## 4 Calculation Formula of Shear Behavior Capacity
The ACI 440.2R-08 code gives the calculation formula for the shear contribution of externally bonded FRP materials based on the effective strain theory:  
\[V_{g}=\frac{A_{g}f_{fe}(sina+cos\alpha)d_{g}}{s_{g}} \tag{17}\]  
where \(V_{g}\) is the shear behavior capacity of the FRP grid; \(A_{g}\) is the total cross-sectional area of all FRP grid limbs, \(A_{g}=2 nt_{g}w_{g}\); n is the number of grid layers; \(t_{g}\) is the grid thickness; \(w_{g}\) is the width of a single grid limb; \(f_{fe}\) is the effective stress of the FRP grid; \(f_{fe}=\varepsilon_{fe}E_{g}\), \(\varepsilon_{fe}\) is the effective strain of the FRP grid, that is the strain of the FRP grid when a failure occurs during strengthening; \(\alpha\) is the angle between the main stress-bearing direction of the FRP grid and the axis of the beam; \(d_{g}\) is the effective height of the FRP grid, and the effective height of the beam is selected; and \(s_{g}\) is the spacing of the FRP grid. The calculation expression for the shear behavior capacity of the strengthened beam is obtained by superimposing the shear contributions of concrete, PCM, stirrups, and the FRP grid as follows:  
\[V=\frac{1.75}{\lambda+1}f_{t}bh_{0}+\frac{1.75}{\lambda+1}f_{Pt}b_{p}h_{P}+y _{w}\frac{A_{sv}}{s}h_{0}+\frac{A_{g}\varepsilon_{fe}E_{g}d_{g}(sina+cos \alpha)}{s_{g}} \tag{18}\]  
where \(b\) and \(h_{0}\) are the cross-sectional width and effective height of the beam, respectively; \(b_{p}\) and \(h_{p}\) are the cross-sectional width and effective height of the PCM, respectively. Let \(h_{p}\) be taken as \(h_{0}\); \(f_{t}\) and \(f_{Pt}\) is the tensile strength of concrete and PCM; \(y_{w}\) is the tensile strength of stirrups; and \(A_{sv}\) and s are the cross-sectional area and spacing of stirrups. Based on the model of the RC beam specimen strengthened with the FRP grid-PCM composite established in this paper, finite element analysis was carried out by taking the research content in Chapter 3 as the variable parameters. Through the regression analysis of the strain data of the FRP grid in the model, it was found that there is a power function relationship between the FRP grid and the combination of concrete strength, the shear reinforcement ratio, shear span ratio, grid spacing and thickness, and grid elastic modulus. The fitting process is shown in Figure 14 to determine the coefficients A and B. Therefore, the effective strain of the FRP grid can be calculated by Equation (19). The value of \(R^{2}\) is 0.944, the value of A is 75.410, and the value of B is 0.856. All the above coefficients are retained in three significant figures. The expression for the FRP effective strain is as follows:  
\[\epsilon_{fe}=75.410\bigg{(}\frac{f_{c}\rho_{sv}s_{g}}{t_{g}\lambda E_{g}} \bigg{)}^{0.856} \tag{19}\]  
where \(f_{c}\) is the compressive strength of concrete, with the unit of N/mm\({}^{2}\); \(\rho_{sv}\) is the shear reinforcement ratio; \(s_{g}\) is the spacing of the FRP grid, with the unit of mm; \(t_{g}\) is the thickness of the FRP grid, with the unit of mm; \(\lambda\) is the shear span ratio of the beam; and \(E_{g}\) is the elastic modulus of the FRP grid, with the unit of N/mm\({}^{2}\). When using the above formula for calculations, each variable should be directly substituted into the formula with the specified units, and no unit conversion is required. In this paper, a total of 25 beams were simulated. Among them, 24 beams suffered from shear-compression failure and 1 beam experienced concrete crushing failure. Since this paper mainly focuses on the shear resistance performance of beams, when deriving the formula for the shear resistance bearing capacity of reinforced concrete beams strengthened with FRP grid-PCM composite materials, the beams that suffered shear failure were selected as the research objects for data fitting. Therefore, this formula is mainly applicable to beams that have undergone shear failure.