Experimental and Numerical Simulation Study on Shear Performance of RC Corbel Under Synergistic Change in Inclination Angle

Abstract

The purpose of this paper is to study the shear performance of reinforced concrete corbels under a synergistic change in the main stirrup inclination angle to explore the synergistic mechanism of the main reinforcement and the stirrup inclination angle, and to evaluate the applicability of existing design specifications. The shear performance test was carried out by designing RC corbel specimens with an inclination angle of the main reinforcement and stirrup. The test results show that a 15° inclination scheme significantly improves the shear performance: the yield load is increased by 28.3%, the ultimate load is increased by 23.6%, the strain of the main reinforcement of the 15° specimen is reduced by 51.3%, the stirrup shows a delayed yield (the yield load is increased by 11.6%) and lower strain level (250 kN is reduced by 23.7%), and the oblique reinforcement optimizes the internal force transfer path and delays the reinforcement yield. A CDP finite element model was established for verification, and the failure mode and crack propagation process of the corbel were accurately reproduced. The prediction error of ultimate load was less than 2.27%. Based on the test data, the existing standard method is tested and a modified formula of the triangular truss model based on the horizontal inclination angle of the tie rod is proposed. The prediction ratio of the bearing capacity is highly consistent with the test value. A function correlation model between the inclination angle of the steel bar and the bearing capacity is constructed, which provides a quantitative theoretical tool for the optimal design of RC corbel inclination parameters.

1. Introduction

The reinforced concrete corbel is a commonly used cantilever bearing member in industrial buildings, water conservancy structures, and power plant facilities. The advantage of a high construction efficiency is particularly prominent in prefabricated concrete structures. The bracket connection can greatly simplify the on-site assembly of prefabricated beams and columns. Usually, the RC bracket is integrally cast with the lower support column or wall and is designed as a short cantilever form with a shear span ratio (a/d) not exceeding 1. Due to the concentrated load of the upper structure of the bracket and the specificity of the geometric shape, the force is complex [1,2,3] and does not meet the plane section assumption, the boundary between the tensile stress zone and the compressive stress zone is not clear, and the strain distribution is significantly nonlinear [4,5,6], which poses a major challenge in design safety. The design method given by the current national codes [7,8,9,10] based on the triangular truss theory and the different calculation theories of the strut-and-tie model (STM) has also become a difficult problem in structural design.

Although experimental research [11,12,13,14,15,16,17,18,19] has long been the cornerstone of studying the performance of corbels, the development of computing technology and finite element software has made the nonlinear analysis of reinforced concrete corbels more convenient and in-depth. Numerical analysis can not only save on research costs, but also obtain the internal mechanical response that is difficult to measure directly in the experiment. Although numerical analysis is of great value, simulation research on reinforced concrete corbels is still insufficient. It is urgent to construct a reliable numerical model verified by experiments to provide a basis for engineering design.

The recent literature suggests that the arrangement of inclined steel bars improves the shear performance of RC corbels. Özkal et al. [20] used the comprehensive strut-and-tie model method to design the reinforcement of ordinary concrete corbels. The results show that compared with the traditional horizontal reinforcement arrangement, the oblique reinforcement arrangement enhances the connection between the corbel and the column and makes a more effective contribution to its overall shear performance. Lei et al. [21] experimentally studied the shear strength of RC corbels with different inclination angles of main reinforcement. The test results of shear cracking and ultimate bearing capacity show that the corbels with a 15° inclination angle of main reinforcement show a higher shear strength.

Under the condition of the main stirrup inclination angle being varied, the influence on the shear performance of the corbel has not been discussed. In order to fill the research gap of the shear capacity of the corbel under the synergistic effect of the inclination angle, this study used ABAQUS 2022 software to simulate the corbel specimens. The simulation results of ABAQUS are in good agreement with the experimental data. On this basis, the applicability of the Chinese and American standard methods under the condition of the main stirrup inclination angle being varied is discussed to help design a corbel with a higher bearing capacity.

2. Experimental Program

2.1. Design of Corbel

In order to study the structural performance of concrete corbels under the synergistic change in main reinforcement stirrup inclination angle, two double corbel specimens were designed and tested. Figure 1 shows the geometric dimensions and reinforcement details of the corbel specimens. The thicknesses and heights of all corbels were 200 mm and 600 mm, respectively. The cross-sectional dimension of the central column of the specimen was 200 mm × 600 mm. The main reinforcement of each side of the corbel specimen is 2Φ22 (HRB400), and the stirrup is Φ12 @ 100 mm. The longitudinal reinforcement of the middle column is 4Φ16 and the stirrup is Φ12 @ 100 mm. We have encrypted the stirrups at the bottom of the column to prevent local crushing at the bottom of the corbel column, which in turn affects the transmission of load. In addition, in order to avoid the premature failure of the structure caused by local stress concentration, a pressure-bearing steel plate is set at the load position, and its geometric size is 300 mm × 200 mm × 30 mm. The variable parameters of the test are the angle between the main stirrup and the horizontal direction. The angle between the main stirrup and the horizontal direction is designed to be 0° and 15°. In the naming of the specimens, ‘A’ represents the inclination angle between the main stirrup and the horizontal direction, and ‘S’ represents the shear span ratio of the specimen, as shown in

Table 1. Design parameters of RC corbels.

Specimen	$\mathit{b}$ (mm)	$\mathit{a}$ (mm)	$\mathit{d}$ (mm)	$\mathit{a}/\mathit{d}$ (-)	Main Reinf	$\mathit{\rho }$ (%)	Stirrup	${\mathit{\rho }}_{\mathit{s}\mathit{v}}$ (%)	Inclined Angle (°)
A0-S0.61	200	350	570	0.61	2$\mathsf{\Phi }$22	0.667	Φ12@100	1.131	0
A15-S0.61									15

.

