## 4 Case Study
### Liquefaction Fragility Analysis  
According to the flowchart of liquefaction hazard analysis shown in Figure 3, Equation (21), and the site condition of the BH-29 site, the fragility curves and the corresponding probability masses of the liquefaction potential index (\(I_{L}\)), liquefaction-induced settlement (\(S_{1}\)), and liquefaction probability index (\(P_{W}\)) are shown in Figure 6, Figure 7, and Figure 8, respectively. During the Chi-Chi earthquake, the BH-29 site experienced a PGA of approximately 0.24 g [27]. If the damage status of at least \"Medium damage\" or \"Moderate liquefaction\" is specified according to the on-site liquefaction conditions [24], i.e., \(I_{L}\geq 10\), \(S_{t}\geq 20\) cm, and \(P_{W}\geq 0.4\), then the corresponding probabilities are 0.80, 0.74, and 0.76 from the fragility curves, respectively. Although these probabilities vary slightly, they are all high enough to cause the specific damage state. These results adequately reflect the soil characteristics at the liquefaction site and validate the practicality of the developed fragility curves. Furthermore, it can be found that the damage state will become higher as the PGA becomes larger from the trend of the probability mass for each damage index. In other words, as the PGA increases, so does the likelihood of severe liquefaction damage. Thus, more measures must be taken to mitigate potential liquefaction damages.  
Figure 5: Relationship of PGA between soft and hard site conditions [5].  
Figure 6: Fragility for \(I_{L}\) of the BH-29 site. (**a**) Fragility curves for the liquefaction potential index (\(I_{L}\)). (**b**) The probability mass of fragility for \(I_{L}\).  
Here, a lognormal cumulative distribution function [28] is assumed to model the fragility curve, \(F(x;a)\) in Equation (3), as  
\[F(x;a)=\Phi\bigg{[}\frac{\ln\ (a/m)}{\xi}\bigg{]}_{x} \tag{22}\]  
in which \(a\) represents PGA; \(m\) is the median of the fragility function (i.e., the intensity measure (\(IM\)) level with a 50% probability of causing the site to exceed the specified value of \(x\)) and \(\zeta\) is the standard deviation of the natural logarithm of the \(IM\) (also known as the dispersion of \(IM\)) [28], respectively; and \(\Phi[\cdot]\) = standardized normal distribution function.  
Table 5 shows the two parameters \(m\) and \(\zeta\) for the damage indices \(I_{L}\), \(S_{t}\), and \(P_{W}\) for several damage levels. It fits very well when comparing regression curves and data points, obtained from Equation (21), as shown in Figures 6-8.  
From Table 5, we can find that the median \(m\) is large when the damage level becomes high, whereas the log-standard deviation \(\zeta\) does not have this common trend. It is reasonable that when the damage level is high, the associated \(IM\) level (i.e., PGA) required to cause the site to exceed this level increases.  
Figure 8: Fragility for \(P_{W}\) of the BH-29 site. (**a**) Fragility curves for the liquefaction probability index (\(P_{W}\)). (**b**) Probability mass of the fragility for \(P_{W}\).  
Figure 7: Fragility for \(S_{t}\) of the BH-29 site. (**a**) Fragility curves for liquefaction-induced settlement (\(S_{t}\)). (**b**) Probability mass of the fragility for \(S_{t}\).