Shear Bearing Capacity Prediction of Steel-Fiber-Reinforced High-Strength Concrete Corbels on Modified Compression Field Theory

Abstract

In order to analyze the shear mechanism of the steel-fiber high-strength concrete corbels, a calculation model for the shear bearing capacity of steel-fiber-reinforced high-strength concrete corbels was proposed based on the modified compression field theory. Considering the existence of residual tensile stress in steel-fiber-reinforced concrete at crack locations, the cracked steel-fiber-reinforced concrete was treated as a continuous material. The constitutive relation of cracked steel-fiber-reinforced concrete and the local stress equilibrium equation were modified. It was compared with the results of 34 steel-fiber high-strength concrete corbels, including those in this paper. The predicted results were compared with the experimental values and the predictions of the Fattuhi model, Campione model, and Russo model to validate the rationality of the proposed model. The results revealed that the mean value between the experimental values and the predicted results of the proposed model is 1.104, with a variance of 0.003, showing good agreement. The proposed model can accurately predict the shear bearing capacity of steel-fiber-reinforced high-strength concrete corbels.

1. Introduction

As cantilever support structures, reinforced concrete corbels are important vertical load-bearing components in practical engineering applications such as industrial plants and road bridges. The steel-fiber-reinforced high-strength concrete formed by incorporating steel fibers into high-strength concrete is a type of composite material that not only overcomes the drawbacks of poor ductility and brittleness of high-strength concrete [1,2], but it also enhances the cracking, tensile, and shear strength of concrete. The application in the corbels can effectively improve the shear performance and bearing capacity of the structure [3,4,5].

The internal stress state of corbels becomes complex under the action of loads, the presence of non-linear strain behavior, and significant stress redistribution across the section. They belong to the category of typical D-region members [6]. According to Saint-Venant’s principle [7], the D-region has the characteristics of discontinuity in a geometric structure or disturbances in the flow of forces, which are no longer suitable for design theory assumed by the plane section [8,9]. Khattab et al. [10] conducted experimental research on the corbels with different shear span ratios, and compared the test results with those calculated using the shear friction and strut-and-tie models specified in ACI 318-14 [11]. Malgorzata et al. [12] performed experiments on reinforced concrete corbels. By fitting and analyzing the test results, the formula for calculating the inclined crack of the corbels is proposed. Yang et al. [13] proposed a calculation method for predicting the shear carrying capacity of reinforced concrete corbels, which showed better predictive results compared to other existing models. Wael et al. [14] presented a reinforced concrete corbel bearing capacity calculation method based on the strut-and-tie model. The results indicated that the proposed strut-and-tie method provided sufficiently accurate predictions, leading to the conclusion that it offers safe and reliable approach for calculating the bearing capacity of corbels. Perceka et al. [15] made 10 high-strength steel-fiber concrete beams. Based on the observation results of the test results and the evaluation results of the previous shear strength equations, the prediction equation of the shear strength of steel-fiber concrete beams was proposed. Muhammad et al. [16] conducted pouring tests on 12 BFRP-reinforced concrete beams. In addition, based on the published experimental shear strength and test results of 217 FRP-RC beams without stirrup, the equation for predicting the contribution of concrete to the shear strength of FRP-RC beams was established with good accuracy. However, there is a limited amount of reported research on the calculation models and methods specifically used for steel-fiber-reinforced high-strength concrete corbels, and there is no unified calculation model and method for the calculation of its bearing capacity, which cannot meet the needs of practical engineering design. In 1986, Canadian scholars Vecchio and Collins [17] proposed the modified compression field theory (MCFT) based on the existing theory of the compression field, which considered the equilibrium, compatibility conditions, and the constitutive relationship of cracked concrete. MCFT has been widely recognized for the analysis of membrane, beam, plate, shell, and plane frames. Even the more complex structural analysis results are ideal, and it is also suitable for steel-fiber-reinforced concrete structures [18,19]. Gao et al. [20] and Zhang et al. [21], Zhu et al. [22], Lei et al. [23], used MCFT to study the shear capacity of reinforced steel-fiber concrete beam–column joints, steel-fiber high-strength concrete two-pile cap, and hybrid concrete deep beams, respectively. The results showed that the predicted shear strength is in good agreement with the experimental results. The modified pressure field theory takes into account the tensile stress of cracked concrete and determines the average stress, average strain, and crack angle according to the equilibrium conditions, compatibility conditions, and the stress–strain relationship between reinforced and cracked concrete. Finally, the shear bearing capacity of reinforced concrete members under the action of shear force can be obtained, which is more accurate and reasonable.