2.2. Material Properties

The compressive strength of the concrete cube is measured using a compressive test of six standard concrete cylinder specimens (D = 150 mm × H = 300 mm) $f_{\mathrm{c}0}$ = 40.83 MPa. HRB400 hot-rolled ribbed steel bars are used, for which 400 represents the standard value of yield strength (MPa). The mechanical properties of the steel bar are measured using a tensile test, and the specific parameters are shown in

Table 2. Mechanical properties of main reinforcement.

Diameter ${\mathit{d}}_{\mathit{S}}$ (mm)	Yielding Strength ${\mathit{f}}_{\mathbf{y}}$ (MPa)	Ultimate Strength ${\mathit{f}}_{\mathbf{u}}$ (MPa)	Yielding Strain ${\mathit{\epsilon }}_{\mathbf{y}}$ (-)	Elongation Ratio ${\mathit{\delta }}_{\mathbf{t}}$ (-)	Elastic Modulus ${\mathit{E}}_{\mathbf{s}}$ (GPa)
22	461.8	633.3	0.0022	0.24	205.8
16	488.3	688.23	0.0023	0.235	212.1
12	418.2	598.56	0.00205	0.23	204.1

.

2.3. Test Measurement Scheme

The test loading device is mainly composed of reaction support, a load sensor, a servo hydraulic jack system, a bilateral loading pier, and a dual channel displacement monitoring unit (LVDT). A vertical load is applied at the top of the specimen at a constant rate of 0.2 mm/min by using the displacement control loading mode. In order to optimize the force flow transmission path, a high-strength-bearing steel plate (30 mm thick, Q345 steel) was set between the specimen and the buttress. Two sets of high-precision LVDT displacement sensors are symmetrically arranged at the bottom edge of the specimen to collect the vertical deformation data under load in real time, as shown in Figure 2.

3. Test Results and Analysis

3.1. Failure Process and Crack Distribution

Through experimental observation, all specimens showed a typical diagonal compression failure mode. The failure mechanism is as follows: after the longitudinal tensile steel bar reaches the plastic yield strength, a concrete crushing zone is formed along the direction of the main compressive stress trace, accompanied by the dense development of oblique shear cracks.

Specimen A0-S0.61 is shown in Figure 3a, when V = 269.8 kN, and the crack first appears in the concave corner between the column and the right corbel. When V = 332 kN, oblique cracks appear at the concave corner of the left corbel. The cracks on both sides continue to develop upward from the concave corner to the column. When V = 545 kN, there are oblique cracks in the right corbel. Subsequently, the cracks extended to the bracket and the root of the bracket, and dense oblique cracks were generated at the bottom of the bracket. When V = 1285 kN, the concrete at the root of the right corbel begins to peel off.

Specimen A15-S0.61 is shown in Figure 3b. When V = 271 kN, the crack first appears at the concave corner between the column and the right bracket. When V = 337 kN, oblique cracks appear at the concave corner of the left corbel. The cracks on both sides continue to develop upward from the concave corner to the column. When V = 430 kN, vertical cracks appear on the right side of the corbel. When V = 528 kN, oblique cracks appear on the edge of the left support of the bracket, which develops to the root of the bracket. When V = 652 kN, oblique cracks appear on the right side of the corbel, and the cracks extend to the root of the corbel and the support, and dense oblique cracks are generated at the bottom of the support. When V = 1732 kN, the concrete began to peel off at the root and support of the left corbel.

The failure behavior of the two RC corbel specimens occurs on the right side, which seems to be unreasonable. However, due to the small number of test samples, the randomness of concrete aggregate distribution, and the inherent inhomogeneity of micro-crack failure, it cannot be completely symmetrical in real materials. This asymmetry does not discount the test results, but captures the real and non-ideal behavior of the material.

3.2. Load–Displacement Curves

At the initial stage of loading, the load displacement of specimen A15-S0.61 and specimen A0-S0.61 showed linear growth, but the initial stiffness of specimen A15-S0.61 was slightly larger than that of specimen A0-S0.61. In the linear elastic stage, the inclined main reinforcement of specimen A15-S0.61 produces a beneficial vertical component force, which can directly resist the external vertical load; secondly, the arrangement of the stirrups of specimen A15-S0.61 makes the integrity of the corbel and the middle column better, thus showing higher stiffness at the initial stage of loading. At the same time, the load of specimen A0-S0.61 and specimen A15-S0.61 rebounded many times after reaching the peak load, which accurately characterized the aggregate interlocking effect of the inclined crack surface. The load–displacement curves of specimen A0-S0.61 and specimen A15-S0.61 are shown in Figure 4.

When specimen A0-S0.61 is subjected to an external load of 400 kN, the displacement increment increases significantly, the load growth rate slows down, and the slope of the curve decreases significantly, indicating that the crack enters the steady-state expansion stage and the steel bar gradually dominates the force. After the displacement reaches 3.34 mm, the load fluctuates many times near 570 kN, and the coefficient of variation reaches 1.2%, which confirms that the steel bar enters the plastic deformation stage. When the load reaches the yield point of 643.1 kN, the displacement increment of 1.45 mm only increases the load by 95 kN, and the large deformation reflects the full yield of the steel bar. After the peak load of 738.5 kN, the load dropped by 1.4% to 728.28 kN, and the oblique cracks penetrated. The bearing capacity decreased to 632.2 kN, and the attenuation rate was 14.5%.

