## 1 Introduction
By changing \(\beta_{\textit{-HYS}}\) in a suitable range of values, different design configurations of the roof's connections can be tested up to the determination of the optimal value of \(\beta_{\textit{-HYS}}\) that is able to minimize the top horizontal displacement of the walls, while also keeping the shear stresses in both the roof and the walls in a safe range. Section 2.3 shows two different hysteretic models that can be adopted to describe the cyclic behavior of the steel connections on the basis of the results of cyclic tests performed on panel-to-panel connections [19, 20, 21, 22, 23, 24, 25] or evaluated by [26]. The two considered hysteretic models are (i) the stiffness degrading (SD) and (ii) the strengthand stiffness degrading (SSD) models. In the equivalent FEM, SD is implemented through the Clough model [21; 27] and SSD is implemented with the Takeda Slip model [28]. Section 3.1 describes the EDR evaluation by using the Capacity Spectrum Method (CSM), performing nonlinear static analyses through the coefficient method of FEMA 356 [29] and the equivalent linearization technique of ATC-40 [30]. The EDR values are evaluated as a function of the hysteretic variable \(\beta_{\text{-}HYS}\), allowing for the evaluation and selection of different possible configurations of the roof. Section 3.2 describes a proposed Time History numerical approach for the EDR evaluation. This is based on nonlinear dynamic analyses of the equivalent FEM subjected to transverse earthquake loading, represented by a set of seven spectrum-compatible accelerograms [31; 32; 33]. In this way, different values of the EDR are calculated for different values of the hysteretic variable \(\beta_{\text{-}HYS}\) so to simulate the response of the retrofitted structure for several alternative design configurations of the roof's connections. Section 4.1 describes the case study used for the analyses and Section 4.2 presents the EDR values obtained by FEMA 356, ATC-40, and the Time History method. The results are expressed for different values of the hysteretic variable \(\beta_{\text{-}HYS}\) in order to identify the optimal value \(\beta_{\text{-}HYS\text{-}opt}\) that is able to maximize the EDR. The optimal value of the hysteretic variable corresponds to a design configuration of the connections of the wood roof diaphragm. In Section 4.3, the seismic response of the structure is analyzed in terms of two ratios, considering the lateral top displacement and the shear on the facade, as a function of the hysteretic variable \(\beta_{\text{-}HYS}\) so to evaluate the optimum \(\beta_{\text{-}HYS\text{-}opt}\). In Section 4.4, a comparison between the optimum values of the hysteretic variable is discussed to identify the most suitable configuration of the dissipative roof that is able to (i) maximize the equivalent damping in terms of the EDR, (ii) reduce the lateral top displacement so to prevent the overturning of the lateral walls, and (iii) limit the in-plane shear force on the facade. In this way, excessive computational efforts during the design phase could be avoided, and the configuration of the roof related to \(\beta_{\text{-}HYS\text{-}opt}\) could be subsequently tested and validated in a more detailed FEM 3D model. Final considerations are listed in Section 5. The strategy at the base of the present study is shown in Figure 1.  
Figure 1: Strategy and structure of the proposed evaluation of the EDR.