## 2 Simulation of the Nave Transversal Response by Equivalent FEM
### Simulation of the Nave Transversal Response  
In a single-nave church, it is mandatory to identify the seismic-resisting elements under transversal earthquake in order to describe the naive transversal response. In the FEM, the structural macro-systems capable of withstanding the transversal earthquake are reproduced as equivalent beam elements. These include the head wall, the facade, and the coupled perimeter walls or the triumphal arches. The head wall and facade are directly reproduced in the FEM by vertical equivalent elements having the same geometry, the equivalent cross-section areas and inertia moment, and with a perfect restraint applied to the base node. The features of the coupled perimeter walls or triumphal arches are defined considering the geometrical and mechanical features of the walls of their relative sub-structures, as is explained in [6; 17; 18; 37]. For example, in the case of the perimeter walls, each sub-structure is represented by the masonry pier included between two adjacent openings or the part of the wall belonging to a single abutment. For the triumphal arch, the sub-structures are the two columns or the two abutments with the relative part of the walls, depending on the as-built geometry of the church. Naming the generic transversal seismic-resistant system as the \"frame\", on the basis of the hypothesis made for the mechanical behavior of the structural elements, it behaves as a bilinear curve described by the yielding and ultimate force (\(F_{frame,y}\) and \(F_{frame,\
u}\), respectively) related to the yielding and the ultimate displacements (\(\delta_{frame,y}\) and \(\delta_{frame,u}\)). The frame's stiffness \(k_{frame}\) is obtained as the ratio between \(F_{frame,y}\) and \(\delta_{frame,y}\). The roof structure is characterized by a dissipative hysteretic behavior represented by an elastic-plastic curve, where the yielding and ultimate forces are \(F_{roof,y}\) and \(F_{roof,u}\), respectively, whereas the related yielding and ultimate displacements are \(\delta_{roof,y}\) and \(\delta_{roof,u}\). These values can be determined from experimental tests on the chosen connectors [24]. The stiffness of the roof \(k_{roof}\) is given by the ratio between \(F_{roof,y}\) and \(\delta_{roof,y}\). By changing the features of the roof's connections, \(k_{roof}\) changes as well. On the basis of these hypotheses, the naive transversal response of the church with the dissipative roof diaphragm can be represented by a flag-shaped diagram (Figure 2) in terms of the base shear versus the top displacement. In a design phase, an index of the energy dissipated in the connections of the roof diaphragm is given by the hysteretic variable \(\beta_{\textit{HYS}}\)[17]. When \(\beta_{\textit{HYS}}\) increases, the damping effect also increases. The hysteretic variable links the inelastic properties of the masonry walls to the mechanical characteristic of the roof, as shown in Equation (1). The \(\beta_{\textit{HYS}}\) optimal value is usually comprised in the range \(\beta_{\textit{HYS}}\) = 0-1.5 because, for higher values, the self-centering rocking behavior is partially inhibited by the large residual displacements implied by the flag-shaped load-displacement diagram. In the design phase, it could also be interesting to test the seismic response beyond \(\beta_{\textit{HYS}}\) = 1.5 [3; 6; 37] so to understand the effects of an over-stiffness and over-strength configuration of the roof on the overall structural response. It should be remarked that an over-stiffness and over-strength roof configuration can have a negative effect on the transversal wave response of historic churches. In fact, in this case, the dissipative effect of the connections is reduced, allowing the roof to transfer a high value of in-plane shear force to the masonry of the facade and head wall. Consequently, the in-plane shear force acting on these macro-elements may not be acceptable in relation to the mechanical properties of the masonry.