Numerical Analysis on Flexural Shear Behavior of Reinforced Concrete Beams Strengthened with Fiber-Reinforced Polymer Grid and Engineered Cement Composites

Abstract

Strengthening reinforced concrete (RC) beams with fiber-reinforced polymer (FRP) grids and engineered cement composites (ECCs) can significantly enhance their shear capacity. However, the specific contributions of the components in reinforced RC beams remain unclear, necessitating further investigation into the flexural shear performance of RC beams. The numerical model was used to analyze the flexural shear performance of RC beams strengthened with an FRP grid and ECCs. Subsequently, the parameters affecting the flexural shear performance of beams were discussed. This included the compressive strength of concrete prism, the shear span ratio, the tensile strength of ECCs, the thickness of the ECC cover, the cross-sectional area of the FRP grid, and the number of FRP grid layers. Finally, a calculation formula was established to predict the shear capacity and verified by the outcomes from numerical models and experimental data. The findings indicated that the ECC-strengthened layer significantly contributed to increasing the shear capacity. Additionally, the FRP grids helped to reduce stress concentration in the flexural shear zone, thereby preventing premature concrete cracking. The max load increased by 8.06% when the ECC’s tensile strength increased from 4 MPa to 10 MPa. In addition, increasing the cover thickness from 8 mm to 20 mm caused the peak load to increase by 14.42%. The calculation formula introduced in this research accurately predicts the shear capacity of the oblique section of RC beams.

1. Introduction

Reinforced concrete (RC) has become a prevalent structural form in building engineering, renowned for its high compressive strength and cost-effectiveness [1,2,3]. However, due to factors such as inadequate design schemes and increased loading conditions, some existing concrete structures fail to meet the required performance criteria. Therefore, there is an urgent need for a simple and rational strengthening scheme for RC structures.

Recently, fiber-reinforced polymer (FRP) materials have become popular for repairing damaged RC structures because of their high tensile strength, light weight, and exceptional durability [4,5]. The application of FRP can enhance the cracking resistance and load-bearing capacity of RC without adding excessive weight to the structure. Unlike traditional reinforcement techniques, FRP reinforcement offers versatility in adapting to various structural forms and members. However, the effectiveness of FRP reinforcement techniques often relies on the bonding ability of organic resins for force transfer between the FRP grid and the original structure [6,7,8]. Additionally, the performance of organic resins is adversely affected by wet and high-temperature environments, limiting the complete utilization of the mechanical characteristics of FRP materials [9,10]. Therefore, the bond strength and construction conditions of organic resins significantly impact the effectiveness of FRP reinforcement. To address this limitation, cement-based materials have been proposed as alternatives to organic resins. For instance, Ye et al. [11] introduced a novel approach using high-performance concrete composite plates strengthened with an FRP grid, which demonstrated tensile elastic strain-hardening behavior and potential application in practical engineering. Kumar et al. [12] investigated the flexural behavior of geopolymer concrete sandwich walls reinforced with FRP grids, whereas Liu et al. [13] focused on RC’s resistance to impact structures enhanced with FRP grids and UHPC. Sridhar et al. [14,15,16,17] performed an investigation on the behavior of RC components reinforced with FRP using both experimental and numerical analyses. In a separate study, Imjai et al. [18] examined how near-surface mounted CFRP rods affect the flexural performance of low-strength RC beams, employing both experimental and numerical techniques. These investigations highlight the effectiveness of the composite system comprising FRP grids and cement-based materials, as cement-based materials exhibit favorable bonding properties with concrete structures.

The analysis indicates that the success of FRP reinforcement technology is reliant on the bonding performance of organic resins, which can be affected by changes in temperature and humidity. To study the actual reinforcement effect of FRP grids on RC beams, it is necessary to find a material with good bonding performance and economic benefits to replace epoxy resin as an adhesive. The use of engineered cement composite (ECCs) has emerged as a viable option due to its strain-hardening behavior, excellent bonding properties, and tensile strength [19,20,21,22]. Zheng et al. [23] reported on experimental investigations on the flexural behavior of corrosion-damaged RC beams, which were reinforced using ECCs and FRP grids. Their findings demonstrated that this reinforcement technique significantly enhanced the flexural capacity without experiencing interface slip. Likewise, Zhang et al. [24] employed FRP grids and ECCs to strengthen GFRP-reinforced RC beams, observing negligible interfacial slippage prior to reaching the peak load in most beams.

While the strengthening effects of ECCs and FRP grids on the flexural behavior of RC beams have been extensively studied, there is limited research discussing the enhancement effect of this reinforcement technology on the flexural shear behavior. Moreover, the shear failure of RC beams with inclined sections represents a brittle failure mode that should be avoided in design. Guo et al. [25] explored how this reinforcement technology affects the flexural shear performance of RC beams. However, their study did not comprehensively investigate the contribution of various components to shear capacity, nor did their formula consider the influence of RC beam dimensions on shear capacity. Therefore, a further numerical analysis and refinement of calculation formulas are necessary to address these research gaps.

