#### Abstract

The use of externally bonded fiber-reinforced polymer (EB-FRP) composites for shear strengthening of reinforced concrete (RC) beams presents many challenges given the complex phenomena that come into play. Premature bond failure, the behavior of the interface layer between FRP composites and the concrete substrate, the complex and brittle nature of shear cracks, and the adverse interaction between internal steel stirrups and EB-FRP are some of these phenomena. Compared to experimental investigations, the finite element (FE) technique provides an accurate, cost-effective, and less time-consuming tool, enabling practicing engineers to perform efficient, accurate nonlinear and dynamic analysis as well as parametric studies on RC beams strengthened with EB-FRP. Since 1996, many numerical studies have been carried out on the response of RC beams strengthened using FRP. However, only a few have been related to RC beams strengthened in shear using EB-FRP composites. In addition, the analytical models that have been reported so far have failed to address and encompass all the factors affecting the contribution of EB-FRP to shear resistance because they have mostly been based on experimental studies with limited scopes. The aim of this paper is to build an extensive database of all the studies using finite element analysis (FEA) carried out on RC beams strengthened in shear with EB-FRP composites and to evaluate their strengths and weaknesses through various studied parameters.

#### 1. Introduction

Given its complexity and its propensity to brittle failure without warning, the shear behavior of reinforced concrete (RC) beams has long been a major concern in the field of structural engineering. Therefore, practicing engineers often privilege the sequence by which flexural failure occurs before shear failure. The lack of shear resistance in RC beams can be due to various interacting factors. Underestimating the real applied loads in the design process, lack of accuracy in the construction phase, and damage due to winds and earthquakes are examples of such factors. In recent years, fiber-reinforced polymer (FRP) composites for rehabilitation and strengthening of RC beams have gained in popularity and have reached worldwide acceptance since their first use as externally bonded (EB) fabrics/laminates to strengthen existing deficient structures in the late 1990s. Their success has been due to the high strength-to-weight ratio and the tensile strength they offer, which can compensate for the shear resistance deficiency of RC beams. Researchers have investigated various FRP shapes such as fabrics, laminates, bars, and rods. However, the EB method consisting of bonding FRP fabrics/laminates to the substrate of RC beams is the most common approach. Other techniques for shear strengthening of RC beams include the embedded through-section (ETS) method as well as the near-surface (NSM) method. If the tensile strength of the concrete substrate is insufficient, mechanical anchorage systems in addition to resin epoxy are generally required. The interaction between concrete and EB-FRP composites depends on various complex interacting factors.

There have been some finite element analysis (FEA) studies of RC beams strengthened in flexure. In contrast, due to the brittle nature of shear cracks and the complex behavior of the bond between concrete and EB-FRP laminates/fabrics, very few studies have considered FEA on RC beams strengthened in shear with EB-FRP. Because it is more cost-effective and less time-consuming than experimental studies, FEA has gained increasing attention in the last few years. Research conducted on RC beams strengthened in shear using EB-FRP has been very restricted. Some of previous studies also used simplistic assumptions, such as perfect bonding between components and a restrictive definition of shear cracks, leading to inaccurate prediction of the number and angles of shear cracks in RC beams affecting the effective bond length, further leading to inaccurate results. Finally, the other merit of FEA is that the response of all specimen components can be recorded during loading, resulting in an insightful comprehension of the complex relations between concrete, steel stirrups, longitudinal tensile reinforcement, and EB-FRP fabrics/laminates, which is impossible to attain by laboratory testing. The database built in the present study evaluates all the FEA carried out on RC beams strengthened in shear using FRP composites, including all the EB, NSM, and ETS strengthening techniques, with a special emphasis on the externally bonded method (EB). An evaluation has also been carried out on the parameters used in previous research studies as follows: type of FRP materials and shear strengthening configurations; size effect; interaction between components and types of interface element; analytical approach in FE simulations; number, size, and type of elements in simulation; ratio of EB-FRP, longitudinal, and transverse steel reinforcement; failure modes in concrete and EB-FRP (debonding, delamination, or rupture); effective stress, strain, and bond length; stress and strain distributions along the shear cracks; shape function of the crack, crack width, and crack pattern; and load-deflection response. Emphasis will be placed on showing the paramount importance of these parameters for the development of an analytical model to calculate the contribution of FRP laminates/fabrics to the ultimate shear capacity of RC beams shear-strengthened with EB-FRP.

#### 2. Important Issues in Modeling RC Beams Strengthened with EB-FRP

To assess the crucial issues related to RC beams strengthened in shear with FRP composites, particularly their failure modes, a review of previous studies has been carried out. Among the few FE research studies related to this type of beam, only those that exhaustively described the main simulation assumptions and validated them with experimental tests were considered in this study. Modeling concrete and its cracks as an inhomogeneous material in FE models has always been a controversial issue. However, the development of FEM has paved the way for other solutions, such as implementing concrete damage plasticity theory (CDP) in the model programming. The plastic-damage model in concrete depends on the models developed by Lubliner et al. [1] and Lee and Fenves [2]. The CDP model can analyze concrete structures subjected to dynamic loading. Furthermore, it is appropriate for evaluating quasi-brittle materials like rock, mortar, and ceramics. The two main failure modes in concrete are cracking in tension and crushing in compression. The constitutive model of CDP can capture the influence of irretrievable damage related to the failure mechanisms that happen in concrete as well as in quasi-brittle materials. Modeling RC beams and their cracks on the one hand and the interaction between concrete, steel reinforcement, and FRP composites on the other are critical parts of a simulation. Generally, two models are used to describe concrete cracking: (i) the discrete crack model and (ii) the smeared crack model.

