## 4 FE-Based Method
### Application of the Method to a Case Study
#### 4.4.3 Discussion of Results  
The elastic properties of the homogenized material constituting the equivalent 2D elements of the case study obtained with the operating procedure described in Section 4.3 are reported in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \(E_{1}\) & \(E_{2}\) & \(E_{3}\) & \(G_{12}\) & \(G_{23}\) & \(\
u_{12}\) & \(\
u_{21}\) \\ (MPa) & (MPa) & (MPa) & (MPa) & (MPa) & (MPa) & \\ \hline
6970 & 4000 & 4000 & 0.1 & 0.1 & 0.2 & 0.115 \\ \hline \end{tabular}
\end{table}
Table 1: Elastic properties of the homogenized orthotropic material constituting the equivalent solid bricks used in the 3D model to reproduce the real hollow ceiling bricks.  
From the comparison of these properties with those of C25/30 concrete (cf. Section 4.4.1) adopted for the RC elements of the considered case study, it can be concluded that the contribution of joists and ceiling bricks to the axial in-plane stiffness of the floor cell is greater in the direction of the joists (X), as expected.  
In this direction, the Young's modulus of the homogenized material, \(E_{x}\), is equal to 42,000 MPa, roughly 33% higher than that of concrete (31,476 MPa). Since in the 2D model, the thickness of membrane elements is equal to that of the floor's slab, it follows that the 33% increase is due to the axial in-plane stiffness of joists and ceiling bricks.  
In the analysis performed, the contribution of ceiling bricks to the in-plane shear stiffness of floor elements was neglected, reasonably assuming that, in real conditions, ceiling bricks are probably not fully adhered to the RC joists and the slab. Hence, only the contributions of the slab and joists were considered. Comparing the shear modulus of the homogenized orthotropic material (Table 2) with that of concrete C25/30, we can observe that the former is about 10% higher than the latter. Since, as stated above, in the 2D model, the thickness of membrane elements is equal to that of the floor's slab, it follows that the 10% increase is due to the joists' shear stiffness. Therefore, it can be gathered that the main contributor to the floor in-plane shear stiffness the RC slab. Since under seismic actions, floors are subject mainly to shear strains, it can be concluded that the simplified analytical-based method described in Section 2.1 is adequate to reproduce the in-plane stiffness of RC floors with lightening elements when the thickness of shell elements is set equal to that of the RC slab of the floor.  
To compare the numerical procedure with the simplified analytical-based method proposed in Section 2.1, it was applied to the case study described in Section 2.2. The elastic properties of the equivalent shell elements with a thickness equal to 8 cm are reported in Table 3.  
Comparing the stresses in structural elements obtained from the FE model built with shell elements with the properties described in Table 3 to those obtained from the model described in Section 2.2, we can observe that the differences are practically negligible. In particular, the reduction of stresses in the circular-shaped RC beam was 14.8%, while with the simplified method, a 15% reduction was obtained. With regard to the stresses in the top beams of the radial trusses, both in the concrete and in HEA200 steel profiles, a reduction of 6.7% instead of 7% was obtained. Therefore, the results obtained for the considered case study from the numerical procedure are fundamentally equal to those obtained from the simplified analytical-based method.