## 3 Hazard-Consistent Fragility Curves of Soil Liquefaction
### Liquefaction Hazard and Fragility Curves  
A diagram that illustrates the steps for evaluating the hazard of liquefaction is presented in Figure 3. The process begins by randomly creating a set of seismic parameters that are compatible with the hazard level, namely the earthquake magnitude, \(M\), and the corresponding peak ground acceleration at stiff sites. These parameters are then used to assess the liquefaction potential.  
In the following, after accounting for the soft site effects and the uncertainties in the associated geological parameters, the factor of safety against liquefaction (\(FS\)) and the liquefaction probability (\(P_{Lig}\)) for every depth considered are first computed. Then, the liquefaction potential index (\(I_{L}\)), liquefaction-induced settlement (\(S_{i}\)), and liquefaction probability index (\(P_{W}\)) are calculated as follows to evaluate the potential liquefaction damage of the site during an earthquake.  
The system will only process the samples with \(\text{PGA}_{\text{R}}\geq 0.05\;\text{g}\) to assess the liquefaction damage level of the site. Soil is considered non-liquefiable if it has a safety factor greater than 1.0 and a zero liquefaction probability for samples with \(\text{PGA}_{\text{R}}<0.05\;\text{g}\). The simulation continues until the number of samples with \(\text{PGA}_{\text{R}}\geq 0.05\;\text{g}\), denoted as Ng, exceeds 100,000.  
If we denote the liquefaction damage index as \(X\) and assume that earthquake occurrences follow a Poisson process, then the annual probability of the damage index exceeding a specified state \(x\) can be expressed as  
\[P\left[X>x\right]_{1 year}=1-\exp\;\left\{-\sum_{i=1}^{n}P\left[X>x|E_{i} \right]\cdot v_{i}\right\} \tag{20}\]  
The symbols above are the same as those in Equation (18).  
Based on the flowchart for liquefaction hazard analysis shown in Figure 3 and Equation (20), the damage state of the liquefaction for any specified return period \(T\) can be determined as a reference for engineering design. Additionally, to fulfill the requirements of seismic scenario simulations and early warning systems for liquefaction damage, it is crucial to establish the fragility curves for the site. Regarding how to build the fragility curves, the partial procedures are the same as those in Figure 3. When the pair of seismic parameters (\(M\), \(\text{PGA}_{\text{R}}\)) is generated, the effects of soft soil conditions and the uncertainties in the geological data parameters are taken into account, and the damage indices (\(I_{L}\), \(\text{S}_{t}\),and \(P_{W}\)) are calculated, the following equation is then used to evaluate the liquefaction fragility corresponding to a specified peak ground acceleration value of \(a\):  
\[\hat{F}\left(x;a\right)=\hat{P}\left[\ X\ >\ \ x\ \left|\ IM=a\ \right.\right]= \frac{\textit{No of }\left(\ X_{i}>\ \ x\ \right)}{\textit{No of }\left(a-\frac{\Delta a}{2}<\text{IM}_{i}\ <\ a+\frac{\Delta a}{2}\ \right)} \tag{21}\]  
in which \(IM_{i}\) and \(X_{i}\) are the \(i\)th Monte Carlo simulation values of PGAR and the corresponding damage index, respectively. \(\Delta a\) is the interval of PGAR considered. The value of 0.02 g is adopted in this study.  
The advantages of using Monte Carlo simulation in this study to construct hazard-consistent fragility curves for soil liquefaction are that the method can consider all uncertainties and variabilities in the input parameters. Furthermore, the seismic parameters of earthquake magnitude (\(M\)) and peak ground acceleration (PGAR) used to assess the liquefaction potential and damage are correlated, unlike traditional evaluation methods where \(M\) and PGAR are typically determined separately. Consequently, the method proposed in this study is more reasonable.