# Experimental Study on Shear Strengthening of RC Beams with an FRP Grid-PCM Reinforcement Layer
## 4 Analytical Model of Shear Capacity  
To predict the shear capacity of RC beams with a CFRP-PCM reinforcement layer, the following Equation (3) is usually used as the analytical system to obtain the shear capacity of strengthened beams (\(V\)), which is introduced by the Japan Road Association [67], given as follows:  
\[V=V_{\text{con}}+V_{\text{PCM}}+V_{\text{st}}+V_{\text{g}} \tag{3}\]  
where \(V_{\text{con}}\), \(V_{\text{PCM}}\), \(V_{\text{st}}\), and \(V_{\text{g}}\) are defined as the shear resistance contribution of the concrete, PCM, stirrups, and CFRP grid, respectively. It should be pointed out that the design method of PCM on shear contribution is very similar to concrete, according to previous experiments together with standards suggested [66; 67], and this corresponding equation for PCM hence would not appear over and over;  
Figure 11: Load–strain (\(P\)–\(\varepsilon\)) responses: (**a**) N0\({}^{\prime}\) and RHWP; (**b**) RHWC4; (**c**) RHWC6; (**d**) RHWC8; (**e**) RHWC4\({}^{\prime}\); (**f**) RHWC64. The insert at each panel indicates the location of the strain gauges measured and shown in each plot. rather, a small amount of difference would still be noted. All the mentioned parameters can be expressed as:  
\[V_{\text{con}}(V_{\text{PCM}})=\frac{\beta_{\text{d}}\cdot\beta_{\text{p}}\cdot \beta_{\text{n}}\cdot f_{\text{vcd}}\cdot b_{\text{web}}\cdot h_{0}}{\gamma_{ \text{b}}} \tag{4}\]  
where \(\gamma_{\text{b}}\) is defined as the reduction factor of member coefficient and the recommended value is equal to 1.00; while the calculation methods for \(f_{\text{vcd}}\) and for coefficients \(\beta_{\text{d}}\) and \(\beta_{\text{p}}\) are:  
\[\beta_{\text{d}} =\sqrt[4]{\frac{1000}{h_{0}}} \tag{5}\] \[\beta_{\text{p}} =\sqrt[3]{100\rho_{\text{w}}}\] (6) \[f_{\text{vcd}} =0.20\sqrt[3]{f_{\text{c}}}, \tag{7}\]  
where \(f_{\text{c}}\)' is defined as the design compressive strength of concrete or PCM; \(\rho_{\text{w}}\) is calculated by the Equation (2), as well as \(h_{0}\), both mentioned in Section 2.1. The parameter \(\beta_{\text{n}}\), depending on the bending moment and the stress produced by axial forces, has value \(\beta_{\text{n}}=1\), since in this study no prestress is introduced. The stirrup contribution is given by:  
\[V_{\text{st}}=\frac{A_{\text{web}}\cdot f_{\text{y0}}\cdot(\sin\alpha+\cos \alpha)\cdot z}{s\cdot\gamma_{\text{b}}} \tag{8}\]  
where, \(\alpha\) is defined as the angle between the stirrups and the longitudinal direction of the member; \(f_{\text{y0}}\) and \(A_{\text{web}}\) are defined as the yielding strength and cross-sectional areas in terms of internal stirrups here; \(z\) is defined as the ratio of the effective depth to the constant value 1.15; and coefficient value \(\gamma_{\text{b}}\) is equal to 1.15. The CFRP grid contribution is given by:  
\[V_{\text{g}}=\frac{A^{\prime}_{\text{web}}\cdot E_{0}\cdot\varepsilon_{\text {u}}\cdot(\sin\alpha_{\text{s}}+\cos\alpha_{\text{s}})\cdot z}{s_{\text{g}} \cdot\gamma_{\text{b}}} \tag{9}\]  
where \(A^{\prime}_{\text{web}}\) and \(E_{0}\) are defined as the cross-sectional areas and elastic modulus with respect to vertical FRP grids regarded as the shear reinforcement here; \(\alpha_{\text{s}}\) is defined as the angle between the CFRP grid and transverse direction of the member; \(\varepsilon_{\text{u}}\) is defined as the rupture strain of the CFRP grid; \(s_{\text{g}}\) is defined as the spacing of the vertical grids, which are generally considered as external stirrups for better understanding; and the coefficient value \(\gamma_{\text{b}}\) is equal to 1.15. The strengthened beam specimens are subjected to shear force during the whole loading process, and it is worth noting that some scholars have proposed that the debonding failure of FRP composites generally occurred at the interface prior to concrete crushing [49; 68; 69]. It can be observed from the experimental results that the local debonding failure occurred at the interface between the CFRP grid and the concrete in advance of the rupture of the CFRP grid.