Volume 7  Year 2024 Pages 6272
DOI: 10.11159/ijci.2024.007
DataFitted Shear Design Equations for EB FRPRC Beams
H.H.N.D. Haggalla^{1}, SangWook Bae^{1}
^{1}Texas Tech University, Department of Civil, Environmental, and Construction Engineering,
2500 Broadway W, Lubbock, Texas, United States 79409
hhaggall@ttu.edu; sangwook.bae@ttu.edu
Abstract  Based on a data fitting method applied to 490 experimental test data that are publicly available in the literature, this study provides simplistic and straightforward equations to determine the shear capacity of FRP bondedRC beams. Complete wrap, Uwrap, and side wrap schemes pertaining to Carbon Fiber Reinforced Polymer (CFRP) were analysed separately. Current design codes follow a customary approach where the nominal shear capacity is calculated by simply accumulating the shear contribution of concrete, transverse reinforcement, and FRP. The interaction between concrete, transverse reinforcement, and FRP is usually not taken into consideration. While the modulus of elasticity of FRP, transverse steel ratio, and FRP ratio all have an inverse interaction with the effective strain of FRP, the concrete compressive strength, longitudinal steel ratio, and the shear spantodepth ratio are positively linked with the effective FRP strain. This investigation further showed that as transverse reinforcement is increased, the influence of FRP on shear contribution significantly decreases in the complete wrap scheme. ACI 440.2R17, CSA S80602 and its latest version CSA S80612 are among the regularly used shear design codes in North America and they were used to compare the performance of the proposed equations. The obtained results show that the proposed equations predict the experimental results more accurately than ACI 440.2R17, CSA S80602, and CSA S80612.
Keywords: Data fitting; Fiber reinforced polymers; Shear strength; RC beams; Curve fitting; Repair; Strengthening.
© Copyright 2024 Authors  This is an Open Access article published under the Creative Commons Attribution License terms. Unrestricted use, distribution, and reproduction in any medium are permitted, provided the original work is properly cited.
Date Received: 20231005
Date Revised: 20240529
Date Accepted: 20240611
Date Published: 20240702
1. Introduction
Over the last three decades, numerous analytical and experimental studies have been conducted to analyse the behavior of Reinforced Concrete (RC) beams strengthened in shear using externally bonded fiberreinforced polymer (FRP) sheets. Due to the high strengthtoweight ratio, noncorrosive characteristics, design flexibility, and extended service life, FRP is gaining popularity over traditional strengthening techniques [1]. While shear behavior is extremely complex and unprecedented, shear failure is comparatively brittle and can lead to sudden, catastrophic failure [2]. Bonding of FRP reinforcement orthogonal to the shear crack plane has been revealed to provide higher loadcarrying capacities [3]. This means that by aligning the external reinforcement so that the principal fiber direction is as parallel as possible to the maximum principal tensile stresses, shear strengthening of reinforced concrete members using fiberreinforced polymer can be optimized. This alignment enhances the effectiveness of the FRP. In practical applications, such as beams under gravity loads or columns under seismic forces, the trajectories of maximum principal stress in shearcritical zones typically form an angle of about 45 degrees with the member axis. However, it is usually more practical to attach the external FRP reinforcement with the principal fiber direction perpendicular to the member axis. The idealization of FRP materials is similar to that of internal steel stirrups, suggesting that FRP enhances shear capacity by bearing tensile stresses at a relatively consistent strain. This strain could be the ultimate tensile strain of FRP, ϵ_{fu}, or a lower value. In this context, the external FRP is expected to stretch in the principal fiber direction to a strain level typically below the ultimate tensile strain, ϵ_{fu}. This strain is referred to as the effective strain ϵ_{fe}. When the effective strain is multiplied by the elastic modulus of FRP in the principal fiber direction, E_{f}, and the available FRP crosssectional area, A_{fv}, it represents the total force the FRP can sustain, V_{frp}, at the point of shear failure of the element. Determining the effective FRP strain through rigorous analysis is extremely difficult, if not impossible. Nevertheless, understanding this concept is essential for comprehending how FRP materials enhance the shear capacity of RC members and for designing effective shearstrengthening strategies.
2. Related Work
With the growing demand for accurate prediction of shear, numerous analytical formulations have been derived to estimate the shear capacity of RC beams strengthened with FRP [12], [49]. However, most of these shear equations are semiempirical and the experimental data available to study the shear behavior of RC structures strengthened using FRP is limited. Out of these, the prediction model developed by Triantafillou and Antonopoulos [4] is based on the assumption that FRP only carries normal stresses and therefore, develops an ultimate strain at the ultimate strength. This model is adopted by the current Eurocode. The model proposed by Khalifa [5] based on the FRP fiber orientation and an assumed crack pattern is the basis for the ACI 440.2R17 [10]. In addition to that, the Canadian Standards Association (CSAS80602/CHBDC2006 and CSAS80612) the Japan Society of Civil Engineers (JSCE 1997), the Norwegian Standard for Design of concrete structures (NS3473), and the Institution of Structural Engineers, UK (IStructE) have also developed design equations and guidelines for shear strengthening of RC structures using FRP. However, almost all these design equations are highly conservative which would potentially lead to costly and uneconomical designs. The National Cooperative Highway Research Program (NCHRP) report 678: Design of FRP systems for strengthening concrete girders in shear [11] summarizes all the design codes available.
