## 4 Seismic Risk Assessment
### Collapse Fragility Curves  
The NTHA results were used to derive the collapse fragility function for each structural configuration, considering the uncertainty due to record-to-record variability. For each IM level, the number of collapses was divided by the total number of records to compute the probability of collapse. A lognormal distribution, through the maximum likelihood method outlined by [PERSON] [43], was then fit to the collapse probability data points, leading to a median collapse intensity, \(\theta\), and a logarithmic standard deviation, \(\beta_{\text{RTR}}\), associated  
Figure 8: Median of peak floor acceleration (MPFA) profiles for the case study school building at IM level 6: (**a**) completely infilled configuration; (**b**) partially infilled configuration (X direction).  
Figure 7: Median interstorey drift ratio (MIDR) profiles for the case study school building at IM level 6: (**a**) completely infilled configuration; (**b**) partially infilled configuration (X direction).  
with the record-to-record variability, listed in Table 3, for each structural configuration. Furthermore, the fragility curves for each structural configuration are provided in Figure 9, demonstrating the high variability surrounding them when the different material properties of infills are considered.  
The dispersion due to record-to-record variability (\(\beta_{\text{RTR}}\)) varies from 0.16 to 0.22, whereas the median collapse intensity (\(\theta\)) can go from 0.41 g to 0.85 g. Going from weak to strong masonry infills, the median intensity of collapse increases by almost 70% in the completely infilled configuration and almost 80% in the case of the partially infilled configuration. Furthermore, the analyst's choice to account or not for the presence of openings also has a relevant impact on the fragility curves, given that accounting for the presence of openings led to a reduction of \(\theta\) in the range of 5 to 18% with respect to the completely infilled structural configuration [44]. It is also worth mentioning that the trends highlighted by the fragility curves, i.e., reduction of the probability of collapse by increasing the masonry infill strength, should not be generalized. The increase in the strength of the masonry infill might induce shear failure in the columns (especially in existing buildings with poorly detailed structural components) with a consequently higher probability of collapse for the same IM level.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scenario** & **Infill Type** & \(\vartheta\)**[g]** & \(\beta_{\text{RTR}}\) \\ \hline \multirow{4}{*}{**Completely Infilled**} & 1 & 0.50 & 0.20 \\  & 2 & 0.51 & 0.19 \\  & 3 & 0.60 & 0.22 \\  & 4 & 0.78 & 0.18 \\  & 5 & 0.85 & 0.17 \\ \hline \multirow{4}{*}{**Partially Infilled**} & 1 & 0.41 & 0.17 \\  & 2 & 0.48 & 0.20 \\ \cline{1-1}  & 3 & 0.53 & 0.22 \\ \cline{1-1}  & 4 & 0.74 & 0.16 \\ \cline{1-1}  & 5 & 0.71 & 0.20 \\ \hline
**Bare** & - & 0.33 & 0.13 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Collapse fragility curve parameters, as a function of both building and masonry infill typologies.  
Figure 9: Collapse fragility curves for the case-study school building: (a) completely infilled and (b) partially infilled structural configurations.