## 1 Introduction
### Basic Assumptions and Purposes of the Study  
The present paper proposes to evaluate the intensity of a scaled ground motion by referring to the earthquake spectral acceleration taken at the inelastic elongated period, the latter being obtainedfrom a time-history analysis. Two different methods are proposed to derive the elongated period. The first approach derives an inelastic peak period \(T_{inP}\) from the Fourier amplitude spectrum of the acceleration of a building's reference point; the second method obtains an inelastic smooth peak period \(T_{inSP}\) from the smooth Fourier spectrum derived from the original spectrum by applying a suitable filtering technique. In practice, the _IM_ parameter is calculated by means of the spectral acceleration (for a dampening factor \(\xi=5\%\)) taken at the structure's elongated period \(T_{in}\) instead of the elastic fundamental period \(T_{el}=T_{1}\). The value of \(T_{in}\) is derived from the Fourier amplitude spectrum (obtained through the fast Fourier transform) relevant to the structural acceleration taken at the center of gravity of the building's roof. Two alternative strategies will be adopted to obtain the elongated period. The first strategy derives the value of \(T_{inP}=2\pi/\omega_{inP}\) from the predominant frequency \(\omega_{inP}\) taken at the peak amplitude of the Fourier spectrum (red diagram in Figure 1). This period will hereafter be referred to as the \"inelastic peak period\". The following formula is thus considered to obtain the value of _IM_ in this case:  
\[IM_{inP}\ =\ S_{a}(T_{inP},5\%), \tag{3}\]  
It should be observed, however, that the higher the scale factor of the input record, the more difficult it may be selecting with confidence the right value of the predominant frequency \(\omega_{inP}\) from the Fourier amplitude spectrum, due to the several peaks that may occur in the range where the predominant frequency falls (see Figure 1). This was found to lead to the inconsistency of the values of a period that does not coherently elongate as the intensity of the scaled ground motion increases, as shown in the following sections. Therefore, a second strategy to obtain the elongated period has been considered. It is based on a smooth Fourier amplitude curve derived from the original Fourier spectrum by applying a suitable filtering technique (blue curve in Figure 1). The value of the predominant frequency \(\omega_{inSP}\) is taken at the curve peak of the smooth Fourier spectrum, and the elongated period \(T_{inSP}=2\pi/\omega_{inSP}\) is obtained accordingly. This period will hereafter be referred to as the \"inelastic smooth peak period\". By referring to \(T_{inSP}\), the value of _IM_ can be obtained as follows:  
\[IM_{inSP}\ =\ S_{a}(T_{inSP},5\%). \tag{4}\]  
Figure 1: Predominant frequencies taken from the Fourier spectrum (red curve) and from the smooth Fourier spectrum (blue curve) of the structural acceleration. In the present study, the smooth Fourier spectrum has been obtained by applying an appropriate \"smoothing window\" in the range 0-25 Hz, which is the range where the fundamental frequencies of buildings typically fall. The smoothing window \(S_{w}\) is determined according to:  
\[S_{w}=\frac{1}{\left(f_{c}\,\Delta_{t}\right)}, \tag{5}\]  
where, \(f_{c}\) is the cut-off frequency of the window while \(\Delta_{t}\) is the time step of the acceleration record. The aim of this paper is to assess whether and to what extent the adoption of either the intensity measure introduced in Equation (3) or that in Equation (4) can lead to an improvement of the IDA procedure, by reducing the dispersion of curves and improving the reliability of the results compared to other literature methods.