After the load of specimen A15-S0.61 exceeds the cracking load of 135.5 kN, the slope of the curve decreases significantly for the first time, indicating that the crack propagates stably. When the displacement is 3.42 mm, the load fluctuates around 700 kN, and the coefficient of variation reaches 0.8%, which characterizes the plastic change in the component. After the yield load of 825.0 kN, the slope of the curve approaches zero, the displacement increases sharply, and the load increases slightly. After reaching the peak load of 912.8 kN, it dropped by 6.3% to 855.18 kN, and the residual strength was 678.0 kN when the final displacement was 8.41 mm, and the attenuation rate was 14.5%. It is worth noting that the slope of the load–displacement curve of specimen A15-S0.61 rises briefly in the range of 841.1–912.8 kN after the yield point.

The ductility performance of concrete members is usually characterized by the ductility coefficient, which mainly includes three kinds of indexes of displacement ductility coefficient. Among them, the displacement ductility coefficient is widely used because of its convenience in measurement and universality in engineering. Based on this, this study selects the displacement ductility coefficient as the core evaluation index to study the influence of the ductility degradation law of corbel members. The coefficient calculation formula is as follows:

$$\mu ={\displaystyle \frac{{∆}_u}{{∆}_y}}$$

(1)

Table 3. Test results.

Specimen	${\mathit{V}}_{\mathit{c}\mathit{r}}$ (kN)	${\mathit{V}}_{\mathit{y}}$ (kN)	${\mathit{V}}_{\mathit{u}}$ (kN)	${∆}_{\mathit{c}\mathit{r}}$ (mm)	${∆}_{\mathit{y}}$ (mm)	${∆}_{\mathit{u}}$ (mm)	$\mathit{\mu }$ (-)
A0-S0.61	134.9	643.1	738.5	0.76	3.68	4.79	1.43
A15-S0.61	135.5	825.0	912.8	0.56	4.85	6.58	1.36

Note: The load in the table is the load on one side of the corbel; $V_{cr}$ is the cracking load; $V_y$ is the yield load; $V_u$ is the ultimate load; ${∆}_{cr}$ is the cracking displacement; ${∆}_y$ is the yield displacement ${∆}_u$ is the ultimate displacement; $\mu$ is the displacement ductility coefficient.

shows the characteristic load, characteristic displacement, and displacement ductility coefficient of the corbel specimens.

Specimen A15-S0.61 showed significant advantages in yield load and ultimate load, which were about 28.3% and 23.6% higher than that of A0-S0.61, respectively. This shows that the reinforcement angle design of specimen A15-S0.61 effectively improves its bearing capacity. Specimen A15-S0.61 also has significant advantages in yield displacement and ultimate displacement, which are about 31.8% and 37.4% higher than A0-S0.61, respectively. It shows that it can achieve greater deformation while achieving a higher load. Although the absolute deformation of A15-S0.61 is larger, its displacement ductility coefficient is slightly lower than that of A0-S0.61. This reflects that the plastic deformation reserve from yield to final failure is relatively small. However, the ductility coefficients of both are greater than 1, indicating that both have certain ductile failure characteristics.

In general, compared with specimen A0-S0.61, specimen A15-S0.61 shows significant advantages in bearing capacity and overall deformation capacity. However, the displacement ductility factor is slightly lower than A0-S0.61. This reflects the common phenomenon of the ‘trade-off relationship between strength and ductility’ in structural engineering.

3.3. Strain Development of Steel Bar

Figure 5a show that when the load is 400 kN, the strain of the main reinforcement of specimen A15-S0.61 is 620 με, and the strain of the main reinforcement of specimen A0-S0.61 reaches 1273 με; compared with specimen A0-S0.61, the strain of the main bar of A15-S0.61 with 15° inclined main bar is significantly reduced by 51.3% under the same load. When the main reinforcement yields, the load of specimen A0-S0.61 is 678.5 kN, while the load of specimen A15-S0.61 reaches 826.2 kN; in contrast, the load of specimen A15-S0.61 with 15° inclined main reinforcement is significantly increased by 21.7% when the same yield strain is reached. This shows that the change in the inclination angle of the corbel main reinforcement optimizes the stress path of the concrete and forms a more efficient composite stress mechanism of reinforced concrete.

As shown in Figure 5b, the first layer of stirrups of both corbel specimens reached yield. Under a load of 250 kN, the stirrup strain of specimen A0-S0.61 is 1078 με, while that of specimen A15-S0.61 is only 832 με. Under the same load, the stirrup strain of the first layer of A15-S0.61 with 15° inclined stirrups is reduced by about 23.7%. When the stirrups yield, the yield load of specimen A0-S0.61 is 620.5 kN, while that of specimen A15-S0.61 is increased to 692.7 kN, with a load increase of 11.6%.

The load–strain relationship of the second-layer stirrup is shown in Figure 5c, and the stirrups are not yielded. At the initial stage of loading, the strain of stirrups A0-S0.61 and A15-S0.61 fluctuated around zero, and local compression occurred. The stirrups of A0-S0.61 and A15-S0.61 began to enter a stable tensile state when the load reached 116.8 kN and 221.1 kN, respectively. When the load is in the range of 248.8–293.3 kN, the stirrup strain of specimen A0-S0.61 increases rapidly. When the stirrup strain is 250 με, the load of specimen A0-S0.61 is 250.9 kN, while the load of specimen A15-S0.61 is increased to 304.9 kN, and the load increase is 21.5%. Under the load of 300 kN, the stirrup strain of specimen A15-S0.61 is 245 με; the stirrup strain of specimen A0-S0.61 is 735 με, and the stirrup strain is reduced by about 66.6%. With the increase in load, the stirrup strain of specimen A0-S0.61 reaches the maximum value of 995 με, which is lower than the yield strain. The stirrup strain of specimen A15-S0.61 reaches the maximum value of 772 με, which is much lower than the yield strain. This further verifies that the inclined stirrups help to reduce the local stress concentration, thereby improving the stress distribution of the component and delaying the yield process.

