## 3 Theoretical Basis  
The nodes at the beams' intersection in the FE model of the one-way joists floor are referred to, in the following, as vertex nodes, and are marked with a circle in Figure 5.  
In the structural analysis, the floor behavior is fully described by the relationship between forces applied at vertex nodes and their displacements. For this reason, to adequately define the membrane elements' properties, the equivalence between the solid FE model (3D model) and the 2D FE model (2D model) of the floor must be imposed in terms of the vertex nodes' force-displacement relation. With restricted attention to the in-plane behavior, the kinematics of each vertex node of is fully described by two translations \(D_{|X}\) and \(D_{|Y}\), which are parallel to the X and Y directions, respectively, and the rotation \(D_{i\theta}\) around the axis orthogonal to the floor middle plane--as shown in see Figure 5--and collected in the 3-vector  
\[\mathbf{D}_{i}=[D_{iX},\ D_{iY},\ D_{i\theta}]^{\mathrm{T}} \tag{8}\]  
Analogously, the in-plane actions at each vertex node are described by two mutually orthogonal forces \(F_{i}\) and \(F_{iY}\) and by torque \(F_{i\theta}\), collected in the 3-vector  
\[\mathbf{F}_{i}=[F_{iX},\ F_{iY},\ F_{i\theta}]^{\mathrm{T}} \tag{9}\]  
Generalized displacements of all vertex nodes are collected in a generalized displacement vector \(\mathbf{D}\)  
\[\mathbf{D}\text{=} \begin{bmatrix}\mathbf{D_{1}}\\ \mathbf{D_{2}}\\ \vdots\\ \mathbf{D_{N}}\end{bmatrix}  \tag{10}\]  
and, analogously, generalized forces are collected in the generalized force vector \(\mathbf{F}\)  
\[\mathbf{F}\text{=} \begin{bmatrix}\mathbf{F_{1}}\\ \mathbf{F_{2}}\\ \vdots\\ \mathbf{F_{N}}\end{bmatrix}  \tag{11}\]  
The relationship between generalized displacements and forces is given by  
\[\mathbf{F}=\mathbf{K}\mathbf{D} \tag{12}\]  
Figure 5: Schematic representation of a floor in the building FE model. where \(\mathbf{K}\) is the symmetrical 3 N \(\times\) 3 N floor in-plane stiffness matrix. Equation (12) also applies to floor FE models in which floor cells and boundary beams are discretized in finite elements with smaller dimensions (Figure 6) in order to enforce the displacement compatibility. The cell internal nodes are called core nodes while the internal beam nodes are called beams nodes. With label \(V\) for vertex nodes, \(B\) for beam nodes, and \(C\) for core nodes, Equation (12) transforms into  
\[ \begin{bmatrix}F_{\mathbf{V}}\\ F_{\mathbf{B}}\\ F_{\mathbf{C}}\end{bmatrix} = \begin{bmatrix}K_{\mathbf{V}\mathbf{V}}&K_{\mathbf{V}\mathbf{B}}&K_\mathbf{{V}C}\\ K_{\mathbf{B}\mathbf{B}}&K_{\mathbf{B}\mathbf{C}}\\ Sym.&&K_{\mathbf{C}\mathbf{C}}\end{bmatrix}\begin{bmatrix}D_{\mathbf{V}}\\ D_{\mathbf{B}}\\ D_{\mathbf{C}}\end{bmatrix}  \tag{13}\]  
where \(F_{\mathbf{V}}\), \(F_{\mathbf{B}}\), and \(F_{\mathbf{C}}\) are the generalized forces acting at vertex, beam, and core nodes, respectively; \(D_{\mathbf{V}}\), \(D_{\  
Substituting (17) and (18) in the first equation of system (14), we obtain an expression of \(F_{V}\) involving only \(D_{V}\) as  
\[F_{V}{=}\ K*D_{V} \tag{19}\]  
Then, the same method can be applied to the analysis of a single floor cell (Figure 7), where the number of vertex nodes reduces to 4 (Figure 8). Recalling that each node has 3 DOFs, both the displacement vector \(D_{r}\) given by Equation (10), and the force vector \(F\), given by Equation (11), have 12 components. Hence, the statically reduced stiffness matrix \(K\) of the floor cell is a symmetrical 12 \(\times\) 12 matrix. The floor cell in-plane behavior can then be described by the following system of equations:  
\[ \begin{bmatrix}F_{11}\\ F_{12}\\ \vdots\\ F_{43}\end{bmatrix} {=} \begin{bmatrix}K_{11}&K_{12}&\cdots&K_{112}\\ K_{21}&K_{22}&\cdots&K_{212}\\ \vdots&\vdots&\ddots&\vdots\\ K_{41}&K_{42}&\cdots&K_{11}\end{bmatrix}\begin{bmatrix}D_{11}\\ D_{12}\\ \vdots\\ D_{43}\end{bmatrix}