This study validated the precision of the refined numerical models through experimental results. Furthermore, the flexural shear behavior of the typical model was investigated, and the impact of each structural component to shear capacity was evaluated by comparing various numerical outcomes. The quantity of FRP grid layers, the thickness of ECC layers, and the ratio of shear span to depth in the beams all affect the shear performance and strengthening effectiveness of RC beams [25]. This research examined how the following key parameters affect the mechanical behavior of RC beams: concrete prism compressive strength (30 MPa to 60 MPa), shear span-to-depth ratio (1.56 to 3.13), ECC tensile strength (4 MPa to 10 MPa), ECC layer thickness (8 mm to 20 mm), FRP grid cross-sectional area (4 mm² to 10 mm²), and the layer count of FRP grids (0 to 2 layers). Subsequently, based on the contribution of various materials to shear capacity, a calculation formula for the shear capacity of oblique sections in RC beams was developed and validated using both numerical models and experimental results.

2. Finite Element Modeling

A thorough examination of the flexural shear performance of RC beams enhanced with an FRP grid and ECCs was conducted through the creation and utilization of a wide range of numerical simulations in ABAQUS 6.14-1 software. These models consider the mechanical behaviors of different materials, ensuring the accuracy and reliability of the models.

2.1. Specifications of Test Specimens

The strengthening technique was provided and reported by Guo et al. [25]. The use of an FRP grid and ECCs has been shown to greatly improve the flexural shear performance of RC beams. In this research, six reinforced beams and one unreinforced beam were subjected to four-point loading tests to evaluate their shear capacity in diagonal sections. This study examined various factors such as the ratio of shear span, along with the spacing and quantity of layers of FRP grids. The RC beam measured 1500 mm in length, 120 mm in width, and 200 mm in height, with detailed cross-sectional details shown in Figure 1. Stirrups that were 6 mm in diameter and HRB400 grade were used at a spacing of 250 mm to investigate the flexural shear behavior. The strengthened RC beams featured a consistent ECC cover thickness of 15 mm on each side.

2.2. Characteristics of Materials

2.2.1. Steel Bars

For HRB400 ribbed steel bars under compression and tension, an isotropic bilinear constitutive model was applied, with the elastic modulus (E_s) set at 200 GPa and Poisson’s ratio at 0.3. The longitudinal reinforcement had a yield strength of 435 MPa and an ultimate strength of 540 MPa [25]. The longitudinal and stirrup steel bars had cross-sectional areas of 254.5 mm² and 28.27 mm², respectively. Usually, the elastic modulus degrades significantly after the steel bar reaches its yield strength, which can be taken as 0.01 Es.

2.2.2. Engineered Cementitious Composites (ECCs)

The ECC was used to reinforce both sides of the RC beam. The ECC exhibited compressive strength and ultimate tensile strength values of 36.51 MPa and 4.6 MPa. The ECC under the compressive model can be expressed as follows [26,27]:

$$\sigma =\left\{\begin{array}{ll}{E}_0\epsilon & (0<\epsilon <{\epsilon }_{0.4})\\ E_0\epsilon (1-\alpha )& ({\epsilon }_{0.4}<\epsilon <{\epsilon }_0)\end{array}\right.$$

(1)

$$\alpha =a{\displaystyle \frac{\epsilon E_0}{f_{cr}^{\prime }}}-b$$

(2)

$$\sigma =\left\{\begin{array}{l}m(x-x_0)+f_{cr}^{\prime }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(x_0<x<x_l)\\ n(x-x_l)+{\sigma }_l\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(x_l<x<x_{\max })\end{array}\right.$$

(3)

where ${\epsilon }_{0.4}$ represents the deformation at 40% maximum strength, and $\alpha$ is a coefficient used to account for the decrease in the elastic modulus. In order to facilitate the calculation of the mechanical behavior of the ECC under tension, the tensile constitutive of the ECC can be approximately simplified to a trilinear model [27,28]. The tensile model of the ECC is as follows:

$$\sigma =\left\{\begin{array}{ll}{E}_0\epsilon & 0<\epsilon \le {\epsilon }_{crack}\\ f_{crack}+(f_{cr}^{\prime }-f_{crack})({\displaystyle \frac{\epsilon -{\epsilon }_{crack}}{{\epsilon }_{cr}-{\epsilon }_{crack}}})& {\epsilon }_{crack}<\epsilon \le {\epsilon }_{cr}\\ f_{cr}^{\prime }+(f_{crf}-f_{cr}^{\prime })({\displaystyle \frac{\epsilon -{\epsilon }_{cr}}{{\epsilon }_{crf}-{\epsilon }_{cr}}})& {\epsilon }_{cr}<\epsilon \le {\epsilon }_{crf}\end{array}\right.$$

(4)

where $f_{crack}$, $f_{cr}^{\prime }$, and $f_{crf}$ are the first cracking, maximum, and failure tensile strength of the ECC, and ${\epsilon }_{crack}$, ${\epsilon }_{cr}$, and ${\epsilon }_{crf}$ are their corresponding strains, respectively.