##### 2.1. Discrete Crack Approach

The discrete crack approach depends on the geometry of the model in which crack propagation spreads among the existing borders of each element when discontinuities are defined in FE discretization. Therefore, a crack’s growth and angle rely on the size and shape of the mesh in FE programs, so that this method is mesh-objective. The solution to overcome this objectivity is to define auto-remeshing programs, which lead to increased computational challenges by changing the mesh topology [3].

##### 2.2. Smeared Crack Approach

Unlike the discrete crack model that propagates among the discontinuities in an element, the smeared crack model grows through the continuity of the material and consequently through elements, by reducing the stiffness of discretized elements. The smeared crack model can be further divided into two categories: the fixed smeared crack model and the rotated smeared crack model. The former model does not change the crack angle, and as the load increases, a crack propagates during the whole calculation process. In the rotated smeared crack model, on the other hand, the orientation of the crack changes as the load increases, and new orientations are determined based on directions of updated main stresses and strains. Nevertheless, the smeared crack method leads to localization of strain, which means that when the element dimension is close to zero, the energy consumption approaches zero. This issue was solved by introducing the crack band model, which links the fracture energy to the constitutive law of concrete. Therefore, the fracture energy during crack propagation does not rely on the dimension of the element and the mesh, making this technique not mesh-objective [3].

##### 2.3. Interactions between Steel Reinforcement and Concrete

Studies conducted by previous researchers have shown that there is an inverse interaction between EB-FRP and steel reinforcement, particularly steel stirrups, which means that when the number of steel stirrups increases, the contribution of EB-FRP to shear resistance decreases [4]. Furthermore, it has been shown that assuming a perfect bond model between the longitudinal steel reinforcement and the concrete results in more distributed crack patterns, leading to a narrower crack width. This affects the debonding process, which occurs in a later stage than without the assumption and consequently overestimates the ultimate shear resistance of the beam [3]. However, noticeable disparities have been observed on the overall load-displacement curves of the specimens. The bond-slip model introduced in the European CEB-FIP Model Code 1990 [5] is an appropriate indication of interaction between the concrete and the steel reinforcement [3].

##### 2.4. Interface between EB-FRP and Concrete

To predict the ultimate shear capacity of RC beams strengthened in shear using EB-FRP, the interaction between concrete and FRP composites should be defined precisely, because otherwise debonding mechanism between concrete and FRP composites cannot be detected. In addition, an accurate definition of the concrete-FRP interaction affects the angle and distribution of shear cracks. Indeed, the assumption of perfect bonding results in distributed diagonal shear cracks, whereas the correct bond model definition leads to one main diagonal shear crack. Therefore, assuming a perfect bond model between concrete and EB-FRP composites overestimates the ultimate load-carrying capacity of the beam [3]. This demonstrates the importance of defining the bond model between concrete and FRP composites with high precision and accuracy.

#### 3. Review and Synthesis of Previous Work on FE Modeling

When FRP composites were introduced in the construction industry, computer programs and FE packages were not developed as much as today. As a result, most evaluations of the efficiency of RC beams strengthened in shear with EB-FRP composites were based on laboratory tests, and hence, the effect of many parameters on the overall response of these tested beams could not be detected. This was particularly true when some of these parameters interacted with each other, making the behavior of these beams complex and difficult to fully understand. In addition, once FE programs were developed, most studies that used them concentrated on strengthening in flexure of RC beams using EB-FRP composites. In this section, the main FE studies carried out on RC beams strengthened in shear, as well as the major parameters affecting the response of these beams and their components, are gathered and presented in Figure 1.

Figure 1 includes 37 studies from 1996 to 2020, with a total of 239 RC beams strengthened in shear using FRP composites; six of them were subjected to microscopic studies in which the stress and strain distributions along fictional diagonal shear cracks and the effective bond length were examined [6, 7], and the rest (233 beams) were simulated using FE software. The details of these 239 shear-strengthened RC beams subjected to FEA are presented in Tables 1–3. Generally, in terms of depth, RC beams are generally classified into three groups depending on their shear span-to-depth ratio () as follows: (i) less than 1 is a deep beam; (ii) between 1 and 2.5 is a moderately deep beam, with shear failure likely occurring before flexure failure; and (iii) equal to or greater than 2.5 is a flexural slender beam that often exhibits flexure before shear failure.

