sheet is taken in account
European codesꞏFib-TG 9.3 [27]	$V_c=C_{\mathrm{R}\mathrm{d},\mathrm{c}}k{\left(100{\rho }_{\mathrm{l}}{f}_{\mathrm{c}\mathrm{k}}\right)}^{\frac{1}{3}}{b}_{\mathrm{w}}d$ CRd,c = 0.18/γc, the concrete material component coefficient γc = 1.5 under the action of lasting and temporary loads; k = 2.0; ρl is the ratio of longitudinal tensile steel reinforcement; fck is the compressive strength of the concrete axis, unit MPa; bw and d are the width and effective height of the calculated section respectively.	The reduction coefficient CRd,c is very small, so that the calculated value of Vc is small when the reinforcement ratio is small, which is conservative
	$V_{\mathrm{f}}=0.9{\epsilon }_{\mathrm{f}\mathrm{d},\mathrm{e}}{E}_{\mathrm{f}}{\rho }_{\mathrm{f}}{b}_{w}d(\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }+\mathrm{c}\mathrm{o}\mathrm{t}\mathsf{\alpha })\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\alpha }$ ${\epsilon }_{\mathrm{f}\mathrm{d},\mathrm{e}}$ is the effective strain design value of carbon fiber; Ef is the elastic modulus of carbon fiber cloth; α is the angle formed between the force direction of the fiber and the longitudinal axis of the beam; ρf is the fiber ratio; bw and d are the width and effective height of the calculated section, respectively; θ is the inclination angle of the concrete pressure rod and 21.8° ≤ θ ≤ 45°, the corresponding 1.0 ≤ cotθ ≤ 2.5.	The effective strain is taken in account and multiplied by a reduction factor of 0.9
Japanese codesꞏ JSCE: 2001 [28]	$V_{\mathrm{c}}={\beta }_{\mathrm{d}}{\beta }_{\mathrm{p}}{f}_{\mathrm{v}\mathrm{c}\mathrm{d}}{b}_{\mathrm{w}}d/{\gamma }_{\mathrm{b}}$ $f_{\mathrm{v}\mathrm{c}\mathrm{d}}=0.2\sqrt[3]{f_{\mathrm{c}\mathrm{d}}^{\prime }}\le 0.72\left(N/m{m}^2\right);$ $f_{\mathrm{c}\mathrm{d}}^{\prime }$ is the designed compressive strength of concrete; ${\beta }_{\mathrm{d}}=\sqrt[4]{1/d}(d:m)$; ${\beta }_{\mathrm{p}}=\sqrt[3]{100{\mathsf{\rho }}_w}$; γb is the coefficient, take 1.3; bw and d are the width and effective height of the calculated section, respectively.	The design compressive strength of concrete is greatly reduced; divided by the coefficient 1.3, it makes the calculation very small
	$V_{\mathrm{f}}=K{A}_{\mathrm{f}}{f}_{\mathrm{t}\mathrm{f}}\left(\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\alpha }+\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\alpha }\right)\mathrm{z}/{\mathrm{s}}_{\mathrm{f}}{\gamma }_{\mathrm{b}}$ K is the effective coefficient of shear reinforcement, and 0.4 is adopted in this paper; Af is the cross-sectional area of the CFRP plate, that is, Af = 2nfwf tf, nf is the number of layers, wf is the width of CFRP plate, and tf is the thickness of CFRP plate. Sf represents the distance between CFRP plates; ftf is the design tensile strength of the fiber material; α is the angle between the direction of the fiber force and the longitudinal axis of the beam; z is d/1.15; sf represents the distance between CFRP plates; γb is the coefficient, take 1.25.	The formula directly brings the tensile strength of CFRP cloth into the calculation, and although multiplied by the reduction factor K, Vf is greatly estimated
Canadian codesꞏCSA S806-12 2017) [29]	$V_{\mathrm{c}}={\phi }_{\mathrm{c}}\lambda \beta \sqrt{f_{\mathrm{c}}^{\prime }}{b}_{\mathrm{w}}{d}_{\mathrm{v}}$ Φc is the strength coefficient of concrete, which is 0.65; $f_{\mathrm{c}}^{\prime }$ is the compressive strength of concrete, which is less than 64 Mpa; λ is the density correction factor of concrete, ordinary concrete takes 1.0; β is the influence coefficient of section height; bw and dv are the width and effective height of the calculated section, respectively.	Considering the concrete strength coefficient φc, Vc increases nonlinearly with ${\mathrm{f}}_c^{\prime }$
	$V_{\mathrm{f}}={\Psi }_{\mathrm{f}}{A}_{\mathrm{f}}{\epsilon }_{\mathrm{f}}{E}_{\mathrm{f}}\left(\mathrm{c}\mathrm{o}\mathrm{t}\mathsf{\theta }+\mathrm{c}\mathrm{o}\mathrm{t}{\mathsf{\alpha }}_{\mathrm{f}}\right)\mathrm{s}\mathrm{i}\mathrm{n}{\mathsf{\alpha }}_{\mathrm{f}}/s_{\mathrm{f}}$ ${\Psi }_{\mathrm{f}}$ is the reduction coefficient of FRP cloth, which is 0.65; Af is the cross-sectional area of CFRP plate, that is, Af = 2nfwf tf, nf is the number of layers, wf is the width of CFRP plate, and tf is the thickness of CFRP plate. sf represents the distance between CFRP plates; εf is the effective strain of carbon fiber cloth; Ef is the elastic modulus of carbon fiber cloth; θ is the inclination angle of the concrete pressure rod; αf is the angle between the direction of the fiber force and the longitudinal axis of the beam; sf represents the distance between CFRP plates.	Similar to the specification ACI440.2R; a smaller reduction factor is used