The current research proposes a model that calculates the shear bearing capacity of steel-fiber-reinforced high-strength concrete corbels based on MCFT. The proposed model considers the stress transfer of steel fiber as the crack corrects the constitutive relationship of steel-fiber-reinforced concrete, allowing for a better reflection of the shear behavior of steel-fiber-reinforced high-strength concrete corbels. It was used to predict the shear capacity of 31 specimens, including the specimens tested in this study. The predicted results were compared with the experimental results and simultaneously compared with calculation models proposed by other scholars to evaluate its accuracy.

2. Establishment of the Proposed Model

The modified compression field theory incorporates stress equilibrium conditions both between and within cracks, offering a comprehensive and practical theoretical framework, and includes the following aspects:

• The shear stress and normal stress experienced by concrete micro-elements are uniformly distributed when subjected to loads.
• The stress and strain of concrete refer to the average stress and strain in the cracking zone.
• The principal stress and principal strain directions of concrete are consistent.
• There is no bond slip between steel bars and concrete, assuming ideal bond conditions.
• Shear stress in steel bars is not considered: only the axial stress in the steel bars is taken into account.

2.1. Equilibrium Conditions

Assuming that the cracked steel-fiber-reinforced high-strength concrete is a homogeneous composite material (whereby Figure 1 illustrates Mohr’s circle of average stress), the following relationships are determined:

$$f_{\mathrm{x}}=f_1-v\cot \theta +{\rho }_{\mathrm{sx}}{f}_{\mathrm{sx}}$$

(1)

$$f_{\mathrm{y}}=f_1-v\tan \theta$$

(2)

$$f_1+f_2=v(\tan \theta +\cot \theta )$$

(3)

where f₁ is the principal tensile stress of the concrete; f₂ is the principal compressive stress of the concrete; f_x and f_y are the average stresses of the element along the x and y directions, respectively; θ represents the inclination of the diagonal compressive; v is the shear stress by the element; f_sx is the stress in the x direction of the steel reinforcement; and ρ_sx is the reinforcement ratio in the x direction.

After cracking, steel-fiber concrete corbels continue to transmit stress across the cracks. The bond between concrete and steel fibers enhances the stress at the crack, providing a bridging effect. This effectively inhibits crack propagation and improves the transfer of shear forces across the crack interfaces [24]. Considering the steel of fibers on concrete shear resistance under the stress balance condition at the crack, the residual tensile stress generated by steel fiber is as follows:

$${\sigma }_{\mathrm{f}}=K_{\mathrm{f}}\frac{L_{\mathrm{f}}}{D_{\mathrm{f}}}{\tau }_{\mathrm{u}}{\rho }_{\mathrm{f}}$$

(4)

where K_f is the effective distribution coefficient of steel fibers; L_f is the length of the steel fibers; D_f is the diameter of steel fibers; τ_u is the bond strength between steel fibers and concrete, whereby τ_u = 0.6 f_c′2/3 [18]; and ρ_f represents the volume fraction of steel fibers.

Steel fibers are randomly distributed in the concrete matrix, and when the fiber pullout angle exceeds a certain limit, it is considered that the steel fibers have little contribution to the tensile strength of the concrete [25,26]. Therefore, assuming that the x-axis direction is the normal direction of the crack surface, the angle formed between a single steel fiber and the x-axis direction is θ_f, as shown in Figure 2.

