Finite Element Analysis of the Shear Performance of Reinforced Concrete Corbels under Different Design Codes

Abstract

This study aimed to investigate the shear performance of reinforced concrete corbels and to evaluate the accuracy and safety of the Chinese code GB 50010-2010’s triangular truss model and the American code ACI 318-19’s strut-and-tie model under various design parameters with a specified design load. A total of 22 corbel specimens with different dimensions and reinforcement configurations were designed and simulated using the finite element software ABAQUS 2020, incorporating the microplane M7 material model, which was validated against experimental data. The findings reveal that for corbels with high-strength concrete or larger shear spans, the Chinese code offers a higher safety margin. Conversely, the safety margin according to the American code initially increases and then decreases with the enhancement of concrete strength, while changes in the shear span have an insignificant impact on the safety margin, which tends to decrease as the shear span increases. Additionally, the inclusion of stirrup reinforcement significantly improves the load-bearing capacity of corbels, with an increase ranging from 15% to 46% compared to those without stirrups.

1. Introduction

Reinforced concrete corbels, serving as vertical load-bearing elements that provide cantilever support, are extensively utilized in various structural applications such as industrial buildings, road and bridge engineering, hydraulic structures, and power plant facilities. Particularly in precast concrete structures, corbels are favored for their convenient and rapid connection during construction, allowing for the efficient assembly of precast beams and columns through corbel joints. These corbels play a pivotal role in transferring loads from the beams to the columns, acting as a critical force transmission hub.

Despite the widespread use of corbels, their complex force characteristics and susceptibility to shear failure, which is a brittle mode of failure, pose significant challenges in design and safety. The current design codes, such as those from China, the United States, Europe, and Canada [1,2,3,4,5], offer guidance based on different theoretical models, including the triangular truss model, the strut-and-tie model (STM), and shear friction theory. However, these models, often semi-empirical, may not fully capture the influence of key parameters on the strength of corbels, as noted in some studies [6,7,8].

While experimental studies [9,10,11,12,13,14,15,16] have long been the cornerstone of research on corbel behavior, the advancement of computational techniques and finite element software has enabled more accessible and insightful nonlinear analyses of reinforced concrete corbels. Such analyses not only reduce costs but also reveal internal force conditions that are difficult to observe in experiments.

Despite the potential of numerical analysis, there is a dearth of studies focusing on the simulation of reinforced concrete corbels. This highlights the need for an effective and experimentally validated numerical model to predict the load-bearing capacity of corbels and guide engineering design.

The current literature predominantly investigates the influence of various parameters on the load-bearing capacity of corbels and evaluates the accuracy and applicability of different codes and formulas [17,18]. However, the impact of parameter variations on the accuracy of codes, while maintaining the same design load capacity, remains underexplored. In order to explore this issue, this study utilized ABAQUS software and the microplane M7 material model to simulate 13 corbel specimens, with the simulation results closely matching the experimental data in terms of load-bearing capacity, deformation, and failure modes. Building on this foundation, a numerical analysis was conducted on 22 corbel specimens with a design load of 750 kN to assess the accuracy and safety of Chinese and American code methods under varying concrete strengths, shear spans, and corbel heights. Additionally, this study explored the impact of the stirrup configuration on the load-bearing capacity of corbel components and the influence of design parameters on the strut coefficient in corbel components. The uniqueness of this study lies in comparing the responses of different code methods to key design parameter variations while maintaining a consistent design load constraint, aiming to provide insights that may aid in the design of corbels with a higher load-bearing capacity and economic optimization, thereby enhancing the economic benefits and safety standards.

2. The Establishment of a Finite Element Model

2.1. Material Models

A 3D detailed reinforced concrete numerical model was developed using ABAQUS finite element software. The microplane M7 material model [19,20] was utilized to simulate the nonlinear mechanical behavior of concrete materials.

The accuracy of a material’s constitutive model determines the precision and effectiveness of finite element structural analysis. Existing concrete material models can be classified into two categories: (1) the macroscopic model, which simplifies material behavior from complex microstructural stress transfer mechanisms to an average stress–strain relationship at the continuum level, establishing the material’s constitutive relationship by directly providing the correlation between macroscopic stress and strain [21]; (2) the microscopic model, which describes material behavior at the microscopic level, focusing on the phenomena that cause macroscopic behavior, such as the development of micro-cracks, and characterizes material behavior through mathematical modeling of the microscopic mechanisms.

The microplane material model falls into the category of microscopic models. The initial concept of the microplane theory originated from Taylor, who proposed that the constitutive relationship of polycrystalline metals could be established by considering the stress–strain relationship on various planes within the material. Taylor suggested that under static or dynamic conditions, the macroscopic stress or strain tensor is the sum of stress or strain vectors acting on all planes. Bažant et al. [22] extended Taylor’s concept to brittle materials with softening properties, such as concrete, and introduced the microplane model.