Most previous studies have adopted nonlinear statistical analyses on limited experimental results to develop various design equations for RC elements strengthened with FRP in shear. Matthys [12] has derived equations to estimate the effective FRP strain using the curve fitting method which can then be used to calculate the shear contribution of FRP. Colotti and Spadea [13] have developed equations using the truss model concept. Based on the equilibrium of forces in a crosssection of a beam at failure, Pellegrino and Moderna [14] have obtained a theoretical equation for effective FRP strain. Nehdi [1516], and Kara [17] have taken a genetic algorithm approach whereas Hosseini and others [1821] have adopted machine learning and neural networks. Anvari [22] has also used an evolutionary machine learning approach, named genetic expression programming. Shahnewaz and Alam [23] have proposed a genetic algorithm approach for predicting the shear strength of steel fiber RC (SFRC). These studies have proven to predict superior results as compared to the current design guidelines available. Furthermore, Lima and Barros [3], and Zhou [24] have carried out reliabilitybased design analyses of FRP shearstrengthened RC beams. Although there has been a lot of experimental and analytical research on this topic, it is still difficult to precisely determine the shear capacity of RC beams strengthened using FRP on their own. Therefore, the main objective of this study is to utilize a comprehensive database to develop a model that better predicts the shear strength of FRPstrengthened RC beams. Belarbi [25] has developed a database summarizing 50 experiments that led to 375 experimental results that have been done from 1992 to 2009. This is an extension of that work which was originally presented by Triantafillou and Antonopoulos [4] and later updated by Bousselham and Chaallal [26]. The database was developed by reviewing 68 experimental studies resulting in 744 experimental data that were available in the literature.
3. Shear Design Guidelines
The current design code available in the United States for the shear strengthening of RC beams using FRP is the ACI 440.2R17 [10]. The nominal shear strength, V_{n} of a member strengthened with FRP multiplied by a strength reduction factor ϕ should exceed the required shear strength, V_{u} as per the Eqs. (1)(2) given in ACI 31819 [27]. The nominal shear strength is the accumulation of contributions from concrete, V_{c}, transverse steel reinforcement, V_{s}, and externally bonded, V_{frp}. Three types of techniques are considered in applying externally bonded FRP in the design code, namely complete wrap, Uwrap, and sidewrap. Depending on the technique used, the shear contribution from FRP, V_{frp} should further be multiplied by a strength reduction factor ψ_{f}= 0.95 for complete wrap members and ψ_{f}= 0.85 for Uwrap (3 sides) and side wrap (2 opposite sides) members. This method of direct addition or rather superposition of shear components is based on the truss model with a shear crack angle of 45 degrees. However, Colotti and Spadea [13] mentioned that this leads to an underestimation of the shear strength of RC beams:
where, V_{n} is the nominal shear strength of the beam, V_{u} is the required shear strength. Eq.(3) given in ACI 440.2R17 for calculating the shear contribution from FRP is adopted from Khalifa et al. [5].
where, V_{frp} is the shear contribution from FRP, A_{fv} is the crosssection area of FRP, E_{f} is the modulus of elasticity of FRP, ϵ_{fe} is the FRP effective strain, α is the angle of orientation of FRP, d_{fv} and s_{f} are the effective depth and spacing of FRP, respectively. The most critical variable in estimating the shear contribution from FRP, V_{frp} is the effective strain, ϵ_{fe}. The failure mode of both the FRP system and the strengthened RC member determines the maximum strain that can be achieved in the FRP system at the nominal strength, which is known as the effective strain in FRP laminates, ϵ_{fe}. The effective strain, ϵ_{fe} is typically lower than the ultimate strain ϵ_{fu}, because of strain changes along the shear fracture, local debonding on either side along the shear crack, or potential bond failure. Generally, the experimental values obtained for ϵ_{fe} are only approximate. On the other hand, the interaction of the effective strain, ϵ_{fe} with concrete strength, , and transverse reinforcement ratio, ρ_{s} cannot be neglected. Furthermore, when calculating the shear contribution from the transverse reinforcement, it is assumed that they have yielded; however, in reality, this may not always be the case when additional FRP is applied, and the beam fails prematurely due to the debonding of FRP. The equations and the method provided in the ACI 440.2R17 overestimate the shear contribution from transverse reinforcement, do not consider the interaction between the concrete, steel, and FRP, and therefore, underestimate the shear contribution from FRP [14].