According to Foster’s research findings [27], when the fiber pullout angle exceeds 60 degrees, the contribution of steel fibers is not considered. The fiber inclination is between 0 and 60 degrees. The effective distribution coefficient of randomly oriented steel fibers in concrete is calculated by integrating according to the following equation:

$$K_{\mathrm{f}}=\frac{1}{2\pi }{\displaystyle {\int }_{-\pi }^{\pi }{\displaystyle {\int }_0^{{60}^{°}}\cos {\theta }_{\mathrm{f}}\sin {\theta }_{\mathrm{f}}d{\theta }_{\mathrm{f}}d\phi }}$$

(5)

For the steel-fiber concrete corbels, the bridging effect of steel fibers enhances the shear stress. Thus, the balance of transverse reinforcement direction (x direction) force can be expressed as follows:

$${\rho }_{\mathrm{sx}}{f}_{\mathrm{sx}}+f_1{\sin }^2\theta -f_2{\cos }^2\theta =0$$

(6)

Similarly, the balance of forces in the y direction is:

$$f_1-v\tan \theta -{\sigma }_{\mathrm{n}}=0$$

(7)

where σ_n is the axial compressive stress, which is taken as σ_n = nf_c′; n is the axial load ratio (n = N/f_c′bh₀; N is the axial load and b is the width of the corbel section).

From Equation (8), it can be expressed as:

$$f_{1}b{h}_0-V\tan \theta =N$$

(8)

2.2. Constitutive Relations

2.2.1. Stress–Strain Relationship of Concrete in Tension

The concrete is considered to be in the linear elastic stage before cracking, that is, the tensile stress–strain relationship is as follows:

$$f_1=E_{\mathrm{c}}{\epsilon }_1({\epsilon }_1\le {\epsilon }_{\mathrm{cr}})$$

(9)

where E_c is the elastic modulus of concrete; ε₁ is the principal tensile strain.

The tensile stress–strain relationship of cracked concrete is given by [28]:

$$f_1=\frac{0.33\sqrt{f_{\mathrm{c}}^{\prime }}}{1+\sqrt{500{\epsilon }_1}}({\epsilon }_1>{\epsilon }_{\mathrm{cr}})$$

(10)

where f_c′ is the compressive strength of concrete cylinder; ε_cr is the cracking strain of concrete, given by ε_cr = 0.33 f_c′^(1/2)/E_c [17].

For the principal tensile stress and strain relationship of steel fiber concrete, the reinforcement effect of steel fiber on concrete is considered. The tensile stress in cracked steel-fiber-reinforced concrete is equal to the residual tensile stress in concrete plus the tensile stress in steel fibers at the crack location [29]. The expression is given by:

$$f_1=\frac{0.33\sqrt{f_{\mathrm{c}}^{\prime }}}{1+\sqrt{500{\epsilon }_1}}+{\sigma }_{\mathrm{f}}({\epsilon }_1>{\epsilon }_{\mathrm{cr}})$$

(11)

2.2.2. Stress–Strain Relationship of Concrete in Compression

The stress–strain relationship of steel-fiber-reinforced concrete in compression, as suggested by Vecchio and Collins [30,31], is as follows:

$$f_2=f_{2,\max }\left[2\left(\frac{{\epsilon }_2}{{\epsilon }_0}\right)-{\left(\frac{{\epsilon }_2}{{\epsilon }_0}\right)}^2\right]$$

(12)

$$\frac{f_{2,\max }}{f_{\mathrm{c}}^{\prime }}=\frac{1}{0.8-0.34\frac{{\epsilon }_1}{{\epsilon }_0}}\le 1$$

(13)

where f_2,max is the maximum principal compressive stress in the concrete; ε₀ is the compressive strain of concrete corresponding to the maximum principal compressive stress, which is generally −0.002; ε₁ and ε₂ are the strains corresponding to the principal tensile stress and the principal compressive stress.

2.2.3. Stress–Strain Relationship of Reinforcement

The stress–strain relationship for steel in tension and compression is used by an ideal elastic–plastic model, which can be expressed as:

$$f_{\mathrm{s}}=E_{\mathrm{s}}{\epsilon }_{\mathrm{s}}\le f_{\mathrm{y}}$$

(14)

where f_s and ε_s represent the stress and strain of steel reinforcement, respectively; E_s denotes the steel reinforcement elastic modulus; and f_y represents the yield strength of the steel reinforcement.