2.2.3. Concrete

The low-strength concrete utilized for RC beams has compressive and tensile strengths of 31.6 MPa and 2.1 MPa, respectively. The compressive model of concrete can be expressed as follows [29]:

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

(5)

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

(6)

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

(7)

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

(8)

Equations (9) and (10) display the stress–strain relationship of the concrete.

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

(9)

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

(10)

in which ${\alpha }_c$, ${\alpha }_t$, $f_{c,r}$, and $f_{t,r}$ are the parameter values of descending sections and the ultimate strength under compression and tension.

2.2.4. FRP Grid

Typically, when FRP reaches its peak load, it experiences immediate debonding. As a result, FRP exhibits minimal plastic behavior and is regarded as a linear elastic material. Moreover, the FRP grid solely withstands tensile stress. Hence, it is appropriate to utilize Equation (11) to describe the constitutive relationship of the FRP grid.

$${\sigma }_f=\left\{\begin{array}{ll}\epsilon E_f& 0<\epsilon <{\epsilon }_{fu}\\ 0& \epsilon >{\epsilon }_{fu}\end{array}\right.$$

(11)

where ${\sigma }_f$, $E_f$, and ${\epsilon }_{fu}$ are the ultimate stress, elastic modulus, and ultimate strain of the FRP grid.

2.3. Element, Boundary Conditions, and Loading Process

As illustrated in Figure 2, the model utilized consisted of concrete, steel bars, FRP grids, and ECCs. Concrete and ECC sections were modeled with 3D solid elements with reduced integration (C3D8R), while the steel bars and FRP grids were represented by truss elements (T3D2). To improve the model’s accuracy, a mesh size of 10 mm was employed. Additionally, boundary conditions were imposed on the bearing pads of the RC beams to simulate the loading conditions during the experimental process. Rotation around the X-axis was allowed at reference point RP-1, while translation along the Z-axis and rotation around the X-axis were permitted at reference point RP-2. Additionally, a displacement along the Y-axis was applied to the upper pad of the beam to simulate the actuator loading illustrated in Figure 3. Tie constraints were used between the concrete and ECC interfaces, as no apparent interface failure or slip was observed during the tests. The steel bars and FRP grids were embedded within the concrete and ECCs, respectively, to account for their interactions.

2.4. Validation of Numerical Modeling

Figure 4 compares the failure modes observed in the experimental results with those predicted by numerical models. With the increase in vertical load, cracks first develop in the pure bending region of the RC beams and then spread across a larger area within the flexural shear zone. The concrete cracks in the flexural shear zone primarily manifest as oblique cracks that extend from the upper portion of the RC beam to the lower portion. Upon the failure of the specimen, a principal crack develops through the oblique section of the RC beam, accompanied by multiple micro-cracks in the lower part of the beam. This indicates that shear failure predominates in the flexural shear zone. By comparing the failure modes observed in the tests with those predicted by the numerical models, it is evident that both the experiments and numerical models exhibit pronounced shear failure in the oblique section within the flexural shear zone, accompanied by concrete cracking in the pure bending zone. Thus, the developed numerical model accurately predicts the failure modes observed in the experiments.

Figure 5 compares the curves obtained from both the experimental and numerical results, emphasizing parameters such as initial stiffness, ultimate load, and initial cracking load. The initial cracking load obtained from the numerical model aligns closely with the value of 30 kN reported in ref. [25]. The numerical model closely matches the experimental data. Additionally,

Table 1. Comparison of ultimate load of numerical model and experiment.

Specimen	Matric	n	S (mm × mm)	a/d	Nu/kN	Nu,FE /kN	Nu/Nu,FE
F0S0	-	-	-	3.13	115	115	1.00
EF1S0	ECC + FRP grid	1	50 × 50	3.13	161	160	1.01
EF1′S0	ECC + FRP grid	1	75 × 75	3.13	153	163	0.94
EF1S2	ECC + FRP grid	1	50 × 50	3.44	150	143	1.05
All						Mean	1.00
						COV	0.0016

Note: a, d, n, and S are, respectively, the shear span, effective depth, number of FRP layers, and interval of grids. N_u and N_u,FE are the peak load of the experiment and numerical model, respectively.

provides the mean and coefficient of variation (COV) of the parameter $N_u$/$N_{u,FE}$, which are calculated as 1.00 and 0.0016, respectively. These values indicate that the finite model accurately predicts the load–displacement curve observed in the experiments. Through a comparison of the failure modes and load–displacement curves between the numerical model and experimental data, it is evident that the developed numerical model in this study exhibits accuracy and can be effectively utilized for subsequent numerical analyses and theoretical investigations.

