## 4 Performance-Based Fragility Curves  
As mentioned previously, the log-normal CDF was used to define the fragility function [9; 35; 36]. Thus, Equation (1) can be expressed as follows:  
\[P(LS|IM=x)=\Phi\left(\frac{\ln(x/\theta)}{\beta}\right), \tag{2}\]  
where, \(\Phi(\ )\) is the standard normal cumulative distribution function, and \(\theta\) and \(\beta\) are the median and standard deviation of ln(_IM_), respectively. At each \(IM=x_{j}\), the structural analyses produce some number of collapses out of the total number of ground motions. The probability of observing collapses or no collapses is given by the binomial distribution. Hence, the MLE function is:  
\[Likelihood=\prod_{j=1}^{m}\binom{n_{j}}{z_{j}}p_{j}^{z_{j}}\left(1-p_{j}^{z_{ j}}\right)^{n_{j}-z_{j}}, \tag{3}\]  
where, \(p_{j}\) represents the probability that a ground motion with \(IM=x_{j}\) causes the collapse, \(z_{j}\) is the number of collapses out of \(n_{j}\) ground motions, and \(m\) represents the number of the _IM_ level. In order to perform this maximization, Equation (2) is substituted into Equation (3), so the fragility parameters are explicit in the likelihood function:  
\[Likelihood=\prod_{j=1}^{m}\left( \begin{array}{c}n_{j}\\ z_{j}\end{array} \right)\Phi\left(\frac{\ln\left(x_{j}/\theta\right)}{\beta} \right)^{z_{j}}\left(1-\Phi\left(\frac{\ln\left(x_{j}/\theta\right)}{\beta} \right)\right)^{n_{j}-z_{j}}. \tag{4}\]  
Figure 8: Probability of exceeding uplift. The parameters that maximize this likelihood function also maximize the log of the likelihood. The \"\(\wedge\)\" notation denotes the estimate of a parameter:  
\[\left\{\hat{\beta},\hat{\beta}\right\}=\text{argmax}\sum_{j=1}^{m}\left\{\ln \left( \begin{array}{c}n_{j}\\ z_{j}\end{array} \right)+z_{j}\ln\[PERSON]\left(\frac{\ln\left(x_{j}/\theta\right)}{ \beta}\right)+\left(n_{j}-z_{j}\right)\ln\left(1-\Phi\left(\frac{\ln\left(x_{j} /\theta\right)}{\beta}\right)\right)\right\}. \tag{5}\]  
Figure 9 depicts the three fragility curves built from the definition of Table 3. As a result, the median values of _IM_ (in terms of _PGA_) were 0.487, 0.521, and 0.557\(\times\)_g_ for the minor, moderate, and extensive limit states, respectively. Correspondingly, the standard deviations of \(\ln(PGA)\) were 0.232, 0.235, and 0.240\(\times\)_g_ for the minor, moderate, and extensive states, respectively. To consider the _IM_ in terms of the spectral acceleration during period \(T\) of 1.68 s and damping ratio \(\zeta\) of 1.5%, the median values of \(S_{a}\) were 0.387, 0.413, and 0.442\(\times\)_g_ for the minor, moderate, and extensive states, corresponding to the log-standard deviations of 0.232, 0.235, and 0.240\(\times\)_g_. The standard deviations indicate the slopes of the fragility curves, and increase as the performance levels shift from the minor to extensive damage state. When an uplift coupled with derailment occurs, the crane is repositioned and its functionality restored after landing inside the rails. Failure can occur before or after the uplift threshold depending on the capacity of each crane. In this analysis, failure occurred after the uplift threshold, and thus the probabilities of exceeding IO, LS, and CP performance levels must be considered with the uplift and derailment behavior, as shown in Figure 10. At the same _IM_ level, e.g., at \(S_{a}=0.387\times\)_g_, it can  
Figure 9: Fragility curves for minor, moderate, and extensive limit states; (**a**) _IM_ is _PGA_; and (**b**) _IM_ is _S_a (\(T=1.68\) s, \(\zeta=1.5\%\)). be clearly seen that the 50% probability of exceeding the IO level will be coupled with a probability of approximately 80% of exceeding the uplift. Similarly, a 50% probability of exceeding LS (\(S_{a}\) = 0.413\(\times\)\(g\)) and CP (\(S_{a}\) = 0.442\(\times\)\(g\)) levels will occur with an approximately 85% and 90% probability of uplift, respectively.