Using the secondary development platform of ABAQUS finite element software, the microplane M7 material model subroutine [20] was integrated into the program for analysis and calculation, utilizing the user-defined interface provided by the program. In the microplane M7 material model, the values of k₁, k₂, k₃, and k₄ are taken to be 0.00035, 50, 25, and 65, respectively, with all other parameters adopting their default values. The Poisson’s ratio for concrete is taken as 0.18. In the absence of an experimentally determined elastic modulus, the elastic modulus E is calculated using the formula E = 4700 × f_c^0.5, where f_c is the cylinder compressive strength. The reinforcement constitutive model adopts a bilinear stress–strain relationship, with the slope of the second section being 0.01 of the slope of the first section. The elastic modulus is 2.0 × 10⁵ MPa, Poisson’s ratio is 0.3, and the yield strength and ultimate strength are determined through testing.

2.2. Element Types and Meshing

The concrete was modeled using a reduced integral, eight-node solid element (C3D8R). A two-node, linear, three-dimensional truss element (T3D2) was employed to model the reinforcement.

Since softening damage is involved, the crack band theory [23] is often utilized to manage crack propagation and post-peak compression behavior, thereby avoiding mesh sensitivity arising from strain localization [24,25]. In the crack band theory, softening damage is assumed to exist within a band area of finite width. In the finite element model, the crack zone width is equal to the element size. During modeling, the element size must be adjusted to ensure that the area under the stress–strain curve equals the material’s fracture energy divided by the crack width. For different element sizes, the stress–strain relationship needs to be readjusted accordingly. To simplify the application of complex material models, the model can be calibrated using predefined model parameters and element size, with the cell size remaining unchanged in subsequent simulations [26,27]. The embedding method is employed to simulate the bond between concrete and reinforcement, assuming a perfect bond. This assumption of perfect bonding between reinforcement and concrete in finite element analysis is a modeling simplification that increases the stiffness of the load–displacement response without significantly affecting the failure mechanism and ultimate capacity of the model. In practical engineering, this assumption is justified when anchorage failure is not the dominant mode of test failure, as reinforcement slippage is limited [28]. Given that no anchorage or pull-out failure of reinforcement occurred in the tests, the assumption of a perfect reinforcement connection is viable.

2.3. Boundary and Loading Conditions

Figure 1 presents the finite element model of the corbel, with three reference points precisely positioned at the geometric centers of the loading surface and the two support surfaces. Utilizing the software’s Constraint Manager, the references are constrained to their respective surfaces by setting the type to Rigid Body and selecting the Tie (nodes) region type. This configuration ensures that the boundary conditions applied to the reference points allow the entire surface to bear forces collectively, effectively functioning as a pad.

At the support reference points, the left side is assigned Encastre conditions for complete fixation, while the right side is set with Pinned conditions to allow for rotational freedom. A controlled displacement load with a smooth amplitude option is applied at the top loading surface reference point. The strategic selection of these boundary conditions simulates common support configurations in structural analysis, ensuring stable and controllable model responses.

3. Verification of the Finite Element Model

3.1. Experimental Program

Thirteen reinforced concrete corbel specimens, as described and produced in Yin Wenmeng’s paper [29], were employed for simulation verification. The specific details and ultimate bearing capacities of the corbels are outlined in

Table 1. Summary of corbel specimens.

Specimen No.	h/mm	fc/MPa	fy/MPa	As/mm2	ρs/%	ρst/%	av/mm	Vt/kN	VFEM/kN	Vt/VFEM
C30-S0.42-C	500	23.8	478	393	0.276	0.213	200	572.5	586.61	0.98
C45-S0.50-C	425	54.2	458	452	0.378	0.262	200	623.5	619.25	1.01
C60-S0.57-C	375	44.2	458	565	0.540	0.300	200	593.0	594.55	1.00
C45-S0.31-C	350	54.2	478	339	0.349	0.208	100	631.5	649.51	0.97
C45-S0.67-C	475	50.4	459	615	0.458	0.350	300	678.5	673.51	1.01
C30-S0.57-A	375	27.7	422	804	0.772	0.452	200	491.5	518.78	0.95
C45-S0.67-A	325	54.2	425	942	1.064	0.532	200	729.5	727.88	1.00
C60-S0.73-A	300	44.2	492	1017	1.251	0.579	200	754.0	767.44	0.98
C45-S0.33-A	325	54.2	459	615	0.688	0.527	100	824.0	849.41	0.97
C45-S0.80-A	400	50.4	492	1017	0.914	0.564	300	662.0	589.43	1.12
C45-S0.33-A0	325	54.2	459	615	0.688	0	100	658.5	693.30	0.95
C45-S0.67-A0	325	54.2	425	942	1.064	0	200	611.5	606.33	1.01
C45-S0.80-A0	400	50.4	492	1017	0.914	0	300	445.0	479.58	0.93
								Average		0.99
								Coefficient of variation		0.05

Note: h is the corbel height; f_c is the cylinder compressive strength; f_y is the yield strength of reinforcement; A_s is the area of tensile reinforcement; ρ_s is the ratio of tensile reinforcement; ρ_st is the ratio of stirrup reinforcement; a_v is the shear span; V_t is the measured value of the single corbel bearing capacity; V_FEM is the finite element simulation value of the single corbel bearing capacity.

. The corbels have a width of 300 mm, a height of 200 mm at the end of the corbels, and a concrete protective layer thickness of 20 mm.

3.2. Effect of Mesh Sizes

When conducting finite element analysis of reinforced concrete structures, the selection of mesh size is crucial for the accuracy of the simulation results. Quasi-brittle materials such as concrete exhibit strain softening behavior after being subjected to tension or compression beyond their elastic limits. This phenomenon leads to a descending branch in the stress–strain relationship post-peak, and simulating this behavior is highly sensitive to the discretization of the mesh.

