## 4 Finite Element Analysis  
### Model Description  
Concrete damage plasticity model in ABAQUS was used to model concrete as it can accurately describe the elastic-plastic behavior as well as damage of concrete. The constitutive relationships of resin concrete material at five days were used (see Figures 2b and 3b). Constitutive relationships of normal concrete recommended in GB 50010-2010 [33] were adopted, as shown in Figure 14a. Three parts of the compressive constitutive relationship are identified. The first part is the elastic range, and the proportional limit stress can be determined to 0.4\(\mathrm{f_{c}}\), where \(\mathrm{f_{c}}\) is the cubic compressive strength. The second part is the nonlinear ascending range, and it is determined by Equation (7). \[\sigma_{\mathrm{c}}=\frac{\rho_{\mathrm{c}}n}{n-1+\mathrm{x_{c}}^{n}}\mathrm{ E_{c}}\varepsilon_{\mathrm{c}} \tag{7}\]  
where \(\sigma_{\mathrm{c}}\) and \(\varepsilon_{\mathrm{c}}\) are the compressive stress and strain, respectively; \(\rho_{\mathrm{c}}=\mathrm{f_{c}}/(\mathrm{E_{c}}\varepsilon_{\mathrm{cr}})\); \(\mathrm{E_{c}}\) is the Young's modulus; \(\varepsilon_{\mathrm{cr}}\) (=0.0021) is the strain when the \(\sigma_{\mathrm{c}}\) reaches \(\mathrm{f_{c}}\); and, \(n=\mathrm{E_{c}}\varepsilon_{\mathrm{cr}}/(\mathrm{E_{c}}\varepsilon_{ \mathrm{cr}}-\mathrm{f_{c}})\); \(\mathrm{x_{c}}=\varepsilon_{\mathrm{c}}/\varepsilon_{\mathrm{cr}}\) and \(\mathrm{x_{c}}\leq 1\) for the ascending range. The third part is the descending part, as expressed Equation (8). \[\sigma_{\mathrm{c}}=\frac{\rho_{\mathrm{c}}}{\alpha_{\mathrm{c}}(\mathrm{x_{c }}-1)^{2}+\mathrm{x_{c}}}\mathrm{E_{c}}\varepsilon_{\mathrm{c}} \tag{8}\]  
where \(\alpha_{\mathrm{c}}\) is a parameter related to the shape of descending part of compressive constitutive relationship. \(\mathrm{x_{c}}>1\) for the descending part. The tensile constitutive relationship of concrete can be defined in two parts. The first part is the elastic part, and the stress is from zero to the ultimate value. The second part is the descending part, and it can be determined by Equation (9). \[\sigma_{\mathrm{t}}=\frac{\rho_{\mathrm{t}}}{\alpha_{\mathrm{t}}(\mathrm{x_{t }}-1)^{1.7}+\mathrm{x_{t}}}\mathrm{E_{c}}\varepsilon_{\mathrm{t}} \tag{9}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Designation**} & **Cracking Load** & **Load Capacity** & **Peak Deflection** & **Maximum Crack Width** \\  & **(kN)** & **(kN)** & **(mm)** & **(mm)** \\ \hline (0) S-0 & 73 & 399 & 9.7 & 4.2 \\ (1) S-1 & 143 & 492 & 11.8 & 7.1 \\ (2) S-2 & 155 & 516 & 10.8 & 2.4 \\ (3) S-3 & 158 & 539 & 10.1 & 1.8 \\ (1)/(0) & 1.96 & 1.23 & 1.22 & / \\ (2)where \(\sigma_{t}\) and \(\varepsilon_{t}\) are the tensile stress and tensile strain, respectively; \(\rho_{t}=f_{t}/(E_{c}\varepsilon_{tr})\); \(f_{t}\) is the tensile strength; \(\varepsilon_{tr}\) (=0.0001) is the strain when the \(\sigma_{t}\) reaches \(f_{t}\); \(\alpha_{t}\) is a parameter related to the shape of descending part of tensile constitutive relationship; and, \(x_{t}=\varepsilon_{t}/\varepsilon_{tr};x_{t}>1\) for the descending part. The simple elastic-plastic model (Figure 14b) and the strengthening elastic-plastic model (Figure 14c) were used for the constitutive relationship of steel bar and steel mesh, respectively. The normal concrete and resin concrete were modeled using C3D8R elements. C3D8R is an eight-node linear element and there is only one integral point in the center of the element and the displacement solution is more accurate using this element, as compared with C3D8 element; the reinforcing bars and steel mesh were modeled using T3D2 element. T3D3 is a two-node linear three-dimensional (3-D) truss element, and is validated to simulate the steel bar effectively [38]. Mesh convergence study was carried out and the global mesh size was determined to be 20 mm combining the calculation time and accuracy. With this mesh size, the finite element model had 10,244 C3D8R elements and 5427 truss elements (S-3).