## 4 Finite Element Modeling Approach
### Modal Assurance Criterion  
Once the elastic modulus and frequencies had been determined both experimentally and numerically, it was possible to calculate the Modal Assurance Criteria (MAC) to compare the FEA model with the experimental one. The MAC is the most commonly used method in the literature to calculate the correlation between two vectors. In this case, these vectors represent the mode shapes obtained experimentally and numerically. Mathematically, the MAC calculates the scaler product of two normalized vectors [33]. The MAC compares the values of any number, n, entries in two 1 x n matrices, and could therefore directly be used to compare the displacement vectors between the numerical and experimental results for n nodes. Of note is the fact that the formulation of the MAC in this case is based on the vectors used representing only one-dimensional space, since the accelerometers are all placed in an out-of-plane direction. The displacement vector outputs from the numerical model for each of the relevant nodes were therefore expressed only in the out-of-plane direction. One displacement value for each node, at each point that an accelerometer was placed, was then entered into a vector based on the node's ID so that relevant pairs of numerical and experimental displacement vectors could be compared.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Property** & **Symbol** & **Unit** & **Value** \\ \hline Density & \(\rho\) & kg/m\({}^{3}\) & 1900 \\ Elastic Modulus & E & GPa & 2.0 \\ Poisson’s Ratio & \(\
u\) & - & 0.2 \\ \hline Compressive Strength & f\({}_{\mathrm{c}}\) & MPa & 3.0 \\ Compressive Fracture Energy & G\({}_{\mathrm{c}}\) & N/mm & 7.40 \\ Tensile Strength & f\({}_{\mathrm{t}}\) & MPa & 0.1 \\ Tensile Fracture Energy & G\({}_{\mathrm{f}}^{\mathrm{I}}\) & N/mm & 0.02 \\ \hline \end{tabular}
\end{table}
Table 4: Linear elastic and non-linear properties used for masonry.  
The output of the MAC is a single number ranging from 0 to 1 depending on how similarly the methods express the explored mode shape; 1 represents complete correlation, while 0 represents complete disagreement of the values and therefore of the mode shapes. The formulation is written without the need for normalization:  
\[\mathrm{MAC}(\mathrm{x_{e}},\mathrm{x_{n}})=\frac{\mathrm{x_{e}^{T}x_{n}}}{ \left\|\mathrm{x_{e}}\right\|_{2}\left\|\mathrm{x_{n}}\right\|_{2}} \tag{6}\]  
where \(\mathrm{x_{e}}\) is a vector of experimental displacement vectors, and \(\mathrm{x_{n}}\) is a vector of numerical displacement vectors [34]. These vectors were obtained from DIANA directly at the nodes specified using the mesh by imprinting and considering only the vectors in the out-of-plane directions.  
Initial MAC calculations for the system revealed an overall good agreement between the mode shapes obtained, with all values above 0.8. The calculated MAC value was 0.96 for the first mode, 0.84 for the second mode, and 0.94 for the third mode. A graphical comparison between the vectors obtained through both methods revealed that the points of maximum displacement, although similar, occurred in different points between the models. The experimental behavior of the structure did not match the numerical one at the points closest to the north tower. This suggested the need to reduce the stiffness of the tower due to the presence of an arched room which represented a significant void in the structure (Figure 12). The ultimately successful approach was the addition of an approximation of the void room in the structure, which improved the MAC values for the first mode to 0.98 and the second mode to 0.86 while slightly reducing that of the third mode to 0.91. This also slightly increased the frequency error in the first mode to \(1.1\%\) while reducing that of the second mode to 0.3% and that of the third mode to 8.8% (Table 5).  
Figure 11: Mode shapes for west wall: (**a**) initial three experimentally determined mode shapes; (**b**) initial three numerically determined mode shapes.