To explore the impact of mesh size on the overall structural response and to identify an appropriate mesh size, this study considered four different mesh sizes. Figure 2 illustrates the effect of various mesh sizes on the load–deflection curve and the structural failure load. It can be observed that a mesh size of 30 mm yields better results, showing closer agreement with experimental outcomes. In the model presented within this paper, a mesh size of 30 mm is adopted.

3.3. Load–Displacement Curve

Figure 3 presents a comparison of the load–displacement curves of some corbels between the test and simulation analysis. The numerical results of the simulated bearing capacities of all corbels are summarized in Table 1. The average ratio of the tested ultimate bearing capacity to the simulated ultimate bearing capacity is 0.99, with a coefficient of variation of 0.05. Overall, the simulation results align well with the experimental data. The simulated peak load and peak displacement closely match the experimental results, although the stiffness in the simulation is slightly higher. This is attributed to the assumption of a perfect bond between reinforcement and concrete, as well as the idealized boundary conditions, which do not account for the influence of fixtures, test device elasticity, clearance, and other factors. In finite element analysis, the post-peak gradient of the load–displacement curve may appear steeper than observed in experiments. Corbels, being susceptible to brittle failure, show minimal deformation prior to failure, after which deformation increases rapidly and the load drops significantly. The force-controlled loading method used in the experimental process can hinder the precise collection of data during the descending phase. Consequently, the descending segments in experimental load–displacement curves typically serve as an indicative rather than an exact representation. Furthermore, in practical engineering applications, the primary focus is on the load-bearing capacity of the corbel; hence, this paper does not consider or compare the descending segments of the curves.

3.4. Failure Mode

During experimental observation, as the load increased, short bending cracks began to appear at the junction between the two sides of the corbel support and the middle column. These cracks then extended upward. Subsequently, oblique cracks gradually developed along the path from the inner side of the cushion plate to the bottom of the corbel. As the number and width of these cracks increased, they progressed and eventually traversed the entire area, leading to structural damage. At this stage, the longitudinal reinforcement had yielded, often accompanied by spalling of the surface concrete in the oblique area or crushing of the concrete at the bottom of the corbel.

In the finite element model, concrete is treated as a continuous medium even after cracking, and crack development is simulated by reducing the material’s stress. While the model does not allow for the direct definition of cracks, the trend of crack development can be inferred from the true strain distribution map of the concrete. For instance, in specimen C45-S0.33-A, the strain and stress distribution map of the finite element model at different loading stages is shown in Figure 4. The simulation’s initial cracking load is 207.85 kN, compared to the test value of 251 kN. The slight discrepancy is acceptable because subtle cracks may be challenging to observe directly, potentially leading to a delay in crack identification in experiments, and actual cracking may occur earlier. Figure 4c,d display the surface and internal strain distribution maps of the corbel member at the maximum load, respectively, with simulated crack patterns aligning well with experimental observations. Figure 4e,f show stress distribution maps of concrete and rebar under the maximum load, revealing stress distribution characteristics consistent with the STM model’s assumptions in the code [2]. Although finite element analysis has limitations in capturing all fracture details, it remains highly accurate and practical for evaluating the failure pattern and fracture development path of structures at a macro level.

4. Finite Element Parameter Analysis and Results

4.1. Parameter Design

In the realm of structural engineering design, the width of reinforced concrete corbels is typically pre-established. For a specific corbel size and a predetermined target design load, a variety of design options can be formulated by adjusting key parameters such as concrete strength, shear span-to-depth ratio, and corbel height in accordance with current codes. However, the precise impact of these diverse design schemes on the actual bearing capacity and mechanical performance of the corbel remains unclear. Thus, the exploration of the optimal performance and economic benefits in corbel design is deemed a question that merits in-depth investigation.

Given that the corbel’s width is typically consistent with the column width and is predefined, it was not incorporated as a variable in the parameter set for this study. The focus was placed on the concrete strength and shear span-to-depth ratio, as these are the primary factors influencing the performance of the corbel and are explicitly considered in design codes [9,13]. Additionally, to thoroughly evaluate the impact of the corbel cross-sectional size on the bearing capacity, variations in corbel height were incorporated into the parameter set. When certain parameters were altered, other variables not included in the parameter set, such as reinforcement ratio and stirrup ratio, were also adjusted to maintain the constant design load. For variables outside the parameter set, including reinforcement and stirrup ratios, under the premise that the longitudinal reinforcement satisfies the force requirements and the stirrups meet the code requirements for the stirrup ratio, the variations in these parameters had an insignificant impact on the bearing capacity.