3. Analysis and Discussion

In order to investigate the flexural shear behavior of strengthened RC beams, it is necessary to carry out a series of mechanical behavior analyses by means of numerical methods, including the analysis of the contribution of each member to the shear capacity of the oblique section, discussion of the stress distribution in the concrete cross-section, failure mode analysis, and complete curve analysis.

3.1. Complete Curve Analysis

Sridhare et al. examined the shear behavior of RC specimens, showing that the numerical model results aligned with DIC observations [30]. For a deeper analysis of the flexural shear behavior of RC beams, the numerical results were comprehensively discussed. Figure 6 presents the load–displacement curves of different RC beams, where the models FE-F0S0, FE-F0S0-W, FE-EF0S0, and FE-EF1S0 respectively represent un-strengthened RC beams, un-strengthened RC beams without stirrups, RC beams strengthened with ECCs, and RC beams strengthened with ECCs and FRP grids. Points O, A, B, and C on the curves denote the following stages: the beginning of loading, initial cracking of RC beams in the pure bending zone, development of oblique cracks in the flexural shear zone, and ultimate failure of concrete in the flexural shear zone, respectively. These points divide the load–displacement curve of the typical model FE-EF1S0 into three stages: the initial loading stage, the development of concrete cracks occurring within the zone experiencing pure bending, and the development of concrete cracks in the flexural shear zone. To investigate the stress distribution of each member at these feature points, the von Mises stresses of concrete, ECCs, FRP grids, and steel bars are plotted in Figure 7.

Stage 1 (Points O–A): Initial loading stage. In this stage, the relationship between load and displacement shows a good linear pattern, as depicted in Figure 6. From Figure 7a,b, it is evident that the upper part of concrete and ECCs experiences compression, while the lower part undergoes tension, with stress in the flexural shear zone being almost negligible. Figure 7c,d illustrate that stress in the pure bending zone is mainly concentrated on the longitudinal bars, with minimal stress observed in the transverse bars. This indicates that the overall stress in the RC beam is primarily concentrated in the pure bending zone, while the stress in each member of the flexural shear zone is relatively low. Furthermore, the shear failure mode is not prominently observed in this stage. By comparing the load–displacement curves of the four models in Figure 6, it can be observed that the influence of stirrups, ECCs, and FRP grids on the initial cracking load and the initial stiffness is almost negligible. Compared to specimen FE-F0S0, the initial cracking load at point A of the RC beam with ECCs increased by 11.29%.

Stage 2 (Points B–C): Concrete crack development stage in the pure bending zone. Upon reaching point A in the load–displacement curve, evident cracking occurs at the bottom of the RC beam in the pure bending zone, leading to a substantial reduction in stiffness. As the load increases further, the stress of concrete and ECC gradually propagates towards the flexural shear zone, with the stress of the stirrup and transverse FRP grid emerging. Figure 7a,b indicate that the stress of concrete and ECCs in the flexural shear zone shows an inclined trend, with a more concentrated stress distribution than in the pure bending zone. In this stage, the maximum von Mises stresses of concrete and ECCs in the flexural shear area can reach 4.45 MPa and 4.52 MPa, respectively, while the maximum von Mises stresses of FRP grids and steel bars in the flexural shear area can reach 47 MPa and 78 MPa. By comparing the load–displacement curves of the four models, it is evident that the ECC-strengthened layer significantly improves the post-cracking stiffness. Moreover, the ECC-strengthened layer delays the formation of concrete oblique cracks in the flexural shear zone and effectively enhances the deformation capacity of RC beams. Compared to specimen FE-F0S0, the load at point B for specimen FE-EF0S0 increased by 12.39%.

Stage 3 (Points C–D): Concrete crack development stage in the flexural shear zone. At this stage, the oblique cracks in the flexural shear zone of the RC beam begin to fully develop, leading to a degradation in beam stiffness with an increase in the width of the concrete crack. The stiffness degradation trend of the load–displacement curve in Figure 6 indicates that the stirrup, FRP grid, and ECCs have significant effects on delaying cracking in the flexural shear zone of RC beams and improving the peak load. Comparing the peak load of the four models, it can be observed that the stirrup, ECCs, and FRP grids increase the peak load by 11.66%, 30.97%, and 6.34%, respectively. Thus, it can be concluded that the ECC-strengthened layer contributes the most significantly to enhancing the shear capacity of the oblique section of RC beams. The load at point C for specimen FE-EF0S0 showed a 30.97% increase compared to that of specimen FE-F0S0.

