# Homogenization of Ancient Masonry Buildings: A Case Study
## 4 Homogenized Model of a Building Portion  
Here, the homogenized model of the longitudinal arm of the building, consisting of a package of four parallel longitudinal walls, connected transversely by horizontal and vertical elements, is defined. The following assumptions are considered, as a reasonable consequence of the features of the structure:  
* in membrane behavior, the longitudinal walls always undergo the same extensional and shear strain in the two orthogonal directions;
* the slabs are infinitely rigid in their plane, so that the four longitudinal walls actually work in parallel;  
The idea is to substitute an 'equivalent' homogeneous plate for the fine model of the arm, having it unitary thickness, the same in plane trace of the block middle line, as well as corresponding stiffness and mass. The plate is fixed at the base and laterally hinged (see the homogenization scheme in Figure 4). In particular, for each of the fine models described in Section 3, a corresponding homogenized model is defined:  
* the LW homogenized model, composed by an equivalent plate, incorporating the structural characteristics of longitudinal walls and slabs (see the homogenization scheme in the plane \(y-z\) in Figure 5a);
* the LTW homogenized model, composed by an equivalent plate, incorporating the structural characteristics of both longitudinal and transverse walls and slabs;
* the SLTW homogenized model, composed by the equivalent plate of the LTW homogenized model plus equivalent beams, integral with the plate: The number and position of the beams are the same of the transverse walls, as they contribute to simulating the effect of such walls on the structure (see the homogenization scheme in the plane \(y-z\) in Figure 5b). Figure 4: Homogenization scheme. The equivalent plates are orthotropic Mindlin-Reissner plates [41], having the following elastic constitutive laws in the decoupled form:  
\[ \begin{array}{l}M_{x}=D_{x}k_{x}\\ M_{y}=D_{y}k_{y}\\ M_{xy}=D_{xy}k_{xy}\\ \\ T_{x}=S_{x}\gamma_{x}\\ T_{y}=S_{y}\gamma_{y}\\ \\ N_{x}=B_{x}\varepsilon_{x}\\ N_{y}=B_{y}\varepsilon_{y}\\ N_{xy}=B_{xy}\gamma_{xy}\end{array}  \tag{1}\]  
where:  
* \(M_{x},M_{y},M_{xy}\) are the bending and twist moments; \(T_{x},T_{y}\) the shear forces; \(N_{x},N_{y},N_{xy}\) the membrane forces;
* \(k_{x}=-w_{,xx}\), \(k_{y}=-w_{,yy}\), \(k_{xy}=-2w_{,xy}\) are the bending and twist curvature, with \(w\) the orthogonal displacement of the plate; \(\gamma_{x},\gamma_{y}\) are the out of plane sliding; \(\varepsilon_{x},\varepsilon_{y},\gamma_{xy}\) the membrane strains (unit extensions \(\varepsilon_{x},\varepsilon_{y}\) and shear sliding \(\gamma_{xy}\)). * \(D_{x},D_{y},D_{xy}\) are the bending and twist stiffness; \(S_{x},S_{y}\) the out-of-plane shear stiffness; \(B_{x},B_{y},B_{xy}\) the membrane stiffness. Note that, contrary to standard constitutive equations constructed by homogenization, as shown e.g., in [42], here the off-diagonal terms are neglected, by assuming the Poisson factor to be zero. This assumption: (i) is reasonable in the context of the approximations of the problem, in which the Poisson effect plays a certainly minor role; and (ii) remarkably simplifies the homogenization procedures, by allowing to identify separately the plate elastic constants relative to different directions. The identification of the equivalent in-plane and out-of-plane elastic constants represents an important goal of the work. In particular, since the geometry varies in height (due to wall tapering and different kinds of slabs), each homogenized model is actually composed by five equivalent plates, having the corresponding interstory height, numbered from bottom to top as in Figure 6. Each of the five plates has to be identified. The problem is stated as follows: the geometry of the assembly being known as well as the elastic properties of the elements, determining the elastic stiffness coefficients  
Figure 5: Homogenization scheme in the plane \(y-z\) of: (a) the LW model; (b) a couple of localized transverse walls.