This study was designed to examine how variations in these parameters influence the accuracy and safety of codes at a constant design load capacity. In accordance with Chinese and American codes, 22 corbel specimens were systematically designed to support a target load of 750 kN for a single corbel. Concrete strength grades ranged from 30 MPa to 70 MPa, and shear spans varied from 100 mm to 300 mm, incremented at 50 mm. For the American code corbels, the height varied from 400 mm to 600 mm, while for the Chinese code corbels, the height ranged from 550 mm to 725 mm. All specimens had a width of 300 mm and a concrete protective layer thickness of 20 mm. The pad plate dimension was 150 mm. Specimens with shear spans of 100, 150, and 200 mm had a cantilever length of 350 mm, while those with shear spans of 250 and 300 mm had cantilever lengths of 400 mm and 450 mm, respectively. The longitudinal reinforcement and stirrups had diameters of 20 mm and 10 mm, respectively, with a stirrup spacing of 100 mm. All provided stirrups were horizontal. The yield strengths for the longitudinal reinforcement and stirrups were 425 MPa and 478 MPa, respectively, and their tensile strengths were 615 MPa and 613 MPa, based on material performance tests referenced [29]. To ensure computational equivalence under the same design load, the model distributes the calculated area of longitudinal reinforcement evenly among four longitudinal bars; however, this results in a discrepancy between the modeled steel area and that of an actual bar with a 20 mm diameter. Design specifications are detailed in

Table 2. Summary of design corbel specimens.

Specimen No.	h/mm	he/mm	fc/MPa	As/mm2	ρs/%	n	ρst/%	av/mm	VFEM/kN	Vd/kN	α	VFEM0/kN	λ/%	βs
C30-S0.32-A	650	350	30	950	0.511	4	0.338	200	850.90	792.63	1.07	606.23	40	0.55
C40-S0.38-A	550	300	40	972	0.623	4	0.403	200	925.97	750.72	1.23	645.70	43	0.48
C50-S0.43-A	500	250	50	1074	0.762	3	0.334	200	990.34	750.40	1.32	828.98	19	0.52
C60-S0.48-A	450	250	60	1204	0.956	3	0.374	200	956.36	750.39	1.28	812.94	18	0.45
C70-S0.54-A	400	200	70	1373	1.237	3	0.425	200	897.82	750.42	1.20	774.99	16	0.40
C50-S0.27-A	400	200	50	869	0.783	3	0.425	100	973.52	750.29	1.30	842.57	16	0.50
C50-S0.36-A	450	250	50	984	0.781	3	0.374	150	923.61	750.16	1.23	742.04	24	0.45
C50-S0.48-A	550	300	50	1145	0.734	4	0.403	250	1013.99	750.03	1.35	785.93	29	0.50
C50-S0.53-A	600	300	50	1204	0.704	4	0.367	300	1000.13	750.40	1.33	784.88	27	0.51
C50-S0.54-A	400	200	50	1400	1.261	3	0.425	200	747.58	750.06	1.00	649.78	15	0.44
C50-S0.47-A	450	250	50	1214	0.963	3	0.374	200	857.81	750.03	1.14	713.39	20	0.46
C50-S0.38-A	550	300	50	963	0.617	4	0.403	200	1069.01	750.02	1.43	732.42	46	0.45
C50-S0.35-A	600	300	50	874	0.511	4	0.367	200	1111.27	750.24	1.48	863.72	29	0.51
C30-S0.28-C	750	400	30	623	0.267	5	0.364	200	961.24	750.20	1.28	660.11	46	0.61
C40-S0.30-C	700	350	40	623	0.308	5	0.391	200	1071.68	750.20	1.43	749.99	43	0.54
C50-S0.32-C	650	350	50	670	0.360	4	0.338	200	1175.98	750.32	1.57	928.08	27	0.55
C60-S0.34-C	625	350	60	698	0.391	4	0.352	200	1241.41	750.15	1.66	1017.34	22	0.51
C70-S0.35-C	600	300	70	729	0.426	4	0.367	200	1402.73	750.55	1.87	1165.20	20	0.51
C50-S0.19-C	550	300	50	623	0.321	4	0.403	100	1156.96	750.20	1.54	945.47	22	0.53
C50-S0.26-C	600	300	50	623	0.320	4	0.367	150	1146.49	750.20	1.53	873.48	31	0.51
C50-S0.38-C	690	350	50	787	0.397	5	0.397	250	1220.62	750.56	1.63	910.05	34	0.55
C50-S0.43-C	725	350	50	897	0.430	5	0.377	300	1269.74	750.70	1.69	958.05	33	0.59

Note: h_e is the height of the corbel’s outer edge; n is the number of stirrups; V_d is the designed load value for a single corbel; α is the ratio of the simulated load-bearing capacity to the designed load value; V_FEM0 represents the finite element analysis simulated load-bearing capacity of a single corbel without stirrups; λ is the increase in load-bearing capacity of corbels with stirrups compared to those without stirrups; β_s is the strut coefficient determined by the load-bearing capacity calculation for corbels without stirrups.

. Specimen numbering is composed of three parts: the first indicates the concrete cylinder’s compressive strength, the second represents the shear span ratio, and the third identifies the design codes, with “C” for the Chinese code and “A” for the American code. Figure 5 presents a schematic representation of the reinforcement configuration for typical corbels, where analogous reinforcement details are applied to the remaining corbel specimens in this study.

4.1.1. Chinese Code GB 50010-2010 Method

According to the Chinese specification GB 50010-2010 [1], the triangular truss model is employed for short corbels with a shear span ratio less than 1. In this model, the horizontal longitudinal bar at the top of the corbel is idealized as a tension tie, and the concrete in the abdomen is idealized as a compression strut. As depicted in Figure 6, the bending moment balance at point A yields the following equation:

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

(1)

In the formula, F_v is the design value of the vertical load, F_h is the design value of the horizontal load, and f_y and A_s are the yield strength and area of the longitudinal reinforcement, respectively. h₁ represents the internal force arm of the simplified triangular truss model, which can be taken as 0.85 times the effective height of section h₀, and (h₁ + a_s) can be approximated as 1.02h₀. a_v represents the length of the shear span.