summarizes the calculation equations of V_f and V_c in different design codes. It can be found that the influence of the CFRP cloth ratio or concrete strength grade on the shear capacity V in the most of design codes is linear. Figure 1 summarizes the prediction trends of the shear capacity V_c and V_f with the standard value of concrete strength grade and the CFRP cloth ratio ρ_f, respectively. It can be clearly seen that the prediction trends based on design codes increase linearly with the concrete strength grade (i.e., the tensile strength) and the CFRP cloth ratio. However, this is not in line with the research findings in [8,10,16,17,18,19,20,21,22,23]. Experimental results showed that there is no linear relationship between increasing concrete strength grade (or CFRP cloth ratio) and RC beam shear bearing capacity, and with an increase in concrete strength grade (or CFRP cloth ratio), the extent of the increase gradually decreases [16,17,24]. For instance, Ma’en et al. [16] investigated the influences of concrete strength from low (f_c = 17 MPa) to high (f_c = 47 MPa) on the shear bearing capacity of rectangular beams. Results show that the increasing shear capacity is not proportional to the compressive strength of concrete. Jin et al. [17] established a three-dimensional meso-scale simulation method to study the shear performance of RC cantilever beams strengthened by side bonding of CFRP cloth. The shear capacity of beams can be efficiently increased with an increase in the CFRP cloth ratio, however the strengthening impact of CFRP fabric on the nominal shear strength of concrete steadily deteriorates.

Generally, although existing studies have proved that the shear strength of CFRP-strengthened RC beams can be significantly improved with the concrete strength grade and the CFRP cloth ratio, research on the real mechanisms of these two aspects are still limited, such as failure mode, interfacial damage law, and stress–strain relationship. In this paper, the shear performance of CFRP cloth strengthened RC beams is analyzed by using the three-dimensional meso-simulation method. Effects of the CFRP cloth ratio and the concrete strength grade on the shear strength of CFRP cloth strengthened RC beams are studied, and the contribution of CFRP cloth to the shear strength of RC beams with different concrete strength grades is analyzed by discussing the interfacial damage and the variation of CFRP strain.

2. Modeling Approach

2.1. Analytical Model

The three-dimensional meso-scale numerical model of RC beam strengthened with CFRP cloth is illustrated in Figure 2. Concrete is treated as a three-phase heterogeneous composite composed of mortar matrix, aggregate particles, and interfacial transition zones (ITZs) [30,31,32]. Coarse aggregate particles are considered to be spheres on account of the geometric properties of the three components [31,32]. Two kinds of gradation are adopted. Small gradation is considered to be an aggregate particle diameter of 12 mm, while large gradation is considered to be an aggregate particle diameter of 30 mm. Approximately 30% of the volume of concrete is made up of coarse aggregate. In fact, the volume of coarse aggregates is about 40–45%. However, according to the works of Donza et al. [33] and Meddah et al. [34], it is considered that relatively small coarse aggregate particles (about 5 mm) and fine aggregate particles can be equivalent to the mortar matrix. Here in this work, the volume of coarse aggregate was set as about 30% of the concrete volume. With the use of the Fortran program [35], the aggregate particles are randomly dispersed into the mortar matrix, and the thin layer around the coarse aggregate is set as the ITZs. In order to maintain the calculation accuracy and improve the work efficiency [36], the thickness of the ITZs is set as 2 mm [37,38,39,40]. The finite element mesh is projected onto the model, and the element type (such as aggregate element, mortar matrix element, and interface element) and material characteristics are allocated depending on the position of each phase component in the mesh. Steel bars are inserted into the plain concrete beam, and the CFRP cloth is bonded on the surface of the concrete beam [41,42,43]. A pad is attached to the beam’s bottom support and fixed at the bottom. Two reference points are placed at the loading point and coupled to the concrete surface.

2.2. Component Properties

2.2.1. Meso-Scale Compositions of Concrete

Considering the irreparable plastic distortion of concrete materials under external loads, the plastic damage model developed by Lubliner et al. [44] and improved by Lee and Fenves [45] is adopted for the description of concrete-like materials. The mechanical parameters of aggregate, mortar, and interface are described using plastic damage models. As shown in Figure 3, it is assumed that concrete failure includes compression failure and tensile failure. The general stress–strain relationship is:

$$\sigma =(1-d)D_0^{\mathrm{e}\mathrm{l}}:(\epsilon -{\epsilon }^{\mathrm{p}\mathrm{l}})$$

(2)

in which, ε is a strain tensor; ε^pl is the plastic strain tensor; $D_0^{\mathrm{e}\mathrm{l}}$ is the initial elasticity matrix; and d represents the damage variable and represents the linear elastic stiffness of the material when undamaged.

It is generally believed that damage is represented as d_t in the tensile state and d_c in the compressive state. A value of “0” indicates no damage to the specimen, and a value of “1” indicates complete failure of the specimen. Meanwhile, introducing E₀ as the initial elastic stiffness of the material, the stress–strain relationship under tensile and compressive loads can be expressed as:

$${\sigma }_{\mathrm{t}}=(1-d_{\mathrm{t}})E_0({\epsilon }_{\mathrm{t}}-{\epsilon }_{\mathrm{t}}^{\mathrm{p}\mathrm{l}})$$

(3a)

$${\mathsf{\sigma }}_{\mathrm{c}}=(1-d_{\mathrm{c}})E_0({\epsilon }_{\mathrm{c}}-{\epsilon }_{\mathrm{c}}^{\mathrm{p}\mathrm{l}})$$

(3b)

The yield function expression of the constitutive model is expressed as:

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

(3c)

in which, the subscript t and c represent tensile and compression respectively; ${\overline{\sigma }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ represents maximum principal stress effectively; σ_cu and σ_t0 represent uniaxial compression and tensile stress; σ_c0 is compressive elastic limit stress; ${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{c}\mathrm{k}}$ and ${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{i}\mathrm{n}}$ stands for inelastic strain;${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{e}\mathrm{l}}$, ${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{e}\mathrm{l}}$ and ${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{p}\mathrm{l}}$, and ${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{p}\mathrm{l}}$ represent the equivalent elastic strain and the equivalent plastic strain of recovery, respectively; and ${\tilde{\epsilon }}_{0\mathrm{c}}^{\mathrm{e}\mathrm{l}}$ and ${\tilde{\epsilon }}_{0\mathrm{t}}^{\mathrm{e}\mathrm{l}}$ represent the equivalent elastic strain.

