## 3 Theoretical Basis
 \tag{20}\]  
It is easy to recognize that the floor cell stiffness is given by two contributions: the one of the beams and the one of the internal floor elements (i.e., RC slab, RC joists, and lightening elements). Hence, the stiffness matrix can be decomposed as the sum of two stiffness matrices  
\[F{=}\ K{D}{=}\ \Big{[}{K}^{(B)}+{K}^{(I)}\Big{]}{D} \tag{21}\]  
Figure 8: Floor cell FE model. Figure 7: Floor cell. where \(K^{(B)}\) accounts for the beams' contribution while \(K^{(I)}\) accounts for the internal elements' contribution. From a more detailed perspective, the composition of these two matrices can be expressed as  
\[K^{(B)}= \begin{bmatrix}K_{\mathbf{V}\mathbf{V}}^{(B)}&K_{\mathbf{V}\mathbf{B}}^{(B)}&\mathbf{0}\\ &K_{\mathbf{B}\mathbf{B}}^{(B)}&\mathbf{0}\\ Sym.&&\mathbf{0}\end{bmatrix}  \tag{22}\]  
and  
\[K^{(I)}= \begin{bmatrix}K_{\mathbf{V}\mathbf{V}}^{(I)}&K_{\mathbf{V}\mathbf{B}}^{(I)}&K_{\mathbf{V} \mathbf{C}}^{(I)}\\ &K_{\mathbf{B}\mathbf{B}}^{(I)}&K_{\mathbf{B}\mathbf{C}}^{(I)}\\ Sym.&&K_{\mathbf{C}\mathbf{C}}^{(I)}\end{bmatrix}  \tag{23}\]  
and, by static condensation  
\[F_{\mathbf{V}}= \begin{bmatrix}K_{\mathbf{V}\mathbf{V}}^{(B)}+K_{\mathbf{V}\mathbf{V}}^{(I)}-K_\mathbf{{V}\mathbf{V}}^{(BI)}\end{bmatrix} D_{\mathbf{V}}. \tag{24}\]  
Then, if only the stiffness contribution of the beams on the floor cell perimeter is considered, (24) reduces to  
\[F_{\mathbf{V}}^{(B)}= \begin{bmatrix}K_{\mathbf{V}\mathbf{V}}^{(B)}-K_{\mathbf{V}\mathbf{B}}^{(B)}K _{\mathbf{B}\mathbf{B}}^{(B)}-1\\ \end{bmatrix} K_{\mathbf{B}\mathbf{V}}^{(B)}\end{bmatrix}D_{\mathbf{V}} \tag{25}\]  
Now, it is possible to ideally quantify the stiffness contribution given by only the internal floor elements, \(\Delta F_{\mathbf{V}}^{(BI)}\) as  
\[\Delta F_{\mathbf{V}}^{(BI)}=F_{\mathbf{V}}-F_{\mathbf{V}}^{(B)}=\Delta\mathbf{K}^{(BI)}D_\mathbf{{V}} \tag{26}\]  
where  
\[\Delta\mathbf{K}^{(BI)}=K_{\mathbf{V}\mathbf{V}}^{(I)}+K_{\mathbf{V}\mathbf{B}}^{(B)}K_{\mathbf{B}\bm {B}}^{(B)}-1\\ \end{bmatrix}K_{\mathbf{B}\mathbf{V}}^{(B)}-K_{\mathbf{V}\mathbf{V}}^{(BI)} \tag{27}\]  
From (24), it can be observed that the stiffness contribution of the beams and internal floor elements are not uncoupled, and thus the solution cannot be expressed by the sum of the internal floor contribution and the beams' contribution. The method described above can be applied also to the equivalent 2D membrane model. Actually, it is also possible in this case to identify the stiffness contribution given by beams, \(K^{(B)}\), and that of the membrane, \(K^{(M)}\). Thus, the system of equations can be written as follows:  
\[F=K\mathbf{D}= \begin{bmatrix}K^{(B)}+\mathbf{K}^{(M)}\end{bmatrix} D \tag{28}\]  
Similarly to what was derived for the floor cell FE model with solid elements and expressed by Equation (26), the solely internal element stiffness contribution given by the membrane is  
\[\Delta F^{(BM)}=\Delta\mathbf{K}^{(BM)}D_{\mathbf{V}} \tag{29}\]  
The difference between the equivalent 2D membrane model and the 3D solid-element model is that in the 2D model, the internal floor elements (i.e., RC slab, RC joists, and lightening elements) are modeled through the membrane, which is made with homogeneous orthotropic material, while in the 3D model, they are modeled using elements with different material and geometrical properties. In order for the membrane elements to reproduce the in-plane behavior of the internal floor modeled with solid elements, the elastic properties and thickness \(h\) of the membrane elements should be determined by imposing the equality between the stiffness matrices related to the internal floor elements of the 2D and 3D model as follows:  
\[\Delta\mathbf{K}^{(BM)}=\Delta\mathbf{K}^{(BI)} \tag{30}\]This condition is difficult to achieve unless the real floor is an orthotropic slab itself.