As shown in Figure 7, during the initial loading stage, the RC beam primarily exhibits tensile stress at the bottom and compressive stress at the top. With the increase in vertical displacement, the stress gradually develops towards the flexural shear zone. Due to the large spacing between the stirrups, it becomes challenging for them to effectively distribute the stress in the flexural shear zone of the RC beam, making it more susceptible to diagonal cracks. Figure 7 shows that the distribution pattern of stress within the concrete and ECCs in the flexural shear zone is more concentrated compared to the pure bending zone, while the stress of transverse FRP grids and the stirrup increases significantly. The stress distribution in concrete is primarily concentrated in the flexural shear zone, whereas the stress in ECCs is distributed in both the flexural shear zone and the pure bending zone. Additionally, the ECC bears significantly higher tensile stress than concrete, indicating that the ECC can enhance the flexural and shear capacities. This result is consistent with the findings of Imjai et al. [31]. The stress in FRP grids is transmitted through the FRP fibers and is mainly concentrated in both the pure bending and flexural shear zones. FRP grids help to alleviate stress concentration in the flexural shear zone, which aids in preventing the premature cracking of the concrete. This alleviation of stress concentration improves the overall shear capacity. Simultaneously, the FRP grids also enhance the crack resistance. An analysis of the numerical model results reveals that the maximum Mises stress in the FRP grid reaches 289.5 MPa, which does not approach its ultimate strength. Therefore, it can be inferred that the FRP grid has not fully utilized its mechanical properties.

To explore the influence of FRP grids on the mechanical behavior, stress diagrams of the maximum and minimum principal stresses of concrete under the same displacement are shown in Figure 8. By comparing Figure 8a, it is evident that the orange region on the surface of the RC beam in specimen FE-F1S0 is significantly smaller than that in specimen FE-F0S0, indicating that the use of the FRP grid alleviated the tensile stress concentration in the flexural shear zone. In Figure 8b, the stress diagrams for the two specimens are essentially identical, indicating that the FRP grid has a negligible effect on the compressive capacity.

3.2. Load Transfer Mechanism in RC Beam

Figure 9 depicts typical vector symbols representing the maximum stresses of the concrete, the ECC, the FRP grids, and the steel bars. In this study, the direction of stress is indicated by the arrow of the vector symbol, the magnitude of stress is represented by the length of the vector symbol, and the density of the vector symbol illustrates the stress distribution. From Figure 9a,b,e,f, it is evident that the concrete and the ECC primarily endure tensile stress in the flexural shear zone while experiencing minimal compressive stress. The peak tensile stress of the concrete is concentrated at the inclined section where cracking failure occurs, and the maximum tensile stress of the ECC is concentrated at the upper and lower edges of the beam. The presence of the ECC enables the sharing of tensile stress in the concrete, significantly enhancing the shear capacity of the oblique section of the RC beams. The excellent tensile capacity of the ECC suppresses the formation of cracks in the tension zone of RC beams and shares part of the tensile stress. However, the ECC in the compression zone does not significantly enhance the compressive capacity of RC beams. Figure 9c,d show that the stirrup and the transverse FRP grid endure tensile stress, and the bottom longitudinal steel bar and the bottom longitudinal FRP grid also experience tensile stress. Furthermore, the stress of the stirrup and the FRP grids is mainly concentrated in the flexural shear zone of the RC beam.

Based on the load transfer paths of the concrete, the ECC, the steel bars, and the FRP grids in the beam, they can be further simplified into the strut–tie model, as depicted in Figure 10. It is important to note that the compressive stress of the concrete and the ECC is very small, and the upper longitudinal steel bar primarily bears the compressive stress. Hence, it can be assumed that the concrete and the ECC mainly endure tensile stress and hardly bear compressive stress.

4. Parametric Analysis

In this study, numerical models were established to study the influence of key parameters on the flexural shear behavior of the beams in

Table 2. Dimensions of numerical models.

No.	Models	H (mm)	B (mm)	fc (MPa)	Ds (mm)	S (mm)	fECC,t (MPa)	t (mm)	AFRP (mm)	n	λ
1	E1	200	120	30	6	250	4	15	6	1	3.13
2	E2	200	120	40	6	250	4	15	6	1	3.13
3	E3	200	120	50	6	250	4	15	6	1	3.13
4	E4	200	120	60	6	250	4	15	6	1	3.13
5	E5	200	120	30	6	250	6	15	6	1	3.13
6	E6	200	120	30	6	250	8	15	6	1	3.13
7	E7	200	120	30	6	250	10	15	6	1	3.13
8	E8	200	120	30	6	250	4	20	6	1	3.13
9	E9	200	120	30	6	250	4	8	6	1	3.13
10	E10	200	120	30	6	250	4	12	6	1	3.13
11	E11	200	120	30	6	250	4	15	8	1	3.13
12	E12	200	120	30	6	250	4	15	10	1	3.13
13	E13	200	120	30	6	250	4	15	12	1	3.13
14	E14	200	120	30	6	250	4	15	6	0	3.13
15	E15	200	120	30	6	250	4	15	6	2	3.13
16	E16	200	120	30	6	250	4	15	6	1	2.81
17	E17	200	120	30	6	250	4	15	6	1	2.50
18	E18	200	120	30	6	250	4	15	6	1	2.19
19	E19	200	120	30	6	250	4	15	6	1	1.88
20	E20	200	120	30	6	250	4	15	6	1	1.56

, including the compressive strength of concrete (30 MPa–60 MPa), tensile strength of ECCs (4 MPa–10 MPa), cover thickness of ECCs (15 mm–30 mm), cross-sectional area of the FRP grid (4 mm–10 mm), number of FRP layers (0–2), and shear span ratio (1.56–3.13). The following will take model FE-EF1S0 as an example. Figure 10 presents the load–displacement curves for all models with varying parameters.

