Intensity Measure Based on a Smooth Inelastic Peak Period for a More E ﬀ ective Incremental Dynamic Analysis

: The Incremental Dynamic Analysis (IDA) assesses the global collapse capacity of a structure by plotting its maximum inelastic response, obtained through a non-linear time-history analysis, versus the scaled intensity of di ﬀ erent input earthquakes. The seismic intensity is often measured through the spectral acceleration at the fundamental elastic period. However, this can produce highly variable results. An alternative method is presented in this paper that relies on the elongated period, calculated either from the Fourier spectrum of the acceleration at a target building point (inelastic peak period) or from a smooth Fourier spectrum (inelastic smooth peak period). By referring to a reference reinforced concrete building and to a set of 10 spectrum-consistent earthquakes, the paper presents the results of a wide investigation. First, the variation in the elongated period as a function of the seismic intensity is discussed. Then, the e ﬀ ectiveness of the proposed method is assessed by comparing the IDA curves to those obtained through the elastic period or through approximate values of the elongated period given in the literature. The results show that the alternative IDA procedure generates curves with less-dispersed collapse thresholds. A statistical analysis shows signiﬁcant improvements in the results when the inelastic smooth peak period is adopted.


Introduction
Non-linear analyses are the most accurate approach for assessing the seismic behavior of civil structures, since the current codes highly encourage post-elastic dissipative behavior under strong earthquakes. Non-linear static analysis based on horizontal static loads with gradually increasing intensity (pushover analysis) is very commonly applied to ascertain the inelastic capacity of a structure until its collapse [1,2]. On the other hand, non-linear time-history analysis under spectrum-consistent earthquakes may be very useful for studying the evolution of the seismic demand on a given structure [3][4][5]. Stemming from the concept of incremental seismic actions typical of pushover analysis, incremental dynamic analysis (IDA) is a type of non-linear analysis technique under scaled ground accelerations that is currently perceived as the most advanced tool for assessing the seismic response of structures. In the guidelines published by the Federal Emergency Management Agency (FEMA) [6,7], IDA is presented as the fundamental basis with which to determine the performance factors of new types of structures, as well as to obtain the collapse thresholds of existing structures while considering the variability of seismic records. Many authors have carried out IDA to obtain the performance of different kinds of new and existing structures [8][9][10][11][12][13][14][15][16][17]-even structures equipped with dissipation or isolation devices [18].

Recommended Values of the "Inelastic" Period for the Earthquake Intensity Measure
The inelastic elongated fundamental period T in is a key parameter for establishing a rigorous evaluation of structural performance into the non-linear regime; for this reason, T in can represent the intensity measure required for IDA better than the elastic fundamental period T el . Therefore, IM could be evaluated by entering the earthquake response spectrum with the inelastic period T in instead of the elastic period T el .
Evaluating the lengthening of the period is still an open issue in earthquake engineering. Based on many numerical investigations, some empirical formulas have been proposed in the literature. Kadas et al. [24] studied over 100 ground motion records and proposed the following relationship between the elastic initial period T el and the inelastic final period T in [24]: where A y is the yield spectral acceleration. Note that Equation (1) limits the value of the elongated period to twice the elastic period.
To predict the value of the fundamental period shift ratio (T in /T el ) of R/C structures, Di Sarno and Amiri [33] provided the following formula: The above relation produces values of T in ranging from approximately 3.6 T el (for very rigid oscillators) to 1.23 T el (for flexible oscillators). Some corrective parameters are also given in [33] to adjust the "reference" values obtained from Equation (2) to account for some deterioration effects.
By referring to one-degree-of-freedom oscillators equivalent to the considered multi-degree-offreedom systems, Katsanos et al. [32] derived the value of the inelastic period at the maximum amplitude of the Fourier spectrum relevant to the structural acceleration under each scaled earthquake. Both regular and irregular multi-story buildings were considered in [32] under hundreds of recorded earthquakes, two different hysteretic rules, and 11 scaling factors (PGA reaching up to 1.7 g). The results presented by Katsanos et al. [32] showed that the shift ratio T in T el is typically lower than 1.7. It should be noted, however, that such results were obtained by carrying out in-plane analyses on the equivalent one-degree-of-freedom oscillators.