The total cross-sectional area of the longitudinal reinforcement shall meet the following requirements:

$$A_s\ge {\displaystyle \frac{F_v{a}_v}{0.85f_y{h}_0}}+1.2{\displaystyle \frac{F_h}{f_y}},$$

(2)

In the formula, if a_v is less than 0.3h₀, take a_v to be equal to 0.3h₀.

When only a vertical load is applied, the calculation formula of the corbel bearing capacity according to the requirements of the Chinese code is expressed as follows:

$$F_v\le {\displaystyle \frac{0.85f_y{A}_s{h}_0}{a_v}},$$

(3)

The cross-sectional dimensions of the corbel should meet the requirements for crack control:

$$F_{vk}\le \beta \left(1-0.5{\displaystyle \frac{F_{hk}}{F_{vk}}}\right){\displaystyle \frac{f_{tk}b{h}_0}{0.5+a_v/h_0}},$$

(4)

where F_vk is the standard value of the vertical load, taken as 0.67F_v; F_hk is the standard value of the horizontal load, which is 0.67F_h; and f_tk is the standard value of concrete tensile strength. b is the width of the corbel; and β is the crack control coefficient, with a value of 0.65 for the corbel supporting the crane beam, and 0.80 for the other corbel.

The bearing capacity of the corbel is calculated using the moment balance of the simplified triangular truss model, which does not account for concrete crushing during bending.

In Table 2, the nine specimens whose specimen numbers end with C are corbels designed according to the Chinese code. Since the compressive strength of the cylinder is selected in the component’s design, the conversion relationship between the compressive strength of the cylinder and the cube is such that the compressive strength of the cube is 1.25 times that of the cylinder. The value of f_tk can be obtained by linear interpolation against the code table, based on the compressive strength of the cube.

4.1.2. American Code ACI 318-19 Method

The STM model of ACI 318-19 [2] is suitable for corbels with a shear span ratio of no more than two. The code assumes that the model is composed of steel reinforcement as tension ties, concrete compression struts, and nodal regions, simplifying the complex stress flow to uniaxial force rod units. The tension ties, compression struts, and nodes are the three components of the STM, which collectively bear the external load. The code requires strength checks for the compression struts, tension ties, and nodal regions. It considers that when a component reaches its ultimate strength, the member reaches its ultimate bearing capacity V_u. The STM model satisfies the static equilibrium condition and the material yield criterion. That is, each node in the model satisfies the equilibrium equation, and the stress in each member and node in the model is less than the material’s yield stress. In theory, the STM is a lower bound analysis method of plastic mechanics.

Firstly, an STM model is established. According to the principle of minimum strain energy [30], the model with the fewest and shortest tension ties is selected, which is the truss model composed of longitudinal reinforcement and diagonal concrete compression struts.

As shown in Figure 7, the model consists of CCT (two-sided compression and one-side tension) nodal regions A, A’, CCC (three-sided compression) nodal regions B, B’, concrete compression struts AB, A’B’, and steel ties AA’. The width of the CCT nodal region l_a is the width of the supporting plate, and the height h_a is twice the distance from the concrete edge to the center of the longitudinal reinforcement. The width l_b of the CCC nodal region is taken as half of the column width [15], and the height h_b is taken as the equivalent height of the compressed zone, denoted by β₁c.

The angle θ and width w_s of the compression strut are as follows:

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

(5)

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

(6)

We calculated the maximum load that each component of the model can withstand. According to the equilibrium equations, we determined the bearing capacity of the corbel when each component reaches its ultimate strength, and we took the minimum value as the shear bearing capacity of the corbel, denoted as V_u. The formula is as follows:

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

(7)

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

(8)

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

(9)

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

(10)

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

(11)

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

(12)

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

(13)

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

(14)

In the formula, F_u,AB, F_u,AA′, F_u,ba, F_u,be, and F_u,inc represent the maximum loads that the compression struts, tension ties, surfaces perpendicular to the horizontal ties or compression struts in the nodal zone, surfaces perpendicular to the vertical compression struts or concentrated loads in the nodal zone, and surfaces perpendicular to the inclined compression struts in the nodal zone can withstand, respectively. V_u,AB, V_u,AA′, V_u,NodeA, and V_u,NodeB are the corresponding corbel bearing capacities when each component reaches its ultimate strength. f_ce,s and f_ce,n are the maximum allowable compressive stresses for the concrete in the struts and the nodal zones, respectively; β_s is the strut coefficient, which is taken as 0.75 for struts meeting the confinement requirements, and 0.4 otherwise; and β_n is the nodal zone coefficient, taken as 1.0 for the B node under three-sided compression, and 0.8 for the A node with a single tension tie passing through. When calculating the concrete compression strut elements, the current design codes’ STM model does not consider the specific impact of diagonal cracks on the struts. The strut coefficient used in the design code is a simplified coefficient related to the efficiency of the struts.