The load–strain relationship of the third-layer stirrup is shown in Figure 5d, and the stirrups are not yielded. Under a load of 320 kN, the stirrup strain of specimen A15-S0.61 is 109 με; the stirrup strain of specimen A0-S0.61 is 212 με, and the stirrup strain is reduced by about 48.5%. When the load reaches 400 kN, the tensile strain of the stirrup of specimen A15-S0.61 begins to decrease. As the load increases to 532.9 kN, the stirrup begins to be compressed. When specimen A15-S0.61 reaches the ultimate load, the compressive strain of the stirrup is 416 με. Specimen A0-S0.61 always maintains the tensile strain state with the increase in load and the increase is gentle. The third-layer stirrup of specimen A15-S0.61 experienced a significant transition from tension to compression during the loading process. This may be due to the passive extrusion formed by the lateral expansion of the concrete strut or the large overall rotational deformation.

4. Finite Element Model

ABAQUS 2022 [22] software has advanced capabilities in dealing with significant material and geometric nonlinearity in the model; its solver is very suitable for solving such complex static problems. In addition, the concrete damage plasticity model (CDP) built into the software has been widely recognized and verified in simulating the failure mechanism of reinforced concrete structures.

4.1. Material Models

Firstly, the density and elastic behavior of ordinary concrete are input. According to the test, the average compressive strength of ordinary concrete is 40.83 MPa, and the elastic modulus is converted from the European standard [23]. The density of concrete is 2.45 × 10⁻⁹ t/mm³, the Young’s modulus is 34,096 MPa, and the Poisson’s ratio is 0.2. The conversion formula of elastic modulus is as follows:

$$E_{ci}=E_{c0}{\alpha }_E{\left({\displaystyle \frac{f_{cm}}{10}}\right)}^{1/3}$$

(2)

$E_{ci}$ is the elastic modulus of concrete at 28 days of age; $E_{c0}$ is the standard elastic modulus and the value is 21,500 MPa; ${\alpha }_E$ is the gradation coefficient and the value is 1.0; and $f_{cm}$ is the average compressive strength of the test.

For the plastic behavior of concrete, the concrete damage plasticity model in the ABAQUS 2022 [22] material library is used to simulate the stress–strain relationship of ordinary concrete. This model is often used to simulate the nonlinear behavior of concrete and is suitable for analyzing the damage evolution and plastic deformation of concrete structures under complex stress states.

The model CDP parameters are shown in

Table 4. CDP model parameters.

Dilation Angle (Ψ)	Eccentricity (θ)	${\mathit{f}}_{\mathit{b}\mathit{o}}/{\mathit{f}}_{\mathit{c}0}$	${\mathit{K}}_{\mathit{c}}$	${\mathit{\mu }}_{\mathit{c}}$
38	0.1	1.16	0.6667	0.001

. The expansion angle Ψ is responsible for controlling the dilatancy effect in the plastic deformation of concrete. Generally speaking, increasing the expansion angle will increase the expansion range of concrete under compression, and the value range is controlled from 30° to 40°. The eccentricity θ controls the shape of the plastic potential function on the deviatoric plane, which determines the curvature of the function in the low-hydrostatic-pressure region. The recommended value in the CDP model is 0.1. $f_{bo}/f_{c0}$ represents the ratio of biaxial compressive strength to uniaxial compressive strength, and the value of concrete material is generally 1.16. The yield surface shape parameter $K_c$ controls the shape of the yield surface on the off plane, reflecting the strength characteristics of concrete under a multi-axial stress state. The default value of ABAQUS is 0.6667, which corresponds to Rankine’s maximum tensile stress criterion [24]. The viscosity coefficient ${\mu }_c$ improves the overall convergence of the model after concrete cracking by adding a damping term related to the strain rate, and the value of this model is 0.001.

In order to accurately characterize the uniaxial stress behavior of concrete, the concrete constitutive model recommended by the GB 50010-2010 [7] is adopted in this study. The model describes the stress–strain whole process curve using a piecewise function, and the constitutive relationship of the compression section is defined by the following equation:

$$\sigma =(1-d_c)E_c\epsilon$$

(3)

$$d_c=\left\{\begin{array}{c}1-{\displaystyle \frac{{\rho }_{c}n}{n-1+x^n}} x\le 1\\ 1-{\displaystyle \frac{{\rho }_{c}n}{{\alpha }_c{(x-1)}^2+x}} x>1\end{array}\right.$$

(4)

$${\rho }_c={\displaystyle \frac{f_{c,r}}{E_c{\epsilon }_{c,r}}}$$

(5)

$$n={\displaystyle \frac{E_c{\epsilon }_{c,r}}{E_c{\epsilon }_{c,r}-f_{c,r}}}$$

(6)

$$x={\displaystyle \frac{\epsilon }{{\epsilon }_{c,r}}}$$

(7)

The tensile section of uniaxial stress–strain curve can be calculated by the following formula:

$$\sigma =(1-d_t)E_c\epsilon$$

(8)

$$d_t=\left\{\begin{array}{c}1-{\rho }_t[1.2-0.2x^5] x\le 1\\ 1-{\displaystyle \frac{{\rho }_t}{{\alpha }_t{(x-1)}^{1.7}+x}} x>1\end{array}\right.$$