4. Database Used for this Study
4. 1. Formation of the Database
The most significant aspect of this study is the database that was created to capture data on the shear strengthening of RC beams using FRP spanning from 1992 to 2022. This resulted in 744 test data. For each test data, experimental details were recorded under 25 categories: beam shape, spantodepth ratio, FRP type, compressive strength of concrete, type of compressive strength test (cylinder/cube), width of the beam, the effective depth of the beam, area of longitudinal reinforcement, yield strength of transverse reinforcement, area of transverse reinforcement, spacing, and transverse reinforcement ratio. In relation to FRP, width, thickness, effective depth, number of wraps, spacing, reinforcement ratio, angle of primary fiber direction, modulus of elasticity, tensile strength, and wrapping scheme were recorded. In addition to that, the ultimate shear force at failure, the contribution of FRP to the overall shear strength, and the failure mode of each beam were recorded.
As in any data collection process, biases can be expected here as well. Therefore, steps were taken to refine the database. There were a few potential sources of biases in the dataset. To ensure that the dataset is free from sampling bias, beams that have FRP stirrups or FRP bars as longitudinal reinforcement were omitted as this study is only focused on externally bonded FRP for shear strengthening. Prestressed and posttensioned beams were not considered. In addition, beams with circular holes/openings and inverted Tbeams were also removed. Beams that were constructed with fibermixed concrete and beams with FRP or mechanical anchorage were also omitted. Experiments that were carried out with impact loading in place of static loading were also removed. All these test data were underrepresented compared to the complete dataset. Moreover, in certain literature, key experimental data that was of interest in this study have not been recorded. In such cases, attempts were made to calculate missing data from the data that is already available. However, in cases where missing data were not able to be retrieved by any method, such test data were removed.
After a thorough process of refining the raw data, the final dataset comprised 490 test data. Out of these 490test data, 119 were Tbeams and the rest were rectangular beams. The shear strength of 333 beams was improved using FRP. The number of beams shear strengthened with AFRP, CFRP, and GFRP were 25, 285, and 23, respectively. The number of beams that were completely wrapped, Uwrapped, and sidewrapped were 66, 185, and 82, respectively, while 157 were control beams. Beams that were wrapped with novel techniques were not included in this study as they were underrepresented. Based on the analysis of previous literature, it can be concluded that this is one of the most comprehensive databases available to date. Over a period from 1992 to 2022, fewer than 1000 experimental results have been documented, highlighting a significant gap and underscoring the necessity for more experimental studies in this area.
4. 2. Significance of the Database
Out of the 490test data, 333 test data provided information on beams strengthened in shear with FRP. Three types of FRP were considered namely, Aramid, Glass, and Carbon. However, 285test data pertaining to CFRP were only considered in this study. The data was further divided, depending on the wrapping scheme: completely wrapped, Uwrapped, and sidewrapped. Before performing detailed analyses, the data was further divided based on the provision of transverse reinforcement. A detailed overview of each set of data with FRP for shear strengthening is shown in Table 1.
The descriptive statistics of each category of data are presented in Table 2. Here, f_{c}^{'} is the concrete compressive strength, t_{f} is the thickness of FRP, w_{f} is the width of FRP, d_{f} is the effective depth of FRP, s_{f} is the spacing of FRP, E_{frp} is the modulus of elasticity of FRP, f_{fu} is the tensile strength of FRP, ρ_{f} is the reinforcement ratio of FRP, ε_{fe} is the effective strain of FRP, b_{w} is the width of the concrete beam, d_{ef} is the effective depth of the beam, ρ_{s} is the longitudinal reinforcement ratio, a/d is the shear span to effective depth ratio, f_{yv} is the yield strength of transverse reinforcement, A_{v} is the area of transverse reinforcement, s_{v} is the transverse reinforcement spacing, s_{f} is the transverse reinforcement ratio, α is the angle of primary fiber direction, V_{u} is the ultimate shear force, and V_{frp} is the shear contribution from FRP. Eq. (3) indicates that determining the effective FRP strain is crucial for predicting the shear contribution of the external FRP reinforcement. The crack opening along the shear crack, local FRP debonding on both sides of the shear crack, the development length of FRP, which depends on the bond at the FRPconcrete interface, and the axial rigidity of FRP all play a role in the estimation of this strain [16].
Table 1. Detailed database of shear strengthening using FRP
Beam shape 
Wrapping scheme 
Provision of stirrups 
Failure modes 
Rectangular (223) 
C (40) 
WS (27) 
R (16) and O (11) 
W/OS (13) 
D (1), R (5), and O (7) 

