# Design Proposal for Masonry Infill Walls Subject to Seismic Actions
## 5 Design Proposal
### Out-Of-Plane Modal Frequency  
When computing the out-of-plane load using EC8 [27], it is necessary to know the out-of-plane natural frequency of the wall and this standard does not provide any analytical solution. [PERSON] et al. [14] proposed an analytical solution to compute this parameter, which is also prescribed by the Italian standard [31], given by Equation (30). \[T_{p}=\frac{2}{\pi}\bigg{(}\frac{1}{l^{2}}+\frac{1}{h^{2}}\bigg{)}\sqrt{\frac {Et^{3}}{12m}} \tag{30}\]  
Here, \(l\), \(h\) and \(t\) are the length, height and thickness of the infill wall, \(E\) is the elasticity modulus and \(m\) is the specific mass of the infill. To compare the experimental and the analytical design out-of-plane loads, when using EC8 [27], the frequencies obtained experimentally were used here instead of those from the above equation, as the analytical solution considerably under-estimated values relative to the experimental values, see Table 7. The average error for model 1 was about 80%, while in models 2 and 3 the error was about 90%. These high errors were caused by the formula, which assumes that the infill wall behaves dynamically as a panel simply supported by cylindrical hinges at the side columns. The much higher experimental frequencies obtained show that this simplification is not correct and relates to the wall being connected to the other sides of the frame. In addition, as the infill wall is mostly within the frame (model 1), or completely within the frame (models 2 and 3), the connection of the infill wall to the frame may also have a considerable restraint effect. It would be expected that the infill walls of model 1 presented lower natural frequencies than for models 2 and 3, as the main difference between them was the thickness of the wall. A possible justification for the expectedly high values of model 1 may relate to the boundary conditions of the infills, due to the execution process. In order to better understand the influence of the boundary conditions, and to confirm the above-mentioned assumptions, a simple numerical model of the infill wall was constructed using DIANA [32], based on shell elements. In the case of model 1, the exterior and interior leaves were modelled separately with 8.5 cm and 10.5 cm, respectively, taking the plaster into consideration. As for models 2 and 3, a single 18 cm leaf was modelled. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Wall** & **Type** & \begin{tabular}{c} **Experimental** \\ **Frequency (Hz)** \\ \end{tabular}  & \begin{tabular}{c} **Analytical** \\ **(Hz)** \\ \end{tabular}  & \ \begin{tabular}{c} **Error (\%)** \\ \end{tabular}  \\ \hline \multirow{8}{*}{
\begin{tabular}{c} **CourThe results, see Tables 8 and 9, show it is possible for different boundary conditions to lead to natural frequencies ranging from 4.4 Hz to 56.0 Hz in model 1 and 9.3 Hz to 91.8 Hz in models 2 and 3. It should be noted that the material characterization of the masonry was that done by [PERSON] [15], and different elasticity moduli were computed for the three types of masonry used in the present study (unreinforced, bed joint reinforced, and reinforced plaster). This is not in agreement with the experimental frequencies, which exhibit similar results. Therefore, during the present computations, the adopted value for the elasticity modulus was 3.43 GPa, which corresponds to the average of the three masonry types [15]. Using the same elasticity modulus for all models will not provide the best results, as the infill walls of model 1 presented similar frequency values for a lower thickness. Nonetheless, and analysing the results for models 2 and 3, to achieve higher experimental values, the infill wall needs to be connected to the frame on all four borders, with a flexible connection intermediate between a simply supported and a fully constrained option. For design purposes and standard provisions, analytical solutions should be simple, and not difficult to quantify the parameters (as a rotational stiffness).