## 4 Analysis of the Relationship Between the Number of Input Ground Motions and Seismic Fragility  
### Example Facility: Six-Story Commercial Building  
A seismic fragility analysis was performed on a six-story steel special moment frame structure designed in 1976 [28, 29, 30]. The target facility, located in California, U.S., was designed according to the 1973 Uniform Building Code (UBC). The building has a rectangular plan measuring 36.6 m \(\times\) 36.6 m, with floors consisting of an 8.2 cm-thick lightweight concrete slab and a 7.5 cm metal deck, and the building height is 25.3 m. The structure is a moment frame, with internal frames designed to resist gravity loads. The main structural members use A-36 steel, with a yield stress assumed to be 303 MPa. The first natural period is 1.40 s, and the total weight of the building, excluding live loads, was calculated to be approximately 3533 tons (34,644 kN), consistent with the results presented by [PERSON] and [PERSON] [31]. Figure 2 illustrates the front and plan views, along with the sizes of the beams and columns. Numerical analysis for the target facility was performed using Opensees V3.7.0, software widely used for dynamic analysis of structures [32]. A 2D numerical model was developed, considering the symmetry of the structure, with the model constructed using nonlinear beam-column elements and fiber sections. Regarding using a 2D numerical model instead of a 3D numerical model for the target facility, [PERSON] et al. [28] and [PERSON] and [PERSON] [29] developed 2D and 3D numerical models for the target facility and performed numerical analyses on both. For symmetric structures such as the target facility, a 2D numerical model was developed using half of the assigned mass for each component based on the 3D model. Two- and three-dimensional models of the six-story commercial building were calibrated to give the best match of the first-model period. Then, a comparative analysis of the structural responses from the 2D and 3D numerical models revealed identical responses. Detailed calibration of the numerical model can be found in works by [PERSON] et al. [28] and [PERSON] and [PERSON] [29]. Therefore, this study used a 2D numerical model instead of a 3D model for the numerical analysis. More detailed information on the six-story structure can be found in Table 2.  
For the time history analysis, the Newmark-beta method, which is a popular method of numerical integration, is adopted with 5% Rayleigh damping. Also, average constant acceleration (\(\gamma\) = 0.5 and \(\beta\) = 0.25) is used to obtain the stability. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Type** & **Model** & **Parameter** \\ \hline Element & Beam & nonlinearBeamColumn & Node I \& J, 4 integer points, Section ID \\  & Column & nonlinearBeamColumn & Node I \& J, 4 integer points, Section ID \\  & Fiber Section & fiberSec & 3 patch quadr with no. of fibers and section properties \\ Material & uniaxialMaterial & Steel01(Bilinear) & E(20 GPa), fy(303 MPa), b(0.03) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Detailed information on the numerical model of the six-story building. Figure 2: Plan and front views of the six-story building. (**a**) Plan view of target building. (**b**) Front view of target building. IDA was performed on the target facility using 400 ground motions, considering the ground motion magnitudes (PGA, with 0.01 g increments from 0.01 g to 5.0 g). Based on the structural response at the top floor, seismic fragility curves were derived for three performance levels (Table 3) [33]. Figure 3 illustrates the derived fragility curves for the three performance levels. The black solid line represents performance level 1 with a median of 0.30 g, the blue dashed line represents performance level 2 with 0.96 g, and the red dotted line represents performance level 3 with 1.01 g. These values were used as reference values (Table 4). is 0.30 g for Performance Level 1.