The shape of RC beams is designed based on their applications and the load they carry. For example, beams with an I cross section are generally used by the road and bridge construction industries in which shear strength is of paramount importance, whereas beams with a T cross section can be designed for either the road building or the housing industry. Figure 2 presents a histogram of 233 RC beams strengthened in shear using EB-FRP and studied by FEA to evaluate the shear contribution of FRP to the ultimate shear capacity of beams. The histogram shows that, among the 233 beams, 46 were T-shaped, 175 were rectangular, and 12 were I-shaped cross sections, representing 19.7%, 75.1%, and 5.1%, respectively, of all the shear-strengthened RC beams. Therefore, despite their generalized use in practice, few FE studies focused on T cross-sectional beams (19.7%), indicating the need for more research on the response of these beams and their influencing parameters. Shear span-to-depth ratio plays a crucial role in the behavior of these beams, and, as illustrated in the histogram, 110 (47.2%) and 118 (52.9%) of the beams belong to the moderately deep () and the flexural slender () beam categories, respectively, showing that more research is needed on deep beams () where shear failure usually occurs before flexure failure.

The configuration type of shear strengthening using EB-FRP is to some extent a function of the cross section of RC beams. For instance, the full-wrap technique cannot be used for shear strengthening of T-shaped or I-shaped section beams because the flange of these beams is generally not accessible. To this end, other effective methods of shear strengthening can be used for these beam cross section shapes, such as side-bonded, U-shaped, ETS, or NSM techniques. As for the interaction between stirrups and FRP composites, experimental tests show that the ETS configuration leads to a greater contribution to shear resistance compared to EB-FRP. In addition, because the concrete core is generally stronger than its surface, more confinement is expected on the ETS bond than with externally bonded (EB) methods. Simple installation and high efficiency are some of the merits of this method. Among the 239 RC beams of all studies presented in Figures 1, 221 (92%) beams were subjected to FE studies that focused on shear strengthening using the EB method (side-bonded, U-wrap, and full-wrap EB-FRP configurations) (Table 1), and only 18 (8%) beams corresponded to the ETS and NSM methods (Table 2). In addition, as illustrated in Figure 3, among the 239 RC beams shear-strengthened with FRP, only 39 beams (16.3%) were T-shaped cross sections, indicating the research needs for FEA to study the parameters affecting the response of such beams.

#### 4. Main Studied Parameters Using FEA of Shear-Strengthened Beams

With the advantages of FEA, many of the shear responses of RC beams strengthened using FRP composites that cannot be captured by experiments can be studied from initiation of loading to ultimate failure while recording the whole failure process and its mechanisms. Most of the results derived from laboratory tests were based on the load-deflection response of the beam and the strain on the FRP composite obtained from strain gauges installed on FRP laminates, fabrics, and bars. In contrast, more parameters can be observed and studied using FEA, as illustrated in Figure 4, which highlights the variations in all components of the shear-strengthened beams. In addition, the complex interactive behavior between the components (concrete, longitudinal steel reinforcement, steel stirrups, and FRP composite) of the studied beams and their interrelations can be interpreted to achieve a more precise closed-form model that integrates all these factors.

##### 4.1. Shape Function of the Crack, Crack Pattern, and Crack Width

Because the width of a crack along its propagation path is not constant, the strain and stress distributions on FRP laminates/fabrics are not uniform. This nonuniformity of stresses and strains influences the response of steel stirrups and FRP laminates/fabrics. This means that the steel stirrups crossed by the shear crack reach the yielding point, and then, the FRP laminates/fabrics crossed by the shear cracks reach their maximum tensile strength, leading to rupture. Therefore, Chen and Teng [8] introduced a distribution factor to calculate the effective stress and strain in FRP laminates/fabrics: where is the distribution factor, which is function of the crack shape and varies with the amount of internal steel reinforcement. Therefore, the authors present the following equation to cover all shear crack shapes: where = crack width, = maximum crack width [9], (normalized vertical coordinate where , the effective depth of the beam), and is the factor determining the shape of the strain distribution. Among the 239 studied beams, 102 beams (42.6%) were evaluated for their crack width, crack pattern, and shape function (Figure 4). However, only two studies [6, 7], involving 10 beams (4.1%), considered the effect of crack shape and crack function in their proposed shear model based on FEA. The rest (92 beams) focused on shear crack patterns. This indicates the need for more research related to shear crack shape functions, considering different crack shapes and their effect on the distribution factors and the effective stress and strain, to develop future predictive models.

##### 4.2. Strain, Stress, and Slip Distribution along the Diagonal Crack on EB FRP

Chen and Teng [8] showed that the width of the shear crack varies along its length, confirming thereby that the strain and stress distributions along the FRP laminates/fabrics are nonuniform. As for the issue that FRP laminates/fabrics crossed by the diagonal shear crack experience different ranges of strain and stress as the crack widens, it could be concluded that the amount of stress/strain in the fibers is influenced by the crack width, and hence, the distribution factor is not constant. Therefore, depends on the location of the FRPs because the fibers located at the end of the crack experience more strain than those situated at the tip of the crack. Nevertheless, as mentioned in the previous section, the shape of the shear crack relies on the steel reinforcement, and hence, the shear crack is not necessarily linear. The maximum width of the crack can be in the middle of the beam if the beam contains a high ratio of longitudinal steel reinforcement [7]. The other factor affecting the fiber strain distribution is the FRP configuration type. For instance, when assuming U-shaped configurations, fibers located below the shear crack experience more strain than those located near the top of the crack because there is enough bond length on the lower side of crack compared to the upper side. However, for side-bonded configurations, with the same bond length at the bottom and top of the crack, the fibers on both sides of the crack experience the same strain distribution. Therefore, it is of paramount importance to consider the strain profile on the fibers to obtain the effective strain in FRP laminates/fabrics. On the other hand, studying slip profiles on the interface layer gives an insight into how the interface layer responds to increasing load and crack propagation. The slip distributions make it possible to understand how shear cracks form because fiber debonding occurs near the shear crack. Therefore, through a slip profile, shear crack propagation can be predicted by FEA, which is not possible by experimental tests. Among the 239 beams, 84 (35.1%) beams were subjected to FEA that evaluated the strain distribution along the fibers and the slip profile along the interface layer, of which 25 beams (10.4%) were T-shaped cross sections (Figure 4). This indicates the need for more FE studies on the strain and slip profile for T cross section beams.