2.2.2. Steel Bar

The stress–strain relationship of steel bar is shown in Figure 4a. The ideal constitutive model of elastic-plasticity is utilized to describe the mechanical behavior [46]. The expression of the stress–strain relationship is:

$${\mathsf{\sigma }}_{\mathrm{s}}=E_{\mathrm{s}}{\epsilon }_{\mathrm{s}}{\epsilon }_{\mathrm{s}}<{\epsilon }_{\mathrm{y}}$$

(4a)

$${\mathsf{\sigma }}_{\mathrm{s}}=f_{\mathrm{y}}{\epsilon }_{\mathrm{y}}\le {\epsilon }_{\mathrm{s}}\le {\epsilon }_{\mathrm{s}\mathrm{u}}$$

(4b)

in which, ε_s, ε_y, and ε_su represent the strain, yield strain, and ultimate strain of the steel bar, respectively; E_s represents the elastic modulus; σ_s represents the stress; and f_y represents the yield strength.

2.2.3. CFRP Cloth

Figure 4b shows the stress–strain relationship of the failure behavior of CFRP cloth. The behavior of CFRP cloth can be described by Lamina material in ABAQUS, which is consistent with that in the efforts of [47]. According to Obaidat et al. [48], before the failure of CFRP cloth, the relationship between stress and strain in CFRP material is quite similar to the ideal elasticity. In order to streamline the calculation, CFRP can be set as the ideal linear elastic material. After the CFRP cloth fails, the damage criterion of Hashin [49] is adopted to represent the failure behavior of CFRP cloth. The damage criterion of Hashin considers four possible failure modes and four corresponding indexes, which is fiber fracture and matrix cracking in tensile and fiber buckling and matrix breakage in compression [50]. The equations corresponding to the four failure modes are as follows:

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

(5a)

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

(5b)

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

(5c)

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

(5d)

in which, X^T and Y^T represent the longitudinal and transverse tensile strength, respectively; X^C and Y^C stand for the longitudinal and transverse compressive strength, respectively; S^L and S^T represent the longitudinal and transverse shear strength, respectively; σ represents effective stress; and coefficient α represents the contribution of shear stress to fiber tensile fracture.

2.2.4. Interactions

As shown in Figure 5a, referring to the “Code for Design of Concrete Structures” (GB50010-2010) [51], the bond-slip curve is used to describe the relationship between stress and bond slip between steel bars and concrete. The nonlinear spring element is placed between the steel bar element and the concrete element after the steel bar has been buried in the concrete. Here, the adhesion between reinforcing steel and concrete was modeled through the nonlinear spring elements. Spring elements are introduced in ABAQUS using Force vs. Relative Displacement curves to link the springs with mesoscale concrete. The specific expression is as follows:

$$\tau =\left\{\begin{array}{l}{k}_{1}s0\le s\le s_{\mathrm{c}\mathrm{r}}\\ {{\tau }_{\mathrm{c}\mathrm{r}}+k}_2(s-s_{\mathrm{c}\mathrm{r}})s_{\mathrm{c}\mathrm{r}}\le s\le s_{\mathrm{a}}\\ {{\tau }_{\mathrm{a}}+k}_3(s-s_{\mathrm{u}})s_{\mathrm{a}}\le s\le s_{\mathrm{r}}\\ ss>s_{\mathrm{r}}\end{array}\right.$$

(6)

where k₁, k₂, and k₃ are the linear slope, the slope of split section, and the downward slope, respectively; τ is the bonding force between steel bars and concrete; and s is for the relative slide between steel bars and concrete. The parameters of feature points are shown in

Table 2. Parameters utilized in the bond-slip model.

Key Point	Splitting Value (cr)	Peak Value (u)	Remnant Value (r)
Bonding 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 means the diameter of steel bar; and f_t means the tensile strength of concrete.

.

CFRP cloths generally present stripping failure in the experimental studies [52]. In order to meet the reality, a bilinear bond-slip model is accepted to consider the stripping behavior. Lu et al. [53]’s simplified bilinear model is adopted in the present study:

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

(7)

in which, $\tau$ and ${\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are stress and maximum stress, respectively; s, s₀, and s_u represent relative slip, initial slip, and maximum slip, respectively.

Based on this bilinear model, Yuan [54] and Lu et al. [53] proposed a theoretical formula for calculating interface bearing capacity:

$$G_{\mathrm{f}}=0.308{\beta }_{\mathrm{w}}^2\sqrt{f_{\mathrm{t}}}$$

(8)

$$P_{\mathrm{u}}={\beta }_{\mathrm{l}}{b}_{\mathrm{f}}\sqrt{{2E}_{\mathrm{f}}{t}_{\mathrm{f}}{G}_{\mathrm{f}}}$$

(9)

in which, β_w is the influence coefficient of CFRP-concrete width, take β_w = 1.0; β_l is the influence coefficient of anchorage length, according to the bond strength formula suggested by Yuan [54], the calculated value here is 1.0; b_f is the width of CFRP cloth; t_f represents the thickness of FRP material; and E_f represents the elastic modulus of FRP material.

In numerical simulation, considering the interaction between CFRP cloth and concrete, a cohesive force unit is used to simulate the adhesive layer interface, that is, the contact unit is set. The stress–slip relationship is shown in Figure 5b, and the interface parameters of the bond-slip model are collected in

Table 3. The interface parameters of bond-slip model.

