## 2 Materials and Methods
### Fragility Analysis  
Fragility functions correlate structural capacity with demand and are considered one of the most significant tools in assessing vulnerability of structures before and after an extreme event such as an earthquake [36]. We obtained capacity parameters from the pushover curve using the approach formulated by [PERSON] and [PERSON] [37]. The ultimate and yield displacements were estimated from the pushover curve using the approach suggested by [PERSON] and [PERSON] [36]. Based on yield (\(d_{y}\)) and ultimate \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Earthquake** & **Station** & **Date** & **Event** & **Magnitude** & **PGA (g)** \\ \hline \multirow{3}{*}{Challart Valley} & Station: 54428 Chalafant, CA-Zack Bros & 20 July 1986 & Foreshock & 5.9 & 0.23 \\ \cline{2-6}  & Station: 54428 Chalafant, CA-Zack Bros & 21 July 1986 & Mainshock & 6.4 & 0.40 \\ \cline{2-6}  & Station: 54171 Bishep, CA-LADWP & 21 July 1986 & Aftershock & 6.4 & 0.17 \\ \hline \multirow{3}{*}{Chamoli} & Station: IIT8 gope Gopeshwar, India & 28 March 1999 & Foreshock & 6.6 & 0.19 \\ \cline{2-6}  & Station: IIT8 gope Gopeshwar, India & 28 March 1999 & Mainshock & 6.6 & 0.35 \\ \cline{2-6}  & Station: IIT8 uhiu Lithuan, India & 28 March 1999 & Aftershock & 6.6 & 0.09 \\ \hline \multirow{3}{*}{Northwest China} & Station 19001 Jashi, China & 6 April 1997 & Foreshock & 5.9 & 0.14 \\ \cline{2-6}  & Station: 19001 CBS Jashi, China & 11 April 1997 & Mainshock & 6.1 & 0.27 \\ \cline{2-6}  & Station: 19001 CBS Jashi, China & 15 April 1997 & Aftershock & 5.8 & 0.23 \\ \hline \multirow{3}{*}{Kozani} & Station: ITSAK Chromio Anapsiktirio, Greece & 15 May 1995 & Foreshock & 5.1 & 0.15 \\ \cline{2-6}  & Station: ITSAK Kozani, Greece & 13 May 1995 & Mainshock & 6.4 & 0.21 \\ \cline{2-6}  & Station: ITSAK Chromio Anapsikrio, Greece & 17 May 1995 & Aftershock & 5.3 & 0.12 \\ \hline \multirow{3}{*}{Mammoth Lake} & Station: 54099 CA-SNARL & 25 May 1980 & Foreshock & 6.0 & 0.18 \\ \cline{2-6}  & Station: 54099 CA-SNARL & 25 May 1980 & Mainshock & 6.1 & 0.40 \\ \cline{2-6}  & Station: 54099 CA-SNARL & 25 May 1980 & Aftershock & 5.7 & 0.34 \\ \hline \multirow{3}{*}{Northridge} & Station: 5303 CA- White Oak Covenant Church & 17 January 1994 & Foreshock & 6.7 & 0.36 \\ \cline{2-6}  & Station: 5303 CA- White Oak Covenant Church & 17 January 1994 & Mainshock & 6.7 & 0.47 \\ \cline{2-6}  & Station: 5353 Canoga Park CA & 17 January 1994 & Aftershock & 6.7 & 0.41 \\ \hline \multirow{3}{*}{Utarkashi} & Station: ghan Ghanisali, India & 19 October 1991 & Foreshock & 7 & 0.11 \\ \cline{2-6}  & Station: utta Utarkashi India & 19 October 1991 & Mainshock & 7 & 0.30 \\ \cline{2-6}  & Station: bhat Bhatwari, India & 19 October 1991 & Aftershock & 7 & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Earthquake sequence used for time history analysis (retrieved from [[https://www.strongmotioncenter.org/](https://www.strongmotioncenter.org/)]([https://www.strongmotioncenter.org/](https://www.strongmotioncenter.org/)), accessed on 15 March 2022). Figure 7: Seismic sequence history for the Northridge earthquake. displacement (\(d_{u}\)) demands, four damage states, slight (D1), moderate (D2), extensive (D3), and collapse (D4), were characterized. The characterization scheme for the four damage states is presented in Equation (1) as follows:  
\[\left\{ \begin{array}{c}\text{D1}=0.7d_{y}\,\\ \text{D2}=1.5d_{y}\,\\ \text{D3}=\frac{1}{2}\big{(}d_{y}+d_{u}\big{)}\,\\ \text{D4}=d_{u}\end{array} \right. \tag{1}\]  
Fragility functions for the four damage states were constructed using the formulation presented by [PERSON] et al. [38]. The cloud-analysis-based fragility derivation approach is a regression-based probability model that estimates the \(DCR_{LS}\) (demand-to-capacity ratio for a damage state) for the given value of the IM, e.g., Sa (T1, \(\xi\)).