(9)

$$x={\displaystyle \frac{\epsilon }{{\epsilon }_{t,r}}}$$

(10)

$${\rho }_t={\displaystyle \frac{f_{t,r}}{E_c{\epsilon }_{t,r}}}$$

(11)

$$d_t=1-\sqrt{{\displaystyle \frac{\sigma }{E_t{\epsilon }_t}}}$$

(12)

In the above equations, ${\alpha }_c$ is the parameter value of the descending section of the uniaxial compression stress–strain curve of concrete; $\sigma$ is the actual uniaxial stress value of concrete; and $d_c$ and $d_t$ are the damage evolution parameters of concrete under uniaxial compression and tension, respectively. ${\epsilon }_{c,r}$ and ${\epsilon }_{c,t}$ are divided into the peak strain of concrete corresponding to the uniaxial compressive strength ($f_{c,r}$) and tensile strength ($f_{c,t}$) of concrete; ${\alpha }_t$ is the parameter value of the descending section of the uniaxial tensile stress–strain curve of concrete; and $E_c$ is the elastic modulus of concrete.

The uniaxial stress–strain curve of ordinary concrete is shown in Figure 6. Since the concrete strain input in the finite element software is inelastic strain, the curve obtained at this time cannot be directly input, and the inelastic strain needs to be calculated.

The inelastic strain ${\tilde{\epsilon }}_c^{in}$ represents the irreversible strain produced by the material during plastic deformation or damage, which is essentially the residual strain component after the total strain is deducted from the elastic strain. In the ABAQUS concrete CDP damage plasticity model, the material parameters need to be input based on the constitutive relationship defined by inelastic strain. The compressive inelastic strain can be calculated using the following formula:

$${\tilde{\epsilon }}_c^{in}={\epsilon }_c-{\tilde{\epsilon }}_{0c}^{el}$$

(13)

$${\tilde{\epsilon }}_{0c}^{el}={\displaystyle \frac{{\sigma }_{un}}{E_0}}$$

(14)

Similarly, the tensile inelastic strain ${\tilde{\epsilon }}_t^{in}$ in can be calculated using the following formula:

$${\tilde{\epsilon }}_t^{in}={\epsilon }_t-{\tilde{\epsilon }}_{0t}^{el}$$

(15)

$${\tilde{\epsilon }}_{0t}^{el}={\displaystyle \frac{{\sigma }_{un}}{E_0}}$$

(16)

where ${\tilde{\epsilon }}_{0c}^{el}$ is the elastic compressive strain under the initial stiffness; ${\epsilon }_c$ is the actual compressive strain; ${\sigma }_{un}$ is the stress value of a point on the curve; ${\tilde{\epsilon }}_{0t}^{el}$ is the elastic tensile strain under the initial stiffness; ${\epsilon }_t$ is the actual tensile strain; and $E_0$ is the initial elastic modulus.

The definition of damage factor in ABAQUS is the loss ratio of elastic modulus, while the damage evolution parameters in the GB50010-2010 [7] are used to describe the trend of the descending section of the curve. Sidiroff [25] used the principle of energy equivalence to calculate the damage factor. The formula is as follows:

$$d_{ac}=1-\sqrt{{\displaystyle \frac{{\sigma }_{tr}}{E_0{\epsilon }_c}}}$$

(17)

$$d_{at}=1-\sqrt{{\displaystyle \frac{{\sigma }_{tr}}{E_0{\epsilon }_t}}}$$

(18)

where $d_{ac}$ and $d_{at}$ represent the compressive damage factor and the tensile damage factor, respectively, and ${\sigma }_{tr}$ is the true stress.

The results of transforming the uniaxial stress–strain relationship of ordinary concrete are shown in Figure 7.

In this paper, according to the stress–strain constitutive relation curve of the steel bar provided by GB50010-2010 [7], it is a double diagonal elastic–plastic model. Specifically, this model shows that the steel bar presents an ideal elastic behavior before yielding, and presents a broken line after yielding. This can more realistically simulate the plastic strengthening effect of steel bars during tension, and the curve is shown in Figure 8. The stress–strain constitutive relation is calculated according to the following formula:

$${\sigma }_s=\left\{\begin{array}{c}{E}_s{\epsilon }_s {\epsilon }_s\le {\epsilon }_y\\ f_y+E_s^{\prime }({\epsilon }_s-{\epsilon }_y) {\epsilon }_s>{\epsilon }_y\end{array}\right.$$

(19)

In the formula, ${\sigma }_s$ is the stress of the steel bar; ${\epsilon }_s$ is the steel strain; $E_s$ is the initial elastic modulus of the steel bar; $E_s^{\prime }$ is the elastic modulus of the steel bar after yielding; and ${\epsilon }_y$ is the yield strain of the steel bar.

4.2. Element Types and Meshing

In the ABAQUS model, the three-dimensional solid linear reduced integral element (C3D8R) is selected for the concrete element type. The element shows high accuracy in displacement calculations and is suitable for various mesh subdivision situations. Although its calculation accuracy is not as good as that of the complete integral element, it significantly reduces the time required for calculation. In order to facilitate the convergence of the model, the three-dimensional truss element (T3D2) is used for column longitudinal reinforcement, the column stirrup, the bracket main reinforcement, and the stirrup.

Because the mesh density of the model has a great influence on the static analysis results, the results of the model are different with the change in the mesh size. In order to achieve a more accurate simulation in a shorter time, the optimal grid size is sought. Three grid sizes of 25 mm, 30 mm, and 35 mm were set for specimen A0-S0.61. As shown in Figure 9, the influence of different mesh sizes on shear capacity is not obvious, and the overall trend of the early curve is basically the same. When the grid size increases to 35 mm, the peak point of the curve obviously lags behind, and the analysis deviation is larger than that of the fine grid. In order to improve the computational efficiency of the model and the accuracy of the analysis results, this paper uses a 30 mm grid for global seeding.