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 obtained Appl. Sci. 2020, 10, 8632 4 of 21 from 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 ξ = 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 = 2π/ω inP from the predominant frequency ω 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: in this case: 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 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 is taken at the curve peak of the smooth Fourier spectrum, and the elongated period = 2 / is obtained accordingly. This period will hereafter be referred to as the "inelastic smooth peak period". By referring to , the value of can be obtained as follows: 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 is determined according to: where, is the cut-off frequency of the window while ∆ is the time step of the acceleration record.  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 ω 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 ω inSP is taken at the curve peak of the smooth Fourier spectrum, and the elongated period T inSP = 2π/ω 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: Appl. Sci. 2020, 10, 8632

of 21
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: where, f c is the cut-off frequency of the window while ∆ 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. For this purpose, the results of an extensive numerical investigation are presented to compare the IDA curves derived by entering the response spectrum with different values of the period: the elastic fundamental period; two values of the inelastic period, namely those in Equations (1) and (2); and the two values of the inelastic elongated period, namely T inP and T inSP , proposed in this paper.

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.

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Step 1. A group of uniformly increasing SF s are selected.

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Step 2. A set of strong accelerograms (two horizontal components for each earthquake) are chosen.

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Step 3. For any considered ground motion and for each SF, the scaled xand y-accelerations are obtained.

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Step 4. The scaled components of the ground motion are applied in the xand ydirections to perform the time-history analysis.

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Step 5. The acceleration at the center of gravity of the building's roof is obtained through the time-history analysis.

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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.

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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.

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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).

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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: 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.

Reference Building to Test the Efficacy of the Method
The 3D two-story R/C building considered in the present investigation is schematically represented in Figure 3. All the structural details of this building are given in [2,3]. The seismic behavior of this building, designed according to an old Italian regulation, was assessed in [2,3], where the effectiveness of the seismic retrofit of the structure with fiber-reinforced polymer (FRP) was also evaluated. Non-linear time-history analyses (with spectrum-consistent earthquakes) were performed in [2], while the results of both pushover analysis and standard multi-record IDA were presented in [3].

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).

Reference Building to Test the Efficacy of the Method
The 3D two-story R/C building considered in the present investigation is schematically represented in Figure 3. All the structural details of this building are given in [2,3]. The seismic behavior of this building, designed according to an old Italian regulation, was assessed in [2,3], where the effectiveness of the seismic retrofit of the structure with fiber-reinforced polymer (FRP) was also evaluated. Non-linear time-history analyses (with spectrum-consistent earthquakes) were performed in [2], while the results of both pushover analysis and standard multi-record IDA were presented in [3].
The building has a rectangular plan (9.80 m × 15.50 m) with a fixed elevation height between floors (3.00 m) and fixed column sections (0.40 m × 0.40 m). An exterior R/C staircase gives the structure an irregular configuration. The slabs are ribbed in the y-direction, which defines two types of frames: frames that support predominantly seismic loads (in the y-direction) and frames that support both gravitational and seismic loads (in the x-direction). The characteristics of the structural materials are shown in Table 1. Further details about the building geometry, its steel reinforcement, and the seismic retrofit with FRP can be found in [2,3].  A 3D finite element model was implemented with SeismoStruct [34], and the building non-linear behavior was introduced through a distributed plasticity model (fiber model). Each element section was split into at least 200 fibers. Structural members were divided into four elements to identify different steel confinement zones, while the areas wrapped by FRP sheets to improve the member behavior (namely, at the ends of the ground-floor columns, see Figure 3) are considered separately, cf. [2,3]. The constitutive models of Mander et al. [11] and Menegotto and Pinto [12] were adopted for concrete and reinforcing steel, respectively. The properties of the first six modes (namely, mode type, frequency, period, effective mass in the x-and y-direction) of the building are provided in Table  2.  The building has a rectangular plan (9.80 m × 15.50 m) with a fixed elevation height between floors (3.00 m) and fixed column sections (0.40 m × 0.40 m). An exterior R/C staircase gives the structure an irregular configuration. The slabs are ribbed in the y-direction, which defines two types of frames: frames that support predominantly seismic loads (in the y-direction) and frames that support both gravitational and seismic loads (in the x-direction). The characteristics of the structural materials are shown in Table 1. Further details about the building geometry, its steel reinforcement, and the seismic retrofit with FRP can be found in [2,3]. A 3D finite element model was implemented with SeismoStruct [34], and the building non-linear behavior was introduced through a distributed plasticity model (fiber model). Each element section was split into at least 200 fibers. Structural members were divided into four elements to identify different steel confinement zones, while the areas wrapped by FRP sheets to improve the member behavior (namely, at the ends of the ground-floor columns, see Figure 3) are considered separately, cf. [2,3]. The constitutive models of Mander et al. [11] and Menegotto and Pinto [12] were adopted for concrete and reinforcing steel, respectively. The properties of the first six modes (namely, mode type, frequency, period, effective mass in the xand ydirection) of the building are provided in Table 2.

