## 2 Simulation of the Nave Transversal Response by Equivalent FEM
### Simulation of the Nave Transversal Response
Furthermore, in the absence of adequate dissipation, the shear force acting in the plane of the roof may be compatible with the mechanical properties of the steel connections, but not with the wood technology characterizing the pitched panels of the roof. It may therefore be necessary to increase the thickness of these wooden panels (whether they are double planks or cross-laminated timber), thereby increasing the masses involved in the seismic response and, consequently, with higher inertial forces. By varying \(\beta_{\textit{HYS}}\), and knowing \(F_{frame,y}\) from the mechanical properties of the lateral walls, \(F_{roof,y}\) is evaluated by Equation (2). Consequently, the stiffness of the roof \(k_{roof}\) can be  
Figure 2: Representation of the naive transversal response by the flag-shaped diagram. expressed in terms of \(\beta_{\text{-}HYS}\) using Equation (3). Defining \(\Delta\) as the ratio between the yielding displacements of the roof and the frame (Equation (4)), the roof's stiffness \(k_{\text{roof}}\) is given by Equation (5). Thus, the roof structure designed in this way allows us to obtain a controlled rocking of the building characterized by a global seismic response with a flag-shaped diagram (Figure 2). In a preliminary design phase, it is useful to consider the role of the ratio \(\Delta\). If \(\Delta<1\), in the nave transversal response, the roof reaches its yielding point before the activation of the rocking of the masonry walls. This means that the energy dissipation begins with the walls in their elastic field, while achieving possible damages in the structural elements of the roof. Therefore, it is preferable not to consider this option in the design phase. If \(\Delta=1\), the roof diaphragm and perimeter masonry walls will reach their relative yielding points simultaneously; thus, the energy dissipation due to the roof starts only if the rocking of the walls is uncontrolled. Otherwise, if \(\Delta>1\), the perimeter walls achieve the yielding point before the plasticization of the roof diaphragm. Notice that \(\Delta\) values greater than one could be acceptable even if, for large \(\Delta\) values, the energy dissipation in the roof elements will be activated only after the onset of dangerous lateral displacements at the top of the perimeter walls. In the following analyses, \(\Delta=1\) is considered as a preliminary design criterion. \[\beta_{\text{-}HYS}=2\ \frac{F_{\text{roof},y}}{F_{frame,y}} \tag{1}\]  
\[F_{\text{roof},y}=\frac{\left(F_{frame,y}\cdot\beta_{\text{-}HYS}\right)}{2} \tag{2}\]  
\[k_{\text{roof},y}=k_{frame,y}\cdot\frac{\beta_{\text{-}HYS}\cdot\delta_{frame,y}}{2\cdot\delta_{\text{roof},y}} \tag{3}\]  
\[\Delta=\frac{\delta_{\text{roof},y}}{\delta_{frame,y}} \tag{4}\]  
\[k_{\text{roof}}=\frac{\beta_{\text{-}HYS}\cdot k_{frame}}{2\cdot\Delta} \tag{5}\]  
Different configurations of the roof diaphragm are tested by performing nonlinear dynamic analyses [26; 37]. The optimum roof stiffness should be calibrated by changing \(\beta_{\text{-}HYS}\) up to its optimal value \(\beta_{\text{-}HYS-opt}\), which is reached when the top lateral displacement is lower than the design value (i.e., 0.5% of the height of the perimeter walls according to [6; 36]) and the in-plane stresses on the wooden roof diaphragm and the facade are compatible with the corresponding strengths. When \(\beta_{\text{-}HYS-opt}\) is determined, the corresponding roof's stiffness is calculated with Equation (6), where \(k_{\text{df}}\) is the bending stiffness and \(k_{\text{df}}\) is the shear stiffness. In Equation (6), given the presence of wooden elements and steel connections, the equivalent elastic modulus \(E_{w}\)*, the equivalent shear modulus \(G_{w}\)*, and the ideal inertia moment of the section \(J_{\text{df}}\)* are introduced and evaluated by Equations (7)-(9) [16; 36], respectively.