2.3. Compatibility Conditions

Assuming no slippage between the steel reinforcement and concrete, the deformation amount of reinforcement and concrete is exactly the same. Therefore, the incremental strain values of the concrete and steel are also same. At the instant of cracking, the incremental strain and stress values of the steel reinforcement can be neglected, resulting in the following equation:

$${\epsilon }_{\mathrm{cx}}={\epsilon }_{\mathrm{sx}}={\epsilon }_{\mathrm{x}}$$

(15)

$${\epsilon }_{\mathrm{cy}}={\epsilon }_{\mathrm{sy}}={\epsilon }_{\mathrm{y}}$$

(16)

where ε_cx and ε_cy are the concrete strain in x and y directions, respectively; ε_sx and ε_sy are the strain in the x and y directions of the reinforcement, respectively; ε_x and ε_y are the strains in the x and y directions of the element.

The geometric relationship between the principal strains of concrete and Mohr’s circle of average strain (Figure 3) is given as follows:

$$\tan 2\theta =\frac{{\gamma }_{\mathrm{xy}}}{{\epsilon }_{\mathrm{y}}-{\epsilon }_{\mathrm{x}}}$$

(17)

$${\epsilon }_1+{\epsilon }_2={\epsilon }_{\mathrm{x}}+{\epsilon }_{\mathrm{y}}$$

(18)

$${\tan }^2\theta =\frac{{\epsilon }_{\mathrm{x}}-{\epsilon }_2}{{\epsilon }_{\mathrm{y}}-{\epsilon }_2}=\frac{{\epsilon }_1-{\epsilon }_{\mathrm{y}}}{{\epsilon }_{\mathrm{y}}-{\epsilon }_2}$$

(19)

where ε₁ and ε₂ are the principal tensile and compressive strains of the element, respectively, while γ_xy is the shear strain.

From (18)–(20), we can obtain:

$${\epsilon }_{\mathrm{x}}={\epsilon }_1{\cos }^2\theta +{\epsilon }_2{\sin }^2\theta$$

(20)

$${\epsilon }_{\mathrm{y}}={\epsilon }_1{\sin }^2\theta +{\epsilon }_2{\cos }^2\theta$$

(21)

$${\gamma }_{\mathrm{xy}}=2({\epsilon }_1-{\epsilon }_2)\sin \theta \cos \theta$$

(22)

2.4. Stress Equilibrium of Steel Fibers at the Crack

In the MCFT, the average stress is assumed to be in equilibrium with the stress at the crack location using the static equivalence method. This allows for a reasonable simplification of the actual complex stress state, considering it as an ideal series of parallel cracks. Figure 4 shows the stress distribution of the cracked unit at the crack location and between cracks. Because the compressive stress f_ci at the crack is relatively small, its influence can be ignored. Considering the stress equilibrium in the direction of the stirrup, the following can be obtained:

$${\rho }_{\mathrm{sx}}{f}_{\mathrm{sx}}\sin \theta +f_1\sin \theta ={\rho }_{\mathrm{sx}}{f}_{\mathrm{sxcr}}\sin \theta -v_{\mathrm{ci}}\cos \theta +{\sigma }_{\mathrm{f}}\sin \theta$$

(23)

Thus,

$$f_1=({\rho }_{\mathrm{sx}}{f}_{\mathrm{sxcr}}-{\rho }_{\mathrm{sx}}{f}_{\mathrm{sx}})-v_{\mathrm{ci}}\cot \theta +{\sigma }_{\mathrm{f}}$$

(24)

where f_sxcr is the reinforcement stress in the x direction of the crack, and v_ci is the shear stress at the crack, which can be calculated by the following formula:

$$v_{\mathrm{ci}}=\frac{0.18\sqrt{f_{\mathrm{c}}^{\prime }}}{0.3+\frac{24w}{a+16}}$$

(25)

The size of the crack spacing mainly depends on the distribution of longitudinal steel bars and stirrups. The denser the arrangement of steel reinforcement, the closer the cracks, and the corresponding spacing decreases. The crack width can be calculated using the following equation:

$$w={\epsilon }_1{s}_{\theta }$$

(26)

$$s_{\theta }=\frac{1}{\frac{\sin \theta }{s_{mx}}+\frac{\cos \theta }{s_{my}}}$$

(27)

where a is the maximum aggregate size, w is the crack width, s_θ is the average crack spacing, s_mx is the crack spacing in the x direction, and s_my represents the crack spacing in the y direction. To simplify, the maximum spacing between longitudinal reinforcement and stirrups is used in the above calculation.

