## 3 Refinement 2: Review of the NTC-2018 Formula for Estimating the Equivalent Viscous Damping  
After establishing the two refined performance targets described in the previous Section, the need to use accurate formulations for the estimation of the equivalent viscous damping developed by both the bare frame and the hysteretic dampers becomes evident. In this regard, the official explanatory notes [26] of the Italian Building Code [25] suggest the following formula for the estimation of the equivalent viscous damping provided by hysteretic structural elements:  
\[\xi_{h}=k\cdot\frac{63.7\cdot\left(F_{y}\cdot d_{max}-F_{max}\cdot d_{y} \right)}{F_{max}\cdot d_{max}} \tag{3}\]  
where: (i) \(F_{y}\) and \(d_{y}\) are the yielding force and displacement; (ii) \(F_{max}\) is the plastic strength at the maximum displacement amplitude \(d_{max}\); and (iii) \(k\) is a reduction factor that takes into account possible degradation of the energy dissipated per cycle. In this regard, three possibilities are provided (see Figure 3): (1) \(k=1.00\) for high-dissipative structures that are characterized by stable elastic-perfectly plastic (EPP) loops (see Figure 3a) such as code compliant moment-resisting steel frames or BRB dampers; (2) \(k=0.66\) for mid-dissipative structures featuring hysteresis loops with moderate reduction in area, such as RC frameswith columns subjected to low compression loads and adequate shear reinforcements )the F-d response of these structures can be predicted through the \"Large Takeda\" hysteretic rule; see Figure 3b); (3) \(k=0.33\) for low-dissipative structures with hysteretic loops that are affected by a marked pinching and a substantial reduction in area, such as RC frames either featuring columns under high compression loads or characterized by significant crack blocking at load reversals (the force-displacement response of these structures can be predicted through the \"Small Takeda\" hysteretic rule; see Figure 3c). Concerns arise for the selection of the most suitable value of the factor \(k\) for structural systems such as \"X braced\" steel frames [41], which, in addition to even more marked pinching, are characterized by a lack of stiffness at the motion reversal (i.e., the direction from the maximum displacement to the origin). It is worth noting that Equation (3) has been derived from a formulation proposed by [PERSON] et al. (1930) [33]; for bilinear hardening systems, as later demonstrated by [PERSON] et al. (1964) [42], it can be written as follows:  
\[\xi_{h}=k\cdot\frac{2}{\pi}\cdot\frac{(\mu-1)(1-r)}{\mu(1+r\mu-r)} \tag{4}\]  
where: (i) \(r=k_{pl}/k_{el}\) is the hardening ratio and (ii) \(\mu=d_{max}/d_{y}\) is the maximum ductility level achieved during the seismic excitation. In the case of elastic perfectly-plastic (EPP) dissipative systems (\(r=0\)):  
\[\xi_{h}=k\cdot\frac{2}{\pi}\cdot\frac{(\mu-1)}{\mu} \tag{5}\]  
Figure 4 shows the trend of the damping ratio \(\xi_{h}\) as a function of the ductility level of the dissipative system. It should be noted that the Italian Building Code (NTC-2018, SS 7.10.5.2) establishes \(\xi_{h}\leq 30\%\) as upper threshold for linear equivalent analyses. Because steel dissipative systems (case k = 1.00) are designed to work at ductility levels between 4 and 12 [28; 29], this limit is exceeded at \(\mu\cong 1.9\) (see Figure 4a). Analogously, for medium dissipative capacity systems (k = 0.66) the limit is overcome at the rather low ductility of \(\mu\cong 3.7\) (see Figure 4b), making the adoption of Equation (1) ineffective. Only in case of low-dissipative systems (see Figure 4c) is Equation (1) fully applicable, as the cap imposed by NTC-2018 is never exceeded even at high ductility levels. Another drawback of Equations (3)-(5) is related to the non-dependence of the equivalent viscous damping on the effective vibration period of the system (\(T_{eff}\)), as otherwise demonstrated in more recent studies [43; 44].