## 2 Methodology
### Construction of the fragility curve for each road segment. Considering the geometric characteristics of the analyzed road network segment, and comparing the debris graphs related to the interfering buildings constituting the considered structural aggregate along the road edges, for each _IM_ it is possible to determine different width values of the road pavement that remains free after the debris fall (\(w_{fp}\)), using the following expression (Figure 4):  
\[w_{fr}=w_{bb}-max\Big{(}w_{br1};\Big{|}w_{br1}-w_{d1,i}\Big{|}\Big{)}-max\Big{(} w_{br2};\Big{|}w_{br2}-w_{d2,i}\Big{|}\Big{)} \tag{10}\]  
where \(w_{bb}\) is the average distance between the opposite interfering buildings along the road sides, \(w_{br1}\) and \(w_{br2}\) are the average widths of the sidewalk on side 1 and 2 of the road, respectively, \(w_{d1,i}\) is the debris width of the \(i\)-th debris graph related to the structural aggregates on side 1, and \(w_{d2,i}\) is the debris width of the \(j\)-th debris graph pertaining to the structural aggregates on side 2. As shown in Figure 5, by applying Equation (10) for each _IM_ considered, it is possible to identify the total number of analysis points and the number of points where the width of the road pavement is reduced beyond the 3.50 m (\(w_{fp}\leq\) 3.50 m), required for the passage of emergency vehicles (number of closures or collapses). In particular, for these closures or collapse points, the required performance  
Figure 4: Geometric characteristics. level can be defined through the width of the debris heap on the road pavement (\(w_{dr}\)), which can assume the following values:  
\[w_{dr}\geq w_{r}-3.50 \tag{11}\]  
where \(w_{r}\) is the average width of the road pavement expressed in meters. The proposed methodology, therefore, concludes by applying a statistically appropriate method for fitting the fragility curve to all these points. Fragility curves are usually described by a lognormal probability distribution function according to the following equation:  
\[P(PL\geq PL_{i}|IM)=\Phi\left[\frac{1}{\beta}\cdot\ln\left(\frac{IM}{\theta} \right)\right] \tag{12}\]  
where \(P(PL\geq PL_{i}|IM)\) is the probability of exceeding a particular performance level, \(PL\), for a given \(IM\), \(\Phi\) is the standard cumulative probability function, \(\beta\) is the standard deviation of the natural logarithm of \(IM\) for the performance level, and \(\theta\) is the median threshold value of \(IM\) required to cause the performance level. According to Equation (12), the development of fragility curves requires the estimation of two parameters, namely \(\theta\) and \(\beta\). At each intensity level \(IM=x_{j}\), the proposed procedure provides a certain number of closures or collapses out of a total number of analyses. In order to estimate the fragility curve parameters, the appropriate fitting technique for this type of data is to use the method of maximum likelihood estimation [19]. More specifically, the estimate of the fragility function parameters is obtained by maximizing the logarithm of the likelihood function as follows:  
\[\{\theta,\beta\}=\underset{\theta,\beta}{argmax}\sum_{j=1}^{m}\left\{\ln \left( \begin{array}{c}n_{j}\\ z_{j}\end{array} \right)+z_{j}\ln\Phi\left(\frac{\ln(x_{j}/\theta)}{\beta} \right)+\left(n_{j}-z_{j}\right)\ln\left(1-\Phi\left(\frac{\ln\left(x_{j}/ \theta\right)}{\beta}\right)\right)\right\} \tag{13}\]  
where \(m\) is the number of \(IM\) levels and \(z_{j}\) is the number of collapses out of \(n_{j}\) ground motions with \(IM=x_{j}\). To solve Equation (13), [PERSON] [19] implemented simple software tools for the calculations. The user only needs to provide observed data on the \(IM\) levels considered, and the relevant numbers of analyses and collapses, and the software tools estimate the fragility curve parameters.