## 4 Implementation and Case Study
### Selection of M-A Sequences  
M-A sequences recorded from past earthquake events imply that the intensity of aftershocks may be as high as that of the mainshock, even though the magnitude of mainshock is larger than that of aftershocks. The duration of ground-motion accelerations of the aftershocks often is sometimes longer than that of a mainshock [34]. [PERSON] and [PERSON] [35] pointed out that the magnitude of aftershocks is independent of elapsed time after the mainshock, and large aftershocks may happen immediately or several months later. After a mainshock, some damaged building or bridge structures have the potential to be subjected to aftershocks again before any retrofit. Therefore, it is likely that the damaged civil structures may be more prone to collapse under the aftershock loads. The uncertainty of seismic loading has a great influence on the performance of civil structures during an earthquake event. Synthesized M-A sequences and as-recorded M-A sequences are adopted in the aforementioned literature. In this work, the recorded earthquake sequences are used for the seismic fragility analysis, in which the probability of failure is a function of mainshock intensity. The recorded earthquake sequences are listed in Table 1, which are downloaded from the Pacific Earthquake Engineering Research Ground Motion Database [36]. In reality, many aftershocks may follow a mainshock. As similarly done in reference [12], an M-A sequence consisting of a mainshock and an aftershock is used in this work for the sake of simplification. More aftershock events after a mainshock can be computed by using the proposed framework in this work. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Num.**} & \multicolumn{3}{c}{**Mainshock**} & \multicolumn{3}{c}{**Aftershock**} & \multirow{2}{*}{**Earthquake Event**} \\ \cline{2-2} \cline{4-7}  & **Sequence** & **M** & & **R\(\mathbf{rup}\) (km)** & & **Sequence** & **M** & **R\(\mathbf{rup}\) (km)** \\ \hline
1 & 1180 & 7.62 & 24.96 & 3863 & 6.3 & 50.27 & CHI-CHI \\ \hline
2 & 1181 & 7.62 & 47.32 & 3851 & 6.2 & 61.84 & CHI-CHI \\ \hline
3 & 1182 & 7.62 & 9.76 & 3865 & 6.3 & 40.64 & CHI-CHI \\ \hline
4 & 1183 & 7.62 & 40.43 & 3866 & 6.3 & 70.74 & CHI-CHI \\ \hline
5 & 1184 & 7.62 & 19.96 & 3867 & 6.3 & 48.43 & CHI-CHI \\ \hline
6 & 1185 & 7.62 & 59.04 & 3868 & 6.3 & 89.11 & CHI-CHI \\ \hline
7 & 1186 & 7.62 & 34.18 & 3258 & 6.3 & 59.98 & CHI-CHI \\ \hline
8 & 1187 & 7.62 & 38.13 & 3259 & 6.3 & 66.88 & CHI-CHI \\ \hline
9 & 1188 & 7.62 & 66.64 & 3260 & 6.3 & 96.03 & CHI-CHI \\ \hline
10 & 1189 & 7.62 & 59.09 & 3261 & 6.3 & 87.57 & CHI-CHI \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Recorded ground motion sequences. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Num.**} & \multicolumn{3}{c}{**Mainshock**} & \multicolumn{3}{c}{**Aftershock**} & \multirow{2}{*}{**Earthquake Event**} \\ \cline{2-2} \cline{4-7}  & **Sequence** & **M** & **Rrup (km)** & **Sequence** & **M** & **Rrup (km)** & \\ \hline  
\hline \end{tabular}
\end{table} 
Table 1: Cont. In this work, the intensity of aftershock is scaled in two manners. First, the same scaling value is used for the aftershock as the mainshock, which keeps the intensity relationship between the mainshock and aftershock. The second is that the aftershock is scaled to an aimed value without considering its intensity relationship with mainshock. Figure 3 shows one of the M-A sequences with the mainshock intensity (peak ground acceleration, PGA) scaled to 0.8 g, and the corresponding PGA accordingly scaled to 0.5 g by using the second method. It is noted that, for the purpose of demonstrating the proposed fragility modeling, the PGAs of the mainshock and aftershock are scaled to different intensity levels to generate the seismic loads to perform incremental dynamic analysis. Both the intensity of the mainshock and the aftershock can affect the performance of a structure; however, for a future earthquake event, the intensity of mainshock and aftershock is difficult to predict.