## 4 Analysis of the Relationship Between the Number of Input Ground Motions and Seismic Fragility
### Example Facility: 13-Story Commercial Building  
The second example facility is a 13-story steel special moment frame structure with one basement level and 13 above-ground floors [28; 29; 30]. The facility is located in California, U.S., and was designed based on the 1973 UBC and constructed in 1975. The facility has a rectangular plan measuring 48.8 m \(\times\) 48.8 m, with a height of 57.45 m. The structure is a moment frame, with internal frames designed to resist gravity loads [28; 29; 30; 37]. The first natural period is 3.04 s.  
Figure 9 illustrates the front and plan views, along with the sizes of the beams and columns. Similar to the previous example, the target facility was analyzed using a 2D numerical model. For the 13-story structure, the same three performance levels were considered as for the six-story structure [33]. Figure 10 presents the results of the seismic fragility curves for the three performance levels; the black line represents Performance Level 1 with a median of 0.4137 g, the blue dashed line represents Performance Level 2 with 1.1322 g, and the red dotted line represents Performance Level 3 with 1.2440 g, all of which were assumed as reference values (Table 10). Figure 9: Plan and front views of the 13-story building. (**a**) Plan view of target building; (**b**) front view of target building. Figure 10: Seismic fragility curve of the thirteen-story building. **(a)** Fragility curves; **(b)** Max. displacements. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Performance Level** & **Median (g)** & **STD** & \
\begin{tabular}{c} **PGA at** \\ **16\% Level (g)** \\ \end{tabular} &
\begin{tabular}{c} **PGA at** \\ **84\% Level (g)** \\ \end{tabular} \\ \hline
1—IO & 0.4137 & 0.9013 & 0.1688 & 1.0139 \\
2—LS & 1.1322 & 0.8779 & 0.4729 & 2.7105 \\
3—CP & 1.2440 & 0.8568 & 0.5306 & 2.9165 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Parameters of seismic fragility curve of the thirteen-story building. Similar to the six-story example, LHS was used to select ground motions for various sizes of samples from the total 400 ground motions. The seismic fragility curve variables were statistically analyzed after repeating the process 100,000 times for each sample size. When considering 10 ground motions, the variables for the fragility curve for the three performance levels were calculated (Figure 11), and a comparison was made with the reference values presented in Table 10. For Performance Level 1 (immediate occupancy), the reference median value was 0.41 g. When 10 ground motions were randomly selected and the fragility curve was derived (after 100,000 repetitions), the median ranged from \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Performance Level**} & \multicolumn{4}{c}{**Median (g)**} & \multicolumn{2}{c}{**Difference**} \\ \cline{2-6}  & **Ref** & **Min** & **Max** & **Avg** & **(\%)** \\ \hline
1—IO & 0.4137 & 0.2827 & 0.5855 & 0.4152 & 0.3517 \\
2—LS & 1.1322 & 0.7984 & 1.5579 & 1.1363 & 0.3652 \\
3—CP & 1.2440 & 0.8865 & 1.6866 & 1.2484 & 0.3537 \\ \hline \multirow{2}{*}{Performance Level} & \multicolumn{4}{c}{Log-standard deviation} & Difference \\ \cline{2-6}  & Ref & Min & Max & Avg & (\%) \\ \hline
1—IO & 0.9013 & 0.6315 & 1.1376 & 0.8949 & 0.7168 \\
2—LS & 0.8779 & 0.6195 & 1.1253 & 0.8732 & 0.5263 \\
3—CP & 0.8568 & 0.6032 & 1.1228 & 0.8524 & 0.5202 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Range of seismic fragility curve parameters of the 13-story building (N = 50). Figure 12: _Cont._