## 5 Safety Assessment
### Structural Assessment
#### 5.2.2 Numerical Analysis
Table 1 presents a comparison between the experimental and numerical modal frequencies for the four experimentally identified modes along with the corresponding MAC values. The computed modal frequencies for all the modes are very close to the experimental ones, with a relative error between the numerical results and the average of the different test setups equal or lower than 2% for all cases. The high MAC values indicate a good correlation between the computed modal shapes and the ones identified by the analysis of the dynamic identification results. It is noted that the calibration of a numerical model of an existing structure with significant damage is not trivial, and the calibration procedure can be based on a single mode representing the most vulnerable part of the structure [31]. In this case, a very good representation of the tower's dynamic response by the numerical model is achieved due to the detailed information on the geometry and crack locations provided by the laser scanning survey as well as on the material properties, masonry quality and dynamic behaviour obtained from the non-destructive experimental campaign. Figure 22 presents the experimental and numerical modal shapes for Modes 1 to 4. Figure 21: Model of the simulated tower showing the location of the different types of masonries and cracks. The elastic properties of the final model are presented in Table 2. The ratio between the value of the dynamic Young's Modulus (computed through the sonic-pulse velocity tests) and the final selected value varies between 1.0 and 1.4 for the different masonry typologies and is in agreement with the values for quasi-brittle materials reported in the literature [10; 32; 33]. A Young's Modulus of 0.4 Mpa was used for all cracks, except for Crack-5 and the bottom part of Crack-2. This implies that the at cracks 1-4, the stiffness is much lower but there is a level of interconnection between the adjacent walls. The elastic properties used at the location of Crack-5 are much higher than the other cracks, referring to a closed crack at the base with smaller degradation at the top. Table 2 presents the material properties assigned to the different masonry zones and cracks. The compressive strength (\(\mathrm{f_{c}}\)), defined in Equation (1), considers a linear relationship with the Young's Modulus, E, where the coefficient \(\alpha\) was taken as 550, in line with recommendations for historic masonry structures [34; 35]. The tensile strength (\(\mathrm{f_{t}}\)) was assumed as 5% of the compressive strength [35]. A [PERSON]'s ratio of 0.2 was adopted for all materials. \[\mathrm{E}=\alpha\mathrm{f_{c}} \tag{1}\]  
The nonlinear mechanical behaviour of the masonry was simulated using a continuum damage mechanics model with damage-induced orthotropic behaviour along the principal axes [36; 37] implemented in an in-house FE software. The model uses two distinct damage indices to distinguish between tensile (i.e., cracking) and compressive (i.e., crushing) damage, taking values between 0 (undamaged state) and 1 (completely damaged state). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \begin{tabular}{c} **Experimental Range** \\ **of All Setups** \\ \end{tabular} & \begin{tabular}{c} **Numerical** \\ **Model** \\ \end{tabular} & \
\begin{tabular}{c} **Absolute Error in** \\ **Frequencies** \\ \end{tabular} &
\begin{tabular}{c} **MAC** \\ \end{tabular} \\ \hline
**Modes** & **Freq. [Hz]** & **Freq. [Hz]** & **[\%]** & **[-]** \\ \hline Mode 1 & 4.92 & 4.82 & 2.03 & 0.88 \\ Mode 2 & 6.85 & 6.88 & 0.44 & 0.78 \\ Mode 3 & 9.57 & 9.54 & 0.31 & 0.75 \\ Mode 4 & 12.39 & 12.42 & 0.24 & 0.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Frequencies and MAC values of the four modes obtained through the dynamic identification campaign and numerical modal analysis. Figure 22: Comparison of experimental (dynamic identification) and numerical (eigenvalue analysis) modal shapes. Black arrows in the numerical results indicate the direction of movement.