4.2. Safety Evaluation of Chinese and American Code Design Methods

To evaluate the precision and safety of corbel designs under varying parameters, in accordance with Chinese and American code methodologies, the structural load-bearing safety ratio, denoted as α, is defined. This ratio is calculated as the simulation result of the member’s bearing capacity divided by its design load value, thereby reflecting the design safety margin of the member:

$$\alpha ={\displaystyle \frac{V_{FEM}}{V_d}}$$

(15)

Higher values of α denote a greater safety margin for the component, whereas lower values suggest a reduced safety margin. The calculated results for α are detailed in Table 2. Figure 8 demonstrates the impact of variations in concrete compressive strength and shear span on the safety margin of corbels, as designed by methods outlined in the Chinese and American codes.

The results indicate that when the design load is 750 kN, the safety margin of the corbel designed according to the Chinese code increases from 1.28 to 1.87 with an increasing concrete compressive strength. The data in Table 2 demonstrate that to maintain the same design load, as concrete strength increases, the section height of the corbel decreases, while the longitudinal reinforcement area increases correspondingly, with a minimal change in section size. In the Chinese code, only the standard value of concrete tensile strength is considered in crack control, and its relationship with the compressive strength of concrete cubes is shown in Table 4.1.3-2 [1]. As illustrated in Figure 9, with an increasing cube compressive strength, the standard value of concrete tensile strength also increases, but the growth rate gradually slows. This suggests that when using high-strength concrete, to meet crack control requirements, despite the slight increase in tensile strength, it is still necessary to maintain a larger section size, thereby exhibiting a higher safety margin in the design.

Under the American code, as the concrete strength grade increases, the section height of the corbel decreases, the reinforcement ratio increases, and the safety margin initially increases before decreasing. This phenomenon is attributed to the fact that the increase in concrete strength and steel bar area contributes more to the improvement in shear strength than the decrease in bearing capacity caused by the reduction in corbel height. However, as the height of the corbel decreases further, the bearing capacity eventually decreases.

Under the Chinese code, as the shear span increases, the section height and longitudinal reinforcement area of the corbel gradually increase, leading to a safety margin increase from 1.54 to 1.69. Conversely, under the American code, although the section height of the corbel increases with the shear span, the longitudinal bar reinforcement ratio decreases, and the safety margin is roughly maintained at about 1.3, showing a slight decreasing trend with an increasing shear span.

Figure 10 illustrates the impact of height variation on the safety margins of corbels designed according to Chinese and American codes. For designs adhering to the American code, Figure 10a presents three sets of data. In the first set, with the concrete strength consistently maintained at 50 MPa and a shear span of 200 mm, it can be observed that as the corbel height increases, the required area of longitudinal reinforcement decreases, leading to an increasing trend in safety margin. This margin increases from 1.0 at a height of 400 mm to 1.48 at 600 mm. The second set of data involves a constant shear span with the concrete strength reduced from 70 MPa to 30 MPa. The safety margin peaks at 1.32 at a height of 500 mm, then exhibits a declining trend. This suggests that the benefits of an increased section depth on the load-bearing capacity are offset by the reduced reinforcement area and the decrease in concrete strength. The third set of data shows a constant concrete strength with an increase in the selected shear span from 100 mm to 300 mm. As the corbel height increases, the shear-span-to-depth ratio also increases, with no significant change observed in the safety margin. For designs based on the Chinese code, Figure 10b presents two sets of data. In the first set, with the shear span held constant and the concrete strength reduced from 70 MPa to 30 MPa, the safety margin decreases from 1.87 to 1.28 as the corbel height increases. In the second set, with a constant concrete strength, the shear span is adjusted from 100 mm to 300 mm, resulting in an increase in safety margin from 1.54 to 1.69. According to the crack formula specified by the Chinese code, when the corbel width and concrete strength remain constant, the shear span must be adjusted correspondingly, with changes in corbel height to maintain the target load. It is therefore not feasible for both the shear span and concrete strength to remain unchanged simultaneously.

In general, the differences in calculation methods between the Chinese and American codes lead to varying impacts of concrete strength, shear span, and corbel height on the safety margins of corbels. According to the Chinese code, the safety margin of corbels increases significantly with the enhancement of concrete strength, while the effect of the shear span is relatively minor. When the shear span is constant, opting for a smaller corbel height with a higher concrete strength can result in a design with a higher safety margin. Conversely, when the concrete strength remains unchanged, selecting a larger corbel height with a larger shear span can also yield a higher safety margin.

In contrast, under the American code, the safety margin first increases and then decreases with the increase in concrete strength and is not sensitive to changes in the shear span. When the concrete strength and shear span remain constant, a higher corbel height contributes to enhancing the safety margin of the designed corbel. However, as the corbel height increases and the concrete strength gradually decreases, the safety margin will first increase and then decrease. When the concrete strength is fixed, the increase in corbel height has little impact on the safety margin across different shear spans.

Therefore, designers should select appropriate design parameters based on specific engineering conditions and design requirements to optimize the structural load-bearing capacity and ensure an adequate safety margin.

