## 3 Refinement 2: Review of the NTC-2018 Formula for Estimating the Equivalent Viscous Damping
Among period-dependent formulations (i.e., \(\xi_{h}=\xi_{h}\Big{(}T_{eff},\ \mu\Big{)}\)), [PERSON] et al., in 2007 [45] proposed  
Figure 3: Qualitative force–displacement (F-d) loops of hysteretic systems with (**a**) high, (**b**) medium, and (**c**) low dissipation capacity. \[\xi_{h}= \begin{cases}\left[a+b\cdot\left(1-T_{eff}\right)\right]\left(\frac{\mu-1}{ \pi\mu}\right)&T_{eff}<1s\\ a\left(\frac{\mu-1}{\pi\mu}\right)&T_{eff}\geq 1s\end{cases}  \tag{6}\]  
where \(a\) and \(b\) are constitutive parameters that depend on the type of hysteretic response: (i) \(a=85\) and \(b=60\) for EPP response; (ii) \(a=65\) and \(b=50\) for \"large Takeda\" response; and (iii) \(a=50\) and \(b=40\) for \"small Takeda\" response. Note that independently of the formulation and from the value of \(T_{eff}\) the equivalent hysteretic damping is set to \(\xi_{h}=0\) when \(\mu\leq 1.0\). A comparison between the equivalent viscous damping (\(\xi_{h}\)) calculated according to the NTC-2018 and [PERSON]'s formula is provided in Figure 4: for all types of considered hysteretic response, Equation (3) always leads to overestimated values of \(\xi_{h}\). For instance, when considering EPP systems (see Figure 4a) operating in the ordinary ductility range \(\mu=4\div 10\), the discrepancy is higher than \(55\%\) for stiff structural systems of low vibration period \(T_{eff}=0.3s\), while the difference grows beyond \(130\%\) for flexible structures having \(T_{eff}>1.0s\). Similar trends can be appreciated for mid-dissipative (see Figure 4b) and low-dissipative (see Figure 4c) structural systems, causing further doubts as to the reliability of Equations (3)-(5). For calculation of the reduction factor (\(\eta\)) to be used in iterative linear spectral analyses such as those of \"Method-B\", the Italian Building Code (NTC-2018) adopts the Eurocode-8 (EC-8) formula [46]:  
\[\eta=\sqrt{\frac{0.10}{0.05+\xi_{tot}}}\geq 0.55 \tag{7}\]  
Figure 4: Comparison between the equivalent viscous damping calculated according to the Italian Building Code and the [PERSON]’s formula for (**a**) high-dissipative EPP systems (k = 1.00); (**b**) mid-dissipative systems (“large-Takeda”, k = 0.66); and (**c**) low-dissipative systems (“small-Takeda”, k = 0.33). where: \(\xi_{tot}=\xi_{el}+\xi_{h}\) is the total damping obtained (adding to the contribution of hysteretic systems \(\xi_{h}\) the conventional elastic one \(\xi_{el}=0.05\)). In 2007 [PERSON] [30] proposed a refined formulation of \(\eta\) with higher accuracy that was verified through comparison with NLTH analyses:  
\[\eta=R_{\mu}=\left(\frac{0.07}{0.02+\xi_{tot}}\right)^{\alpha} \tag{8}\]  
where the exponent \(\alpha\) depends on the features of the seismic input: (i) \(\alpha=0.50\) for oscillatory seismic events (far-field sites) and (ii) \(\alpha=0.25\) for \"pulse-like\" seismic events caracterized by directivity effects (near-fault sites). Figure 5 shows a comparison between the NTC-2018 and [PERSON]'s formulation of the spectral reduction factor \(\eta\). We can note the following differences: (1) the NTC-2018 establishes a lower bound \(\eta\geq 0.55\) on the spectral reduction factor, more or less equivalent to the upper bound of the equivalent viscous damping \(\xi\leq 30\%\), while no limits are provided by the [PERSON] formulation; (2) for oscillatory seismic inputs (\(\alpha=0.50\)), the second formulation provides lower values of \(\eta\) (e.g., for \(\xi=0.20\) the discrepancy is about \(-11\%\)); (3) for \"pulse-like\" seismic inputs (\(\alpha=0.25\)), the second formulation provides significantly higher values of \(\eta\) (e.g., for \(\xi=0.20\) the discrepancy is about \(+19\%\)). Focusing on this final point, a dedicated study by [PERSON] and [PERSON] [47] proved that the Priestley formula is more reliable in cases of structural systems subjected to near-fault events, which are usually characterized by directivity effects (near-fault sites). [PERSON] et al.