## 1 Introduction
### Inelastic Period vs. PGA  
The PGA of a scaled record may be an effective measure of the seismic intensity. As a function of the PGA of the scaled accelerograms considered in the present study (Table 2), the values of the inelastic peak period \(T_{inP}\) obtained from the Fourier amplitude spectrum relevant to the acceleration of the center of gravity of the building's roof are plotted in Figure 5a,b for the \(x\)- and \(y\)-directions, respectively. The diagrams in Figure 5 show very abrupt changes in the inelastic peak period curves (sharp increases and decreases) regardless of the considered record and at both low and high PGA values. Similar diagrams for the inelastic smooth peak period \(T_{inSP}\) are provided in Figure 6a,b. These figures show a much more gradual growth of \(T_{inSP}\) for increasing PGA values. While guaranteeing that the predominant frequency is found in the frequency band containing the highest Fourier amplitudes, the smooth Fourier spectrum avoids abrupt changes in the period (see the blue curve in Figure 1), which leads to a more suitable relationship between the elongated structural period and the intensity measure, namely, the PGA, of the earthquake (Figure 6). \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Earthquake** & **Date** & **Station ID** & _Mw_ & **Epicentre Distance (km)** & **PGA (g)** \\ \hline Kalamata & 13 September 1986 & ST163 & 5.9 & 11 & 0.272 \\ Friuli (aftershock) & 15 September 1976 & ST24 & 6.0 & 14 & 0.346 \\ South Iceland & 17 June 2000 & ST2562 & 6.5 & 21 & 0.278 \\ Kozani (aftershock) & 19 May 1995 & ST1372 & 5.2 & 16 & 0.265 \\ South Iceland & 17 June 2000 & ST2482 & 6.5 & 15 & 0.477 \\ Montenegro & 15 April 1979 & ST67 & 6.9 & 16 & 0.375 \\ Erzincan & 13 March 1992 & ST205 & 6.6 & 13 & 0.213 \\ Coquimbo & 16 September 2015 & CO03 & 8.3 & 92 & 0.691 \\ Maule & 27 February 2010 & ANGO & 8.8 & 209 & 0.928 \\ Valparaiso & 24 April 2017 & VA01 & 6.9 & 39 & 0.906 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Set of input ground motions consistent with the target elastic spectrum (see Figure 4). Figure 5: Evolution of the inelastic peak period \(T_{inP}\) in the \(x\)-direction (**a**) and in the \(y\)-direction (**b**). The curves refer to the input ground motions considered in Table 3.  
A comparison between Figures 5 and 6 highlights that much more regular curves are obtained when considering the smooth peak period \(T_{inSP}\) instead of the peak period \(T_{inP}\). Furthermore, the curves in Figure 6 show a slow and gradual increment in the elongated inelastic period as the seismic intensity increases. A smooth transition of the period is an expected occurrence in a structure subjected to gradually scaled accelerograms. As obtained from a modal analysis, the values of the elastic fundamental period of the building in the \(x\)-direction (\(T_{el-x}=0.288\) s) and in the \(y\)-direction (\(T_{el-y}=0.292\) s) are plotted for comparison in Figures 5 and 6 (solid horizontal lines). For very low PGA values, for instance, lower than 0.3 g (when the structure has not yet entered or has only just entered the plastic range), the value of the period taken from the Fourier amplitude spectrum should be very close to the fundamental elastic period. In contrast, Figures 5 and 6 demonstrate that the conventional periods determined by modal analysis are often (even significantly) lower than \(T_{inP}\) and \(T_{inSP}\) determined by means of time-history analysis and the Fourier spectrum-based procedures. The estimates of the inelastic period given by [PERSON] et al. [24] (see Equation (1)) and by [PERSON] and [PERSON] [32] (see Equation (2)) are also displayed in Figures 5 and 6 (dashed and dotted lines, respectively). The maximum value allowed by Equation (1), \(T_{in}=2T_{d}\), is considered for comparison; although being an upper limit, this value is found to underestimate the actual elongated period in many cases when both the peak period and the smooth peak period are considered (see Figures 5 and 6).