## 2 Finite Element Modeling and Validation
### Numerical Simulation Model
#### 2.2.1 Material Constitutive Model  
In the analysis, the concrete is modeled by the uniaxial stress-strain curve model of concrete given in the Code for the Design of Concrete Structures (GB50010-2010) [16]; the constitutive relationship curves are shown in Figure 2. Since both PCM and concrete belong to cementitious composites with similar stress performance, the same constitutive model as that of concrete is used.  
The constitutive relation of concrete under uniaxial tension can be determined by Equations (1)-(4):  
\[\sigma_{t}=(1-d_{t})E_{c}\varepsilon_{t} \tag{1}\]  
Figure 1: Test beam size and reinforcement drawing (mm).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \multicolumn{2}{c}{**Compressive Strength**} & **Tensile Strength** & **Yield Strength** & **Elastic Modulus** \\  & **(MPa)** & **(MPa)** & **(MPa)** & **(MPa)** \\ \hline Concrete & & 30 & & 30 \\ PCM & 70 & 12 & & 23 \\  & \(\phi\)6 & & 440 & 200 \\ Steel bar & \(\phi\)10 & & 440 & 200 \\  & \(\phi\)13 & & 440 & 200 \\  & 3 mm & & 455.6 & & 17.4 \\ BFRP grid & 4 mm & & 410.0 & & 16.3 \\  & 5 mm & & 382.7 & & 15.4 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mechanical properties of materials.  
Figure 2: Stress–strain relation curve of concrete.  
\[d_{t}=\left\{\begin{array}{ll}1-\rho_{t}[1.2-0.2x^{5}]&x\leq 1\\ 1-\frac{\rho_{t}}{a_{t}(x-1)^{1/2}+x}&x>1\end{array}\right. \tag{2}\]  
\[x=\frac{\varepsilon}{\varepsilon_{t,r}} \tag{3}\]  
\[\rho_{t}=\frac{f_{t}}{E_{c}\varepsilon_{t,r}} \tag{4}\]  
where \(E_{c}\) is the elastic modulus of concrete, \(\rho_{t}\) is the parameter values of the descending section of the stress-strain curve under uniaxial tension, \(f_{t}\) is the uniaxial tensile strength of concrete, \(\varepsilon_{t,r}\) is the peak tensile strain of concrete corresponding to the peak stress fault, \(\alpha_{t}\) is the parameter values of the descending part of the tensile stress-strain curve, and \(d_{t}\) is the damage evolution parameters of concrete under uniaxial tension.  
The constitutive relation under uniaxial compression can be determined by Equations (5)-(9):  
\[\sigma_{c}=(1-d_{c})E_{c}\varepsilon_{c} \tag{5}\]  
\[d_{c}=\left\{\begin{array}{ll}1-\frac{\rho_{c}n}{n-1+x^{2}}&x\leq 1\\ 1-\frac{\rho_{c}}{a_{c}(x-1)^{2}+x}&x>1\end{array}\right. \tag{6}\]  
\[\rho_{c}=\frac{f_{c}}{E_{c}\varepsilon_{c,r}} \tag{7}\]  
\[n=\frac{E_{c}\varepsilon_{c,r}}{E_{c}\varepsilon_{c,r}-f_{c}} \tag{8}\]  
\[x=\frac{\varepsilon}{\varepsilon_{c,r}} \tag{9}\]  
where \(\sigma_{c}\) is the compressive stress of concrete when the compressive stress of concrete becomes \(\varepsilon_{c0}\), \(f_{c}\) is the design value of the axial compressive strength of concrete, and \(\varepsilon_{c,r}\) is the compressive strain of concrete when the compressive stress reaches fc. When the calculated \(\varepsilon_{c,r}\) value is less than 0.002, it is taken as 0.002. \(\varepsilon_{cut}\) is the ultimate compressive strain of concrete. \(f_{c}\) is the uniaxial compressive strength of concrete. \(n\) is the coefficient. When the calculated value is greater than 2.0, it is 2.0.  
Steel bars were modeled using a bilinear stress-strain relationship with post-yield strain hardening. The expression and the relationship curve are shown in Figure 2(a). FRP grids were modeled using a linear elastic stress-strain relationship. The expression and the relationship curve are shown in Figure 2(b).