Mechanical Property	C30	C40	C50	C60	C70	C80
The initial stiffness K0 (MPa/mm)	70	70	70	70	70	70
The initial slip S0 (mm)	0.050	0.050	0.050	0.050	0.050	0.050
Softening stiffness K (Mpa/mm)	23.330	23.330	23.330	23.330	23.330	23.330
The maximum bond slip Su (mm)	0.200	0.200	0.200	0.200	0.200	0.200
Tensile strength of concrete ft/Mpa	2.01	2.39	2.64	2.85	2.99	3.11
Fracture energy Gf/N/mm	0.4367	0.4762	0.5004	0.5200	0.5326	0.5432
Interfacial bearing capacity Pu/kN	7357.86	7683.38	7876.87	8029.04	8125.88	8206.21
Maximum stress τmax (Mpa)	0.04367	0.04762	0.05004	0.052	0.05326	0.05432

. The fracture energy and stripping bearing capacity are obtained from Equations (8) and (9). Among them, the tensile strength of concrete adopts the design value of “Code for Design of Concrete Structures” (GB50010-2010) [51]. The fracture energy and interface bearing capacity of concrete-CFRP increase with the increase of tensile strength. This is because the higher the strength of concrete, the better the bond performance between CFRP cloth and concrete, the greater the fracture energy required for stripping. Therefore, the bilinear model and formula mentioned above can reflect the damage law of the concrete–CFRP interface well.

2.3. Verifications

A total of 16 geometrically comparable RC beams were created to perform the four-point load shear failure test in the earlier work of Jin et al. [55]. To check the correctness of the meso-scale numerical model developed above, a group of specimens (S-0 and S-0.0835%) were used as the simulation objects in this investigation. In which, “S” refers to the size of the test beam and is 100 mm × 300 mm, and “0” and “0.0835%” refer to the CFRP cloth ratio. The three different types of microscopic concrete components and the primary mechanical properties of steel bars used in the experiments are listed in Table 4.

According to the work of [56,57], aggregate is assumed not to be destroyed during fracture. The properties of mortar and concrete are quite similar, while ITZ is considered to be a weaker mortar matrix [58]. In order to establish the correct mechanical parameters for ITZ, uniaxial compression failure tests were performed on the standard cube test specimen (150 mm side length). Based on the inversion method, it is possible to describe the mechanical characteristics of ITZ using the deteriorating mechanical properties of the mortar matrix. At last, the simulated uniaxial compressive strength of concrete under a set of parameters is basically consistent with the experimental results, indicating that the parameters are reasonable. Figure 6a shows the comparison of uniaxial compressive strength between test results and simulation results. Moreover, meso-scale models with different mesh sizes (1 mm, 2 mm, and 4 mm) were simulated and compared. It is discovered that the failure mode is not much impacted by the mesh size. This is consistent with the previous work [55]. When the three-phase meso-mechanical parameters of concrete listed in Table 4 are used, the calculated uniaxial compressive strength of concrete is 49.2 MPa, which is essentially compatible with the test value of 49.0 MPa, and it is thought that the chosen parameters are suitable given that the value obtained is in line with the test value.

Figure 6b shows the comparison of failure modes between the test results and the simulation results of specimen S-0. In this paper, three mesh sizes (1 mm, 2 mm, and 4 mm) of the model were simulated and verified. As can be observed, there is good agreement across the failure mechanisms, as the test results and the simulated fracture angle, form, and location are comparable. Figure 6d illustrates the comparison of the load-displacement curves between the simulation and test findings of specimen S-0. The predicted trend of the load-displacement curve and peak load are in good agreement with the test data. In fact, although numerical simulation can simulate experiments well, there are some differences between experimental and numerical simulation. For the post-peak behavior in Figure 6b, during the test, the RC beam will produce a large displacement at the moment of failure. At this time, the beam and the loading point are separated, so the post-peak decline trend is steeper. However, the beam and the loading point were always connected during the loading process in the simulation, so the post-peak descent section was slowed down. Therefore, post-peak behavior is inconsistent. By comparing the failure mode and load-displacement curve comprehensively, it is considered that the meso-scale simulation results are in good agreement with the existing test results [55]. In addition, the failure modes and stress–strain curves of the three mesh size models are similar. Thus, considering the calculation efficiency and accuracy comprehensively, all the mesh size of the subsequent models is set as 2 mm.

To confirm the validity of the CFRP-strengthened simulation approach. The S-0.0835% specimen (100 mm × 300 mm × 1800 mm) in the Jin et al. [55] test was further selected for verification. The CFRP cloth is wrapped in a U-shaped package, and one layer of CFRP cloth is bonded. The thickness t_f, width w_f, and spacing s_f (center to center) of the CFRP cloth are 0.167 mm, 40 mm, and 100 mm, respectively. The mechanical parameters of the concrete meso-components adopted by the model match those in Table 4. Moreover, the mechanical parameters of the CFRP cloth and Hashin damage variable parameters are, respectively, shown in

Table 5. Mechanical parameters of the CFRP cloth.

Density ρ (g/m3)	Thickness tf (mm)	Tensile Strength σt (MPa)	Elastic Modulus E (GPa)	Poisson’s Ratio ν
1.8	0.167	3820	232	0.2

and

Table 6. Strength and damage variables of the Hashin model.

Mechanical Parameters	Numerical Value
Longitudinal tensile strength (MPa)	#1188
Longitudinal compressive strength (MPa)	#3.96
Transverse tensile strength (MPa)	#3.96
Transverse compressive strength (MPa)	#3.96
Longitudinal shear strength (MPa)	#3.96
Transverse shear strength (MPa)	#3.96

Note: “#” is the data of direct reference experiment [47].

.

Table 4. Mechanical parameters of the meso-components of concrete and reinforcing bars in the numerical model for shear failure of the RC beam.

Mechanical Property	Mortar Matrix	ITZ	Aggregate
Elastic modulus E/GPa	* 33.1	#26.5	^ 70.0
Poisson’s ratio ν	* 0.2	#0.2	* 0.2
Dilatancy angle ψ (°)	18	15	18
Fracture energy Gc/J/m2	50	30
Compressive strength σc/MPa	* 49.0	#39.2	* 80
Tensile strength σt/MPa	* 2.77	#2.22	* 6

Note: The data marked with “*” were measured in the test [59], the data marked with “#” were measured in the test [55], the data marked with “^” were the parameters of the repeated test, the other parameters were default values, and the fracture energy values were quoted from [60].

