## 4 Case Study  
### Description of the Case Study  
The nonlinear dynamic analyses and EDR evaluation are performed for a case study represented by a single-nave church with masonry walls and a wooden roof modeled with the equivalent FEM. The church has a 33 \(\times\) 12 m rectangular plan, 90 cm thickness head and facade walls, and 60 cm thickness perimeter walls, while the roof structure is composed of hardwood trusses (horizontal tie beam elements with a 17 \(\times\) 17 cm section, inclined principal rafters with a 30 \(\times\) 18 cm section, struts with a 12 \(\times\) 17 cm section, and a king post with a 12 \(\times\) 12 cm section), joists (12 \(\times\) 16 cm section), and 3.5 cm thick planks. The 3D initial model and the equivalent FEM with inelastic features are shown in Figure 7. The masonry is characterized by Young's modulus \(E\) = 1200 MPa and a density \(\rho\) = 18 kN/m\({}^{3}\). The hardwood elements are characterized by Young's modulus parallel to the fibers \(E_{0,mean}\) = 10,000 MPa, a shear modulus \(G_{mean}\) = 630 MPa, a tensile strength parallel to the fibers \(f_{t,0,k}\) = 20 MPa, and a density \(\rho_{w}\) = 5.00 kN/m\({}^{3}\). Starting from the design target behavior, in which the yielding of the steel connections of the roof and the yielding of the inelastic hinges of the walls are activated simultaneously (\(\Delta\) = 1; Equation (4)), the properties of the roof inelastic hinges vary as a function of the hysteretic variable \(\beta_{\textit{-HYS}}\). Table 1 reports the inelastic hinge properties for the resistant frames (corresponding to the pairs of the lateral walls) and the roof. To consider several possible configurations of the proposed retrofitting of the roof, seven different cases were analyzed. Starting from the capacity of the masonry walls, evaluated according to [6; 17; 18],which cannot be changed, different values of the hysteretic variable \(\beta_{\text{-}HYS}\) were taken into account and, consequently, the yield load of the roof connections was defined according to Equation (2). Then, by assuming \(\Delta=1\) (Equation (4)), the stiffness of the roof can be determined. Consequently, by knowing the stiffness of the roof, the mechanical properties of the wood panels, and the geometrical features of the church, the design of the steel connections can be carried out using Equations (6)-(10). The ultimate over-yielding force ratio of the roof elements was assumed to be 1.25 on average according to experimental tests performed by [24]. Points P1-D1 and P2-D2 represent the yielding and ultimate points, respectively. For the inelastic hinges of the roof, if the SD model is adopted (the Clough model), P1 and P2 indicate the yielding and the ultimate force of the roof's inelastic hinges, respectively, and D1 and D2 are the dimensionless drifts given by the ratio between the yielding or the ultimate displacements and the length of the church. Otherwise, if the SSD model is adopted (Takeda slip model), three points, P1-D1, P2-D2, and P3-D3, should be defined to describe the inelastic hinge behavior. The Takeda model is normally used to represent the hysteretic behavior of reinforced concrete structures [43].