## 3 Methods for EDR Evaluation  
### Equivalent Damping Ratio (EDR) in Capacity Spectrum Method (CSM)  
The equivalent damping ratio is a parameter used to evaluate the equivalent viscous damping of the dissipative roof structure. Equivalent viscous damping was first proposed in [8] and extended to the yielding of single-degree-of-freedom systems in [44,45]. Furthermore, several methods for the EDR evaluation have been proposed in [40,46]. The EDR can be evaluated by nonlinear static analyses (pushover analysis, PO), a type of analysis capable of evaluating the capacity of the structure and identifying the most stressed structural elements. In PO, the approaches of the coefficient method of FEMA 356 [7,29] and equivalent linearization technique of ATC-40 [30] are considered. In FEMA 356, the EDR is indicated by the symbol \(\beta_{\alpha q}\), while, in ATC-40, the EDR is referred to as the Effective Viscous Damping (\(\beta_{\textit{eff}}\)). In both FEMA 356 and ATC-40, the hysteretic behavior of the structure is responsible for the dissipation of the energy due to the earthquake [47]. The energy dissipated in the hysteresis loops can be measured by the area enclosed in the hysteresis loops themselves. This area can be used for the determination of the Equivalent Viscous Damping. For the EDR evaluation, the equivalent FEM, described in Section 2, can be adopted, and the inherent damping of the structure should also be considered. PO allows us to obtain the capacity curve of the structure in terms of the base shear versus the top displacement of the structure. This curve can then be converted into its corresponding spectral acceleration (\(S_{a}\)) versus spectral displacement (\(S_{d}\)) diagram by adopting the Capacity Spectrum Methods (CSMs) of FEMA and ATC-40. The CSM graphically plots the capacity curve in the \(S_{a}\)-\(S_{d}\) plane and is very useful because it gives a clear graphical representation of the nonlinear behavior of the structure. To obtain the capacity curve in the \(S_{a}\)-\(S_{d}\) plane, the equivalent linearization method [30] or coefficient method [29] can be adopted. When the capacity curve is plotted in the \(S_{a}\)-\(S_{d}\) plane, the following two response spectra could be represented in the same plane: (i) the elastic spectrum (considering only the inherent viscous damping of the structure) and (ii) the inelastic spectrum (related to the total damping). The value of the inherent damping depends on the materials properties, but, in [48], it is considered equal to 5% for all types of structures, although it could be set to 3% for steel structures [49] and between 3% and 5% for unreinforced masonry structures [50]. In this work, 5% will be considered. A graphical representation of the use of the CSM for the evaluation of the equivalent viscous damping is given in Figure 4, where \(a_{y}\) and \(d_{y}\) define the linear branch of the bilinearized capacity curve, while \(a_{pi}\) and \(d_{pi}\) represent the coordinates of the performance point (PP, the intersection between the capacity curve and the inelastic spectrum), with \(d_{pi}\) being the maximum inelastic displacement of the structure. Figure 3: The Clough model used for the SD model (**left**) and the Takeda Slip model used for the SSD model (**right**). When the equivalent linearization method of ATC-40 is adopted, the EDR is evaluated by Equation (11), where \(E_{d}\) is the dissipated plastic energy (corresponding to the area enclosed by the hysteresis loop) and \(E_{S0}\) is the maximum elastic strain energy (graphically represented by the area of the triangle defined by the spectral displacement \(d_{pi}\) and the spectral acceleration \(a_{pi}\) of the PP). The EDR is obtained by considering the sum of the inherent damping (5%) and the hysteretic damping represented as the equivalent viscous damping (\(\beta_{0}\)).