When the reinforcement at the crack reaches its yield strength, the control condition for the principal tensile stress f₁ can be obtained via Equation (24):

$$f_1\le {\rho }_{\mathrm{sx}}(f_{\mathrm{sxy}}-f_{\mathrm{sx}})-v_{\mathrm{ci}}\cot \theta +{\sigma }_{\mathrm{f}}$$

(28)

At the same time, the control condition for the reinforcement stress at the crack is given by:

$$f_{\mathrm{sxcr}}=f_{\mathrm{sx}}+(f_1+v_{\mathrm{ci}}\cot \theta -{\sigma }_{\mathrm{f}})/{\rho }_{\mathrm{sx}}\le f_{\mathrm{sxy}}$$

(29)

By combining Equations (3) and (6), we can obtain:

$$v={\rho }_{\mathrm{sx}}{f}_{\mathrm{sx}}\tan \theta +(v_{\mathrm{ci}}\tan \theta +{\sigma }_{\mathrm{f}})\tan \theta$$

(30)

The shear capacity of steel-fiber-reinforced high-strength concrete corbels is obtained via:

$$V=A_{\mathrm{sv}}{f}_{\mathrm{y}}\frac{h_0}{s}\tan \theta +(v_{\mathrm{ci}}\tan \theta +{\sigma }_{\mathrm{f}})b{h}_0\tan \theta$$

(31)

2.5. Calculation Process

The abovementioned equilibrium conditions, constitutive relations, compatibility conditions, and stress equilibrium of steel fibers at the crack are solved simultaneously to calculate the shear capacity of steel-fiber-reinforced high-strength concrete corbels. Figure 5 illustrates the flowchart of the solution process, which involves the following main steps:

• Input corbels’ geometry and material parameters.
• Assume ε₁, and the increment of ε₁ is 0.0001.
• Assume θ.
• Calculate v_ci and w using Equations (25) and (26).
• Assume f_sx0.
• Calculate f₁ using Equation (11), and satisfy Equation (28).
• Calculate v using Equation (30).
• Calculate f₂ using Equation (3), and f₂ is limited by f_2,max.
• Use Equation (14) to determine whether it is equal to the assumed f_sx0. If it is, continue to the next step; otherwise, return to (6) to assume the stirrup stress again.
• Calculate axial load N using Equation (8), and whether the axial force is equal to 0. If it is true, the analysis of the corresponding principal tension strain state is completed. If not, return to (4) to assume the crack inclination again until it is equal.
• Calculate f_sxcr using Equation (29), if f_sxcr is less than f_sxy, the calculation ends. Otherwise, return to (2) to re-assume the main tensile strain of concrete.

3. Model Verification

3.1. Experimental Work

With the aim of verifying the above shear bearing capacity calculation model, the shear test of nine steel-fiber high-strength concrete corbels under concentrated load was completed, with a width of 200 mm, a height of 450 mm at the root cross-section, and a width of 200 mm. The design parameters of the shear span-to-depth ratio (λ), the steel fiber dosage (ρ_f), the main reinforcement ratio (ρ_s), the stirrup reinforcement ratio (ρ_sh), and the main parameters and reinforcement are shown in

Table 1. Specimen parameters.

Specimens	Main Reinforcement	Ratio of Reinforcement	Stirrup	Stirrup Reinforcement Ratio	Shear Span Ratio	Effective Depth	Steel Fiber Dosage
MC01	4C12	0.55%	A10@100	0.785%	0.2	400	1.5%
MC02	4C12	0.55%	A10@100	0.785%	0.3	400	1.5%
MC03	4C12	0.55%	A10@100	0.785%	0.4	400	1.5%
MC04	4C12	0.55%	A10@100	0.785%	0.3	400	1.5%
MC05	4C12	0.55%	A10@100	0.785%	0.3	400	1.5%
MC06	4C12	0.75%	A10@150	0.390%	0.3	400	1.5%
MC07	4C12	0.98%	A10@200	0.520%	0.3	400	1.5%
MC08	4C12	0.55%	A10@100	0.785%	0.3	400	0
MC09	4C12	0.55%	A10@100	0.785%	0.3	400	0.75%

and Figure 6.

