## 4 FE-Based Method
### Operating Procedure  
The proposed procedure is articulated in the following steps:  
1. Realization of the floor cell model, including the perimeter beams, using solid finite elements. Each element constituting the floor cell is modeled with its actual dimensions and with the elastic properties of the material of which it is made. This model is built to reproduce the actual behavior of the floor cell. 2. Realization of the floor cell model, including the perimeter beams, using frame elements for the beams and 2D elements with the membrane behavior for the floor. The geometry is congruent with that of the 3D model. In particular, the side length of the 2D element is equal to the distance, in the 3D model, between the longitudinal axes of the perimeter beams orthogonal to that side. This condition guarantees that the distances between the longitudinal axes of the perimeter beams in the 2D model are equal to those in the 3D model. Perimeter beams are modeled with their actual cross-section and with the elastic properties of the concrete of which they are constituted, used also in the 3D model. The thickness of the 2D elements has to be chosen, and a homogenized material with orthotropic elastic behavior has to be defined, with initial values of the unknown equivalent elastic properties being set. Herein, for RC floors with joists, the thickness is set equal to that of the RC slab, while the initial values of the homogenized material elastic properties are set equal to those of the concrete constituting the real slab. 3. Application to loads and restraints of the static scheme of MODE 1 to both the 2D and 3D model. The applied forces are equal to 707 kN so that the resultant load applied to each vertex node is equal to 1000 kN. Then, the values of the displacement components parallel to Y at vertex nodes B and C obtained from the two models are compared. If the differences between these displacement components are greater than a fixed tolerance, which is herein taken to be equal to 1%, it is necessary to modify the elastic shear modulus \(G_{xy}\) of the 2D elements by raising or decreasing its value. Then, the analysis of the 2D model is carried out again and similarly to what was  
Figure 12: Mesh nodes in a vertical cross-section of the parallelepiped, to which loads and restraints are applied at intersection regions. done previously, the comparison of displacement components is performed. Thus, the value of \(G_{xy}\) is determined through an iterative procedure ended once the differences between the displacement components mentioned above are smaller than the fixed tolerance. The 2D model is then updated, with the value of \(G_{xy}\) obtained at the end of the iterative procedure. 4. Similarly to what done in step 3 for MODE 1, the 2D and 3D models of the floor cell are now analyzed under the static scheme of MODE 2. In this case, the load magnitude applied to nodes B and C is taken equal to 500 kN in order to apply, in X direction, a total force equal to 1000 kN. The values of the displacement components parallel to X at vertex nodes B and C obtained from the two models are compared. If the percentage difference between these displacement components is greater than a fixed tolerance, which is herein taken to be equal to 1%, it is necessary to modify the elastic modulus along X of the homogeneous material, \(E_{x}\), by raising or decreasing its value. At the end of this second iterative procedure, the 2D element is characterized by the equivalent values of \(G_{xy}\) and \(E_{x}\). 5. Finally, in consideration of the static scheme of MODE 3 and to search for the equality of the displacement components parallel to Y at vertex nodes C and D obtained from the two FE models, the equivalent value of \(E_{y}\) is determined. Moreover, from the comparison of displacement components parallel to X at vertex nodes B and C obtained from the two FE models, the value of \(v_{yx}\) is determined. Similarly to what was done for MODE 2, the load magnitude applied to nodes C and D can be taken to be equal to 500 kN.