U (112) 
WS (57) 
D (54) and O (3) 

W/OS (55) 
D (30), R (8), and O (17) 

S (71) 
WS (40) 
D (29) and O (11) 

W/OS (31) 
D (28), R (1), and O (2) 

Tbeam (62) 
C (0) 
WS (0) 
 
W/OS (0) 
 

U (60) 
WS (38) 
D (11), R (11), and O (16) 

W/OS (22) 
D (10), R (7), and O (5) 

S (2) 
WS (1) 
D (1) 

W/OS (1) 
D (1) 
Note 2: WS=with stirrup, W/OS = without stirrup
Note 3: D=Debonding, R=Rupture, and O=Shear and other modes of failure
Note 4: The number in parenthesis represents the number of beams in each category
The effective strain, ϵ_{fe} is therefore calibrated using a function of which indicated a power function as shown in Figure 1 (inclusive of first and third quartiles). Initially, a correlation analysis was run for the full dataset which indicated that out of all 3 wrapping schemes considered, complete wrapping is the most efficient, while the twoside wrapping is the least efficient. Therefore, it was decided to analyse complete wrap, Uwrap, and sidewrap schemes separately. This analysis further indicated an inverse correlation between the shear contribution from FRP and the transverse steel ratio, which further supports the fact that as the transverse reinforcement is increased, the effect of FRP on shear contribution becomes less [15]. ACI 31819 recommends that concentrated loads should be placed within a distance of 2h from the face of the support, given that h is the beam height. Consequently, the lower bound for the shear spantodepth ratio, a/d, is taken as 2, although in practice, this ratio might be slightly higher because the effective beam depth, d_{ef}, is usually somewhat less than the overall beam height, h. This distinguishes between deep beams (a/d<2) and regular beams (a/d≥2). There is a greater increase in shear resistance due to FRP for slender beams compared to deep beams, likely due to the arch action exhibited by deep beams. Therefore, the shear contribution of externally bonded FRP is less significant for deep beams than for slender beams, indicating that the shear spantodepth ratio, a/d, is positively correlated with the effective FRP strain. Similarly, the concrete compressive strength, f_{c}^{'}, and the longitudinal reinforcement, ρ_{s}, due to dowel action also exhibit a positive trend with respect to effective FRP strain. Conversely, for transverse reinforcement, ρ_{v}, this trend is negatively correlated. Similar behavior has been observed by Lima and Barros [3] and Collins [28] for regular beams. In particular, the dataset of beams strengthened using the Uwrap scheme, which was also reinforced with stirrups, was used to study the correlation in depth between the variables in consideration concerning the effective strain, ϵ_{fe} as it is the largest sample data set available.
Table 2. Descriptive statistics of the input parameters
Parameter 
Min. 
Max. 
Mean 
f_{c}^{'} [MPa] 
11 
57 
31 
t_{f} [mm] 
0.05 
25 
0.46 
w_{f} [mm] 
1 
800 
98.55 
d_{f} [mm] 
85 
1028 
289 
s_{f} [mm] 
1 
800 
141.73 
E_{frp} [GPa] 
5.3 
390 
196.3 
f_{fu} [MPa] 
106 
4902 
3152 
ρ_{f} 
0.000138 
0.17 
0.0053 
ϵ_{fe} [mm/mm] 
0.00013 
0.049 
0.0053 
b_{w} [mm] 
63.5 
600 
181 
d_{ef} [mm] 
85 
1028 
317 
ρ_{s} 
0 
18.72 
3.03 
f_{yv} [MPa] 
0 
653 
240 
A_{v} [mm^{2}] 
0 
142 
40 
s_{v} [mm] 
0 
800 
148 
ρ_{v} 
0 
0.011 
0.0013 
α 
0 
90 
80 
V_{u} [kN] 
18.8 
1584.5 
226.1 
V_{frp} [kN] 
0.98 
492.9 
74.5 
The parameters which had the most significant correlation with the effective strain, ϵ_{fe} of FRP are shown in the correlation plot in Figure 2. The model error of current ACI 440.2R17 concerning the experimental results obtained in terms of the shear contribution from FRP was measured using which resulted in 1.5 for rectangular beams and 0.66 for Tbeams respectively. This suggests that the ACI 440.2R17 underestimates the shear contribution of FRP, V_{frp}, for rectangular beams and overestimates for Tbeams. In general, the difference between V_{frp,exp} (the experimentally measured shear strength) and V_{frp,calc} (the calculated shear strength) highlights the accuracy and reliability of the ACI 440.2R17 used to predict the shear strength of FRPreinforced beams. An overestimation occurs when V_{frp,calc} is significantly higher than V_{frp,exp} indicating that ACI 440.2R17 predicts a higher shear strength than what is observed experimentally. This could lead to unsafe designs, as the actual strength of the beams would be lower than predicted, potentially resulting in structural failures. In contrast, underestimation happens when V_{frp,exp} is considerably lower than V_{frp,exp}, meaning ACI 440.2R17 predicts a lower shear strength than what is measured experimentally. While this conservative approach is safer, it may result in overdesign, causing unnecessary use of materials and increased costs. Understanding and minimizing these differences is crucial for refining the design codes to ensure that calculated shear strength values closely match experimental results, improving the safety, reliability, and costeffectiveness of FRPreinforced beam designs.