##### 4.3. Load-Deflection Curve

Most laboratory and FE results on RC beams strengthened in shear using FRP composites have been based on the load-deflection curve, particularly in experimental tests. Generally, the load-deflection response has become the way to evaluate the ultimate load-carrying capacity as well as the ductility and the behavioral features of EB-FRP shear-strengthened RC beams. In fact, the load-deflection response has become a criterion to validate the accuracy of FEA results against experimental results. However, it cannot be the only criterion for validating FEA and developing analytical models because it has been observed that, even if the load-deflection curves of shear-strengthened RC beams using EB-FRP were compatible when subjected to two different tests, their failure modes could be different. In addition, it has been established that specimens featuring the same load-deflection responses from different tests may present many discrepancies with regard to formation and number of shear cracks, strain profile along fibers, and slip profiles along the interface, as well as fiber strain distribution along the horizontal axis of the specimens. Nevertheless, the load-deflection response can be one of the indications to validate whether a simulated model is accurate. Among the 239 beams, 185 (77.4%) FE beams were studied in terms of load-deflection response, of which 125 (52.3%) were rectangular, 48 (20%) were T-shaped, and 12 (5%) were I-shaped cross sections (Figure 4).

##### 4.4. Number, Size, and Types of Elements in Simulation

Selecting the right element in simulation for each component of the beam is of paramount importance in FE methods, which are highly sensitive to the size and type of the elements. As the size of elements is reduced, the time and cost for FEA will increase. Modeling EB-FRP RC beams can be classified into two categories: (i) three-dimensional simulation, which is time-consuming but provides results that are representative of the real model and (ii) two-dimensional modeling, which takes advantage of the plane stress behavior that EB-FRP RC beams can exhibit. The two-dimensional model ignores the deflection of the beam into the normal direction to its plane and makes some further simplifying assumptions, reducing considerably the time and cost of FEA. In the three-dimensional model, the brick element is chosen for concrete, whereas for steel reinforcement, either the one-dimensional bar or the three-dimensional brick element can be considered. For EB-FRP laminates/fabrics, the two-dimensional shell element for EB-FRP and the one-dimensional link or bar elements would be appropriate. As for concrete-to-FRP and steel reinforcement-to-concrete interface elements, there are many options affecting the right choice, depending on the type of fibers in the EB-FRP (one-directional, two-directional) and ranging from the one-dimensional link element to three-dimensional cohesive elements depending on whether the model in question is three- or two-dimensional. Few investigations considered the effect of element type as a studied parameter in FE simulations. In fact, only 17 (7.1%) of the 239 beams underwent this study, of which 12 (5%) were rectangular and 5 (2%) were T-shaped cross sections (Figure 4).

##### 4.5. Effective Stress and Strain, Bond Length, and Distribution Factor ()

The effective stress and strain experienced by FRP are essential components to calculate FRP contribution to shear resistance, which depends on the stress and strain distribution along the shear cracks, which in turn relies on the shape functions of cracks. As for the effective bond length of EB-FRP, which is the length of FRP that has not debonded and hence still contributes to the shear resistance, it is directly related to the shear crack distribution in the concrete. Indeed, the more the shear cracks are distributed, the shorter is the effective bond length of the FRP, and the more likely the beam will exhibit a premature failure [10]. Many models based on the effective bond length have been reported in the literature. The models introduced by Neubauer and Rostasy [11] and Chen and Teng [12] are some of the most reliable models, which are expressed in the following Equations (3)–(4), respectively:

where is the tensile strength of concrete [13]:

Among the 239 RC beams strengthened in shear using FRP composites, only 41 (17.1%) were considered to study the effective stress and strain, the bond length, and distribution factor (), of which 29 (12.1%) were rectangular and 12 (5%) were T-shaped cross sections (Figure 4).

