# Homogenization of Ancient Masonry Buildings: A Case Study
## 5 LW Homogenized Model Identification
### In-Plane (Membrane) Behavior  
Assuming each wall of the fine model to behave as a homogeneous and isotropic Cauchy continuum, of elastic Young modulus and Poisson factor \(E_{m},\
u_{m}\), respectively, the membrane elastic constants \(B_{x}\),\(B_{y}\),\(B_{xy}\) are determined for the equivalent plates at the different stories as follows.  
On any of the stories, the longitudinal walls are characterized by almost identical windows, periodically repeated along the length. Therefore, it is easy to identify Representative Volume Elements (RVE), whose assembly reproduces the walls: A rectangular cell of dimensions \(a_{x}\times a_{y}\), where \(a_{x}\) is the center distance of the windows, \(a_{y}\) the interstory height, \(x\) and \(y\) the horizontal and vertical directions, respectively, is used as RVE; see details in Figure 7a. In each of the four longitudinal wall, the rectangle cell circumscribes a I region, centered on the cell and derived by the intersection between a male wall and part of two horizontal bands. The four I regions are distinguished by the different wall thickness \(h_{i}\) (with \(i=1,...,4\)) and by different void dimensions that are \(2b_{x}\times 2b_{y1}\) in the two external facades and \(2b_{x}\times 2b_{y2}\) in the two internal facades; see Table 1. Note that, whereas the dimensions along \(x\) are constant, those along \(y\) vary.  
Figure 6: Scheme of the equivalent plates of the homogenized model.  
The identification of the membrane elastic stiffness coefficients of each plate is made by equating the strain energy of the four I cells and of the equivalent rectangular cell, of dimensions \(a_{x}\times a_{y}\) and unitary out-of-plane thickness, once the same in-plane strain is applied, assumed constant on the cell domain. Longitudinal strain along \(x\)- and \(y\)-directions are applied to determine the extensional elastic coefficients:  
\[\begin{split} B_{x}&=E_{m}\sum_{i=1}^{4}h_{i}\left[1- \frac{2b_{x}b_{yi}}{a_{x}a_{y}}-\frac{\left(a_{x}-2b_{x}\right)\left(a_{x}-2b_{ x}-2b_{yi}\right)}{2a_{x}a_{y}}\right]\\ B_{y}&=E_{m}\sum_{i=1}^{4}h_{i}\left(1-\frac{2b_{x} b_{yi}}{a_{x}a_{y}}-\frac{2b_{x}^{2}}{a_{x}a_{y}}\right),\end{split} \tag{2}\]  
and shear strain is applied to determine the membrane shear stiffness coefficient:  
\[B_{xy}=G_{m}\sum_{i=1}^{4}h_{i}\left(1-\frac{2b_{xi}b_{yi}}{a_{x}a_{y}}\right) \tag{3}\]  
where \(b_{y3}=b_{y2}\) and \(b_{yi}=b_{y1}\). It is worth noticing that these identified coefficients are proportional to the full-void difference of the cell, namely:  
\[1-\frac{2b_{xi}b_{yi}}{a_{x}a_{y}} \tag{4}\]  
As a further comment to get Equation (2), the regions at almost zero-strain (according the real strain distributions) are subtracted to the domain of the I region, as shown in Figure 7b.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Homogeneous Plate** & \(a_{x}\) & \(a_{y}\) & \(b_{x}\) & \(b_{y1}\) & \(b_{y2}\) & \(h_{1}\) & \(h_{2}\) & \(h_{3}\) & \(h_{4}\) \\ \hline
1 & 3.8 & 5.90 & 0.76 & 2.72 & 2.72 & 1.49 & 0.62 & 0.62 & 1.66 \\
2 & 3.8 & 7.05 & 0.76 & 2.73 & 5.61 & 1.3 & 0.57 & 0.57 & 1.4 \\
3 & 3.8 & 7.05 & 0.76 & 2.78 & 2.40 & 1.26 & 0.57 & 0.57 & 1.1 \\
4 & 3.8 & 5.85 & 0.76 & 2.53 & 3.75 & 0.94 & 0.52 & 0.52 & 0.98 \\
5 & 3.8 & 5.05 & 0.76 & 1.28 & 3.77 & 0.52 & 0.49 & 0.49 & 0.52 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Cell geometry; numerical values in meters.  
Figure 7: (a) cell geometry; (b) modified I domain under extensional strains: subtraction of the regions at almost zero-strain.  
Validation of the identification procedure comes from good agreement in comparing the outcomes of finite element models, when both the fine and homogeneous structures are subjected to the same in-plane solicitations. The numerical results are reported in Table 2.