Further insights into the model error fluctuation are depicted in Figure 3, revealing the need for substantial improvement in ACI 440.2R17 guidelines. The analysis shows that the maximum overestimation reaches about 80%, while the maximum underestimation is approximately 50%. These notable discrepancies reveal that the estimated values in ACI 440.2R17 do not closely match the experimental values, failing to fall within a range of at least ±20%. Such large errors highlight critical areas where the ACI 440.2R17 significantly overestimates or underestimates the actual shear strength of FRPreinforced beams. This emphasizes the need to enhance ACI guidelines for more accurate shear strength predictions, ensuring safer and costeffective structural designs.
5. Proposed Model
The correlation analysis in Figure 2 revealed that the compressive strength of concrete, , has a positive correlation with effective strain, ϵ_{fe} while longitudinal steel ratio, ρ_{s}, transverse steel ratio, ρ_{v}, FRP ratio, ρ_{f}, and modulus of elasticity of FRP, E_{f} all have a negative correlation with ϵ_{fe}. With the scatter plot obtained for effective strain, ϵ_{fe} along with the variables in consideration that indicated a power function (see Figure 1), it was evident that the data calls for a nonlinear regression analysis. With the influence of the studies done by Pellegrino and Modena [14] and Nehdi and Nikopour [16], since the distribution of the data was generally known, it was decided to derive a custom equation based on experimental data fitting in the form of Eq. (4) to find the coefficients that provide the best fit. The coefficients C1 and C2 are unknown. An exponential curve fitting method in the commercially available software MATLAB 2023 version was used for this purpose. A similar approach was taken for the side wrapping and complete wrapping schemes as well. For beams without transverse reinforcement, Eq. (5) was considered:
While the preceding formulae for effective strain, ϵ_{fe} do not explicitly consider the strain distribution along the shear fracture and the FRP bond behavior, they have the advantage of avoiding complicated expressions that would otherwise be provided by methods such as genetic algorithms or machine learning. The results for ϵ_{fe }obtained from the above equations were used to calculate the shear contribution from FRP as given in Eq. (6).
Moreover, it is interesting to note the dependency of the effective strain of FRP, ϵ_{fe} on transverse steel ratio, longitudinal steel ratio, and the shear spantodepth ratio and not just on the stiffness of FRP, ρ_{f} E_{f} (refer to Figure 2). This prevails the importance of considering the interaction of FRP with concrete, longitudinal, and transverse reinforcement when investigating the behavior of shear. The estimated new effective strain values were then used to calculate the shear contribution of FRP as shown in Eq. (6) where ϵ_{fe}^{* } is the estimated effective FRP strain.
6. Results and Discussion
Since the database contained different types of FRP sheets, initially, the ratio R = ϵ_{fe}/ϵ_{fu} was considered to examine the percentage of effective strain, ϵ_{fe} with respect to the ultimate strain of FRP, ϵ_{fu} which should ideally be ≤ 1, assuming any unrealistic data has been omitted already during the database refining process. The resulting ratios obtained are shown in Table 3. For both Uwrap and sidewrap schemes, the results indicate that beams with stirrups have reached a similar effective strain compared to beams without stirrups. However, for the complete wrap scheme, beams without stirrups have reached a higher effective strain as compared to beams with stirrups.
Table 3. Values for R, C_{1}, and C_{2} obtained to estimate ϵ_{fe}
R = 
C_{1} 
C_{2} 

