## 2 Analytical Fragility
### Selection of Seismic Demands  
Based on the authors' previous study [24], the sensitivity of a crane structure was analyzed and used for identifying the sources of uncertainty that are significant to the response of the structure. A series of random variables including ground motion intensity, ground motion profiles, mass, damping, and elastic modulus of steel, were considered. The results demonstrated that among all the sources of uncertainty considered, the ground motion intensity and profile played the most important role in the overall response of the container crane in terms of portal drift. In addition, the elastic modulus was recommended to be considered as a source of uncertainty in a seismic analysis. Thus, to minimize the computational cost, the ground motion intensity and profile were considered as the demand randomness, and the elastic modulus of steel was considered as the capacity randomness of the container crane. Other uncertain parameters in the finite element analysis were evaluated to be deterministic at their mean/median values. To consider the site conditions where the crane was located, many boreholes were drilled to investigate some major physical characteristics of the ground, especially the shear wave velocity (\(v_{s}\)) of the top 30 m from the ground surface. The report revealed that the minimum and maximum of \(v_{s}\) were approximately 247 and 447 m/s, respectively, indicating that the soil classes ranged from S2 to S3 according to the KBC, as presented in Table 1.  
For the crane's site, seismic zone I (with a seismic zone factor of \(Z=0.11\times g\)) was considered. By assuming a return period of 500 years as recommended for port structures, that the crane works under the Immediate Occupancy (IO) performance level [23], and the chosen soil class is S3, the 1.5% damping horizontal acceleration response spectrum was developed, as illustrated in Figure 2(a). Ten recorded ground motions (denoted as RGMs) were selected to consider the effect of the realistic ground motion profile, as presented in Table 2. Then, a spectrum matching process was conducted on the design response spectrum illustrated in Figure 2(b), to generate 10 design spectrum-matched ground motions, as depicted in Figure 3(a). The reason for using the spectrum matching technique is to reduce the variability (jaggedness) observed in the acceleration spectra for individual ground motions [25]. To ensure that the ground motions had a number of peaks of similar amplitude, 10 artificial ground motions (denoted as AGMs) were generated based on the prescribed response spectrum of Figure 2(a). The QuakeGem program was used [26], which is dependent on the spectral-representation-based simulation algorithm proposed by [PERSON] in 1996 [27; 28], with a filtered white noise signal by three different trapezoidal envelope functions. The trapezoidal functions include the rise time (1-2 s), vibration duration (8-14.5 s), and descent time (17-28 s). Figures 2(c) and 3(b) depict 10 individual artificial response spectra and the corresponding artificial time-history accelerations, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Soil Classes**} & \multirow{2}{*}{**Name of Ground Type**} & \multicolumn{2}{c}{**Classification Criteria**} \\ \cline{3-4}  & & **Bedrock Depth, H (m)** &
\begin{tabular}{c} **Soil Average Shear Wave** \\ **Velocity, \(v_{s}\) (m/s)** \\ \end{tabular} \\ \hline S1 & Rock & Less than 1 & \(\geq\)760 \\ \hline S2 & Shallow and hard ground & (1–20 or less) & \(\geq\)260 \\ \hline S3 & Shallow and soft ground & & \(<\)260 \\ \hline S4 & Deep and hard ground & & \(\geq\)180 \\ \hline S5 & Deep and soft ground & Above 20 & \(<\)180 \\  & & & (\(v_{s}\)\(\leq\) 120 m/s is not classified as S5, regardless of bedrock depth) \\ \hline S6 & Grounds that require site-specific characterization and ground response analysis \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Site classification (soil classes) of Korean Building Code [21].