## 2 Development and Validation of the Scaled-Down PCS Model
### Scaled-Down PCS Model Construction
In the CDP model, concrete failure is represented by tensile cracking following tensile yield and compressive crushing following compressive yield. The yield function incorporates equivalent plastic strain (\(\widetilde{e_{t}^{pl}}\)) and compressive equivalent plastic strain (\(\widetilde{e_{c}^{pl}}\)). The model uses non-elastic strains, including tensile cracking strain (\(\widetilde{e_{t}^{kl}}\)) and compressive inelastic strain (\(\widetilde{e_{c}^{in}}\)). The compressive inelastic strain, as illustrated in Figure 5a, is computed as follows:  
\[\left\{ \begin{matrix}\widetilde{e_{c}^{in}}=\varepsilon_{c}-\varepsilon_{0c}^ {el}\\ \widetilde{e_{0c}^{pl}}=\widetilde{\varepsilon_{0}}\\ \widetilde{e_{c}^{pl}}=\widetilde{\varepsilon_{c}^{in}}-\frac{d_{c}}{(1-d_{ c})}\times\frac{\varepsilon_{c}}{E_{0}}\end{matrix} \right. \tag{5}\]  
where \(\sigma_{c}\) and \(\varepsilon_{c}\) are the compressive stress and strain, respectively; \(\varepsilon_{0c}^{el}\) is the elastic deformation under compression; \(\widetilde{e_{c}^{pl}}\) is the equivalent plastic strain under compression; and \(d_{c}\) is the damage factor. The tensile inelastic strain, shown in Figure 5b, is calculated as follows:  
\[\left\{ \begin{matrix}\widetilde{e_{t}^{in}}=\varepsilon_{t}-\varepsilon_{0t}^ {el}\\ \widetilde{e_{0t}^{pl}}=\frac{\sigma_{t}}{E_{0}}\\ \widetilde{e_{t}^{pl}}=\widetilde{\varepsilon_{t}^{in}}-\frac{d_{t}}{(1-d_{ t})}\times\frac{\sigma_{t}}{E_{0}}\end{matrix} \right. \tag{6}\]  
where \(\sigma_{t}\) and \(\varepsilon_{t}\) are the tensile stress and strain, respectively, \(\varepsilon_{0t}^{el}\) is the elastic deformation under tension, \(\widetilde{e_{t}^{pl}}\) is the equivalent plastic strain under tension, \(d_{t}\) is the damage factor. The method for calculating the concrete damage factor follows [PERSON] and [PERSON] [43], expressed as follows:  
\[d_{torc}=1-\sqrt{\frac{\sigma_{torc}}{E_{0}\cdot\varepsilon_{torc}}} \tag{7}\]  
where \(d_{torc}\) represents the tensile or compressive damage factor, \(\sigma_{torc}\) is the tensile or compressive stress of the concrete, and \(\varepsilon_{torc}\) is the tensile or compressive strain of the concrete. Figure 5: Stress–strain curve of CDP model for concrete. (**a**) Compressive behavior; and (**b**) tensile behavior. In this study, C30 concrete is modeled as an elastic material, whereas C50 concrete is treated as elasto-plastic. For C50 concrete, the compressive and tensile strains are inserted into Equations (1)-(6) to derive the corresponding stresses, thereby accurately representing its stress-strain relationship. Additionally, Equation (7) is used to calculate the damage factors associated with non-elastic strains, which account for the material's degradation under loading conditions. The detailed damage data for C50 concrete are summarized in Tables 1 and 2, where the compressive strength (\(f_{c,r}\)) is 50 MPa with a corresponding strain (\(\epsilon_{c,r}\)) of 0.00192, and the tensile strength (\(f_{t,r}\)) is 3.3 MPa with a strain (\(\epsilon_{t,r}\)) of 0.000128. The steel plates in the structure are made of Q345 steel with a yield strength of 345 MPa, and the reinforcing bars are HRB400. Both the steel plates and reinforcing bars are modeled using a bilinear hardening model, as follows:  
\[\sigma_{s}=\left\{ \begin{array}{ccc}E_{s}\cdot\epsilon_{s}&0\leq\varepsilon_ {s}\leq\varepsilon_{y}\\ k\cdot E_{s}\cdot\epsilon_{s}&\varepsilon_{y}<\varepsilon_{s}\leq\varepsilon_{u} \end{array} \right.