## 2 Finite Element Modeling and Validation
### Establishment of the Finite Element Model
#### 2.2.1 Material Constitutive Model  
When simulating RC beams strengthened with an FRP grid-PCM composite, the constitutive model provided in the Code for Design of Concrete Structures (GB50010-2010) [18] is adopted. Among them, the uniaxial tension constitutive relation of concrete can be determined according to Equations (1) to (4), and the uniaxial compression constitutive relation can be determined according to Equations (5) to (9). The constitutive relation curve is shown in Figure 2. In view of the fact that there is relatively little research on the constitutive relation of PCM, considering that it mainly plays the role of bonding with the FRP grid and transferring forces in the shear span zone, and it belongs to cement-based composite materials just like concrete with similar mechanical properties, the same constitutive model as that of concrete is selected for calculation as follows:  
\[\sigma_{t}=(1-d_{t})E_{c}\varepsilon_{t} \tag{1}\]  
\[d_{t}= \begin{cases}1-\rho_{t}\big{[}1.2-0.2x^{5}\big{]}&x\leq 1\\ 1-\frac{\rho_{t}}{a_{t}(x-1)^{12}+x}&x>1\end{cases}  \tag{2}\]  
\[x=\frac{\varepsilon}{\varepsilon_{t,r}} \tag{3}\]  
\[\rho_{t}=\frac{f_{t}}{E_{c}\varepsilon_{t,r}} \tag{4}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multicolumn{2}{c}{**Compressive Strength (MPa)**} & \multicolumn{1}{c}{**Yield Strength (MPa)**} & \multicolumn{1}{c}{**Elastic Modulus (GPa)**} & \multicolumn{1}{c}{**Ultimate Strain (\(\mu\)\(\varepsilon\))**} \\ \hline Concrete & \multicolumn{2}{c}{37} & \multicolumn{2}{c}{32} \\ \hline PCM & \multicolumn{2}{c}{61.4} & \multicolumn{2}{c}{26} & 210 \\ \hline \multirow{3}{*}{Steel bar} & \(\phi\)6 & \multicolumn{2}{c}{461} & 200 \\ \cline{2-5}  & \(\phi\)12 & \multicolumn{2}{c}{464} & 200 \\ \cline{2-5}  & \(\phi\)30 & \multicolumn{2}{c}{470} & 200 \\ \hline BFRP grid & 5 mm & \multicolumn{2}{c}{21} & 24,500 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of the tested materials. Figure 1: The dimensions and reinforcement of the test beams. where \(\sigma_{t}\) is the tensile stress of concrete when it reaches \(\varepsilon_{t0}\); \(\alpha_{t}\) is the parameter value of the descending branch of the uniaxial tensile stress-strain curve of concrete; \(f_{t}\) is the representative value of the uniaxial tensile strength of concrete; \(\varepsilon_{t,r}\) is the peak tensile strain of concrete corresponding to the peak stress; and \(d_{t}\) is the damage evolution parameters of concrete under uniaxial tension as follows:  
\[\sigma_{c}=(1-d_{c})E_{c}\varepsilon_{c} \tag{5}\]  
\[d_{c}= \begin{cases}1-\frac{\rho_{c}n}{n-1+x^{n}}&x\leq 1\\ 1-\frac{\beta_{c}^{2}}{\alpha_{c}(x-1)^{2}+x}&x>1\end{cases}  \tag{6}\]  
\[\rho_{c}=\frac{f_{c}}{E_{c}\varepsilon_{c,r}} \tag{7}\]  
\[n=\frac{E_{c}\varepsilon_{c,r}}{E_{c}\varepsilon_{c,r}-f_{c}} \tag{8}\]  
\[x=\frac{\varepsilon}{\varepsilon_{c,r}} \tag{9}\]  
where \(\sigma_{c}\) is the compressive stress of concrete when it reaches \(\varepsilon_{t0}\); \(E_{c}\) is the elastic modulus of concrete; \(\alpha_{c}\) is the parameter value of the descending branch of the uniaxial compressive stress-strain curve of concrete; \(f_{c}\) is the design value of the axial compressive strength of concrete; \(\varepsilon_{c,r}\) is the peak compressive strain of concrete corresponding to the peak stress; \(d_{c}\) is the damage evolution parameter of concrete under uniaxial compression; and \(n\) is the coefficient. To avoid the difficulty in the convergence of finite element calculations caused by stress concentration, which may affect the numerical calculation results, rigid pads are placed at the supports and loading points, respectively, during the modeling process. The rigid pads are set as steel materials, and their stress-strain relationship is regarded as a completely linear elastic relationship. The yield stress is taken as 1000 MPa, and the elastic modulus is the same as that of the steel bars, which is 200 GPa.