# Homogenization of Ancient Masonry Buildings: A Case Study
## 3 Fine Model of a Building Portion  
First, a _fine_ finite element model of a building portion is developed to study some structural mechanisms and, then, to calibrate some parameters of the homogenized model. The studied portion consists of the long longitudinal arm described in the case study, believed to be the most vulnerable seismically portion. Moreover, the long longitudinal arms are very similar to each other and the modest local differences can be considered such as not to significantly change the dynamic behavior. All the FEM models are implemented in the commercial software SAP 2000 [40].  
The structural elements composing the studied arm are modeled as follows:  
* The longitudinal and transverse walls and the vaults are modeled by homogeneous isotropic shell elements, having both a membrane and flexural behavior. Any element is a Cauchy continuum body in plane stress, defined in a 2D space, and obeys a hyperelastic isotropic linear law according to the Young modulus \(E_{m}=2.2\times 10^{6}\) kN/m\({}^{2}\) and the Poisson factor \(\
u_{m}=0.15\) of the block masonry.
* The slab beams are modeled by one-dimensional Timoshenko beam elements, axially, bending and shearing deformable. They obey to a hyperelastic isotropic linear law according the Young modulus \(E_{s}=2\times 10^{8}\) kN/m\({}^{2}\) and the Poisson factor \(\
u_{s}=0.3\) of the steel.  
Figure 2: Long longitudinal arm details: (**a**) plan of the first floor; (**b**) perspectives of the two external facades. All the units are in meters.  
An equivalent averaged mass density \(\rho_{f}=3000\) kg/m\({}^{3}\) (1.6 times that of the masonry) is assigned to the only longitudinal shell elements; all the other elements are mass-free. The equivalent mass is obtained by allocating the total mass (deriving by the own weight of the vertical elements, according the masonry mass density 1900 kg/m\({}^{3}\), and the slab-bearing load equal to 11-15 kN/m\({}^{2}\)) to the only longitudinal walls.  
The geometric and mechanical characteristics of the vertical and horizontal elements have been assigned on the basis of information obtained from knowledge campaigns. Details about the geometry can be found in Section 2. The soil-structure interaction was not taken into account, so that external fixed constraints were applied to all the base nodes. The interaction with the other building arms is modeled by hinge constraints applied to the nodes of the vertical sides of each facade.  
In order to investigate the influence of the localized and distributed transverse walls, three finite element models are defined:  
* the Longitudinal Wall (LW) fine model, composed only by a _package of four parallel longitudinal walls and horizontal slabs;_
* the Longitudinal and Transverse Wall (LTW) fine model, composed by a _package of four parallel longitudinal walls, horizontal slabs, and transverse distributed walls;_
* the Stiffened Longitudinal and Transverse Wall (SLTW) fine model, composed by _package of four parallel longitudinal walls, horizontal slabs; and distributed and localized transverse walls._  
In Figure 3a, a perspective view of the SLTW finite element model, made up of 39,353 nodes and 40,183 elements, is shown. Different colors are used to distinguish different thicknesses of the shell elements, modeling: The longitudinal walls, in green scale, according the wall tapering; the vaults in red; the localized and distributed transverse walls, respectively, in dark and light blue. The model includes all the levels, with the exception of the last one, considered only as mass. Figure 3b shows some details about slabs and transverse walls.  
Figure 3: Fine finite element model: (**a**) overview; (**b**) details of slabs and transverse walls.