## 3 Seismic Damage Model
### Model Verification and Discussion  
A comparative analysis of three damage models, namely the M-Usami damage model, the damage model of Usami, and Park-Ang damage model, was carried out. Their viability for the seismic damage evaluation was considered by using the previous testing results, from which a corresponding damage index range was determined according to a damage grade. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Specimen**} & **Axial** & **Slendeness** & **Transverse Reinforcement** & **Loading Angle 0** & **Constant \(c\)** & **Loading Angle \(\theta\)** & **Test Constant c** \\  & **Ratio \(n\)** & **Ratio** & **Ratio \(\rho_{v}\)/\(\gamma\)** & **Ratio \(\rho_{v}\)** & **in X Direction** & **in X Direction** & **in Y Direction** \\ \hline B1 & 0.2 & 13.1 & 1.06 & \(\pi/4\) & 1.64 & \(\pi/4\) & 2.28 \\ B2 & 0.1 & 13.1 & 1.06 & \(\pi/4\) & 1.79 & \(\pi/4\) & 2.40 \\ B3 & 0.05 & 13.1 & 1.06 &According to the failure state of quasi-static test results, four performance levels (I to IV) were defined to describe the seismic damage state of RC bridge piers (Table 4), and the damage index range from less than 0.1 to more than 1.0 was also given based on each damage state. The four-stage definition of damage limit states in this paper is based on the traditional defines from literature and codes such as [26; 31], where the damage limit states are described from these four qualitative damage states: slight, moderate, extensive, and complete. The quantitative damage index values for 14 specimens corresponding to different qualitative damage levels were obtained by using experimental displacements and hysteresis energy values in Section 2. These experimentally determined damage index values were compared with those from the damage model as shown in Figure 12. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Damage Index**} & \multirow{2}{*}{**Performance Level**} & \multirow{2}{*}{**Performance Objective**} & \multirow{2}{*}{**Damage Level Description**} & \multirow{2}{*}{**Qualitative Description of the Performance Level Division (Failure Phenomenon)**} \\  & & & & & \\ \hline \multirow{2}{*}{\(DD\)\(<\) 0.1} & \multirow{2}{*}{I} & \multirow{2}{*}{Can use after simple repair} & Most in good condition, minor damage only & No damage or local concrete slight cracking \\ \hline \multirow{2}{*}{\(0.1\leq DD\)\(<\) 0.40} & \multirow{2}{*}{II} & Much damage needs to be repaired & Moderate damage & Cracking phenomenon is common and lateral longitudinal reinforcement yielded \\ \hline \multirow{2}{*}{\(0.4\leq DD\)\(<\) 1.0} & \multirow{2}{*}{III} & Structure with unrepairable damage & Heavy damage & Concrete crushed and the ultimate bearing capacity reached \\ \hline \multirow{2}{*}{\(DD\)\(>\) 1.0} & \multirow{2}{*}{IV} & \multirow{2}{*}{Structure collapse} & \multirow{2}{*}{Collapse} & Ultimate compressive strain of core concrete, ultimate strength of longitudinal reinforcement reached; 85\% of the limited load. \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Damage limit states of the piers. Figure 12: Damage index values of three damage models with experimental data. (**a**) the M-Usami model; (**b**) the Usami model; (**c**) the Park–Ang model. It can be seen that the M-Usami model and the Usami model can model the destruction of the experimental specimen well, especially at levels of minor damage and moderate damage. However, for specimens B4, B8, B10, B3, and B14, the values of the damage index from the Usami model are too small in the serious damage state when compared to the experiment but can represent the moderate damage state. This observation shows that the effect of the hysteretic energy dissipation on damage will increase at large displacements as the Usami model fails to consider this in its calculation. For the Park-Ang model, the damage level of components is over-estimated, especially for the specimens B3, B5, and B7. Overall, the damage index calculated from the M-Usami agrees well with the real damage state.