# Probabilistic Seismic Safety Assessment of Railway Embarnments
## 2 Methodology to Perform Fragility Curves  
Fragility curves are usually described by a cumulative distribution function (CDF) of lognormal distribution, and thus they can be expressed as follows:  
\[F=P[ds|IM=x]=\Phi\left[\frac{ln\left(\frac{x}{\theta}\right)}{\beta}\right] \tag{1}\]  
where \(F\) denotes a seismic fragility, \(P[ds|IM=x]\) is the probability of reaching or exceeding a particular damage state with \(IM=x\), \(\Phi()\) is a cumulative distribution function (CDF) of lognormal distribution, \(\theta\) is the median of fragility function and \(\beta\) is the standard deviation of \(lnIM\). There are two common statistical approaches for estimating parameters (\(\theta\) and \(\beta\)). The parameters are obtained by the method of moments so that the distribution has the same moments (e.g., mean and standard deviation) as the sample moments of the observed data. The obtained results can be utilized to identify the fragility variables considering the maximum likelihood approach, as suggested by [29; 33]. Maximum likelihood determines the parameters that produce the distribution with the highest likelihood of having produced the observed data. Assuming the independence of the perception generated for several ground motions and a lognormal distribution of fragility function Equation (1), the likelihood function is specified as the product of binomial probabilities as multiple \(IM\) levels, as follows:  
\[Likelihood=\prod_{i=1}^{m}\binom{n_{i}}{z_{i}}\Phi\left(\frac{ln\left(\frac{ x_{i}}{\beta}\right)}{\beta}\right)^{z_{i}}\left(1-\Phi\left(\frac{ln\left(\frac{ x_{i}}{\beta}\right)}{\beta}\right)\right)^{n_{i}-z_{i}} \tag{2}\]  
where \(m\) is the number of \(IM\) levels, \(z_{i}\) denotes the number of cases exceeding a certain damage state at the \(m\)th intensity level, \(n_{i}\) defines the number of ground motions, and \(\prod\) is the product of all individual likelihoods over \(i\) values from 1 to m [33]. The fragility function variables are estimated by maximizing the likelihood function from Equation (2). The framework to drive fragility curves is presented in Figure 1, and the following steps should be taken into account to develop fragility curves.  
1- Select several categories of railway embankments with different geometry.
2- Select earthquake ground motions (GMs) considering a set of near-field and far-field ground motions. Each earthquake is scaled from 0.1 g to 1 g to specify intensity measure (IM) levels.
3- Define damage indexes, intensity measure and damage limit state. In this study, certain levels of the maximum permanent vertical displacement (PVD) are selected as  
Figure 1: The framework of the proposed method.  
a limit states, and PVDs are calculated for each nonlinear response history analysis. Moreover, peak ground acceleration is considered the intensity measure. Based on the literature, threshold values were set for assessing three damage states, including slight, moderate and extensive deformation of the embankment [27].
* Evaluate the performance level for each embankment, considering a given ground motion earthquake record, and determine the damage limit state.
* Plot calculated probabilities versus PGAs for each damage state.
* Develop fragility curves based on the most likelihood method.  
The whole process may be divided into three main blocks, as shown in the flowchart presented in Figure 1. The first block contains the embankment modeling subjected to a set of near- and far-field ground motions, and obtains the maximum vertical displacement for each embankment, while the second block aims to obtain damage states with respect to the parameters available in the literature [27]. An example of performing a fragility curve for an embankment using the explained approach is shown in the last block.