## 1 Introduction
As the first step, the displacement at the PP is set equal to \(d^{*}_{max}\), correspondent to the target seismic displacement of the analyzed structure, where \(F^{*}_{max}\) is the corresponding force determined on the SDOF capacity curve. The equivalent bilinear curve of the equivalent SDOF system is then obtained by imposing same initial stiffness as the initial stiffness of the SDOF capacity curve and identifying the yield point (\(d^{*}_{y}\); \(F^{*}_{y}\)) through the equivalence of areas A\({}_{1}\) and A\({}_{2}\), as shown in Figure 2.  
Figure 1: Equivalent Single-Degree-of-Freedom structural system. Through the equivalent bilinear capacity curve, the equivalent viscous damping ratio \(\xi_{eq}\) is determined with Equation [C7.3.10] of [31]:  
\[\xi_{eq}=k\frac{63.7\left(F_{y}^{*}d_{max}^{*}-F_{max}^{*}d_{y}^{*}\right)}{F_{ max}^{*}d_{max}^{*}}+5 \tag{4}\]  
where parameter \(k\) accounts for the actual energy dissipation capacity of the structural system. In principle, \(k\) should be calibrated based on experimental data, but when relevant data are missing, it can be taken as 1.0 for structures with high damping capability (characterized by wide and stable hysteresis loops), 0.66 for structures with moderate damping capability (moderate change of the hysteresis loops), and 0.33 for structures with low damping capability (hysteresis loops affected by substantial pinching and decrease in area). Through \(\xi_{eq}\), the damping reduction factor \(\eta\) is determined to evaluate the damped demand spectrum according to the equation [3.2.4] of the IBC [30]:  
\[\eta=\sqrt{10/\left(5+\xi_{eq}\right)}\geq 0.55 \tag{5}\]  
The equivalent bilinear curve and the damped spectrum demand are compared in the ADRS plane; their crossing point is the new PP, and Method B is iteratively applied until the difference between two consecutive PPs is sufficiently small. According to Equation (5), the minimum value of the damping factor \(\eta\) is 0.55, which corresponds to an equivalent viscous damping ratio \(\xi_{eq}=28\%\). In a structure equipped with damped braces, \(\xi_{eq}\), and consequently \(\eta\), depends upon the damping capacity of the dampers. In this regard, it must be emphasized that steel hysteretic dampers available nowadays, such as Buckling Restrained Brances (BRBs), friction and extrusion dampers, are characterized by high damping capacity, with \(\xi_{eq}>28\%\)[3, 4, 5, 6, 32, 33, 34, 35, 36]. Therefore, the contribution of such dampers to increase the overall damping \(\xi_{eq}\) of the combined structural system may be not completely considered in the analyses due to the limitation on \(\eta\)[30]. Non-Linear dynamic, or Time-History, analyses (NLTHAs) are acknowledged, by far, to be the most reliable method for predicting the seismic response of structures. However, they involve the formulation of accurate mathematical models of the real structure, are time consuming and the accuracy of the results depends on the assumptions made on the seismic action. Therefore, for ordinary structures, NLTHAs are usually replaced by simpler and computationally less onerous static analyses. Regarding the representation of the seismic input in NLTHAs, the IBC [30] SS3.2.3.6 allows for consideration of either natural or artificial accelerograms. The use of artificial accelerograms was attractive in the past years since it allowed for the generation of spectrum compatible time-series even in those cases where the elastic design spectrum was the only available information [37]. Today, thanks to the growing availability of strong motions databases, the use of real accelerograms recorded during earthquakes is preferred [38]. According to the IBC [30] SS3.2.3.6, natural  
Figure 2: Definition of the equivalent bilinear curve according to Commentary to IBC [31].