## 3 FEM Model Validation
### Model Updating  
A preliminary eigenvalue analysis was performed under the assumption of full continuity among all the structural elements, which is representative of the undamaged state. The values of [PERSON]'s modulus adopted for this analysis are those listed in Table 1. The first six vibration modes of the numerical model are shown in Figure 9. Mode 1 (2.45 Hz) represents the fundamental transverse mode of the structure, which is dominated by the in-phase out-of-plane movement of the lateral walls of the naive and provides the highest participation mass of approximately 25%. Modes 2 (3.28 Hz) and 3 (4.25 Hz) also involve the motion of the lateral walls in the X direction but with increasingly complex deformation shapes, achieving a cumulative mass participation of about 40% in Mode 3. In higher modes, mass participation in the X direction drops significantly. Mode 4 (4.60 Hz) is characterized by out-of-plane displacement of the rear facade, which is associated with a 12% participation mass. The subsequent higher-frequency shapes are associated with local \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Mechanical Properties**} & \multicolumn{1}{c}{**Adobe**} & \multicolumn{1}{c}{**Rubble Stone**} & \multicolumn{1}{c}{**Timber**} \\  & & **Masonry** & **Masonry** & **Timer** \\ \hline Density & \(\rho\) (kg/m\({}^{3}\)) & 1900 & 1900 & 420 \\ Young’s modulus & E (MPa) & 340 & 1080 & 5700 \\ Poisson’s ratio & v (\(\cdot\)) & 0.2 & 0.2 & 0.3 \\ Compressive strength & \(f_{c}\) (MPa) & 0.45 & 0.60 & - \\ Tensile strength & \(f_{c}\) (MPa) & 0.05 & 0.06 & - \\ Compressive fracture energy & \(G_{c}\) (N/mm) & 1.38 & 1.80 & - \\ Tensile fracture energy & \(G_{f}^{I}\) (N/mm) & 0.01 & 0.01 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of adobe masonry, rubble stone masonry, and timber. Figure 8: Adopted stress–strain curves for compression (parabolic) and tension (exponential). modes and low masses, predominantly in the Y direction, e.g., the motion of the lateral walls of the sacristy and baptistery. The in situ experimental campaign also included dynamic identification testing using ambient vibration measurements, through which 13 tentative mode shapes and associated natural frequencies were identified [19]. Figure 10a depicts the first three experimental modes. Their comparison against the preliminary numerical vibration modes revealed discrepancies in both the frequencies and mode shapes. Therefore, the numerical model was calibrated by adjusting the Young's modulus of adobe and rubble stone masonry and by testing various structural conditions. The goal was to match the frequencies and mode shapes of the first three modes obtained from the ambient vibration tests. Several scenarios were considered during the calibration process, namely the following: reduced stiffness in the buttresses of the nave; exclusion of the pilasters of the triumphal arch; reduced stiffness in the rear facade; the presence of cracks at the inner corners of the rear facade; absence of tie beams in the presbytery; and a fully undamaged condition. Cracks were simulated by introducing a vertical gap in the upper half of the internal corners, with a depth equal to half the thickness of the lateral walls. The Modal Assurance Criterion (MAC) was employed to compare the mode shape vectors from the experimental and numerical analyses, as defined in Equation (2) [34]. In this formulation, \(\varphi^{u}\) and \(\varphi^{d}\) denote the respective mode shapes and \(n\) represents the degrees of freedom corresponding to the accelerometer positions and associated nodes in the FEM model. The MAC values range from 0 (indicating no correlation) to 1 (perfect correlation), indicating the degree of similarity between the mode shapes. Typically, a value of 0.7 and higher is considered an acceptable correlation.