4.3. Corbels without Stirrups

The performance of the designed reinforced concrete corbel specimens under two types of loading conditions, with and without stirrups, was analyzed and compared. The results presented in Table 2 indicate that the bearing capacity of specimens with stirrups is enhanced by 15% to 46% compared to specimens without stirrups. This demonstrates that stirrups have a significant effect on improving the bearing capacity of the corbel, emphasizing that the design should avoid the absence or minimal use of stirrups.

As illustrated in Figure 11, stirrups play a particularly crucial role in increasing the bearing capacity, especially when using low-strength concrete or facing large shear spans. To demonstrate this, specimens C40-S0.38-A and C60-S0.48-A are taken as examples, for which numerical model analysis revealed the change in maximum stress of the first layer of stirrups during loading. Figure 12 shows that under the same load, the stirrup in the low-strength concrete corbel member experiences greater stress, thus more effectively strengthening the structure. This is because in low-strength concrete, the material’s tensile properties are weak, leading the stirrup to play a more significant role in controlling crack development.

For specimens with a large shear span ratio, an increase in the ratio leads to a reduction in the angle between the compression strut and the horizontal, thereby decreasing the vertical component of force of the compression strut. Under the same shear force, to compensate for the reduced vertical force, the compression strut needs to provide a greater supporting force. Consequently, the demand for horizontal tensile force in the member also increases. Stirrups, by providing necessary lateral restraint, not only enhance the shear bearing capacity of the concrete but also play an active role in resisting horizontal tension.

Figure 13 clearly demonstrates the significant impact of stirrups on the enhancement of the corbel load capacity at various corbel heights. As the height of the corbel increases, the reinforcing effect of the stirrups becomes more pronounced. Specifically, for corbels 400–500 mm in height, the use of stirrups can increase the load capacity by about 15% to 20%. For corbels exceeding 500 mm in height, this enhancement effect may exceed 20%. At even greater corbel heights, the effect of stirrups on load capacity enhancement further strengthens, potentially reaching 40% or more.

In taller corbels, the likelihood of material non-uniformity and internal defects increases, leaving the cross-section more susceptible to cracking and stress concentration. Consequently, as the corbel height increases, a greater number of stirrups need to be arranged over a larger area. Stirrups effectively improve the ductility and load-bearing capacity of the corbel by restraining the lateral expansion of concrete and controlling the development of cracks.

4.4. Strut Coefficient of the Compression Strut

The STM model in the ACI 318-19 code [2] utilizes the strut coefficient β_s to calculate the strength of compression struts, reflecting the reduction in concrete’s compressive strength when subjected to transverse tensile strains. Under conditions without stirrups, the code recommends a value of 0.4 for β_s. However, this simplified assignment overlooks the potential influence of other parameters on β_s, which may lead to significant discrepancies between calculated and experimental results [31], thereby affecting the safety of structural load-bearing capacity predictions.

Based on the structure’s ultimate load-bearing capacity, the value of the effective strut coefficient βs is re-determined through static equilibrium conditions and the STM model according to Equations (7) and (11). This process is conducted only when the longitudinal reinforcement has not reached the yield state and no stirrups are present, ensuring that the load-bearing capacity is primarily controlled by the compression strut. Table 2 provides the calculated results of the strut coefficient for the ultimate load-bearing capacity in the absence of stirrups.

Figure 14 illustrates the variation in the strut coefficient with changes in concrete compressive strength and shear span. Figure 15 shows the variation in the strut coefficient with changes in corbel height. The results indicate that corbel specimens with lower concrete strength or larger shear span ratios or greater corbel heights have a higher strut coefficient. Conversely, in cases of high concrete strength, the strut coefficient is lower, which may lead to unsafe predictions of the structural load-bearing capacity under conditions without stirrups and unyielded longitudinal reinforcement, with the risk of code-calculated results exceeding experimental load-bearing capacities. Therefore, to enhance the accuracy and safety of structural load-bearing capacity predictions, the value of the strut coefficient β_s should be determined considering a broader range of design parameters.

5. Comparison with Previous Studies

To ascertain the reliability of the test outcomes presented in this paper, a comparative analysis was conducted with existing research findings [11,13,15,29,32,33]. The comparison encompassed 36 corbels from the literature, whose parameter variables were closely aligned with those of the specimens examined in this study, thereby ensuring the comparability of the test data.

Table 3. Details of corbels in previous studies.