Set of Spectrum-Consistent Records
A set of severe ground motions compatible with the seismic conditions at the building's location (Messina, Italy) was considered in the present investigation. Related to the building location, the design response spectrum for the no-collapse requirement (horizontal component) was considered according to Eurocode 8 [35] and Italian code [36]. Figure 4 plots this design spectrum (black curve) and the parameters adopted to obtain it. Consistent with this reference design response spectrum, 10 pairs of accelerograms were considered, some characteristics of which are listed in Table 3. In particular, the earthquake name, the date, the station identifier, the moment magnitude M w , the epicentre distance and the peak ground acceleration (PGA) are provided in the table. Note that seven of the selected accelerograms are taken from a European database and correspond to near-source earthquakes with a moderate moment magnitude M w ; they are obtained by means of REXEL software [37] according to EC8 [36]. The remaining three ground motions listed in Table 3 are near-intermediate source Chilean records obtained through SeismoSpect [38]. The response spectra of the 10 couples of records (light blue curves) with their average spectrum (blue curve) are plotted in Figure 4 together with the reference design spectrum (black).

Set of Spectrum-Consistent Records
A set of severe ground motions compatible with the seismic conditions at the building's location (Messina, Italy) was considered in the present investigation. Related to the building location, the design response spectrum for the no-collapse requirement (horizontal component) was considered according to Eurocode 8 [35] and Italian code [36]. Figure 4 plots this design spectrum (black curve) and the parameters adopted to obtain it. Consistent with this reference design response spectrum, 10 pairs of accelerograms were considered, some characteristics of which are listed in Table 3. In particular, the earthquake name, the date, the station identifier, the moment magnitude , the epicentre distance and the peak ground acceleration (PGA) are provided in the table. Note that seven of the selected accelerograms are taken from a European database and correspond to near-source earthquakes with a moderate moment magnitude ; they are obtained by means of REXEL software [37] according to EC8 [36]. The remaining three ground motions listed in Table 3 are nearintermediate source Chilean records obtained through SeismoSpect [38]. The response spectra of the 10 couples of records (light blue curves) with their average spectrum (blue curve) are plotted in Figure 4 together with the reference design spectrum (black).  (Table 2), average spectrum, and reference design spectrum. Table 3. Set of input ground motions consistent with the target elastic spectrum (see Figure 4).   (Table 2), average spectrum, and reference design spectrum. Table 3. Set of input ground motions consistent with the target elastic spectrum (see Figure 4). In the present investigation, the seismic records have been scaled through non-negative scale factors SF, according to the approach proposed in Vamvatsikos and Cornell [19]. Starting from 0.3, the scale factors have been progressively increased every 0.3 until the dynamic response reached the collapse threshold defined in [19].

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 xand 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.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 23 In the present investigation, the seismic records have been scaled through non-negative scale factors , according to the approach proposed in Vamvatsikos and Cornell [19]. Starting from 0.3, the scale factors have been progressively increased every 0.3 until the dynamic response reached the collapse threshold defined in [19].

