## 1 Introduction
### Procedure Description  
Based mainly on the classical formulation [19], an alternative procedure is proposed herein to determine the IDA curves. The proposed approach considers the \(IM\) parameter given either by Equation (3) or Equation (4). Both the inelastic peak period \(T_{inP}\) and the inelastic smooth peak period \(T_{inSP}\) are considered to check which produces better results within the IDA procedure.  
The following eight steps are followed to carry out the alternative procedure proposed herein, as summarized in Figure 2.  
* Step 1. A group of uniformly increasing \(SF_{s}\) are selected.
* Step 2. A set of strong accelerograms (two horizontal components for each earthquake) are chosen.
* Step 3. For any considered ground motion and for each \(SF\), the scaled \(x\)- and \(y\)-accelerations are obtained.
* Step 4. The scaled components of the ground motion are applied in the \(x\)- and \(y\)- directions to perform the time-history analysis.
* Step 5. The acceleration at the center of gravity of the building's roof is obtained through the time-history analysis.
* Step 6a. The Fourier amplitude spectrum of the acceleration at the center of gravity is obtained, and the frequency at the maximum spectral amplitude is taken (see the red curve in Figure 1). From this frequency value, the period \(T_{inP}\) is calculated.
* Step 6b. A smooth Fourier spectrum is derived from the original spectrum by applying a suitable filtering technique. The value of the predominant frequency is taken at the peak of the smooth curve (see the blue curve in Figure 1), and the corresponding elongated period \(T_{inSP}\) is found.
* Step 7a. The response acceleration spectrum (with 5% critical damping) is entered with \(T_{inP}\), and the spectral acceleration \(S_{a}(T_{inP},5\%)\) is obtained. The intensity measure is thus calculated according to Equation (3).
* Step 7b. The earthquake acceleration spectrum (with 5% critical damping) is entered with \(T_{inSP}\), and the spectral acceleration \(S_{a}(T_{inSP},5\%)\) is obtained. The intensity measure (\(IM\)) is thus calculated according to Equation (4).
* Step 8. The parameter chosen to quantify the demand on the structure (\(DM\)) is obtained from the time-history analysis for each of the considered cases. The intensity measure \(IM\) is then plotted versus the demand measure \(DM\) to obtain the IDA curve.  
Figure 2: Steps from 1 to 8 followed to carry out the incremental dynamic analysis (IDA) procedure (the activity related to each step is briefly recalled in the eight pictures).  
Please note that Steps 6 and 7 are divided into one of two components to consider either the inelastic peak period \(T_{inP}\) or the inelastic smooth peak period \(T_{inSP}\).  
In the present study, the maximum interstory drift ratio (\(ID_{max}\)) is adopted as the damage measure \(DM\):  
\[DM=ID_{max}, \tag{6}\]  
As is usually assumed in the literature [19]. Once the values of \(IM\) and \(DM\) are both given, a point is drawn on the diagram where the IDA curves will be displayed. The procedure will be carried out repeatedly from Steps 3 to 8 until all the points of each single-record curve are obtained. Then, the algorithm will continue iterating to obtain the curves for the other ground motions in the set. In the present case, a multi-record IDA diagram is obtained for each of the two different inelastic elongated periods.