Source	Specimen No.	h0/mm	b/mm	av/mm	fc/MPa	fy/MPa	As/mm2	fyt/MPa	Ast/mm2	Vt/kN	Vd,ACI/kN	αACI	Vd,GB/kN	αGB
1976 Mattock [11]	B1	226	152	102	25	335	258	449	129	209	103	2.02	89	2.35
	B2	226	152	152	24	321	400	462	129	173	116	1.50	70	2.46
	B3A	225	152	229	29	357	639	452	258	187	115	1.63	61	3.08
2015 Al- Shaarbaf [32]	LNC1	215	150	108	35	532	339	510	113	186	214	0.87	96	1.94
	LSCC1	215	150	108	36	532	339	510	113	218	220	0.99	97	2.24
	HNC1	215	150	108	46	532	339	510	113	210	228	0.92	108	1.94
	HSCC1	215	150	108	48	532	339	510	113	229	229	1.00	110	2.08
	LNC8	215	150	108	35	532	339	510	57	193	214	0.90	96	2.01
	HNC8	215	150	108	46	532	339	510	57	209	228	0.92	108	1.93
2018 Wilson [15]	C0	559	356	368	37	506	2039	506	1032	1427	1005	1.42	527	2.71
	C1	559	356	330	45	487	2039	487	774	1678	1238	1.36	585	2.87
	C2	559	356	330	47	487	2039	487	1032	1785	1242	1.44	619	2.88
1996 Foster [13]	SB2	740	150	250	56	430	1548	420	465	1200	856	1.40	476	2.52
	SC2–1	600	125	300	62	430	2322	420	465	980	629	1.56	277	3.54
	SD1	600	125	300	95	430	2322	420	465	1000	808	1.24	315	3.17
	SD2	600	125	300	65	430	2322	420	465	1000	661	1.51	281	3.56
	PG2	500	150	300	94	415	2322	490	387	1050	957	1.10	286	3.67
2010 Othman [33]	C11	239	180	135	40	415	452	415	101	351	227	1.55	128	2.74
	C12	239	180	135	50	415	452	415	101	383	230	1.67	140	2.73
	C13	239	180	135	60	415	452	415	101	424	232	1.83	148	2.87
	C21	239	180	135	40	415	452	415	151	373	227	1.64	128	2.91
	C22	239	180	135	50	415	452	415	151	406	230	1.77	140	2.90
	C23	239	180	135	60	415	452	415	151	476	232	2.05	148	3.22
	C31	239	180	135	40	415	452	415	201	402	227	1.77	128	3.13
	C32	239	180	135	50	415	452	415	201	425	230	1.85	140	3.03
	C33	239	180	135	60	415	452	415	201	476	232	2.05	148	3.22
2021 Yin [29]	C30-S0.57-A	347	300	200	28	422	804	478	471	492	372	1.32	253	1.94
	C45-S0.67-A	295	300	200	54	425	942	478	471	730	408	1.79	267	2.73
	C60-S0.73-A	271	300	200	44	492	1017	478	471	754	453	1.67	217	3.48
	C45-S0.33-A	298	300	100	54	459	615	478	471	824	464	1.78	381	2.16
	C45-S0.80-A	371	300	300	50	492	1017	478	628	662	469	1.41	296	2.24
	C30-S0.42-C	475	300	200	24	478	393	541	303	573	314	1.82	345	1.66
	C45-S0.50-C	399	300	200	54	458	452	478	314	624	295	2.12	351	1.78
	C60-S0.57-C	349	300	200	44	458	565	478	314	593	318	1.87	322	1.84
	C45-S0.31-C	324	300	100	54	478	339	541	202	632	295	2.14	428	1.48
	C45-S0.67-C	448	300	300	50	459	615	478	471	679	329	2.06	358	1.89

Note: h₀ is the effective height of the corbel section; b is the width of the corbel; f_yt is the yield strength of the horizontal stirrups; A_st is the area of the horizontal stirrups; V_d,ACI is the load-bearing capacity calculated using the strut-and-tie model according to ACI 318-19 [2]; α_ACI is the ratio of V_t to V_d,ACI; V_d,GB is the load-bearing capacity calculated using the triangular truss model according to GB50010-2010 [1]; α_GB is the ratio of V_t to V_d,GB.

provides a detailed overview of the corbels, including the calculated safety margins. Figure 16 illustrates the impacts of various parameters on the safety margins when employing the Chinese and American code design methods. The trends in the safety factors observed in this study were consistent with those reported in the majority of the referenced research, indicating a satisfactory level of congruence.

6. Conclusions

In this study, a total of twenty-two reinforced concrete corbels were designed to evaluate the accuracy and safety of the triangular truss model according to GB 50010-2010 [1] and the strut-and-tie model according to ACI 318-19 [2] under various design parameters with a given design load. The main research findings are as follows:

(1)

Increasing the compressive strength of concrete can significantly enhance the safety level in the triangular truss model. For instance, when the compressive strength of concrete increases from 30 MPa to 70 MPa, the safety margin increases from 1.28 to 1.87. In contrast, in the strut-and-tie model, the safety level first increases and then decreases with the enhancement of concrete compressive strength. When the compressive strength of concrete increases from 30 MPa to 50 MPa, the safety margin rises from 1.07 to 1.32, and then it drops to 1.2 when further increased to 70 MPa.

(2)

In the triangular truss model, the safety level also increases correspondingly with the enlargement of the shear span. For example, when the shear span increases from 100 mm to 300 mm, the safety margin increases from 1.54 to 1.69. In comparison, the safety level in the strut-and-tie model does not show significant changes and tends to decrease with the increase in the shear span.

(3)

With the shear span held constant, the safety level in the triangular truss model gradually decreases as the corbel height increases and the concrete strength gradually reduces. The safety level in the strut-and-tie model first increases and then decreases.

(4)

The configuration of horizontal stirrups significantly improves the load-bearing capacity of the corbel, especially in cases of low-strength concrete, a large shear span, or larger corbel heights. Compared to specimens without stirrups, the load-bearing capacity of specimens with stirrups increased by 15% to 46%.

(5)

Specimens featuring low-strength concrete, large shear spans, or greater corbel heights exhibit a higher strut coefficient. For corbel members with high-strength concrete and low stirrup ratios, the simplified values in existing codes may lead to unsafe prediction results as they ignore the potential influence of other parameters.