## 4 Case Study
### Results and Discussion  
The seismic response of the tank was evaluated through IDA, employing the selected suite of 28 ground motions (Section 4.2) applied to the 2 DoF system (Section 2) modeled in OpenSeesPy [44] with the impulsive and convective characteristics determined from the FEM modal analysis (Section 4.1). Each ground motion was scaled incrementally from a \(PGA\) of 0.005 g up to 2.0 g, with a step size of 0.05 g. A total of 11,200 simulations were conducted to capture the tank response across various seismic intensities. For each simulation, the key \(EDPs\) were extracted: (a) the sloshing wave height \(\delta_{g}\), (b) the base shear \(FS\), (c) the overturning moment \(OM\), and (d) the meridional stress \(\sigma_{m}\). Figure 9 shows the variation of these parameters as a function of \(a_{g}\). For the sloshing wave height, the increase is nearly linear at lower \(a_{g}\) values but becomes nonlinear at higher intensities (\(a_{g}>0.5\) g) due to complex fluid-tank interactions. Similarly, the base shear and overturning moment increase consistently with \(a_{g}\), with the base shear showing a noticeable jump at moderate intensities. The meridional stress increases steeply at moderate \(a_{g}\) levels, indicating that this parameter is more sensitive to moderate seismic intensities, which is particularly relevant for assessing the risk of elephant's foot buckling (\(DS_{EFB}\)). These results were processed to derive fragility curves for the four identified damage states: liquid overspill \(DS_{LO}\), base shear exceedance \(DS_{BS}\), overturning moment exceedance \(DS_{OM}\), and elephant's foot buckling \(DS_{EFB}\), as presented in Figure 10. These curves represent the probability of exceeding each damage state as a function of seismic intensity. The fragility curve for liquid overspill (\(DS_{LO}\)) (blue line) remains flat for lower \(a_{g}\) values, rising only after \(a_{g}\approx 0.2\) g. This indicates that liquid overspill is highly resistant to low seismic intensities due to sufficient freeboard, with significant probability of occurrence under moderate seismic demands. The curve for base shear exceedance (\(DS_{BS}\)) (green line) shows a steep rise at relatively low \(a_{g}\) values. This suggests that base shear is the most sensitive damage state, with high failure probabilities under low seismic demands. For overturning moment exceedance (\(DS_{OM}\)) (red line), the fragility curve remains flat until \(a_{g}\approx 0.8\) g and only begins to rise noticeably after that value. This behavior highlights the tank's resistance to overturning, likely due to its squat geometry and low center of gravity. Finally, the fragility curve for elephant's foot buckling (\(DS_{EFB}\)) (yellow line) exhibits a steep slope, with failure probabilities rapidly increasing in the range of \(a_{g}\approx 0.1\) g to 0.4 g.  
Figure 8: Pseudo-acceleration response spectra of selected ground motions. This steep rise suggests that elephant's foot buckling is the second dominant failure mode at moderate seismic intensities, driven by stress concentrations near the base of the tank. The comprehensive seismic fragility assessment of the tank is summarized in Table 4, where key statistical parameters, such as the median (\(\mu\)) and standard deviation (\(\beta\)), are presented. These parameters are essential for constructing the fragility curves, as they quantify the central tendency and variability of the tank's response to seismic loads. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{\(DS_{LO}\)} & \multicolumn{2}{c}{\(DS_{BS}\)} & \multicolumn{2}{c}{\(DS_{OM}\)} & \multicolumn{2}{c}{\(DS_{FEB}\)} \\ \hline \(\mu\) & \(\beta\) & \(\mu\) & \(\beta\) & \(\mu\) & \(\beta\) & \(\mu\) & \(\beta\) \\ \hline
0.461 & 0.149 & 0.118 & 0.056 & 1.141 & 0.126 & 0.249 & 0.058 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mean (\(\mu\)) and standard deviation (\(\beta\)) of the fragility curves.