## 3 FEM Model Validation
### Model Updating
\[MAC_{u,d}=\frac{\left|\sum_{i=1}^{n}\varphi_{i}^{u}\varphi_{i}^{d}\right|^{2}}{ \sum_{i=1}^{n}\left(\varphi_{i}^{u}\right)^{2}\sum_{i=1}^{n}\left(\varphi_{i} ^{d}\right)^{2}} \tag{2}\]  
Lower and upper bounds for the Young's modulus of both masonry types were initially established. Following several iterations, the final value for the rubble stone masonry was set at 1400 MPa. Subsequently, additional analyses were conducted to determine the most suitable value for adobe masonry across all models. Among the tested scenarios, the model incorporating discrete cracks, referred as to Model-A, achieved the highest correlation with the first three target modes, particularly improving the match with the third mode,  
Figure 9: Mode shapes of the model with no damage and average values for [PERSON]’s modulus obtained from the experimental campaign. as illustrated in Figure 10. These results suggest that cracks have a greater impact on the out-of-plane behavior of the rear facade rather than variations in the elastic properties of the adobe masonry. Table 2 summarizes the frequencies obtained from both the numerical modal analyses and the experimental campaign, as well as the corresponding MAC values for the first three modes. The average and maximum errors in frequency are 2.4% and 4.6% respectively, which are rather low, whereas the average and minimum MAC are 0.91 and 0.85, respectively, which are rather high. These values indicate an excellent match. The calibrated Young's modulus values were set at 335 MPa for adobe and 1400 MPa for rubble stone masonry. Since crack repair is the minimum intervention necessary for structural safety and should be, ideally, executed regardless of any additional strengthening intervention, the analyses were conducted based on the undamaged model (Model-B) calibrated with the selected elastic material properties. A final modal analysis was performed to verify the dynamic characteristics of the church. As illustrated in Figure 11, Modes 1 and 2 are \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Numerical Model**} & \multicolumn{3}{c}{**Dynamic Identification Tests**} \\ \hline \multirow{2}{*}{Parameter} & Adobe masonry (MPa) & 310 & **335** & 340 & \\  & Rubble masonry (MPa) & 1400 & **1400** & 1400 & \\ \hline \multirow{2}{*}{Mode 1} & Frequency (Hz) & 2.39 & **2.47** & 2.49 & 2.48 \\  & Error & \(-3.7\%\) & \(-0.4\%\) & 0.2\% & \\  & MAC & 0.96 & **0.96** & 0.96 & \\ \hline \multirow{2}{*}{Mode 2} & Frequency (Hz) & 3.16 & **3.26** & 3.28 & 3.12 \\  & Error & 1.2\% & **4.6\%** & 5.2\% & \\  & MAC & 0.92 & **0.92** & 0.92 & \\ \hline \multirow{2}{*}{Mode 3} & Frequency (Hz) & 3.80 & **3.87** & 3.89 & 3.79 \\  & Error & 0.1\% & **2.2\%** & 2.5\% & \\ \cline{1-1}  & MAC & 0.85 & **0.85** & 0.85 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical properties of adobe masonry and rubble stone masonry for calibration. Figure 10: Model validation: (**a**) Experimental mode shapes; (**b**) Mode shapes of the numerical model with discrete cracks, assuming a Young’s modulus of 335 MPa for adobe and 1400 MPa for rubble stone masonry. similar to those observed in the experimental results, with MAC values exceeding 0.90. The frequency discrepancies are 0.1% for Mode 1 and 6.2% for Mode 2. Following repair of the existing cracks, the out-of-plane mode shape of the rear facade shifts from Mode 3 to Mode 4. While the MAC value remains at 0.85, the associated frequency increases significantly, resulting in a deviation of 22%. This frequency shift highlights the sensitivity of the rear facade's dynamic response to existing damage and emphasizes the structural impact of the observed cracks. The substantial increase in frequency following crack repair indicates stiffer out-of-plane behavior. Accordingly, the repair of these cracks is a necessary measure to restore partial structural integrity and to improve the seismic response of the rear facade.