## 2 Liquefaction Fragility
### Damage Evaluation of Soil Liquefaction  
In engineering practice, concerns are not only focused on the liquefaction potential at a specific depth, but there is also a need to predict the overall potential of liquefaction to cause foundation damages. In this study, three liquefaction damage indices are adopted. These are the liquefaction potential index (\(I_{L}\)), liquefaction-induced settlement (\(S_{t}\)), and liquefaction probability index (\(P_{W}\)). Each index and associated damage states are described in the following. [PERSON] et al. (1982) [PERSON] et al. (1982) proposed a liquefaction potential index (\(I_{L}\)) using the factor of safety (\(FS\)) against liquefaction and the depth weighting factor (\(W\)) to quantify the severity of the degree of liquefaction at depths ranging from 0 to 20 m beneath the ground as below:  
\[I_{L}=\int_{0}^{20}F\cdot W(z)dz \tag{7}\]  
or in discrete form:  
\[I_{L}=\sum_{i=1}^{n}F_{i}\times W_{i}\times H_{i} \tag{8}\]  
where  
\[F_{i}= \begin{cases}1-FS_{i}&for\ \ 0\leq FS_{i}\leq 1\\ 0&for\ FS_{i}>1\end{cases}  \tag{9}\]  
and  
\[W_{i}=10-0.5\,z_{i} \tag{10}\]  
in which \(z\) is the depth of the soil in meters; \(H\) is the thickness of the soil layer considered. The severity of liquefaction was classified into four classes called \"very low\", \"low\", \"high\", and \"very high\", depending on the value of \(I_{L}\) shown in Table 1, proposed by [PERSON] et al. (1982) [PERSON] et al. (1982). The second damage index utilized in this study is liquefaction-induced settlement, which can lead to the collapse or partial collapse of a structure, particularly when there is substantial differential settlement between adjacent structural elements. The total settlement, \(S_{t}\), for sandy soil is given by:  
\[S_{t}=\sum_{i=1}^{n}\varepsilon_{vi}\ H_{i} \tag{11}\]  
where \(H_{i}\) = thickness of layer \(i\); \(\varepsilon_{vi}\) = volumetric strain of layer \(i\); \(n\) = number of soil layers. In 1992, [PERSON] and Yoshimine [PERSON] and [PERSON] (1992) developed a practical method that correlates volumetric strain with relative density and associated penetration resistances (SPT-N value and CPT tip resistance), and the factor of safety (\(FS\)) of the sand against the state of liquefaction. It can be used to compute the settlements in Equation (11). The relationship between the extent of the damage and the approximate settlements is shown in Table 2. \begin{table}
\begin{tabular}{c c} \hline \(I_{L}\) & **Liquefaction Potential Class** \\ \hline
0 & Very low \\ \hline
0 \textless{} \(I_{L}\leq 5\) & Low \\ \hline
5 \textless{} \(I_{L}\leq 15\) & High \\ \hline \(I_{L}\) \textgreater{} 15 & Very high \\ \hline \end{tabular}
\end{table} 
Table 1: Liquefaction potential classes based on \(I_{L}\)([PERSON] et al., 1982) [PERSON] et al. (1982). When evaluating \(I_{L}\) and \(S_{t}\) in Equations (8) and (11), the simplified method [11; 12; 17] is used first to calculate the factor of safety against liquefaction, denoted by \(FS_{\text{Seed}}\). Then Equation (12) is adopted to convert \(FS_{\text{Seed}}\) to \(FS\) (as mentioned above) based on the research by [PERSON] [17] to determine the factor of safety in the case of equal probability of liquefaction. \[FS=1.1417\,FS_{\text{Seed}}\,^{0.5433} \tag{12}\]  
The third damage index used in this study is the liquefaction probability index (\(P_{W}\)). Because the probability of liquefaction, \(P_{Lip}\), is essential for making risk-based design decisions, this study employs the expression for \(P_{Lip}\) derived by [PERSON] (2008) [18] using logistic regression to evaluate the potential for liquefaction:  
\[P_{Lip}=\frac{1}{1+\exp\{-[10.097-0.245(N_{1})_{60 cs}+3.757\ln(CSRN)]\}} \tag{13}\]  
where  
\[CSRN=CSR/MSF=\ 0.65\cdot\ \left(\frac{a_{\text{max}}}{g}\right)\left(\frac{ \sigma_{v0}}{\sigma_{v0}^{\prime}}\right)\,r_{d}\,/\,\left(\frac{M_{w}}{7.5} \right)^{-256} \tag{14}\]  
in which \(\left(N_{1}\right)_{60 cs}\) represents the clean sand equivalence of the corrected SPT blow count as defined by [PERSON] et al.