Table 4. Mechanical parameters of the meso-components of concrete and reinforcing bars in the numerical model for shear failure of the RC beam.

Mechanical Property	Mortar Matrix	ITZ	Aggregate
Elastic modulus E/GPa	* 33.1	#26.5	^ 70.0
Poisson’s ratio ν	* 0.2	#0.2	* 0.2
Dilatancy angle ψ (°)	18	15	18
Fracture energy Gc/J/m2	50	30
Compressive strength σc/MPa	* 49.0	#39.2	* 80
Tensile strength σt/MPa	* 2.77	#2.22	* 6

Note: The data marked with “*” were measured in the test [59], the data marked with “#” were measured in the test [55], the data marked with “^” were the parameters of the repeated test, the other parameters were default values, and the fracture energy values were quoted from [60].

As shown in Figure 6b, the approximate position, angle, and shape of the inclined crack in the simulation results of specimen S-0.0835% are in good agreement with the test results [55]. Figure 6c compares the relationship between the CFRP strain and the shear force of CFRP-strengthened beams, and the variation trend of the strain–shear is consistent between the test results and the simulation results. The peak position and trend of the load-displacement curve (see Figure 6d) also match the test results [55], which verifies the applicability and rationality of the CFRP cloth reinforcement simulation method.

3. Results and Discussion

3.1. Test Design

The function of CFRP in enhancing the shear resistance of RC beams is to improve the stress distribution inside RC beams and either augment or replace the shear force carried by stirrups [47,61,62]. Moreover, presence of CFRP cloth can restrict the propagation rate of inclined cracks in RC beams, thereby improving their displacement ductility. In the work of Jin et al. [55], shear tests were carried out on RC beams of the same size with different CFRP cloth ratios. Taking the size L series as an example, as the CFRP cloth ratio increases from 0 to 0.2505%, the width of the main oblique crack decreases by 30–54%. It shows that the displacement ductility of RC beam is improved. This section aims primarily at investigating how fiber content affects the shear bearing capacity of CFRP-strengthened RC beams with varying concrete strength grades. Figure 7 shows the loading diagram of a CFRP-strengthened RC beam. The CFRP cloth ratio ρ_f of CFRP can be expressed as:

$${\rho }_{\mathrm{f}}=\frac{2n{w}_{\mathrm{f}}{t}_{\mathrm{f}}}{b{s}_{\mathrm{f}}}$$

(10)

in which, w_f represents the width of CFRP cloth; b represents the width of the beam section; t_f represents the thickness of CFRP cloth; and s_f represents the distance between CFRP cloths.

To study how concrete strength grade and the CFRP cloth ratio affect the shear properties of RC beams, using the aforementioned modeling approach, a total of 30 beams with six groups of various concrete strength grades were established. Each group of specimens includes CFRP-RC beams and RC beams. The concrete strength grades of the specimens are C30, C40, C50, C60, C70, and C80, respectively. The CFRP cloth ratios of the specimens are 0.0%, 0.1336%, 0.2672%, 0.4008%, and 0.5344%, respectively. Additionally, the longitudinal bars ratio was set at 1%, and the shear-span ratio for each specimen was set at 2.0.

Table 7. Physical parameters of the CFRP cloth strengthened RC beam.

Specimen Name	Effective Cross-Section Height h0 (mm)	CFRP Layers n	The Width of the CFRP wf (mm)	The Spacing of the CFRP sf (mm)	Fiber Ratio ρf
CUB-C30-0	270	0	0	100	0.0%
CUB-C30-0.1336	270	1	40	100	0.1336%
CUB-C30-0.2672	270	2	40	100	0.2672%
CUB-C30-0.4008	270	3	40	100	0.4008%
CUB-C30-0.5344	270	4	40	100	0.5344%
CUB-C40-0	270	0	0	100	0.0%
CUB-C40-0.1336	270	1	40	100	0.1336%
CUB-C40-0.2672	270	2	40	100	0.2672%
CUB-C40-0.4008	270	3	40	100	0.4008%
CUB-C40-0.5344	270	4	40	100	0.5344%
CUB-C50-0	270	0	0	100	0.0%
CUB-C50-0.1336	270	1	40	100	0.1336%
CUB-C50-0.2672	270	2	40	100	0.2672%
CUB-C50-0.4008	270	3	40	100	0.4008%
CUB-C50-0.5344	270	4	40	100	0.5344%
CUB-C60-0	270	0	0	100	0.0%
CUB-C60-0.1336	270	1	40	100	0.1336%
CUB-C60-0.2672	270	2	40	100	0.2672%
CUB-C60-0.4008	270	3	40	100	0.4008%
CUB-C60-0.5344	270	4	40	100	0.5344%
CUB-C70-0	270	0	0	100	0.0%
CUB-C70-0.1336	270	1	40	100	0.1336%
CUB-C70-0.2672	270	2	40	100	0.2672%
CUB-C70-0.4008	270	3	40	100	0.4008%
CUB-C70-0.5344	270	4	40	100	0.5344%
CUB-C80-0	270	0	0	100	0.0%
CUB-C80-0.1336	270	1	40	100	0.1336%
CUB-C80-0.2672	270	2	40	100	0.2672%
CUB-C80-0.4008	270	3	40	100	0.4008%
CUB-C80-0.5344	270	4	40	100	0.5344%

displays the geometrical properties of the specimens. The specimen naming rules are: the letter “CUB” indicates CFRP U-shaped Bonded RC beam, “C30” indicates the concrete strength grade, and “0” indicates that the CFRP cloth ratio is 0.