The longitudinal reinforcement and stirrup reinforcement of the specimens were made of HRB400- and HPB300-grade steel bars, respectively. The measured mechanical properties of various types of steel reinforcement are presented in

Table 2. Measured mechanical properties of reinforcement.

Reinforcement Type	Diameter (mm)	Cross-Sectional Area (mm2)	Yield Strength (MPa)	Elastic Modulus (GPa)	Ultimate Strength (MPa)
HPB300	10	78.5	333.7	195.95	535.8
HRB400	12	113.1	425.2	172.6	541.4
HRB400	14	153.9	467.8	185.0	582.3
HRB400	16	201.1	427.3	192.5	587.7

.

The main test results and failure patterns of the specimens are shown in

Table 3. Test results and failure patterns of specimens.

Specimens	${\mathit{V}}_{\mathbf{cr}}^{\mathbf{N}}$/kN	${\mathit{V}}_{\mathbf{cr}}^{\mathbf{D}}$/kN	Vu/kN	Failure Patterns
MC01	270	400	879	Diagonal shear patterns
MC02	220	290	822	Diagonal compression pattern
MC03	166.5	255	695	Diagonal compression pattern
MC04	220	355	874	Diagonal compression pattern
MC05	193	386.5	981.5	Diagonal compression pattern
MC06	192	234.5	670.5	Diagonal compression pattern
MC07	164	240	751	Diagonal compression pattern
MC08	150	331	775	Diagonal compression pattern
MC09	180	381	767	Diagonal compression pattern

. All specimens experienced shear failure, and can be divided into three stages, namely initial cracking, critical cracking, and ultimate failure. Due to the difference in shear span ratio, the typical failure modes of the specimen include diagonal shear failure and diagonal compression failure. The top longitudinal tensile reinforcement gradually reaches the yield strength, and the penetrating diagonal crack appears at the connection surface between the loading plate and the column, forming the concrete failure mode caused by shear force. As shown in Figure 7a, the failure section is close to the vertical and slightly oblique shear crack, which resulted in the patterns of diagonal shear failure. Figure 7b shows that the middle part of the diagonal direction is perpendicular to the direction of the main tensile stress, and several oblique ventral shear cracks that are roughly parallel to each other appear successively, and oblique prisms appear in the abdomen of the corbel. After the occurrence of the main oblique crack, when the ultimate load is approaching, the yield strength of the longitudinal tensile reinforcement part reaches the vertical part, which led to the patterns of diagonal compression failure.

3.2. Comparison with Test Results

A comparative analysis was conducted by selecting shear capacity models proposed based on the Fattuhi model [32], Campione model [33], and Russo model [34] and by comparing them with the model presented in this paper. The introduction is presented below.

3.2.1. Fattuhi Model [32]

The formulation takes into account the contributions of both traditional bar reinforcement and fibrous reinforcement, considering their respective volumes and their influence on the mechanical behavior of the corbels. Fattuhi proposed an empirical formulation based on experimental data:

$$V=k_{1}bh{(f_{\mathrm{t}})}^{k_2}{(\frac{f_{\mathrm{y}}}{f_{\mathrm{cu}}})}^{k_4}{(\frac{d}{h})}^{k_5}{(\rho )}^{k_6}$$

(32)

The values of the constants k₁ to k₆ are 57.292, 0.315, −0.812, −0.049, 0.678, and 0.626, where b is the corbel’s width; h is the corbel’s height; d is the corbel’s effective height; f_y is the yield strength of longitudinal bar; f_t is the concrete tensile strength; and ρ is the longitudinal reinforcement ratio.