##### 4.6. Failure Modes in Concrete and EB-FRP (Debonding, Delamination, Rupture)

Failure of RC beams strengthened in shear using EB-FRP can be mainly related either to reinforced concrete (crushing of concrete struts, tensile shear failure, yielding and fracture of steel reinforcement) or to EB-FRP composites (debonding or rupture). In fact, the failure modes observed in EB-FRP specimens are (i) rupture of FRP laminates/fabrics, which occurs usually in a fully wrapped FRP configuration; (ii) debonding of FRP due to lack of effective bond length near shear cracks; and (iii) failure of RC beams due to delamination of EB laminates/fabrics from the concrete substrate when EB-FRP still contributes to the shear resistance [9]. Failure modes related to loss of concrete strength also depend on the shear span-to-depth ratio, according to Teng et al. [14]. They can involve compressive or tensile shear failure of the RC beams, as well as failure of deep beams. The failure modes of 124 (51.8%) beams were subjected to FE studies, of which 113 (47.2%) were rectangular and only 11 (4.6%) were T-shaped cross sections (Figure 4). The advantage of FE studies over experimental tests is that detection of failure modes using FEA is much easier than laboratory testing of shear-strengthened RC beams using EB-FRP. In addition, the postfailure response of the specimens can be captured using FEA, which is impossible in an experimental protocol.

##### 4.7. Shear Strengthening Configurations, FRP Materials, and Fiber Orientation

There is an array of configurations for strengthening RC beams in shear, which are associated with the beam’s cross section, shear crack orientation, and accessibility to the surfaces of the beam to be strengthened. Three EB configurations are used in practice for strengthening RC beams in shear: (i) the side-bonded configuration where FRP laminates/fabrics are installed on the two lateral sides of the beam. The corresponding failure mode is mainly by FRP delamination from the concrete substrate; (ii) the U-shaped configuration where the FRP laminates/fabrics are installed on three surfaces of the beam (lateral sides and soffit of the beam). The corresponding failure mode is mainly by FRP debonding from the top side of the shear crack due to lesser effective bond length compared to the bottom side of crack, and (iii) the full-wrap configuration is usually labeled by the symbol W and involves wrapping FRP laminates/fabrics over the whole surface of the beam. The full-wrap configuration is more applicable to beams with easy access to all beam surfaces, such as rectangular RC beams and columns. All three configurations can be installed to the concrete substrate in the form of continuous or discontinuous (strips) bonded FRP composites. If bond length is insufficient, anchorages can be used to prevent premature debonding failure. Furthermore, various types of fibers are used for FRP composites, such as carbon, glass, and aramid. The fiber orientation in FRP laminates/fabrics can be horizontal, vertical, or diagonal at any angle to the axis of RC beams. When feasible, the optimal fiber orientation would cross the shear crack perpendicularly, providing more contribution to shear resistance. 140 (58.5%) of the 239 beams were subjected to FEA to evaluate the effect of shear strengthening configurations, type of FRP composites, and fiber orientation, of which 127 (53.1%) were rectangular and only 13 (5.4%) were T-shaped cross sections (Figure 4).

##### 4.8. Analytical Approach in FE Simulations

Solving problems using FEA can be classified into two techniques: (i) static solvers and (ii) dynamic solvers, which can further be divided into dynamic implicit and dynamic explicit. Because ABAQUS/CAE 2017 was used in the majority of studies, the following description is concentrated on ABAQUS solvers. A dynamic analysis is to be privileged over static analysis because of many reasons and considerations, such as the brittle behavior of concrete, debonding phenomenon, delamination of concrete substrate, problem of reaching convergence in analysis, and the postpeak behavior of RC beams strengthened in shear using EB-FRP laminates/sheets. Nevertheless, results obtained from a dynamic approach should be verified against a general static analysis. When performing dynamic analysis on specimens, certain parameters should be considered to improve the accuracy of the results. These include the loading pattern (smoothing, stepping, or ramping), loading duration on the structure, amount of damping for EB-FRP RC beams, and time increment, among others [9]. Figure 4 reveals that among all the FEA studies, none of them mentioned how these influencing parameters were selected to solve their models. Therefore, when the dynamic solver is used, an explanation should be provided of how these parameters were selected in the implicit or the explicit analysis.

##### 4.9. Interaction between Components and Types of Interface Elements

Steel stirrups and EB-FRP composites are the two main factors that contribute to shear resistance in EB-FRP strengthened RC beams. Nevertheless, Bousselham and Chaallal [15] showed an inverse interaction between EB-FRP and steel stirrups. The contribution of EB-FRP to shear resistance was found to decrease as the ratio of rigidity of steel stirrups to EB-FRP increased, confirming thereby the interaction between internal and external reinforcement. Chen et al. [6] introduced a model that explained this inverse shear interaction between steel stirrups and EB-FRP and demonstrated that because of this inverse interaction, neither internal nor external reinforcement reaches its full capacity. Therefore, two mobilization factors were proposed by Chen et al. [6] for determining the contribution of steel stirrups and FRP in EB-FRP RC beams: where and are the mobilization factors accounting for FRP and steel stirrups and varying between 0 and 1. Chen et al. [6] investigated the response of these two mobilization factors and found that and could be functions of crack width. The authors then created artificial cracks and observed the response of the two mobilization factors as the cracks grew, using two configuration types (U-shaped and side-bonded). They showed that as the crack width increases, was always greater than , which means that when most of the FRP strips were already debonded or had reached their maximum strength, the steel stirrups had not yet reached their maximum strength.

