## 4 Numerical Analysis  
To evaluate the dynamic behaviour of the two churches, an Eigenvalue analysis was carried out in order to obtain the fundamental natural periods and the related vibration mode shapes. Figures 10 and 11 report the four main vibration mode shapes that characterise the dynamic behaviour of Santa Maria di Collemaggio and Santa Maria Novella, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Earthquake Name** & **Station** & **Year** & **Mw [-]** \\ \hline \multicolumn{4}{c}{**Santa Maria di Collemaggio**} \\ \hline Irpinia\_Italy-01 & Torre Del Greco & 1980 & 6.90 \\ Irpinia\_Italy-02 & Rionero In Vulture & 1980 & 6.20 \\ New Zealand-02 & Mathain Dam & 1987 & 6.60 \\ Umbria Marche (aftershock 2)\_Italy & Norcia & 1997 & 5.60 \\ Umbria Marche (aftershock 2)\_Italy & Norcia-Zona Industriale & 1997 & 5.60 \\ L’Aquila\_Italy & Celano & 2009 & 6.30 \\ L’Aquila (aftershock 1)\_Italy & L’Aquila—Parking & 2009 & 5.60 \\ \hline \multicolumn{4}{c}{**Santa Maria Novella**} \\ \hline Santa Barbara & Santa Barbara Courthouse & 1978 & 5.92 \\ Coyote Lake & Gilroy Array \#6 & 1979 & 5.74 \\ Mammoth Lakes-01 & Long Valley Dam (Upr L Abut) & 1980 & 6.06 \\ Irpinia\_Italy-01 & Bagnoli Irpinio & 1980 & 6.90 \\ Irpinia\_Italy-01 & Brienza & 1980 & 6.90 \\ Irpinia\_Italy-02 & Rionero In Vulture & 1980 & 6.20 \\ Coalinga-01 & Slack Canyon & 1983 & 6.36 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main characteristics of the seismic signals. Figure 9: Eccentricity between the gravity centre (G\({}_{\rm c}\)) and the stiffness centre (S\({}_{\rm c}\)): (**a**) Santa Maria di Collemaggio; and (**b**) Santa Maria Novella. respectively, where MT indicates the participating mass in the transverse direction and ML is the participating mass in the longitudinal direction. The results obtained from the Eigenvalue analysis show a complex dynamic behaviour that affects both churches, with the total participating mass being divided into several modes. In fact, considering what is shown in Figures 10 and 11, it is possible to notice that only mode 4, which characterises the dynamic behaviour of Santa Maria Novella, presents a participating mass greater than 50%, while mode 1 for Santa Maria di Collemaggio involves about 34% of the total mass of the church. However, it is important to highlight that mode 1 for both basilicas exhibits a vibration mode shape that involves the naive structures. To evaluate the seismic performance of the two churches, a series of non-linear time history analyses (NTHAs) were carried out using the seismic records reported in Table 2 and considering the approach proposed in [39]. The non-linear behaviour of the masonry materials used in the walls, domes, vaults, and triumphal arches (implemented in the FEMs as plate elements) was introduced by means of the concrete smeared cracking (CSC) constitutive law [44; 45]. The adopted parameters, calculated according to the method reported in [46], are outlined in Table 3, where f\({}_{\text{c}}\) is the compressive strength; G\({}_{\text{c}}\) is the compressive fracture energy; h is the average size of the elements; f\({}_{\text{t}}\) is the tensile strength; and G\({}_{\text{f}}^{\text{f}}\) is the fracture energy, which regulates the tensile behaviour. Figure 12 reports the compressive and tensile non-linear behaviours adopted for the CSC approach. To take into account the shear behaviour, a linear model was considered using a shear reduction factor \(\beta=1\). For the columns and the main pillars (introduced in the FEMs using beam elements), appropriate concentrated plastic hinges located at the base and at the top of the elements were used [47]. Figure 10: Main fundamental vibration periods for Santa Maria di Collemaggio: (**a**) mode 1, period: 0.674 s, MT = 37.49%, and ML = 0.00%; (**b**) mode 4, period: 0.400 s, MT = 14.43%, and ML = 0.04%; (**c**) mode 10, period: 0.318 s, MT = 0.19%, and ML = 32.79%; and (**d**) mode 13, period: 0.290 s, MT = 0.53%, and ML = 10.68%.