4.3. Boundary and Loading Conditions

As shown in Figure 10, the constraint conditions include the embedding of concrete and column longitudinal reinforcement, column stirrups, corbel main reinforcement, and corbel stirrups. By establishing boundary conditions, the displacement of 10 mm is distributed to the column section at the top of the corbel. The reference point is established at the support, the left constraint condition is the hinge support, and the right constraint condition is the fixed support. The steel plate adopts the R3D4 three-dimensional quadrilateral rigid element, which can effectively deal with the behavior of a rigid body. According to Reference [26], the friction coefficient between concrete and steel is between 0.4 and 0.7. After a trial calculation, it was found that the convergence of the model is the best when the value is 0.6.

4.4. Finite Element Analyses and Results

By extracting the data of the model results and drawing the load–displacement curve, the test curves of two reinforced concrete corbels are compared with the simulation curves, as shown in Figure 11.

The load–displacement curve obtained using the finite element simulation is similar to the experimental value in the rising section, but the stiffness is slightly different. This is due to the decrease in the effective modulus of concrete due to the existence of micro-cracks, resulting in the elastic modulus of concrete calculated by the standard formula being slightly higher than the actual effective value. Secondly, the boundary conditions are more idealized in the simulation, ignoring the flexibility of the actual support. In the descending section of the curve, the difference between the simulation results and the experimental data is particularly significant. In the test, the cracks propagated tortuously along the aggregate interface, forming a significant interlocking effect [26].

The mechanical interlocking between the crack surfaces provided a residual shear force and delayed the decrease in bearing capacity. However, the tension–compression softening curve used in the concrete constitutive model did not consider this effect, resulting in the underestimation of the bearing capacity after failure. In the test, the shear force is redistributed through the action of the uncracked concrete area and steel bar dowel, which makes the bearing capacity decrease more smoothly. The shear stiffness of the crack surface is not defined in the model, and the friction contribution of the crack interface is not considered, which weakens the resistance after failure. The simulated values of the ultimate shear capacity of the two specimens in this paper were collected and compared with the test results. The results are shown in

Table 5. Comparison of finite element simulation shear bearing capacity and test results.

Specimen	Test Result ${\mathit{V}}_{\mathit{u}}$ (kN)	Finite Element Results ${\mathit{V}}_{\mathit{f}}$ (kN)	Error $\left|({\mathit{V}}_{\mathit{u}}-{\mathit{V}}_{\mathit{f}})\right|$$/{\mathit{V}}_{\mathit{s}}$ (%)
A0-S0.61	738.5	723.5	2.03
A15-S0.61	912.8	892.1	2.27
Average Error			2.15
Standard Deviation			0.12

.

Two specimens were analyzed using the finite element method and compared with the experimental failure mode. The results of the main tendon inclination comparison group are shown in Figure 12. The failure mode of the finite element analysis is highly similar to the test, and the concrete cracking process is accurately simulated. Initial cracks are generated at the concave corner of the bracket. As the load increases, cracks appear at the support, and the width continues to grow. Finally, oblique cracks are generated at the root of the bracket.

As shown in Figure 13, through the extraction and analysis of the finite element results of the reinforcement of RC corbel specimens, it is found that the strain distribution cloud diagram of the main reinforcement of each corbel is basically consistent with the experimental results. The stirrups are pasted at the junction of the corbel and the middle column. The simulation results show that the first-layer stirrups of specimens A0-S0.61 and A15-S0.61 yield at the junction, which is consistent with the data extracted from the test. The yield points of the second and third layers of stirrups are far away from the junction of the bracket and the middle column, resulting in a slight difference from the experimental phenomenon. The concrete stress flow of specimens A15-S0.61 and A0-S0.61 formed an obvious compressive bar force flow. Under the yield load, the range of the high-stress zone of concrete in specimen A15-S0.61 is expanded, which makes the stress distribution more continuous. It is worth noting that the upper flange of specimen A15-S0.61 has a higher stress level and bears a larger bending moment. At the same time, due to the limitation of the size of the corbel, the stress flow is more concentrated at the root of the corbel. The arrangement of the oblique stirrups makes the third layer of stirrups closer to the root of the corbel, which effectively shares the compression of the concrete at the root of the corbel, which is consistent with the strain data obtained from the test.

5. Comparison of STM and TTM Models

5.1. STM

In the field of corbel design, STM is one of the mainstream methods widely used in the world, such as for the American standard ACI 318-19 [8], European standard EC 2 [9], and Canadian standard CSA A23.3-04 [10]. The STM simplifies the corbel into a truss system composed of three parts: a tensile steel bar (tie bar), a concrete compression bar (compression bar), and a joint area. The core idea of the model is that the ultimate bearing capacity of the corbel depends on the state in which any of the three bars, struts, or connection areas reaches its ultimate strength first. Taking the American specification as an example, the definitions and values of the key design parameters are listed in

Table 6. Parameter value of STM model.

Ratio of Shear Span to Depth	Angle Between Strut and Tie ($\mathit{\theta }$)	Concrete Strut AB	Node A (CCT)	Node B (CCC)
$\lambda \le 2$	${25}^{\circ }\le \theta$	$0.64{\beta }_{\mathrm{c}}\left(\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}{\rho }_{\mathrm{h}}\ge 0.0025{/\left(\mathrm{s}\mathrm{i}\mathrm{n}\theta \right)}^2\right)$ $0.34{\beta }_{\mathrm{c}}\left(\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}{\rho }_{\mathrm{h}}<0.0025/{\left(\mathrm{s}\mathrm{i}\mathrm{n}\theta \right)}^2\right)$	$0.65 {\beta }_{\mathrm{c}}$	$0.85 {\beta }_{\mathrm{c}}$

Note: ${\rho }_{\mathrm{h}}$ is horizontal stirrups ratio; ${\beta }_{\mathrm{c}}$ is an improvement coefficient for concrete strength.