4.1. Compressive Strength of Concrete

Figure 11a illustrates the load–displacement curve of the model influenced by the compressive strength of concrete. It can be observed that the compressive strength of concrete enhances the stiffness of the RC beams. As the concrete compressive strength increases from 30 MPa to 60 MPa, the peak load rises by 1.98%, 2.90%, and 3.80%, respectively. The compressive strength of concrete does not exert a significant influence on the peak load of the RC beam, consistent with the results obtained in the analysis of the influence of concrete compressive strength in Section 3.2. This finding reaffirms that the shear capacity of the oblique section of RC beams can be disregarded in relation to the influence of concrete compressive strength.

4.2. Tensile Strength of ECC

The tensile strength of the ECC is a crucial parameter for this material, as the ECC exhibits excellent strain-hardening ability under tension. Figure 10 presents the stress–strain curves for ECCs with different tensile strengths, all exhibiting similar strain-hardening behavior. To investigate the influence of the ECC’s tensile strength on the shear capacity of the oblique section, load–displacement curves of models with varying ECC tensile strengths are presented in Figure 11. As the ECC tensile strength increases from 4 MPa to 10 MPa, the peak load of the model rises by 2.62%, 6.16%, and 8.06%, respectively. Although the effect of the ECC’s tensile strength on the peak load is evident, its impact on stiffness and initial cracking load is minimal. Moreover, RC beams strengthened with higher ECC tensile strength demonstrate superior deformation ability and delayed brittle fracture of concrete, as shown in Figure 11b. To visually support this observation, plastic deformation diagrams of concrete and ECCs in models E1, E5, E6, and E7 under the same displacement are depicted in Figure 12. It is evident that with the increase in the ECC’s tensile strength, the plastic deformation of concrete and the ECC in the flexural shear zone significantly decreases. Therefore, a higher tensile strength of the ECC effectively constrains crack expansion and delays the brittle fracture of RC beams.

4.3. Cover Thickness of ECC

The thickness of the ECC cover is pivotal in influencing the flexural shear behavior of RC beams. If the cover thickness is large, the pure bending failure of RC beams may occur. Conversely, if the cover thickness is small, it becomes challenging to enhance the shear strength of the oblique section of RC beams. Therefore, a numerical analysis of this critical factor is necessary. As the cover thickness of the ECC increases from 8 mm to 20 mm, the peak load of the RC beam increases by 4.77%, 10.43%, and 14.42%, respectively. In the case of model E8, the failure mode of the RC beam approaches pure bending failure, indicating that a 20 mm cover thickness of the ECC can effectively prevent flexural shear failure in this RC beam.

4.4. Cross-Sectional Area of FRP Grid and Number of FRP Grid Layers

Based on the discussion in Section 3.1, it is evident that the primary function of FRP grids is to prevent stress concentration in the concrete and ECC of the flexural shear zone. By collaborating with the concrete and the ECC, the FRP grids effectively delay the brittle failure of RC beams. To promote the advancement of this strengthening technology and evaluate the impact of the FRP grids on the shear capacity of the oblique section of the RC beams, the cross-sectional area and the number of FRP grid layers are studied in Figure 11d,e. It has been noted that increasing the cross-sectional area of the FRP grid does not effectively improve the peak load capacity of the RC beams, as the tensile force in the FRP grid does not reach its ultimate stress level. Similarly, the number of FRP grid layers has little influence on the peak load of the RC beams. Even with an increase in the cross-sectional area and the number of the layers, the tensile force remains almost unchanged. Consequently, for future reinforcement endeavors, it is recommended to utilize single-layer FRP grids to strengthen the RC beam, with the cross-sectional area of FRP grids controlled between 4 mm² and 8 mm².

4.5. Shear Span Ratio

The shear span ratio ($\lambda$) is a parameter that exerts a significant influence on the shear capacity of the oblique section of RC beams.

$$\lambda ={\displaystyle \frac{a}{h_0}}$$

(12)

In this equation, $a$ and $h_0$ are, respectively, the minimum distance a from the point of the applied concentrated load on the beam to the edge of the support and the effective depth of the cross-section.

