## 1 Introduction  
Hysteretic dampers are widely used for seismic passive protection of both new and existing structures [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. According to the European standard EN 15129 [16], hysteretic dampers, which are further classified in steel hysteretic dampers, friction dampers and metal extrusion dampers, belong to the category of Displacement-Dependent Devices (DDDs). These systems are characterized by constitutive behavior which is mainly dependent on displacement. Among DDDs, steel hysteretic dampers are the most popular devices for the seismic protection of ordinary structures, such as school, residential and industrial buildings, as proved by the large number of studies and applications [17; 18; 19; 20; 21; 22; 23; 24; 25], thanks to their good damping capacity and predictable behavior combined with ease of manufacturing and installation [26; 27; 28; 29]. Focusing on the Italian scenario, recommendations on anti-seismic devices are given in Chapter 11 of the Italian Building Code, the IBC [30; 31]. In particular, the code requires that devices designed to satisfy the performance requirements at the ultimate limit state (seismic action with probability of exceedance of 10% over the reference period of the structure V\({}_{\text{R}}\), corresponding to the Life safety Limit State, LLS according to IBC [30]) must be able to resist earthquakes of higher intensity with probability of exceedance of 5% over V\({}_{\text{R}}\), corresponding to the Collapse Limit State, CLS. Therefore, with \(d_{\text{{{{d}}}}{\text{{{{d}}}}}}\) being the design displacement at LLS, the device must be designed to withstand the maximum between \(\gamma_{x}\cdot d_{bd}\) and \(d_{2}\), where \(\gamma_{x}=1.1\) is the amplification factor specified by EN 15129 [16] and \(d_{2}\) is the displacement at the CLS [30; 31]. As hysteretic devices introduce significant non-linear behavior and energy dissipation into the structure, such features should be taken into account in the modelling and analyses of structural systems incorporating such devices [16]. Therefore, the effectiveness of the retrofitted system is verified through non-linear analyses, which can be performed according to either static or dynamic methods. The Commentary to the IBC [31], in paragraph SSC7.3.4.2, provides guidance for the application of Non-Linear Static Analyses (NLSAs). In such analyses, the structure is described through its capacity curve \(F_{b}\)-\(d_{c}\), where \(F_{b}\) is the shear force at the base and \(d_{c}\) is the displacement of a control point, which usually coincides with the center of mass of the top floor. For each limit state, the performance level is determined by comparing the capacity curve to the spectral demand. To perform such comparison, the structural system is replaced by an equivalent Single-Degree-of-Freedom (SDOF) system through Equations (1)-(3), and capacity curve \(F_{b}\)-\(d_{c}\) of the physical Multi-Degree-of-Freedom (MDOF) system is replaced by the \"reduced\" capacity curve \(F^{*}\)-\(d^{*}\) of the equivalent SDOF system shown in Figure 1.  
\[\Gamma=\frac{\varphi^{T}M\tau}{\varphi^{T}M\varphi} \tag{1}\]  
\[d^{*}=\frac{d_{c}}{\Gamma} \tag{2}\]  
\[F^{*}=\frac{F_{b}}{\Gamma} \tag{3}\]  
For practical purposes, according to [31], the amount of energy dissipated by the system, and therefore the equivalent viscous damping ratio \(\zeta_{\alpha\beta}\), is evaluated by replacing the capacity curve of the equivalent SDOF system by an equivalent bilinear curve, determined based on the principle of energetic equivalence. The performance point (PP) of the system can be determined by applying either Method A, based on the principle of equal displacements or equal energies, or Method B, which is based on the Capacity Spectrum Method. In this study, Method B is considered, and the PP is evaluated through an iterative procedure in the acceleration-displacement response spectrum (ADRS) space.