3.2. Shear Failure Patterns

Figure 8 illustrates the ultimate failure mode of simply supported RC beams strengthened by CFRP cloth under different concrete strength grades ranging from C30 to C80. All beams exhibit shear failure, and the main failure mode is shear and pressure failure. Additionally, it depicts the ultimate failure mode of CFRP-strengthened RC beams with varying CFRP cloth ratios (i.e., 0.1336%, 0.2672%, and 0.5344%) at a concrete strength grade of C30. Under shear load, a diagonal shear crack formed along the connection line between the loading and the support points in the shear span region of each specimen. In addition, in the pure moment zone, there are several cracks extending towards the top.

Though all specimens have shear failure, the oblique crack width and damage distribution area of the shear span section decrease gradually with the increase of the concrete strength grade. In addition, for specimens with low concrete strength grade, cracks extend more tortuous paths in the interior, because the strength of the aggregate is much higher than that of mortar, and the main failure mode at this time is intergranular failure. However, when the concrete strength grade is high, the mortar strength is close to the aggregate, the probability of transgranular failure becomes larger, and the crack is closer to a straight line. Compared with beams with lower concrete strength grade, once cracks appear in specimens with higher concrete strength grade, the failure speed is faster and the shear failure brittleness is significantly increased, which aligns with Oh’s [6] conclusion. For C30 concrete beams, as shown in Figure 8, the increase of the CFRP cloth ratio leads to more serious damage because CFRP sheets limit the development of cracks. The higher the number of layers, the slower the crack propagation. This is in line with the conclusions reached by Jin et al. [55].

3.3. Analysis of Shear Capacity of CFRP Cloth

In this section, the F3 plate and simulation results of three different CFRP cloth ratios (0.1336%, 0.2672%, and 0.5344%) are selected to analyze the interface damage and the CFRP strain, aiming to better understand the relationship between shear strength of the RC beam and the concrete strength grade, as well as the CFRP cloth ratio. An oblique crack running through the middle of the plate can be observed, with maximum CFRP strain and damage occurring near its contact with concrete.

3.3.1. Damage Analysis of the CFRP Cloth–Concrete Interface

The interface damage primarily occurs at the top and bottom of the beam, as well as at the intersection of an oblique crack and a CFRP cloth. Due to inadequate bonding between the upper and lower layers, detachment of the CFRP cloth initiates from these two points. Furthermore, the most severely damaged area is located at the intersection of the CFRP cloth and oblique crack, thus warranting a primary focus on analyzing this region.

The damage model for the interface between CFRP and concrete is illustrated in Figure 9. In the diagram, CSQUADS is utilized to denote the extent of interface damage ranging from 0 to 1, while the vertical axis represents height. From the damage mode and degree curves, it is evident that the primary components of interface damage are fundamentally identical. When the concrete strength grade is equal, a higher CFRP cloth ratio results in more severe damage. Similarly, when the CFRP cloth ratio is constant, a greater concrete strength grade leads to increased damage. This is because the higher the strength of concrete, the higher the fracture energy required for failure, the better the fit of CFRP cloth and concrete, and the less easy to peel off [53,54].

3.3.2. Strain Analysis of CFRP Cloth

For beams with various concrete strength grades, Figure 10 depicts the vertical location of the F3 fabric and the vertical strain distribution of CFRP with various CFRP cloth ratios (0.1334%, 0.2672%, and 0.5344%). The abscissa in this graph represents CFRP strain, while the ordinate represents beam height. It can be found that for beams with various concrete strength grades, the strain trend of CFRP is basically the same. These phenomena are also observed in the experiment [55].

It is found that under the same CFRP cloth ratio, with the concrete strength grade increasing, the peak strain of CFRP cloth gradually increases. This is because the higher the strength, the stronger the interaction between CFRP cloth and concrete, the better the bond, and the greater the shear capacity provided by the CFRP cloth. This aligns with the conclusion drawn in article [57]. Under the same concrete strength grade, with the CFRP cloth ratio increasing, the maximum strain in CFRP cloth decreases. This is due to the fact that the CFRP cloth ratio is altered by changing the number of layers of CFRP. When the CFRP cloth ratio increases, the cross-sectional area of CFRP cloth also increases, however, the shear bearing capacity provided by CFRP is not multiplied, thus the CFRP strain will decrease.

3.4. Load-Displacement (P-Δ) Curve

Figure 11 illustrates the load-displacement (P-Δ) curves with an increasing concrete strength grade under different CFRP cloth ratios. Compared with ordinary RC beams (CFRP cloth ratio is 0), the shear bearing capacity of RC beams increases significantly with the concrete strength grade increasing, which is in conformity with the study [7,8]. However, this increase is gradually slowing down, which indicates that the effect of concrete strength grade on shear bearing capacity cannot be predicted with a linear formula. In addition, the higher the strength of the concrete, the greater the initial stiffness, the earlier the peak bearing capacity appears, and the faster the bearing capacity declines. Under certain concrete strength grades, when the CFRP cloth ratio increases, the extent of peak shear bearing capacity increase gradually decreases, obviously. Therefore, the higher the CFRP cloth ratio is, the less intuitionistic the enhancement effect will be. Under different CFRP cloth ratios, the displacement of the RC beam corresponding to the peak load has little change. The results show that the contribution of CFRP cloth to low strength beams is the greatest, and the range of shear contribution of CFRP cloth decreases with the increase of beam strength. This is also observed in the article [17].

4. A Shear Law Considering Both Concrete Strength Grade and CFRP Ratio

The above simulation results illustrate that the shear bearing capacity of beams is significantly improved by the ratio of CFRP and the strength grade of concrete. In this section, based on the two influencing factors of the CFRP cloth ratio and concrete strength grade, a new shear strength formula is proposed, which is able to quantitatively describe the effects of the CFRP cloth ratio and concrete strength grade on the shear strength of CFRP cloth strengthened RC beams.

The nominal shear strength τ_u is firstly utilized to analyze the shear performance of CFRP cloth strengthened RC beams. τ_u is defined as [63]:

$${\tau }_{\mathrm{u}}=\frac{P_{\mathrm{u}}}{b{h}_0}$$

(11)

in which, P_u is the peak load, b is the width of the beam, and h₀ is the effective section height of the beam.

