# Application of Sonic Tests and Modal Identification in the Characterization of Masonry Walls
## 5 Numerical Modeling of the Walls  
Nowadays, one of the main aims of the non-destructive tests is the calibration/adjustment of the numerical models. The numerical models of the wall's tests were developed in finite element software SAP2000 V24 using geometry of the wall using layered shell elements, modeled through 800 4-nodes square shell elements and 861 nodes. The mechanical properties of materials were adjusted to fit the experimental values, in terms of Young model, based on the sonic tests and compared with the results obtained in the dynamic identification tests. The model was developed using shell elements considering only the middle plan of the element. The goodness of the fitting based on the experimental values of the frequencies was based on the minimization of the errors in frequencies, which were numerically determined, and those resulting from the measurements performed, thus updating the model of finite elements. The Young modulus was the parameter selected to adjust this calibration, and the error was calculated based on Equation (3), where \(\mathrm{a_{i}}\) is the modal mass ratio and \(\mathrm{n}\) is the number of the experimental mode shapes, considering the first three modes (see Figure 10). The starting point was the Young modulus obtained with the sonic tests in the vertical direction, and some adjustments were performed to minimize  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Wall**} & **1 st Mode** & **2 nd Mode** & **3 rd Mode** \\  & **(Hz)** & **(Hz)** & **(Hz)** \\ \hline W1 & 9.34 & 26.37 & 33.72 \\ W2 & 9.16 & 26.34 & 33.86 \\ W3 & 8.13 & 26.33 & 39.20 \\ W4 & 9.57 & 26.30 & 39.45 \\ W5 & 9.39 & 26.20 & 43.39 \\ W6 & 9.38 & 26.29 & 43.53 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Wall frequencies obtained experimentally.  
Figure 9: Example of the mode shapes for wall W1: (**a**) f1 = 9.34 Hz; (**b**) f2 = 26.37 Hz; and (**c**) f3 = 33.72 Hz.  
the error in frequencies, which were numerically determined, and those resulting from the measurements performed on site, thus updating the of finite element model.  
\[D_{f}=\frac{\sum_{i=1}^{n}\left|\frac{f_{FEMi}-f_{FDD,i}}{f_{FDD,i}}\right|\cdot a _{i}}{\sum_{i=1}^{n}a_{i}} \tag{3}\]  
In the numerical model, a Poisson coefficient of 0.25 was considered, and a density of 1937 kg/m\({}^{3}\) was considered in the sonic tests. A global agreement between the experimental (Exp) and numerical frequencies (Num) was found. The comparison between the natural frequencies from dynamic identification and those from numerical modeling is reported in Table 4. It is also clear that the main differences observed are related with the third mode. When not considering the third mode, the errors (D\({}_{t}\)) would be lower than 1% in all the walls (with exception of wall 3). The value presented in Table 4 for the Young modulus was the value used in the numerical analysis. Table 4 also presents the values of the average Young modulus obtained with the sonic tests in the vertical and horizontal directions for each wall. It can be seen that there exists an agreement between the Young modulus considered in the numerical analysis and the Young modulus obtained with the sonic tests in the vertical direction.  
The agreement between the numerical and the experimental modal shapes was estimated through the Modal Assurance Criterion (MAC) [29]. The numerical simulation better approximates the experimental measurements as the MAC approaches to 1. Figure 11 shows the MAC correlation matrix between the experimental and numerical modes. The results reveal that the MAC coefficient is greater than 0.8 for modes 1 and 3, and lower than 0.4 in mode 2, showing a poor correlation in the torsional mode.  
Figure 10: Numerical model shapes of the walls obtained experimentally: (**a**) first mode; (**b**) second mode; and (**c**) third mode.