.

As shown in Figure 14, the value of $l_a$ is the width of the support plate; $h_a$ is 2 times the distance from the edge of the concrete to the center of the longitudinal steel bar; and $l_b$ is half of the column length [18]. The maximum load that the specimen can bear can be calculated using the following formula.

$$tan\theta ={\displaystyle \frac{h_0-h_b/2}{a_v+l_b/2}}$$

(20)

$$w_s=min\left\{\right(h_{a}cos\theta +l_{a}sin\theta ),(h_{b}cos\theta +l_{b}sin\theta \left)\right\}$$

(21)

$$V_{u,AB}=F_{u,AB}sin\theta =f_{ce,s}{w}_{s}bsin\theta$$

(22)

$$V_{u,AA\prime }=F_{u,AA\prime }tan\theta =f_y{A}_{s}tan\theta$$

(23)

$$V_{u,NodeA}=min\left\{\begin{array}{c}{V}_{u,ba}=F_{u,ba}tan\theta =f_{ce,n}{h}_{a}btan\theta \\ V_{u,be}=F_{u,be}=f_{ce,n}{l}_{a}b\\ V_{u,inc}=F_{u,inc}sin\theta =f_{ce,n}\left(h_{a}cos\theta +l_{a}sin\theta \right)bsin\theta \end{array}\right\}$$

(24)

$$V_{u,NodeB}=min\left\{\begin{array}{c}{V}_{u,ba}=F_{u,ba}tan\theta =f_{ce,n}{h}_{b}btan\theta \\ V_{u,be}=F_{u,be}=f_{ce,n}{l}_{b}b\\ V_{u,inc}=F_{u,inc}sin\theta =f_{ce,n}\left(h_{b}cos\theta +l_{b}sin\theta \right)bsin\theta \end{array}\right\}$$

(25)

$$f_{ce,s}=0.85{\beta }_s{f}_c^{\prime }$$

(26)

$$f_{ce,n}=0.85{\beta }_n{f}_c^{\prime }$$

(27)

$$h_b={\beta }_{1}c={\displaystyle \frac{f_y{A}_s}{0.85f_c^{\prime }b}}$$

(28)

$$V_u=min\left\{\begin{array}{c}{V}_{u,AB}\\ V_{u,A{A}^{\prime }}\\ V_{u,NodeA}\\ V_{u,NodeB}\end{array}\right\}$$

(29)

5.2. Chinese Code GB 50010-2010 Method

GB 50010-2010 [7] applies to short corbels (shear span ratio ≤ 1). This model idealizes the top longitudinal reinforcement as a tension tie and the web concrete as a compression strut, forming a triangular truss. As shown in Figure 15a, the bearing capacity equation is derived from moment equilibrium at point A as follows:

$$f_y{A}_s{h}_1=F_v{a}_v+F_h\left(h_1+a_s\right)$$

(30)

Under the actual test loading conditions, the horizontal load is zero, and the formula is rewritten as follows:

$$F_v\le {\displaystyle \frac{0.85f_{\mathrm{y}}{A}_{\mathrm{s}}{h}_0}{a_v}}$$

(31)

The maximum load that specimen A0-S0.61 can bear can be calculated using the above formula. Based on the triangular truss model, we adjust the calculation model after the synergistic change in the inclination angle, as shown in Figure 15b. According to the geometric relationship of the triangle, the following formula can be obtained:

$$B_0\mathrm{C}=a_v\cdot \sin \alpha /\sin (\alpha +\theta )$$

(32)

The equation can be obtained by balancing the bending moment at point A as follows:

$$f_y{A}_s{\cdot h}_1{a}_v^{\prime }/B_0\mathrm{C}=F_v^{\prime }{a}_v^{\prime }$$

(33)

Further simplification can be obtained as follows:

$$F_v^{\prime }\le {\displaystyle \frac{{0.85f}_y{A}_s{h}_0}{a_v}}\cdot {\displaystyle \frac{\sin (\alpha +\theta )}{\sin \alpha }}$$

(34)

Through the comparison of Formulas (31) and (34), it can be seen that the size of $F_{v,max}^{\prime }$ and $F_{v,max}$ is equivalent to the size of $\sin (\alpha +\theta )/\sin \alpha$ and 1, while $\alpha$ and $\theta$ are both acute angles due to the limitation of the size of the bracket. When $\alpha +\theta \le 90$°, $\sin (\alpha +\theta )/\sin \alpha$ is always greater than 1; when $\alpha +\theta >90°$, $\sin (\alpha +\theta )/\sin \alpha$ is not necessarily greater than 1. In order to determine the angle of the pressure bar, Formula (20) shows that the angle between the pressure bar and the horizontal angle $\alpha \approx$ 47.5°. Obviously, $\alpha +\theta \le 90$°, and when the reinforcement angle is 15°, the bearing capacity is improved, which is consistent with the test results. By taking $\alpha$ = 47.5°, $\theta$ = 15° into Formula (34), we can obtain: $F_{v,max}^{\prime }$ = 584.7 kN. It is worth noting that $F_{v,max}^{\prime }/F_{v,max}$ = 1.2031, and the ratio of the test results is 1.236. Based on the TTM model, the angle parameter between the tie rod and the horizontal plane is introduced, which can better predict the influence of the inclined tie rod on the bearing capacity of the corbel members. The predicted bearing capacity improvement ratio of the adjusted TTM model is $F_{v,max}^{\prime }/F_{v,max}$ = 1.2031, which is highly consistent with the experimental measured improvement ratio of 1.236. This confirms the feasibility of quantifying the bearing capacity gain by adjusting the horizontal angle of the tie rod, and provides a theoretical basis for the optimization design of the dip angle.