To investigate its impact on RC beams, load–displacement curves of RC beams with shear span ratios ranging from 1.56 to 3.13 were analyzed, as shown in Figure 11f. Compared to model E1, the peak loads of models E16, E17, E18, E19, and E20 increased by 9.6%, 17.63%, 27.58%, 39.45%, and 54.87%, respectively. It is evident that both the peak load and stiffness of RC beams decrease significantly with an increase in the shear span ratio. Figure 13 illustrates the distribution pattern of the maximum tensile stress in the ECC under different shear span ratios. As the shear span ratio decreases, the angle between the tension of the ECC-strengthened layer and the bottom edge of the beam gradually increases, indicating that the change in the shear span ratio alters the angle of load transfer. Additionally, it is observed that the bottom of the ECC in the pure bending zone consistently bears tensile stress.

5. Shear Capacity of Oblique Section Calculation Formula

Proposing an accurate formula for predicting the shear capacity of the oblique section and predicting the peak load is crucial to promoting the reinforcement technology of RC beams. The shear resistance of RC beams strengthened with FRP grids and the ECC consists of four main components: the stirrup, concrete, ECCs, and FRP grids. Hence, it is necessary to consider their individual contributions to shear resistance separately.

5.1. Establishment of Calculation Formula

The shear capacity of the oblique section calculation formula of the RC beam can be expressed as follows:

$$V_u=V_c+V_s+V_{ecc}+V_{frp}$$

(13)

where V_c, V_s, V_ecc, and V_frp are, respectively, the contribution of concrete, the steel bar, ECC, and FRP grid to the shear capacity of the oblique section.

The size effect is an important factor in the shear strength of RC beams. Therefore, it is necessary to consider the influence of the size of RC beams on the shear strength contribution of concrete in the calculation formula. Based on the theory of fracture mechanics, Bažant et al. [32] proposed a size theory suitable for concrete materials.

$${\sigma }_{N_u}={\displaystyle \frac{B{f}_t^{\prime }}{\sqrt{1+D/D_0}}}\cdot \gamma \cdot \beta$$

(14)

$$\gamma ={\displaystyle \frac{1}{\lambda }}$$

(15)

$$\beta =\left\{\begin{array}{ll}1& {\rho }_{sv}\le {\rho }_{sv,\min }\\ (A-1)\cdot \tanh [\alpha ({\rho }_{sv}-{\rho }_{sv.\min })]+1,& {\rho }_{sv}>{\rho }_{sv,\min }\end{array}\right.$$

(16)

in which B and D₀ are empirical coefficients, which can be set as 1.802 and 616, respectively [33]. f’_t and D are, respectively, the tensile strength of concrete and cross-sectional height of the RC beam. A and $\alpha$ are, respectively, the parameters related to the stirrup ratio.

The shear capacity of concrete can be described as follows:

$$V_c={\alpha }_c\cdot {\sigma }_{Nu}b{h}_0$$

(17)

where ${\alpha }_c$ is the modified coefficient of the concrete. This study substituted the axial tensile strength of concrete instead of the splitting tensile strength, so the formula needs to be further modified with the modified coefficient of the concrete. By comparing model FE-F0S0, it is recommended that ${\alpha }_c$ be taken as 1.8. b and h₀ represent the width and effective height of the concrete cross-section, respectively.

The contribution of the stirrup to the shear capacity of the RC beam in the oblique section is shown in Equation (14).

$$V_s=f_{sv}{\displaystyle \frac{n_s{A}_{sv1}}{s}}\cdot h_0$$

(18)

where $f_{sv}$, $n_s$, and $A_{sv1}$ are, respectively, the yield strength of the stirrup, number of limbs of the stirrup, and cross-sectional area of the single-limb stirrup.

The ECC for the RC beam oblique section shear strength calculation formula should be similar to that of concrete; it can be expressed as follows:

$$V_{ecc}=f_{ECC,t}\cdot (H-2h_s)\cdot t$$

(19)

where $h_s$ is the thickness of the concrete protective layer.

$$V_{frp}={\alpha }_{f}n{f}_{cfrp}\cdot {\displaystyle \frac{A_{CFRP}}{s}}$$

(20)

in which it is considered that the FRP grid fails to play a full role in the flexural shear zone of the RC beam, and the maximum stress is close to 0.2 times of its ultimate stress. Therefore, the reduction factor ${\alpha }_f$ can be assumed to be 0.2.

5.2. Verification of Calculation Formula

To verify the accuracy of the calculation formula for the shear capacity of the oblique section, the numerical results and experiment results are used for a comparison with the results of the calculation formula.

Table 3. Verification of calculation formula.