3.2.2. Campione Model [33]

The shear strength of fiber-reinforced concrete corbels, including stirrups, is expressed below, taking into account the influence of concrete grade, fiber volume, and reinforcement ratio:

$$V=\chi {f_{\mathrm{c}}}^{\prime }(b{x}_{\mathrm{c}}\cos \alpha )\sin \alpha +\frac{b}{\sin \alpha \cos \alpha }0.2\sqrt{{f_{\mathrm{c}}}^{\prime }}F\left[(d-x_{\mathrm{c}})(1-\frac{f_{\mathrm{ctf}}}{f_{\mathrm{y}}}\cdot \frac{E_{\mathrm{s}}}{E_{\mathrm{ct}}})+c\right]$$

(33)

$$\chi =0.74(\frac{{f_{\mathrm{c}}}^{\prime }}{105}{)}^3-1.28(\frac{{f_{\mathrm{c}}}^{\prime }}{105}{)}^2+0.22(\frac{{f_{\mathrm{c}}}^{\prime }}{105})+0.87$$

(34)

$$\alpha =\frac{d-\frac{x_{\mathrm{c}}}{3}}{a}$$

(35)

where η is the load distribution coefficient, f_c′ is the concrete tensile strength, α is the angle between the tie and strut, F is the fiber factor, f_ctf is the steel fiber concrete tensile strength, and c is the thickness of the protective layer.

3.2.3. Russo Model [34]

Based on the combination of the strut-and-tie mechanism resulting from cracked concrete and the main tension reinforcement, as well as the strength contribution from secondary reinforcement, the following are obtained:

$$V=0.8(k\chi {f_{\mathrm{c}}}^{\prime }\cos {\theta }_1+0.65{\rho }_{\mathrm{h}}{f}_{\mathrm{yh}}\cot {\theta }_1)bh$$

(36)

$$k=\sqrt{{(n{\rho }_{\mathrm{f}})}^2+2n{\rho }_{\mathrm{f}}}-n{\rho }_{\mathrm{f}}$$

(37)

$$\theta =2\arctan \left[\frac{-1+\sqrt{{(\frac{a}{d})}^2+(1-\frac{k^2}{4})}}{\frac{a}{d}-\frac{k}{2}}\right]$$

(38)

$$n=\frac{42.6}{\sqrt{{f_{\mathrm{c}}}^{\prime }}}$$

(39)

where χ is the softening coefficient; θ₁ is the angle between the concrete strut and the axis of the column; and f_yh and ρ_h are the yield strength and reinforcement ratio of stirrup, respectively.

3.2.4. Model Validity

To validate the rationality of the proposed model in this study, the experimental results of 34 corbel tests from this study and references [35,36,37,38] were compared with the calculation results of various models proposed by researchers, including the model proposed in this study.

Table 4. Comparison of experimental and calculated shear bearing capital.

