## 2 Numerical Modeling
### Ground Motion Selection and Scaling  
In this research, a recorded MSAS ground motion from the 1999 CHICHI earthquake, whose information is summarized in Table 5, was selected as the representative earthquake input for the dynamic analysis. The CHICHI earthquake was a very famous earthquake that occurred in Taiwan. Rich ground motion records were obtained, and, after the main-shock, there were many recorded aftershocks. Many mainshock-aftershock sequences were recorded at the same station in orthogonal directions. Therefore, a representative mainshock-aftershock sequence recorded at the CHY029 station in direction N was selected. The acceleration time history is shown in Figure 5a. In this manuscript, for simplicity and consistency, \"CHICHI-CHY029-N\" is used to designate this earthquake.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{8}{c}{**Eccentricity (\%)**} \\ \cline{2-8}  & **0** & **5** & **10** & **15** & **20** & **25** & **30** \\ \hline
1 & 1.169 & 1.175 & 1.209 & 1.268 & 1.342 & 1.430 & 1.528 \\
2 & 1.152 & 1.161 & 1.165 & 1.166 & 1.166 & 1.166 & 1.166 \\
3 & 0.770 & 0.765 & 0.753 & 0.737 & 0.720 & 0.704 & 0.691 \\
4 & 0.328 & 0.330 & 0.342 & 0.362 & 0.386 & 0.413 & 0.443 \\
5 & 0.323 & 0.326 & 0.327 & 0.327 & 0.327 & 0.327 & 0.327 \\ Translational–torsional period ratio & 1.518 & 1.536 & 1.606 & 1.720 & 1.864 & 2.031 & 2.211 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The first five vibration periods of the bidirectional eccentric structural models.  
Figure 5: The records of CHICHI-CHY029-N mainshock and aftershock: (**a**) The acceleration time history; (**b**) spectral acceleration.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Earthquake Name** & **Type** & **Station** & **Direction** & **Time** & \(\mathbf{M_{w}}\) & **PGA** \\ \hline CHICHI & Mainshock & CHY029 & N & 20 September 1999 & 7.62 & 0.238 g \\  & Aftershock & CHY029 & N & 25 September 1999 & 6.30 & 0.158 g \\ \hline \hline \end{tabular}
\end{table}
Table 5: Information of the selected mainshock and aftershock earthquake records.  
Figure 5b shows the design spectra and acceleration response spectra of the mainshock and aftershock. The design spectra are based on the Chinese seismic design code [29], where the maximum acceleration coefficient is 0.5. The mainshock and aftershock records are scaled to rare earthquake level in degree 7. As shown in Figure 5b, the periods corresponding to maximum spectral acceleration of mainshock and aftershock records are 0.2 s and 0.6 s, respectively.  
The amplitudes of the mainshock and aftershock are scaled according to the following method. The peak acceleration (PGA) of the recorded mainshock earthquake is 0.238 g, and that of the aftershock is 0.158 g. The maximum IDR is more sensitive to the intensity of the aftershock ground motion as compared with other structural response indicators, which is due to the dynamic analysis results that the structural IDR most obviously changes with the structural eccentricity. Therefore, the IDR was selected as an amplification factor for earthquakes. The performance level and the corresponding IDR are listed in Table 6, as recommended in FEMA 2000. When the structural IDR reaches 1%, 2%, and 4%, the mainshock PGA is calculated to be 0.15 g, 0.25 g, and 0.32 g, respectively. For each mainshock PGA, the amplitude of the aftershock is scaled by 0.6, 0.8, 0.9, and 1 times the amplitude of the mainshock. Table 6 shows the proportional PGA values of the mainshock and aftershock at different damage levels. It is worth mentioning that for consistency, all the cases of mainshock amplitudes are calculated and determined based on symmetric structural models. A time interval of 40 s between the mainshock and aftershock is added into the program to calm the model vibration after the mainshock loading.