# Homogenization of Ancient Masonry Buildings: A Case Study
## 5 LW Homogenized Model Identification
### Out-of-Plane (Shear) Behavior  
The identification of out-of-plane behavior implies a higher complexity due to the influence of the collaborating slabs. Therefore, to this aim, the LW fine finite element model described in Section 3, under suitable load and constraint conditions, is used. Here, the geometry of the assembly being known as well as the elastic properties of the elements, the out-of-plane shear elastic constants (\(S_{x}\),\(S_{y}\)) of each homogenized plate are determined as follows. First, coefficient \(S_{x}\) is identified. To this aim, the LW fine model, released to the base (but still hinged at the sides) and subjected to a horizontal, uniformly distributed load of intensity \(p=1000\) N/m\({}^{2}\) on the surface of the most external facade, is analyzed. The deformed configuration of the fine model is shown in Figure 8 in (a) the 3D space and (b) the \(x-z\) plane. It is observed that (a) the out-of-plane deformation is strongly influenced by the stiffness difference along \(y\) and (b) the slab beams rigidly move forward, as the cross-sections of a pure shear beam fixed at the ends. Therefore, the idea is to consider five equivalent pure shear beams, i.e., one for each story, with the axis-line along \(x\), and calibrate the relevant stiffness coefficients \(GA_{i}\) (for \(i=1,...,5\)). The shear beams have a length of \(L=82\) m, i.e., the arm length, are fixed at the ends and are subjected to a uniform load \(pa_{yi}\), with \(a_{yi}\) the interstory height (Table 1); see the scheme in Figure 9a. The identification is made by a displacement equivalence between numerical and analytical models: The out-of-plane displacements of the fine model, evaluated at \(x=\dfrac{L}{2}\) and at half height of each interstory, are equated to the mid-span displacement of the equivalent shear beam, namely \(w_{i}\left(\dfrac{b}{z}\right)=\frac{PL^{2}}{8C\Lambda_{i}}\), which come from the analytical solving of the relevant differential boundary value problem:  
\[GA_{i}w_{i}^{\prime\prime}(x)=-p\] \[w_{i}(0)=0\qquad\qquad\text{for }i=1,...,5 \tag{5}\] \[w_{i}(L)=0\]  
The unknown stiffness coefficient \(GA_{i}\) is thus obtained and, from them, the equivalent plate coefficient comes as \(S_{xi}=\dfrac{GA_{i}}{a_{yi}}\); numerical values are in Table 2. In Figure 9b, a comparison between  
Figure 8: \(S_{x}\) identification. Deformed configuration of the fine model in the: (**a**) 3D space; (**b**) \(x-z\) plane. the equivalent beam (\(i=5\)) and fine model deformed configurations in the \(x-z\) plane is shown, and a very good agreement is observed. As a further step, the out-of-plane shear stiffness coefficient \(S_{y}\) is identified. To this aim, a central vertical strip of width \(a_{x}\) is extracted from the LW fine model. A distribution of horizontal forces, equivalent to the triangular load \(\dfrac{p}{H_{5}}y\) with \(p=2000\) N/m and \(H_{5}=31\) m, which is the total height of the wall, is applied to the external facade; the interaction with the adjacent strips is neglected. The out-of-plane displacement of the fine model is shown in Figure 10a, in the 3D space, and in Figure 10b, in the \(y-z\) plane. It is observed that the deformation is substantially similar to that of a shear-type frame with total height \(H_{5}\), deriving by a _pure shear_ behavior of the interstory walls. Therefore, for the aforementioned purpose, five equivalent pure shear beams, arranged in series and having axis-line along \(y\) and length \(a_{yi}\), are considered. The shear beams are fixed at the base and subjected to the triangular load \(\dfrac{p}{H_{5}}y\); see the scheme in Figure 11a. The identification of the relevant shear stiffness coefficients \(GA_{i}\) (\(i=1,...,5\)) is carried out, as for the previous case, by a displacement equivalence between numerical and analytical models: The out-of-plane displacement of the fine  
Figure 10: \(S_{y}\) identification. Deformed configuration of the fine model in the: (**a**) 3D space; (**b**) \(y-z\) plane.