References	Specimens	b/mm	h0/mm	ρf	Vtest	Vtest/VF	Vtest/VC	Vtest/VR	Vtest/Vn
This study	MC01	200	400	1.50%	879	0.859	1.768	1.120	1.113
	MC02	200	400	1.50%	822	1.116	1.420	1.208	1.041
	MC03	200	400	1.50%	695	1.192	1.156	1.139	1.159
	MC04	200	400	1.50%	874	0.978	1.534	1.201	1.107
	MC05	200	400	1.50%	981.5	0.929	1.750	1.270	1.114
	MC06	200	400	1.50%	670.5	0.911	1.159	1.227	1.102
	MC07	200	400	1.50%	751	1.020	1.298	1.272	1.116
	MC08	200	400	0%	775	1.018	1.342	1.148	1.144
	MC09	200	400	0.75%	767	1.071	1.321	1.130	1.173
Faleh et al. [35]	CF3-0.5	200	250	0.50%	1125	1.25	1.480	1.397	1.162
	CF3-1.0	200	250	1.00%	1220	0.960	1.435	1.356	1.104
	CF3-1.5	200	250	1.5%	1230	0.911	1.281	1.337	1.027
Fattuhi et al. [36]	1	150	125	1.70%	153	0.917	1.436	1.453	1.132
	2	150	125	1.70%	160	0.990	1.449	1.470	1.104
	3	150	129	1.70%	91.2	0.947	1.580	1.130	1.079
	4	150	127	1.70%	93	0.952	1.647	1.208	1.029
	5	150	124	1.70%	103	1.008	1.121	1.325	1.213
	6	150	125	1.70%	95.7	0.940	1.075	1.276	1.003
	9	150	125	1.70%	152.9	0.954	1.558	1.680	1.097
	10	150	124	1.70%	102.9	1.024	1.166	1.418	1.068
	12	150	127	0.70%	92	0.777	1.974	1.258	1.145
	13	150	125	1.70%	111.7	0.920	1.674	1.424	1.143
	16	150	125	1.70%	114.3	0.948	1.622	1.344	1.103
	18	150	125	1.00%	119	0.820	1.921	1.422	1.052
Yang et al. [37]	CW1	200	400	0%	1271	0.964	1.615	1.252	1.115
	CW2	200	400	0.50%	1367	1.005	1.499	1.390	1.073
	CW3	200	400	0.75%	1440	1.043	1.145	1.327	1.110
Fattuhi et al. [38]	20	153	124	1.75%	126	0.851	1.342	1.313	1.056
	21	156	122	1.50%	118	0.821	1.455	1.460	1.041
	23	153	123	2.00%	126.5	0.869	1.267	1.324	1.038
	27	153.5	124	2.50%	171.5	0.850	1.681	1.358	1.204
	32	154	120	2.00%	132.5	0.835	1.131	1.221	1.044
	39	153.5	124	2.25%	144.5	0.843	1.231	1.420	1.219
	49	154.4	122	2.50%	164.5	0.830	1.721	1.466	1.073
Mean						1.453	0.942	1.311	1.103
Variance						0.061	0.009	0.017	0.003

presents the results obtained from the study. V_test is the measured shear capacity values. V_n is the calculated shear capacity values by the model proposed in this study. V_F is the calculated values by the Fattuhi model. V_C is the calculated values by the Campione model. V_R is t-calculated values by the Russo model. The mean value of the test value and the Fattuhi model was 0.942, and the variance was 0.009. The calculated value of the specimen was greater than the test value, and the calculated result of the model was not safe. The mean values of the ratio calculated by the Campione model and Russo model were 1.453 and 1.311, respectively, and the variances were 0.061 and 0.017, respectively. The calculated results were conservative, and the error between the proposed model and the test value was large. From information obtained from various scholars, it can be determined that the calculation results of the proposed model are closer to the experimental values, and the variance is smaller. The shear bearing capacity of the specimen can thus be calculated accurately.

Figure 8 shows the comparison between the experimental values and the calculated values. In the upper left area of the figure, the experimental values are greater than the calculated values, indicating a conservative safety. The lower right area of the figure represents cases where the experimental values are smaller than the calculated values, indicating an underestimation of the safety margin. For the Fattuhi model, there is significant fluctuation in the experimental data. Both the Campione model and Fattuhi model, along with their calculated results, underestimate the experimental values, indicating a conservative estimation, but with high dispersion. In contrast, the calculated values from the proposed model in this paper show a better agreement with the experimental values, with lower dispersion. This illustrates that the model proposed in this paper is more capable of accurately predicting the shear capacity of steel-fiber-reinforced high-strength concrete corbels.

4. Conclusions

This study proposes a calculation model for the shear capacity of steel-fiber-reinforced high-strength concrete corbels based on the modified compression field theory by calculating the shear capacity of 34 steel-fiber high-strength concrete corbels and comparing the results of the models proposed by various scholars. Through comparative analysis, the rationality of the proposed model is confirmed. It provides new ideas for theoretical analysis, leading to the following conclusions:

• Based on the modified compression field theory, the contribution of steel fibers to the tensile strength of concrete at crack locations is considered. The constitutive relationship of cracked steel-fiber-reinforced concrete and the local stress equilibrium equation are modified. A calculation model for the shear bearing capacity of steel-fiber-reinforced high-strength concrete corbels is proposed.
• Compared with other scholars’ models, the predicted value of shear capacity calculated by the proposed model is closer to the experimental value. It can be effectively utilized for predicting and analyzing the shear bearing capacity of steel-fiber-reinforced high-strength concrete corbels, and has a clear mechanical model.