Rectangular beams 
C 
WS 
51.3 % 
0.24 
0.12 
W/OS 
60.8 % 
0.43 
0.67 

U 
WS 
29.7 % 
0.02 
0.45 

W/OS 
29.7 % 
0.33 
0.30 

S 
WS 
15.5 % 
0.0085 
0.55 

W/OS 
15.3 % 
0.48 
0.58 
Note 2: WS=with stirrup, W/OS = without stirrup
As mentioned before, this provides more evidence that the impact of FRP on shear contribution decreases as transverse reinforcement is increased. This could be due to the fact that the complete wrap scheme has a higher force transfer zone on either side of the shear crack, which is the development length of FRP and, therefore, precedes stirrups in shear resistance. Moreover, complete wrapping schemes indicate higher effective strain, ϵ_{fe} overall. It is apparent that FRP wrapping configuration significantly influences the shear strengthening and closedshaped/complete wrapping schemes are likely to provide higher failure loads. The coefficients C_{1} and C_{2} found for each CFRPwrapping scheme are given in Table 3. It is important to note that the results obtained from this analysis are based on U.S. customary units. The available data on AFRP and GFRP were insufficient to perform the analyses and therefore, they were not considered. The final strain level in the FRP sheets corresponds to the coefficients C_{1} and C_{2}. For beams with transverse reinforcement strengthened using CFRP, evidently, the complete wrap scheme without stirrups has the highest C_{1} and the side wrap scheme with stirrups has the lowest. Whereas the complete wrap scheme without stirrups has the highest C_{2} and the complete wrap scheme with stirrups has the lowest. This means that the complete wrap scheme without stirrups reaches closer to its ultimate strain therefore, ultimate capacity provides better efficiency compared to the Uwrap and side wrap bond applications. However, this is not the case for beams without transverse reinforcement strengthened using CFRP. Regardless of the provision of stirrups, both Uwrap and side wrap schemes tend to exhibit similar efficiency. However, the effective strain percentage of the Uwrap scheme is as twice much as the side wrap scheme. On average, the proposed model for ϵ_{fe} well agrees with the experimental results for all the wrapping schemes. With the new estimated effective FRP strains, the new shear contribution from FRP, V_{frp} was calculated and compared against the experimental values, which indicated a similar variation as the estimated effective FRP strain,ϵ_{fe} .
The ability of the stirrups to yield or not, however, could not be determined because the shear contribution from transverse reinforcement was not available for all test data. This prevented a comparison of the experimental and estimated ultimate shear forces, which are the result of the sum of the shear contributions from the concrete, transverse reinforcement, and FRP.
Furthermore, the results were compared against the design codes available to investigate the applicability of the proposed equations (see Table 4). For each wrapping scheme, the mean, and the standard deviation between the ratio of were calculated and tabulated in Table 4. As far as design codes are concerned, the ACI 440.2R17, CSAS80602 [29], and CSAS80612 [30] were considered in this analysis. These codes don’t consider the interaction between RC, transverse reinforcement, and FRP when estimating the shear contribution of FRP. Compared to the ACI 440.2R17, the proposed equations based on data fitting provide comparatively better results. A detailed summary of the design equations for ACI 440.2R17 and CSAS80602 are shown in Table 5. The deviation of the results is comparatively lower in the proposed method when compared to ACI 440.2R17.
Table 4. Performance considerations for the shear design equations

