## 4 Numerical Analysis
The seismic vulnerability was quantified through risk indices defined as the ratio between the value of the peak ground acceleration that leads to the collapse of the first  
Figure 11: Main fundamental vibration periods for Santa Maria Novella: (**a**) mode 1, period: 0.896 s, MT = 44.14%, and ML = 0.00%; (**b**) mode 4, period: 0.525 s, MT = 0.00%, and ML = 58.64%; (**c**) mode 5, period: 0.475 s, MT = 21.77%, and ML = 0.00%; and (**d**) mode 11, period: 0.341 s, MT = 0.00%, and ML = 1.58%. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**-** & **f\({}_{\text{c}}\) [MPa]** & **G\({}_{\text{c}}\) [N/mm]** & **h [mm]** & **f\({}_{\text{t}}\) [MPa]** & **G\({}_{\text{f}}^{\dagger}\) [N/mm]** \\ \hline \multicolumn{6}{c}{**Santa Maria di Collemaggio**} \\ \hline Walls & 1.25 & 0.1 & 300 & 0.15 & 0.1 \\ Vaults & 1.67 & 0.1 & 300 & 0.15 & 0.1 \\ \hline \multicolumn{6}{c}{**Santa Maria Novella**} \\ \hline Columns & 3.86 & 0.1 & 300 & 0.30 & 0.1 \\ Arches & 3.86 & 0.1 & 300 & 0.30 & 0.1 \\ Walls & 1.07 & 0.1 & 300 & 0.10 & 0.1 \\ Vaults & 1.07 & 0.1 & 300 & 0.10 & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: CSC parameters. Figure 12: (**a**) Compressive and (**b**) tensile non-linear behaviour function of CSC constitutive law. structural elements (PGA\({}_{\text{C}}\)) and the design peak ground acceleration (PGA\({}_{\text{D}}\)) calculated as reported in [39] considering the life safety limit state:  
\[RI=\frac{PGA_{\text{C}}}{PGA_{\text{D}}} \tag{1}\]  
The failure mechanism in both churches involves the two main pillars of the transport as the first elements reaching collapse. This is due to the presence of a high eccentricity between the gravity centre and stiffness centre, shown in detail in Figure 9, which causes a torsional effect and a consequent increase in the shear force acting at the base of the main pillars of the transport. It is important to highlight that the above-described collapse mechanism is what affected the Santa Maria di Collemaggio during the 2009 L'Aquila earthquake, which caused the failure of the transport area. Table 4 reports the value of the risk indices obtained from the execution of the non-linear time history analyses. The risk indices obtained from the non-linear dynamic analyses are characterised by values less than 1, which demonstrates how such structures are not adequate to resist the design of seismic action defined by the Italian Building Code [39]. Furthermore, it is possible to notice that both the risk indices obtained are characterised by values less than 0.3, indicating the high seismic vulnerability of the transport main pillars of both churches, due to the mechanical characteristics of the quasi-brittle masonry material. As reported in the previous Section 3, the obtained results have demonstrated the high seismic vulnerability of the analysed case studies. Moreover, several aspects that define the seismic performance of the two churches appear similar. In fact, it is possible to notice that the vibration mode shape of mode 1, which characterised the dynamic behaviour of both churches, shown in Figures 10a and 11a, is in the transverse direction and mainly involves the naive structure with participating masses equal to 37.49% and 44.14% for Santa Maria di Collemaggio and Santa Maria Novella, respectively. Furthermore, both the case studies are characterised by first fundamental natural periods less than 1 s due to their structural configuration and the presence of stiff vertical elements, such as the walls and the facade. These aspects significantly influence the formation of the collapse mechanism under seismic action, described in detail in the previous Sections 3 and 4, which characterises both Santa Maria di Collemaggio and Santa Maria Novella. In particular, the results of the non-linear time history analysis have highlighted that the most critical elements are the main pillars of the transport, as also demonstrated by the partial collapse that affected Santa Maria di Collemaggio during the 2009 L'Aquila earthquake.