## 2 Material Constitutive Models
### Matrix Concrete  
In this analysis, the Concrete Damage Plasticity (CDP) model implemented in FEM ABAQUS software is utilized to define the tensile and compressive constitutive relationships of concrete. This model is particularly suited for simulating the stiffness degradation of concrete during the load-holding process. The stress-strain relationship for the concrete, as defined in the model, is adopted from reference [27]. The corresponding curve is illustrated in Figure 1.  
The uniaxial tensile stress-strain curve for the matrix concrete is determined by applying Equations (9) through (11) [27]:  
\[\sigma=(1-d_{t})E_{c}\varepsilon \tag{9}\]  
\[d_{t}= \begin{cases}1-\rho_{t}\left[1.2-0.2x^{5}\right]&x\leq 1\\ 1-\frac{\rho_{t}}{a_{t}(x-1)^{1.7}+x}&x>1\end{cases}  \tag{10}\]  
Figure 1: Stress–strain curve of matrix concrete. \[x=\frac{\varepsilon}{\varepsilon_{t,r}}\ \ \rho_{t}=\frac{f_{t,r}}{E_{c} \varepsilon_{t,r}} \tag{11}\]  
where \(\alpha_{t}\) is the parameter values for the descending segment of the concrete's uniaxial tensile stress-strain curve; \(f_{t,r}\) is the representative value of concrete's uniaxial tensile strength; \(\varepsilon_{t,r}\) is the peak tensile strain of concrete associated with the representative value of uniaxial tensile strength; \(d_{t}\) is the parameter for the evolution of damage under uniaxial tension in matrix concrete. Similarly, the uniaxial compressive stress-strain curve for the matrix concrete is determined by applying Equations (12) through (14) [27]:  
\[\sigma_{c}=(1-d_{c})E_{c}\varepsilon_{c} \tag{12}\]  
\[d_{c}= \begin{cases}1-\frac{\frac{\rho_{c}n}{n-1-\frac{\kappa^{2}}{\rho_{c}^{2} }}}{1-\frac{\rho_{c}n}{\varepsilon_{c}(x-1)^{2}+x}}&x\leq 1\\ 1-\frac{\rho_{c}n}{\varepsilon_{c}(x-1)^{2}+x}&x>1\end{cases}  \tag{13}\]  
\[\rho_{c}=\frac{f_{c,r}}{E_{c}\varepsilon_{c,r}};\ n=\frac{E_{c}\varepsilon_{c,r}}{E_{c}\varepsilon_{c,r}-f_{c,r}};\ x=\frac{\varepsilon}{\varepsilon_{c,r}} \tag{14}\]  
where \(\alpha_{c}\) is the parameter values for the descending segment of the matrix concrete's uniaxial compressive stress-strain curve; \(f_{c,r}\) is the representative value of concrete's uniaxial compressive strength; \(\varepsilon_{c,r}\) is the peak compressive strain of concrete associated with the representative value of uniaxial tensile strength; \(d_{c}\) is the parameter for the evolution of damage under uniaxial compressive in matrix concrete. When performing structural analysis with the CDP model, it is imperative to ensure the precision of the input parameters. While the material stress-strain curve data can be fitted using the aforementioned equations, the resultant data are not directly applicable as input for the software. Instead, a conversion must be executed according to [PERSON]'s energy equivalence principle, as illustrated in Equations (15) and (16) [28]. This methodology facilitates a more precise simulation of the material's behavior under diverse loading conditions, thereby bolstering the reliability and accuracy of the simulation outcomes. \[d_{k}=1-\sqrt{\frac{\sigma_{k}}{E_{0}\varepsilon_{k}}};\ k=t,c \tag{15}\]  
\[\varepsilon_{t}^{in}=\varepsilon_{t}-\varepsilon_{0t}^{el} \tag{16}\]  
where \(d_{k}\) (\(k=t\),\(c\)) comprises the factors of tension (\(k=t\)) and compression (\(k=c\)) damage; \(\sigma_{k}\) is true stress; \(E_{0}\) is the initial elastic modulus; \(\varepsilon_{k}\) is true strain; \(\varepsilon_{0t}^{el}\) is the tensile cracking strain, which is defined as the total strain minus the undamaged elastic strain of the material; and \(\varepsilon_{t}^{in}\) is the tensile inelastic strain that needs to be input into ABAQUS.