They further showed that there was an opposite behavior between and , indicating an inverse interaction, in which was always less than , and that this inverse interaction reduced the FRP shear contribution more than when the effect of this interaction was not considered. Hence, if the effect of this inverse interaction on just the contribution of FRP strips is considered, based on the equation proposed by Chen et al. [6], the following equation can be derived, in which the effect of inverse interaction is considered by means of a coefficient :

Integrating Equation (6) and Equation (7), it follows that [9]: where:

This indicates that when the amount of steel stirrups, and consequently the value of increases, the value of declines [9, 15]. From the database, approximately 100 EB-FRP RC beams were considered in this study to evaluate the response of inverse shear interaction between steel stirrups and EB-FRP, with different EB-FRP configurations (continuous U-shaped, strip U-shaped, fully wrapped, and side-bonded) and based on steel stirrups ratio, EB-FRP ratio, and gain in shear contribution, as shown in Figures 5(a)–5(d). The figures show that the shear gain due to FRP decreased by increasing the ratio of steel stirrups, which confirms the research findings by Bousselham and Chaallal [15]. In addition to the ratio of steel stirrups and EB-FRP, other factors, like the FRP configuration or the size effect, could have affected this inverse shear interaction because this database considers all types of shear configurations and beam sizes. The FRP gain contribution can be computed as follows: where is the total shear resistance of the beam and is the contribution of FRP to shear resistance.

**(a)**

**(b)**

**(c)**

**(d)**

Among the 239 beams, only 44 (20%) beams were subjected to FEA that considered the interaction between all components simultaneously (concrete, steel reinforcement, and FRP composites), based on the modeling of concrete cracking, of which 12 (5%) beams assumed a properly defined bond-slip law using a smeared crack + crack band model for both concrete-to-FRP and concrete-to-steel interactions, as illustrated in Figure 6. Moreover, for only 16 (6.7%) and 10 (4.1%) of the beams, a bond-slip law with smeared crack + crack band model was assumed for concrete-to-FRP and for concrete-to-steel interactions, respectively. Therefore, if the inverse interaction with concrete is considered as a fact in the overall response of RC beams strengthened in shear using FRP composites, both interactions for internal and external reinforcement should be introduced to the FE package alongside the smeared crack + crack band model. Accordingly, the simulated model would be representative of the real beam, and the results obtained from the FE package would be more reliable. In the current study, as the database shows, only 5% of all specimens were adequately and correctly modeled and simulated. Therefore, there is a need for more studies with precise and accurate simulations of RC beams strengthened in shear using FRP and particularly the EB strengthening technique. The aim of these studies would be to measure the effective strains and stresses using precise FE simulations to incorporate their effects into a reliable closed-form model providing the contribution of FRP composites to shear resistance of FRP-strengthened RC beams.

##### 4.10. Interface Elements between Concrete and FRP Composites

There are few models in the FE software that can be introduced as an interface element between concrete and FRP, ranging from one-dimensional elements (e.g., link, spring, and truss) to 2-dimensional or 3-dimentsional elements. They stand for the behavior of the interface layer (interfacial shear stress, slip profile along the interface layer). As for the 2-dimensional interface elements, cohesive elements existing in all FE software’s could be an appropriate candidate to simulate the response of the interface layer in its plane. Indeed, precision of results obtained from the behavior of interface layer much depends on the model introduced to those elements, which can be obtained from both numerical and experimental tests. Few studies have been conducted on the response of bond-slip models, which are described hereafter. Since the failure modes at the interface layer could be different (debonding, delamination), the one-dimensional elements are not an appropriate representative for presenting the response of the debonding failure at the interface layer. However, most of the research studies carried out on the 2-dimensional interface elements (cohesive elements) concentrated on the behavior of the interface layer in its plane (failure modes 2 and 3), ignoring the delamination failure occurring normal to the cohesive plane (delamination phenomenon, failure mode 1). Therefore, it is of paramount importance that 2-dimensional or 3-dimensional elements should be implemented as an interface layer to consider all types of failure modes (1, 2, and 3), by representing the debonding in 2-directions of the interface elements at its plane and the delamination normal to the interface elements. According to the study conducted by Lu et al. [16], the following are the most reliable models accounting for the response of bond-slip models.

The relationship proposed by Neubauer and Rostasy [11] for the bond-slip model consisted of two parts:

0 If for the descending part of curve,

where

Nakaba et al. [17] proposed the following relationship in which both ascending and descending parts are shown as an integrated curve: where

Monti et al. [18] applied two different equations for the ascending and descending part of the bond-slip curve: where

Savoia et al. [19] employed just a united curve for both ascending and descending parts: Where , and

Dai and Ueda [20] proposed to separate the equations for the ascending and descending sections of the curve: where

Another model proposed by Ueda et al. [21] in which the integrated equation is proposed for both ascending and descending parts of the curve: where

Finally, the bond-slip model introduced by [16], which is the most accepted model used worldwide by researchers, is as follows:

For ascending and softening parts of bond-slip curve, the following models are applied: where

In direction normal to the cohesive layer, which is representative of delamination in the interface, the following is for estimation of initial stiffness of cohesive layer:

##### 4.11. Interface Elements between Concrete and Steel Reinforcement

Unlike the interface layer between concrete and FRP where all 3 modes of failure should be defined (from one-dimensional to 2- and 3-dimensional elements), the interface between concrete and steel bars could be represented by one-dimensional elements (link, spring, truss) because experimental tests have proved that steel bars slip in their own direction (failure mode 2). Therefore, one-dimensional elements could simulate the response of the interface layer between concrete and steel reinforcement. However, 2- or 3-dimensional elements could be defined to the interface layer in a way that the stiffness of cohesive layer should be higher in comparison to mode 2, where there would be no slippage in modes 1 and 3. This later technique increases the time of the FE analysis.