4.1. Shear Contribution Analysis

Figure 12a presents that the variation trend of shear strength of CFRP cloth strengthened RC beams shear with tensile strength under five kinds of CFRP cloth ratios. Taking RC beams with the strength grade of C30 as the control group, the shear capacity increases by 25–50% as the strength grade of concrete increases from C30 to C80. When the concrete strength increases from C30 to C40, the nominal shear strength increases by 24.6%, and when the concrete strength increases from C70 to C80, the nominal shear strength increases by 0.96%. Therefore, the improvement of concrete strength grade against shear capacity is non-linear. It can be seen that under the same CFRP cloth ratio, the shear bearing capacity of RC beams increases with the increasing tensile strength, and the increased amplitude decreases somewhat, which aligns with the conclusion drawn by Yu [1]. For beams with the same strength of concrete, the increase of the CFRP cloth ratio also significantly improves the shear strength. Also taking RC beams with strength grade C30 as the control group, the shear bearing capacity of strengthened beams increased by 9.66%, 16.34%, 20.67%, and 23.13%, respectively, as the CFRP cloth ratio increased from 0.1334% to 0.5344%. This illustrated that the strengthening of CFRP enhances the shear bearing capacity of beams. With the increase of the strength of the concrete, the increased range of shear resistance provided by CFRP was decreased, as shown in Figure 12b. Nevertheless, the weakening of this enhancement is not obvious. In fact, the effects of CFRP are similar to those of stirrup. Thus, the conclusions drawn in this study are in line with the conclusions reached by Liu et al. [64].

According to the research shown above, the strength of the concrete and the CFRP cloth ratio have significant impacts on the ability of simply supported beams to bear shear. Some national codes [25,26,27,28,29,65] are used to calculate the shear strength of concrete under different strengths, as shown in Figure 13a. It is clear that plain concrete beams have shear strengths that are around two times greater than those predicted by the code, which is consistent with the findings of various testing [9,66]. This illustrates that the calculation formula of shear strength given in the code has too much safety reserve, resulting in the waste of concrete materials. In addition, national codes and the conclusions of many scholars [6,7,8,9,10] are mostly linear; however, the results of simulation are shown to be nonlinear.

The shear capacity of CFRP-strengthened C80 concrete beams is compared to the specification in Figure 13b. As can be observed, the curve produced by the codes exhibits linear development as the CFRP cloth ratio rises. Additionally, there is an obvious difference among these codes. Some of them overestimated the contribution of CFRP cloth, such as JSCE: 2001 [28], while others were overly conservative [26,27,29]. This shows that the understanding of RC beams strengthened by CFRP cloth is not clear in different countries, and there are great differences. Compared with Chinese codes GB500367-2013 [25], it can be seen that it is similar to the results of the ABAQUS simulation. Meanwhile, the simulation results are nonlinear. Therefore, we propose revisions to the standard formula in the following form:

$${\tau }_{\mathrm{u}}={\tau }_{\mathrm{c}}{(\phi }_c+{\phi }_{\mathrm{f}})$$

(12)

in the equation, φ_c and φ_f are the influence coefficients of concrete strength grade and CFRP cloth ratio, respectively. Below, we will analyze and determine the specific form.

4.2. Determination of Correlation Coefficient

4.2.1. Determination of φ_c

Figure 14a shows the curve of influence coefficient φ_c with concrete strength grade obtained by fitting multiple simulation results. In this study, the nominal shear strength of a plain concrete beam with a concrete strength grade of C30 (τ_c = 3.636 MPa) was taken as the control group. The other plain concrete beams with concrete strength grades from C40 to C80 are divided by the shear strength of the control group. Thus, the strength enhancement coefficient related to the concrete strength grade under different concrete strength grades is obtained. The scatter plot of the strength coefficient is drawn, and the relationship between concrete tensile strength f_t and concrete strength grade enhancement coefficient φ_c is obtained through logarithmic function fitting. The correlation coefficient of the concrete strength grade R² = 0.99 indicates that the curve is well fitted. The fitting formula is as follows (concrete strength grade: C30–C80):

$${\phi }_{\mathrm{c}}=1.2\mathrm{l}\mathrm{n}\left(f_{\mathrm{t}}\right)+0.2$$

(13)

4.2.2. Determination of φ_f

Figure 14b shows the curve of influence coefficient φ_f with the variation of CFRP cloth ratio obtained by fitting multiple simulation results. The shear capacity provided by CFRP cloth is obtained by subtracting the ultimate shear force of the C30 plain concrete beam from the simulated total shear force. For the influence coefficient of the CFRP cloth ratio, the shear strength of the plain concrete beam with the strength of C30 is also taken as the control group, and the shear strength provided by the CFRP cloth under different CFRP cloth ratios of the C30 concrete beam (ρ_f = 0.1334%, 0.2672%, and 0.5344%) is divided by the shear strength of the C30 plain concrete beam. The shear strength enhancement coefficient φ_f related to the fiber distribution ratio is obtained. The scatter plot is drawn and fitted in logarithmic form. The relationship between the CFRP cloth ratio influence coefficient φ_f and the CFRP cloth ratio ρ_f (CFRP cloth ratio: 0.1336–0.5344%) is obtained:

$${\phi }_{\mathrm{f}}=\left\{\begin{array}{c}0,{\rho }_{\mathrm{f}}=0\\ 0.1\ln \left({\rho }_{\mathrm{f}}\right)+0.3,{\rho }_{\mathrm{f}}>0\end{array}\right.$$

(14)

It should be mentioned that the strength improvement coefficients are calculated by using Equations (13) and (14) solely, and the influence of the concrete strength grade and CFRP cloth ratio on the shear strength of the CFRP cloth strengthened concrete beams is considered, respectively. Further research will be done on the effects of additional elements such as the stiffness and strength of the CFRP, CFRP–concrete interface bonding, the shear span ratio, FRP type, and loading technique on concrete beams. When the CFRP cloth ratio exceeds the optimal value, the shear reinforcement effect is not ideal. The optimal value will be discussed in another study.