Complete wrap 
Uwrap 
Side wrap 

With transverse reinforcement 
Without transverse reinforcement 
With transverse reinforcement 
Without transverse reinforcement 
With transverse reinforcement 
Without transverse reinforcement 

Method 
Mean 
SD 
Mean 
SD 
Mean 
SD 
Mean 
SD 
Mean 
SD 
Mean 
SD 

Proposed Equations 
1.00 
0.40 
1.00 
0.52 
0.97 
0.38 
1.01 
0.45 
0.95 
0.58 
1.01 
0.59 

ACI 440.2R17 
2.01 
0.86 
1.65 
0.80 
1.29 
0.60 
1.20 
0.93 
0.98 
0.97 
0.06 
4.78 

CSA S80602 
1.63 
1.21 
1.66 
0.86 
0.66 
0.47 
0.82 
0.47 
0.92 
0.98 
0.45 
0.35 

CSA S80612 
2.40 
1.01 
3.94 
3.05 
1.07 
0.73 
2.16 
1.20 
0.64 
0.73 
0.96 
0.76 
Evidently, the side wrap scheme should be further investigated, and the design equations provided in the ACI 440.2R17 and CSAS806 ought to be revised. Both ACI 440.2R17 and CSAS80602 underestimate the contribution from FRP in the complete wrap scheme. Meanwhile, the CSA S80612 standard significantly underestimates the shear contribution of FRP for complete and Uwrap schemes, while it overestimates the shear contribution in the side wrap scheme. For the side wrap scheme, all three codes overestimate the results with a higher scatter of results for beams with transverse reinforcement. Perceptibly, beams shear reinforced using the side wrap scheme need much more investigation. In the Uwrap scheme, ACI 440.2R17 underestimates results by about 20%30%, and CSAS80602 overestimates this by a similar percentage. Meanwhile CSAS80612 underestimates results by about 10%15%.
5. Conclusion
Based on the data fitting method, this study developed shear design equations for FRPRC beams with and without FRP stirrups. Results for shear strength computed using the suggested equations are in good agreement with the experimental findings in the database considered. When compared to the majority of research, which has developed complex equations utilizing various statistical methodologies, such as machine learning, etc., these equations are significantly more straightforward and simplistic. Conclusively, the following summary can be made:
 Through this study, an extensive repository containing 490 test data relevant to RC beams shear strengthened utilizing FRP was created. This database underwent a rigorous refining process. When compared to preceding publications, this could be regarded as one of the most comprehensive databases yet created.
 The twosided wrapping is the least efficient, according to the correlation study carried out using the database, while the complete wrapping is the most efficient.
 This analysis also revealed an inverse relationship between the shear contribution from FRP and the transverse steel ratio, further demonstrating that the effect of FRP on shear contribution decreases as transverse reinforcement is increased. Evidently, the completely wrapped beams without stirrups showed higher effective strain than beams with stirrups. This may be taken into account given that transverse reinforcement is preceded in shear resistance by the complete wrap FRPstrengthening design, since it has a higher force transfer zone on each side of the shear crack. Complete wrapping methods, however, generally show higher effective strain and are hence, likely to produce higher failure loads.
 To optimize the equations for the shear design of FRPRC beams, the curve fitting approach can be a useful tool. The results obtained from the analysis are in good agreement with experimental results.
 The suggested equation shows how the FRP effective strain, ϵ_{fe} is dependent on the compressive strength of the RC, the axial stiffness of FRP, the transverse reinforcement, and the longitudinal reinforcement. however, the current design codes do not consider this. When assessing the performance of RC beams shear enhanced using FRP, it is crucial to take this interaction between the parts into account.
 The current ACI 440.2R17 code exhibited subpar performance compared to the equations proposed in this study developed using the curve fitting method. However, the proposed technique should be further examined for the side wrap scheme in beams with transverse reinforcement because it greatly increased the standard deviation of the results for the predicted shear contribution of FRP. The Canadian standards (CSAS80602 & CSAS80612) investigated in this study behave in a similar manner.
 It is recommended that AFRP and GFRP be added to the database of experimental findings for shear strengthening utilizing CFRP in order to conduct a more thorough analysis and produce optimized equations.
References
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