Telford [22] proposed the most accepted model for both plane and deformed bars to implement the interface layer that account for ascending and softening parts. Since the deformed bars are now used for both stirrups and longitudinal bars, the following presents the model proposed by Telford [22] for bond-slip relationship between concrete and deformed steel bars (Figure 7): where , mm, and mm.

##### 4.12. Ratio of FRP, Stirrups, and Longitudinal Bars

The ratio of FRP composites, longitudinal tensile steel reinforcement, and transverse steel stirrups affects the shear resistance of EB-FRP strengthened RC beams. In addition, the ultimate load-carrying capacity of the beam depends on the complex interaction among these reinforcing elements. Therefore, more FE parametric studies are required to clarify these interactions and their underlying mechanisms to achieve an optimized design model for the shear resistance of a beam. FEA is a powerful and cost-effective tool to perform such studies compared to experimental tests. For longitudinal steel reinforcement, it has been shown that assuming a perfect bonding model can reduce the shear resistance of specimens by reducing the shear contribution of EB-FRP in beams where the shear cracks are more distributed and inclined at a high angle with respect to the horizontal axis of the beam. Indeed, FRP laminates/fabrics crossed by shear cracks at a high angle are less solicited and result in less FRP contribution to shear resistance [3]. More than 100 beams were selected from the database to evaluate the interaction between longitudinal reinforcement and EB-FRP, as illustrated in Figures 8(a)–8(d). The figures reveal that the contribution of EB-FRP to shear resistance is reduced by increasing the amount of longitudinal tensile reinforcement, confirming the results reported by Chen et al. [3]. For steel stirrups, it has been demonstrated by Bousselham and Chaallal [15] that the higher the stiffness of the steel stirrups, the less is the EB-FRP contribution to shear resistance. Finally, previous investigations (FEA, experimental tests, and analytical models) have demonstrated that, given the EB-FRP propensity to debonding failure, increasing the stiffness and cross section of EB-FRP could increase the FRP contribution to resistance up to a threshold, beyond which no increase in FRP contribution would occur because it is limited by the effective bond length [9]. 138 (57.7%) of the 239 beams were subjected to FEA that evaluated the ratio of reinforcing components, of which only 33 (13.8%) were T-shaped, and 2 (less than 1%) were I-shaped cross sections (Figure 4). There is a need for more FE studies to develop a reliable closed-form model for calculating the respective contributions of these shapes to shear resistance.

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##### 4.13. Dimension of the Beam (Size Effect)

The size effect is considered as a factor that can have a negative impact on the load-carrying capacity of RC beams. If all parameters are kept unchanged, the shear resistance of RC beams, particularly deep beams, will decrease as the beam’s depth increases [23]. More than 100 specimens from the study database were considered to evaluate the relation between the beam depth and shear span-to-depth ratio versus normalized shear strength, as illustrated in Figures 9(a)–9(c). The figures reveal that, given the shear span-to-depth ratio (), increasing the beam’s depth () reduces the normalized shear strength of the RC beam strengthened in shear using EB-FRP, which confirms the results obtained by Benzeguir et al. [23]. The database shows that few studies have been performed on the size effect of shear-strengthened RC beams using EB-FRP. Two known theories related to the size effect are generally used: the Weibull theory and the theory based on fracture mechanics. The model proposed by Bazant and Planas [24] is the only closed-form model that considers this effect and can be expressed as:

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A complete parametric study on the size effect on EB-FRP RC beams was conducted by Benzeguir et al. [23], where five major factors influencing the size effect were evaluated: the shear-span-to depth ratio , the aggregate size , the ratio of longitudinal tension bars , the ratio of steel stirrups , and the concrete strength . Many numerical studies have been carried out on beams of different sizes. However, the aim of these numerical studies was not to evaluate the size effect in RC beams strengthened in shear using EB-FRP composites. 165 (69%) of the 239 beams in this study were analyzed by FEA to study the effect of beam dimension, of which 149 (62.3%) were rectangular, 14 (5.8%) were T-shaped, and 2 (less than 1%) were I-shaped cross sections (Figure 4).

#### 5. Validation of Numerical FEA and Experimental Tests

The validation of over 200 specimens strengthened with different configurations (side-bonded, U-shaped, fully wrapped, EB with anchorage, ETS, NSM) and unstrengthened (control beams) is carried out to evaluate the accuracy of the FEA results in terms of ultimate load-carrying capacity of the specimens. Figure 10 shows the numerical versus experimental results for total strength achieved by all the specimens. The results reveal that the FE studies predicted the experimental tests with good agreement because the square of the Pearson product-moment correlation coefficient was greater than 99% for all beams.