4.3. Verification of Theoretical Formulas

Figure 15 compares the simulated results with the calculation results of Equation (12) for shear capacity. In fact, when the concrete strength grade or CFRP cloth ratio is involved in the test, the shear strength of the RC beam can be predicted according to the selected CFRP cloth ratio ρ_f and concrete tensile strength f_t. It can be found that according to Equation (12), the three-dimensional curved surface calculated by considering the two parameters of concrete strength grade and CFRP cloth ratio reflects the effect law of CFRP cloth ratio and concrete strength grade on shear strength analyzed previously. Additionally, it can be seen that the error between theory and simulation is between 0% and 8.49%, the simulation values for nominal shear strength are close to the theoretical surface, showing that the predicted results (i.e., the theoretical values) and the simulated results are in good agreement. This confirms the accuracy and logic of the proposed prediction model used to describe the shear strength of RC beams strengthened with CFRP.

In order to further demonstrate the rationality and accuracy of the theoretical formula proposed in this paper, existing experimental data were selected with nominal shear capacity as the index, and the calculated value was compared with the theoretical value, as shown in Figure 16a. When the ρ_f = 0, that is, φ_f = 1, the experimental data of the effect of concrete strength grade on the shear capacity of RC beams made by Ding and Qi [10], Yi et al. [67], Shi and Xu [20], and Mphonde and Frantz [8] are selected. The shear strength is converted to nominal shear strength and compared with theoretical value. The figure shows the theoretical value of nominal shear strength that varies with the tensile strength of concrete calculated according to Equation (12). It can be seen that all the test values fall within the error range of 20%, indicating that the formula could better predict the shear bearing strength of plain concrete beams.

The experimental data on RC beams strengthened with CFRP by Jin et al. [31], Sherif [68], Rousan [69,70], Ma’en et al. [16], and Chen et al. [7] are chosen. The corresponding parameters were converted to the fiber specific ρ_f used in this paper according to Equation (11). The shear strength calculated according to Equation (12) is compared with the theoretical value, as shown in Figure 16b,c. According to the concrete strength grade used in the experiments of these scholars, which is C30 and C60, it can be observed that the experimental value is in conformity with the theoretical curve, and the error is within 20%.

In summary, the novel theoretical formula put forth in this work is well in line with the outcomes of both mesoscale simulations and current experimental findings. The shear strength of RC beams strengthened by externally bonded CFRP can be accurately predicted using a theoretical model that takes the impact of the concrete strength grade and CFRP cloth ratio into account. To further confirm the validity of the theoretical formulas presented to characterize the strength of concrete, however, a significant quantity of experimental data should be provided, particularly for RC beams strengthened by CFRP for ultra-high strength concrete.

5. Conclusions

To explore the shear failure of RC beams strengthened by U-shaped bonded CFRP, a three-dimensional meso-scale simulation approach is constructed. The effects of concrete strength grade and CFRP cloth ratio on the failure mechanism and failure mode of RC beams are modeled and studied. Additionally, analysis and discussion are provided about the impact of the CFRP cloth ratio on the shear strength of RC beams. The novelty of this paper is that it presents a solution to the calculation of the shear capacity of RC beams strengthened by CFRP cloths. The logarithmic function is used to express the influence coefficient of the tensile strength of concrete and the CFRP cloth ratio, and a new formula is formed instead of the linear formula in the specification. Finally, a theoretical formula for CFRP-strengthened RC beams is developed that can describe the quantitative impacts of the concrete strength grade and CFRP cloth ratio. The concluding remarks can be summarized as follows:

(1)

RC beams with higher strength grades can usually withstand greater loads before failure. High-strength concrete tends to have lower shrinkage and creep, which helps control cracks. Under the same load conditions, fewer and narrower cracks may form in high-strength concrete than in low-strength concrete. The shear capacity of RC beams is substantially influenced by the concrete strength grade, and the shear capacity can be significantly increased by increasing the concrete strength grade. In the scope of this study, up to 50% improvement can be achieved, but the improvement of each concrete strength grade is reduced from 25% to 1%;

(2)

The effect of concrete strength grade against shear capacity is not linear. When the strength is below C60, the shear strength is close to linear; when the strength is above C60, the growth is slowed down and increases logarithmically;

(3)

CFRP cloth can improve the ductility of concrete beams, even in high-strength concrete. The external bonding of CFRP can greatly increase the shear resistance of the beam from 10% to 24%. In addition, with the increase of the proportion of CFRP fiber, the strengthening range of the anti-shear strength of CFRP gradually decreased from 10% to 2%;

(4)

The effects of concrete strength grade and CFRP cloth ratio on the shear strength of U-shaped bonded CFRP-strengthened RC beams could be quantitatively described using the suggested shear capacity formula. The error between the theoretical value and the simulated value obtained by this formula is within 10%, and the error between the experimental data and the researchers is within 20%. Therefore, current experimental data and simulation results show that the method is reasonable.

It should be noted that the effects of concrete strength grade and CFRP cloth ratio on the shear strength of U-shaped bonded CFRP-strengthened RC beams are the sole impacts examined in this study. The stiffness and strength of the CFRP, CFRP–concrete interface bonding, the influence of shear span ratio, CFRP cloth bonding mode, FRP type, and loading method on shear capacity will be further studied. Considering these parameters in the analysis of CFRP-strengthened structures allows for a more comprehensive understanding of their behavior. It enables a deeper insight into the distribution of stresses, deformation patterns, and failure modes, leading to more accurate predictions of the shear performance. Additionally, more tests on CFRP-strengthened concrete beams with greater CFRP cloth ratios should be performed in order to validate the fitting methods used to forecast the shear capacity of CFRP-strengthened RC beams.