## 4 Seismic Damage Evaluation  
### Structural Damage Analysis under Earthquake Action  
The El-Centro wave is selected as a representative ground motion input to investigate the damage process. Acceleration-time history and elastic response spectrum of El-Centro wave are shown in Figure 12. The analysis time increment is 0.02 s to ensure the accuracy and convergence. Fundamental period \(T_{0}\) of specimen measured in the above experiment is 0.169 s, the corresponding acceleration in response spectrum is \(S_{\text{a}}=0.67\)g, where g is acceleration of gravity. This intensity of earthquake is appropriate to reflect dynamic characteristics of hybrid structure [42; 43]. Specim experiences different degree of stiffness degradation, which eventually leads to the increase of period. Period-time history curves for the first seven vibration modes are shown in Figure 13. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Order** & **Frequencies of Specim (Hz)** & **Frequencies of FEA (Hz)** & **Absolute Error (\%)** \\ \hline
1 & 5.942 & 5.864 & 1.312 \\
2 & 18.981 & 18.126 & 4.504 \\
3 & 25.130 & 23.974 & 4.601 \\
4 & 37.935 & 36.469 & 3.865 \\
5 & 50.034 & 48.433 & 3.199 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of frequencies, FEA: finite element modal analysis. Figure 12: _Cont._ \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Quantity** \\ \hline Node & 340 \\ “dispBeamColumn” element & 420 \\ “ShellMITC4” element & 220 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Component of model. Displacement-time history curves and acceleration-time history curves of the top node are shown in Figures 14 and 15. Furthermore, the total shear resisted by the columns and the core wall at the bottom are shown in Figure 16. At the initial stage when \(t=1.2\) s, periods of all vibration modes remain constant, no stiffness degradation occurs, and the core wall bears 53.37% of total shear. When \(t=3.2\) s, periods increase in varying degrees. Concrete-encased steel frame slightly damages and shear is redistributed to core wall. Lateral displacement of specimen increases gradually and reaches maximum. Subsequently when \(t=4.8\) s, periods increase rapidly especially in the first and the second vibration modes. Core wall resists 77.36% of the total shear. Figure 14: Displacement–time history curves. Figure 12: (**a**) Time history curve of acceleration of El-Centro wave; (**b**) response spectrum. Figure 13: Time history curve of period for the first seven vibration modes. The dynamic characteristics at each loading time including higher-order vibration modes are obtained by finite element modal analysis. Furthermore, the max story drift angle \(\theta_{\text{max}}\) was chosen to reflect the damage state and to quantify the damage degree. Seismic damage index at the end of each loading point is expressed as \(D_{\theta}=1-\theta_{\text{max},0}/\theta_{\text{max},D}\), where \(D_{\theta}\) is the damage index which considers max story drift angle \(\theta_{\text{max}}\). \(\theta_{\text{max},0}\) and \(\theta_{\text{max},D}\) are the max story drift angles at initial and damage stage, respectively. The frequencies of the first seven vibration modes and \(D_{\theta}\) at each loading stage are shown in Table 6. The results show the degradation of frequencies relating to loading history, which means different degrees of damage. When specimen is subjected to the seismic load, severe damage occurs in frame and core tube. The degeneration of fundamental frequency is an important parameter to reflect structural damage. The common damage assessment based on natural frequency is expressed as Equation (19) [44]. \[D=1-\frac{f_{i}^{2}}{f_{0}^{2}} \tag{19}\]  
where \(f_{i}\) and \(f_{0}\) are the frequencies at damage and initial stage, respectively. However, with the increase of structure height and the increase of irregularity, the influence of higher-order mode shapes cannot be neglected.