As described in Section 5.1, $l_b$ = 300 mm; $h_a$ = 60 mm; $a_v$ = 350 mm; according to the different types of C-C-T and C-C-C nodes, the values of ${\beta }_n$ are 0.8 and 1.0, respectively; and $l_b$ is calculated to be 50.54 mm by Formula (28). The ultimate bearing capacity V = 382.5 kN of specimen A0-S0.61 can be calculated. This is because the contribution of STM to the stirrup is not directly calculated as the main tie rod, but through its confinement effect, the effective strength of the concrete compression bar is significantly improved, thereby indirectly but fundamentally improving the bearing capacity of the entire joint. For specimen A15-S0.61, the inclined arrangement of the stirrups is more conducive to restricting the expansion of the concrete struts. The component of the stirrups arranged horizontally on the vertical compression bar is 0.737$A_h{f}_y$; the stirrup with an inclination angle of 15 reaches 0.887$A_h{f}_y$. The ratio of the two is highly consistent with the experimental data. As a lower bound method based on limit analysis, STM has proposed two force transfer mechanisms: direct strut action and indirect truss action [21]. Although many attempts have been made, it is still difficult to decouple these two effects for RC corbels with oblique ribs. The bearing capacity of specimen A15-S0.61 was estimated by the difference in stirrup contribution.

Using the above calculation method, the bearing capacity of the test specimens was calculated and counted, as shown in

Table 7. The prediction results of STM and TTM models.

Specimen	STM		TTM
	${\mathit{V}}_{\mathit{M}\mathit{a}\mathit{x}}$ (kN)	${\mathit{V}}_{\mathit{M}\mathit{a}\mathit{x}}/{\mathit{V}}_{\mathit{u}}$	${\mathit{V}}_{\mathit{M}\mathit{a}\mathit{x}}$ (kN)	${\mathit{V}}_{\mathit{M}\mathit{a}\mathit{x}}/{\mathit{V}}_{\mathit{u}}$
A0-S0.61	382.5	0.518	486.1	0.658
A15-S0.61	460.4	0.504	584.7	0.641

.

6. Application Limitations and Future Work

In this experiment, two full-scale RC corbel specimens were used, and the sample space was small. The inclination angle parameter setting is limited, and other possible favorable angles (such as 30°, 45°) are not studied. The calculation model has certain limitations for the calculation after the synergistic change in the inclination angle of the main stirrup. The contribution of the adjusted TTM model to the shear performance of the stirrup is not clear, and the role of the load transmitted to the inclined main reinforcement through the concrete is not considered and still needs further discussion. The recommendations for future research, inspired by the findings and limitations of this study, include the following:

• Parametric Expansion: Future studies should explore a broader spectrum of inclination angles and various combinations of inclined main bars and stirrups to identify the global optimum for shear strength and ductility.
• Complex Loading Conditions: Investigating the performance under seismic loading and long-term sustained loading is crucial to understanding the behavior in real-world scenarios
• Durability Aspects: Research into how inclined reinforcement layouts affect crack width development and long-term durability, particularly regarding corrosion resistance, would be highly valuable.
• Development of Design Guidelines: A major goal should be the development of practical design guidelines and simplified analytical methods that incorporate the benefits of optimized reinforcement inclination.

7. Conclusions

In this paper, the influence of the inclination angle of the main reinforcement and stirrup on the shear strength of RC corbels is systematically studied using experimental, high-fidelity finite element simulation, and theoretical calculation methods. The conclusions are as follows:

(1)

The 15° inclined reinforcement scheme (main bars + stirrups) markedly improved shear performance compared with conventional horizontal layouts: Yield load increased by 28.3% (from 643.1 kN to 825.0 kN). Ultimate load increased by 23.6% (from 738.5 kN to 912.8 kN). This confirms that optimizing reinforcement angles unlocks substantial untapped structural potential ignored by current design codes.

(2)

It is revealed that the inclined steel bar reshapes the internal force transmission. At 400 kN, the main rod strain of the 15° specimen is reduced by 51.3%, indicating that the stress concentration at the corner is reduced. The stirrups showed a delayed yield (yield load increased by 11.6%) and lower strain level (decreased by 23.7% at 250 kN), indicating that the confinement of concrete struts was enhanced. The finite element stress cloud diagram further verifies that the oblique steel bar optimizes the stress path of the oblique concrete strut and makes the stress distribution more uniform.

(3)

The ABAQUS CDP model achieved a high accuracy: The ultimate load prediction errors were ≤2.27% for both specimens. The model also produced a faithful reproduction of the failure modes (diagonal compression crushing) and crack propagation patterns.

(4)

While the 15° specimen showed a superior strength and deformation capacity (37.4% higher ultimate displacement), its displacement ductility coefficient was slightly lower than the 0° specimen. This highlights a design balance: inclined angles enhance load-bearing efficiency but marginally reduce the plastic deformation reserves post yield.

(5)

Based on the triangular truss model of the horizontal inclination angle of the tie rod, the parameter of the angle between the tie rod and the horizontal angle is introduced, and the predicted ratio is highly consistent with the ratio of the experimental value, which confirms the feasibility of quantifying the bearing capacity gain by adjusting the horizontal inclination angle of the tie rod. At the same time, compared with the American strut-and-tie model, the predicted value of the triangular truss model is closer to the experimental value.