No.	Models	H (mm)	B (mm)	ft (MPa)	fECC,t (MPa)	t (mm)	Nu (kN)	Nu,m (kN)	Nu,m/Nu	Ref.
1	E1	200	120	2.19	4	15	115	134.12	0.91	[-]
2	E2	200	120	2.19	4	15	161	168.71	0.93
3	E3	200	120	2.19	4	15	153	164.91	0.95
4	E4	200	120	2.19	4	15	156	168.71	0.96
5	E5	200	120	2.19	6	15	150	168.71	0.99
6	E6	200	120	2.19	8	15	167	168.71	0.89
7	E7	200	120	2.19	10	15	115.12	134.12	0.91
8	E8	200	120	2.19	4	20	166.48	164.65	0.86
9	E9	200	120	2.19	4	8	169.78	164.65	0.85
10	E10	200	120	2.19	4	12	171.3	164.65	0.84
11	E11	200	120	2.19	4	15	172.81	164.65	0.83
12	E12	200	120	2.19	4	15	170.84	174.73	0.93
13	E13	200	120	2.19	4	15	176.74	184.81	0.98
14	E14	200	120	2.19	4	15	179.90	194.89	1.04
15	E15	200	120	2.19	4	15	172.49	171.37	0.89
16	E16	200	120	2.19	4	15	150.75	155.24	0.87
17	E17	200	120	2.19	4	15	157.94	160.61	0.87
18	E18	200	120	2.19	4	15	167.12	168.10	0.88
19	E19	200	120	2.19	4	15	167.94	171.56	0.90
20	E20	200	120	2.19	4	15	169.25	175.01	0.91
21	F0S0	200	120	2.19	4	15	164.24	164.65	0.88	[24]
22	EF1S0	200	120	2.19	4	15	172.60	164.65	0.83
23	EF1′S0	200	120	2.19	4	15	182.46	176.62	0.84
24	EF1S1	200	120	2.19	4	15	195.83	191.15	0.83
25	EF1S2	200	120	2.19	4	15	212.39	209.79	0.83
26	EF2S0	200	120	2.19	4	15	232.15	234.57	0.84
27	SB0	200	300	1.8	4	30	296	299.02	1.01	[34]
28	SB1	200	300	1.8	4	30	376	415.60	1.11
29	SB2	200	300	1.8	4	30	449	418.67	0.93
30	SB3	200	300	1.8	4	30	471	422.21	0.90
31	SB5	200	300	1.8	4	30	271	321.54	1.19
32	SB6	200	300	1.8	4	30	381	418.67	1.10
33	SB7	200	300	1.8	4	30	381	418.67	1.10
34	SB8	200	300	1.8	4	30	411	418.67	1.02
								Mean	0.991
								COV	0.015

shows the parameters of the calculation formula of all specimens, and $N_u$, $N_{u,m}$, and $N_{u,m}/N_u$ are, respectively, the results of the peak load of the specimen, the results of the calculation formula, and the ratio of the result of the calculation formula to the peak load of the specimen. It can be found that the mean and COV of $N_{u,m}/N_u$ are, respectively, 0.991 and 0.015, and the error between the predicted results of the calculation formula and the experimental results is less than 20%. Hence, the formula introduced in this study could be regarded as a reliable predictor of the shear strength of RC beam oblique sections.

6. Conclusions

Based on the numerical analyses conducted using ABAQUS software, the following conclusions regarding the mechanical behavior of RC beams can be drawn:

• By comparing the peak load of the four models, it is evident that the stirrup, ECC, and FRP grids increase the peak load by 11.66%, 30.97%, and 6.34%, respectively. Therefore, it can be concluded that the ECC-strengthened layer has the most significant contribution to enhancing the shear capacity of the oblique section of the RC beams.
• The maximum stress of the FRP grids in the flexural shear zone is less than 20% of the peak stress in the flexural shear zone, indicating that the FRP grids do not fully utilize their tensile strength. However, the presence of FRP grids effectively reduces stress concentration in the flexural shear zone and prevents the premature cracking of the concrete. Therefore, it is essential to incorporate FRP grids to share the tension in the flexural shear zone of the RC beams in this strengthening technique, as it helps control the development of concrete cracks and improves the overall performance of the structure.
• By investigating the different influences of the parameters of the RC beam, it is found that the tensile strength of the ECC, ECC cover thickness, and the shear span ratio have significant influence on the shear capacity of the oblique section of the RC beam. With the increase in the ECC tensile strength from 4 MPa to 10 MPa, the peak load of the model increases by 8.06%. When the cover thickness of the ECC increases from 8 mm to 20 mm, the peak load of the RC beam increases by 14.42%. With the shear span ratio in the range of 1.56 to 3.13, the peak load of the RC beam is increased by 54.87%.
• Based on the contributions of various materials to the shear capacity, a calculation formula for the shear capacity of the oblique section of the RC beam was proposed. The accuracy of this calculation formula was validated using the experimental results and numerical models, demonstrating that the proposed formula can effectively predict the shear capacity of the oblique section of the RC beam.
• In future research, numerical models will be further developed and optimized to accurately simulate the interactions between FRP grids, ECCs, and RC beams under various loading conditions. Additionally, the long-term performance and durability of RC beams reinforced with FRP grids and ECCs will be studied, including the effects of environmental factors such as temperature, humidity, and corrosion on the integrity and effectiveness of the reinforcement system. A cost–benefit analysis will also be conducted to evaluate the economic feasibility of using FRP grids and ECCs to reinforce RC beams.