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#### 6. Synthesis, Results, and Recommendations

FEA provides a powerful tool that can replace experimental tests if accurate assumptions are provided to the software and the required in-depth knowledge is obtained regarding parameters that affect dynamic analysis, such as loading pattern (smoothing, stepping, or ramping), impact loading duration, amount of damping for EB-FRP RC beams, and finally time increment when performing nonlinear dynamic analysis. FEA of RC beams strengthened in flexure is well documented. This does not hold true for RC beams strengthened in shear using EB-FRP composites, where clearly there is a need for more research studies. Indeed, first, the brittle behavior of shear cracks in RC beams is still unpredictable and becomes even more complex when RC beams are strengthened in shear with EB-FRP because the type of EB strengthening affects the shear crack pattern. Second, interactions between the components of such strengthened beams have not been fully documented, and indirect interactions between the components and their effects on each other are still not fully understood. Third, selecting the right type of finite element for each component of these beams is of paramount importance and needs a theoretical and experimental understanding of the response of each material when used for shear strengthening.

From existing FE studies on EB-FRP RC beams, it is obvious that early studies assumed perfect bond-slip models for their specimens, and only 18.4% (44 beams) of all the FE studies considered bond-slip laws for both concrete-to-FRP and concrete-to-steel reinforcement interactions. This has led to incorrect results because Bousselham and Chaallal [15] proved that increasing the rigidity of steel stirrups results in a reduction of EB-FRP contribution to shear resistance. Therefore, defining appropriate bond-slip laws between all the various materials involved in a study must be a priority. Furthermore, Chen et al. [6] showed that there is an inverse interaction between internal and external reinforcement and that its effect should be considered in a closed-form model, suggesting a need for research on this phenomenon.

Because of the unpredictable nature of shear cracking, most international guidelines recommend a crack angle of 45°, which is obviously overestimated, but conservative. The crack shape functions presented by Lu et al. [7] are based on four simplified assumptions, leading to unrealistic stress and strain distributions for fibers intercepted by the shear crack. Moreover, from the data evaluated earlier, 35.1% (84 beams) of all FE studies considered the stress and strain distributions along the shear crack. In addition, the assumption of one main shear crack is to some extent unrealistic because experimental tests revealed more marginal distributed shear cracks that contributed to shorter effective bond length. Therefore, more research is needed to encompass the patterns and shapes of shear cracks for EB-FRP strengthened RC beams.

The smeared crack model in conjunction with the crack band model is an appropriate model for defining shear cracks in concrete. These should be considered alongside appropriate bond-slip laws between all components of EB-FRP RC beams to achieve an accurate simulation of the real specimen. It may be worth noting that, from the FE data already gathered, only 5% (12 beams) research studies have observed this phenomenon.

#### 7. Conclusions

This study has concentrated on the evaluation of many factors affecting the accuracy of simulating RC beams strengthened in shear with EB-FRP and of parameters studied by researchers through FEA. To that end, an extensive database consisting of over 200 FE specimens validated by experimental tests was gathered and evaluated in this study. The most relevant features drawn from the FEA and the studied parameters were as follows: (i)To achieve an accurate simulation reflecting the behavior of the real beam, the parameters and elements introduced in FEA should represent the real response of each component. Essential building blocks of FEA include the smeared crack model alongside the crack band model and the bond-slip law to describe concrete-to-FRP and concrete-to-steel reinforcement interactions, and dynamic analysis in which parameters such as the structure-damping ratio, the time increment, the crack pattern, and the loading duration are well characterized. Less than 6% of all FE studies addressed these parameters, and none of them described the implementation details(ii)After FEA is performed on RC beams strengthened in shear with EB-FRP, the study parameters obtained from the output to develop the analytical model should be extracted precisely, and their complex effects on each other should be carefully examined. The angle of the shear crack in concrete, the interaction between components of specimens, the inverse interaction between internal and external reinforcements, the shape of the shear crack, the stress and strain distributions along the shear cracks, the size effect, and the type of FRP configuration are some of these important interacting parameters(iii)The ultimate load-carrying capacity of a specimen is not the only good indication for evaluating the overall specimen response. FEA is a powerful and useful tool for evaluating other important response indicators, considering all the beam components during the process of loading. Therefore, based on the built database, studies are needed to consider the influencing parameters required to develop a reliable closed-form model to calculate with confidence the EB-FRP contribution to shear resistance(iv)FEA can capture the real behavior of a beam, including details that can be important to the research community, such as the number and angle of shear cracks and the stress distribution along the shear cracks and fibers(v)By drawing comparisons between variations in the ratio of EB-FRP and in the ratio of steel stirrups, it is obvious that for each ratio of steel stirrups, there exists only one peak for the ratio of EB-FRP corresponding to the maximum gain due to FRP. This inverse interaction should be considered for optimal design(vi)When drawing comparisons between FRP contributions to shear resistance in RC beams strengthened in shear using EB-FRP, the negative impact of size effect should be included in any final closed-form model

#### Data Availability

All data, models, and code generated or used during the study appear in the submitted article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and from the Fonds de recherche du Québec - Nature et Technologies (FRQ-NT) through operating grants is gratefully acknowledged.