# Experimental Study on Shear Strengthening of RC Beams with an FRP Grid-PCM Reinforcement Layer
## 4 Analytical Model of Shear Capacity
Therefore, the shear capacity evaluation method for concrete beams reinforced by a CFRP grid-PCM reinforcement layer, which is based on the effective strain of FRP rod introduced by JSCE's Concrete Library [70], needs to be utilized, and the expression of corresponding effective strain is represented as follows:  
\[\varepsilon_{\text{u,rod}}=\sqrt[f^{\prime}_{\text{mcd}}\frac{\rho_{\text{w}}E _{\text{s}}}{\rho_{\text{web}}E_{\text{g}}}\cdot\left[1+2\left(\frac{\sigma^{ \prime}_{\text{N}}}{f^{\prime}_{\text{mcd}}}\right)\right]\times 10^{-4} \tag{10}\]  
where \(\rho_{\text{web}}\), \(\rho_{\text{w}}\) are defined in Equations (1) and (2), respectively, \(E_{\text{s}}\) is defined as the elastic modulus of the tensile steel bars; \(E_{\text{g}}\) is defined as the elastic modulus of FRP rods and thus vertical FRP grids are used instead; \(f^{\prime}_{\text{mcd}}\) and \(\sigma^{\prime}_{\text{N}}\) are defined as the design compressive strength of concrete considering the size effect from the specimens and the average value of the axial compressive strength, respectively. The values of \(f^{\prime}_{\text{mcd}}\) and \(\sigma^{\prime}_{\text{N}}\) are determined as:  
\[f^{\prime}_{\text{mcd}}=\left(\frac{h}{0.3}\right)^{-\frac{1}{10}}\cdot f^{ \prime}_{\text{c}} \tag{11}\]  
Figure 11:0  
where \(P_{\text{ed}}\) is defined as the effective tensile force of reinforcement in the axial direction; \(N^{\prime}_{\text{d}}\) is defined as the design bending moment and axis compressive force; \(A\) and \(h\) are defined as the total cross-sectional area and the height of the specimen, respectively. Using the effective strain defined in Equation (10), it is possible to evaluate the shear value \(V\) according to Equation (3). Comparing the calculated shear (\(V_{\text{cal,rod}}\)) with the experimental ultimate shear capacity (\(P_{\text{exp}}\)), a safety ratio is defined. In Table 9, the values of \(V_{\text{cal,rod}}\), \(P_{\text{exp}}\) and safety ratios are listed, corresponding to the experimental values of this study and some collected data from other results [37; 51; 71]. The safety ratio values are in the range of 1.16 to 1.61 considering all strengthened beams. These values are much higher than 1.0, indicating that the method based on the effective strain of the FRP rod can be used to evaluate the shear capacity of RC beams using a CFRP grid and PCM layer. Similarly, the values of the effective strain can be evaluated from the experimental values of maximum strain of the CFRP grid (\(\varepsilon_{\text{p,max}}\)) listed in Figure 11:1 The values increase with the reduction of the cross-sectional area of the CFRP grid. It can be seen that the smaller cross-sectional areas are, the better the deformation performance of the CFRP grid. If these strains are compared with the strains obtained by Equation (10) for the FRP rod (listed in Table 9), the values are much higher. This is because the FRP grid can resist higher shear force by vertical grids, horizontal grids, and grid points (as the intersections of vertical and horizontal grids), meaning that both vertical and horizontal grids have an inherent relation rather than only a single vertical direction of FRP grids working like an FRP rod. For this reason, there is a great difference between their load-transferring mechanisms; this can explain why the previous calculation system can be used to predict the shear capacity, but always being over-predictive. Therefore, a better evaluation method of shear capacity for concrete beams with CFRP grids should be developed to calculate or predict shear capacity more accurately than by using the previous method. To replace the effective strain corresponding to the FRP rod, the ultimate strain value of the FRP grid (\(\varepsilon_{\text{u,grid}}\)) has been defined as the measured ultimate or debonding strain of the CFRP grid