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 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 are provided in Figure 6a,b. These figures show a much more gradual growth of 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).
A comparison between Figures 5 and 6 highlights that much more regular curves are obtained when considering the smooth peak period instead of the peak period . 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. in the x-direction (a) and in the y-direction (b). The curves refer to the input ground motions considered in Table 3. The curves refer to the input ground motions considered in Table 3.
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). Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 23 Figure 6. Evolution of the inelastic smooth peak period in the x-direction (a) and in the ydirection (b). The curves refer to the earthquakes considered in Table 3.
As obtained from a modal analysis, the values of the elastic fundamental period of the building in the x-direction ( = 0.288 s) and in the y-direction ( = 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 and determined by means of timehistory analysis and the Fourier spectrum-based procedures.
The estimates of the inelastic period given by Kadas et al. [24] (see Equation (1)) and by Di Sarno and Amiri [32] (see Equation (2)) are also displayed in Figures 5 and 6 (dashed and dotted lines, respectively). The maximum value allowed by Equation (1), = 2 , 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). On the other hand, the inelastic period estimated by Di Sarno andAmiri (in the present case, ~1.5 ) is almost always defective (at least for PGA > 0.4 g) with respect to the values and calculated through the non-linear time-history analysis and Fourier spectrum-based methods. Moreover, EC8-Part 1 [36] and ASCE/SEI 7-10 [39] suggest estimating the elongated inelastic period as 2 and 1.5 , respectively [32].

Assessing the Effectiveness of the Proposed IDA Procedure
The effectiveness of the IDA procedure proposed in Section 2.1 will be assessed by comparing it with two approaches: the standard procedure that refers to the elastic period and the procedure that refers to literature values of the elongated period calculated according to Equation (1) or to Equation (2). First, the and values are compared through different percentiles. Subsequently, the dispersion achieved by both the and the is analyzed by comparing the standard deviations and the coefficient of variation of the set of IDA curves corresponding to each direction of analysis.
The intensity measure will be given by the = 5% damped spectral acceleration at the structure's first-mode period, that is, = ( , 5%). The value of will be considered either as the elastic fundamental period (standard IDA procedure) or as the elongated inelastic period , estimated following Kadas et al. [24] or Di Sarno and Amiri [32] or calculated as proposed in this paper (alternative IDA procedure).  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 Kadas et al. [24] (see Equation (1)) and by Di Sarno and Amiri [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 el , 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). On the other hand, the inelastic period estimated by Di Sarno andAmiri (in the present case, T in ∼ 1.5T el ) is almost always defective (at least for PGA > 0.4 g) with respect to the values T inP and T inSP calculated through the non-linear time-history analysis and Fourier spectrum-based methods. Moreover, EC8-Part 1 [36] and ASCE/SEI 7-10 [39] suggest estimating the elongated inelastic period as 2T el and 1.5T el , respectively [32].

Assessing the Effectiveness of the Proposed IDA Procedure
The effectiveness of the IDA procedure proposed in Section 2.1 will be assessed by comparing it with two approaches: the standard procedure that refers to the elastic period and the procedure that refers to literature values of the elongated period calculated according to Equation (1) or to Equation (2). First, the IM and DM values are compared through different percentiles. Subsequently, the dispersion achieved by both the IM and the DM is analyzed by comparing the standard deviations and the coefficient of variation of the set of IDA curves corresponding to each direction of analysis.
The intensity measure will be given by the ξ = 5% damped spectral acceleration at the structure's first-mode period, that is, IM = S a (T 1 , 5%). The value of T 1 will be considered either as the elastic fundamental period T el (standard IDA procedure) or as the elongated inelastic period T in , estimated following Kadas et al. [24] or Di Sarno and Amiri [32] or calculated as proposed in this paper (alternative IDA procedure).
According to Equation (6), the maximum interstory drift ratio of the building, referred to as ID max , was adopted as the indicator of the maximum demand on the structure (the DM parameter). All analyses performed herein were referenced to the two-story building displayed in Figure 3 and to the set of spectrum-consistent earthquakes described in Table 2.

Standard IDA Procedure Based on the Elastic Fundamental Period
Referring to the aforementioned building and spectrum-consistent earthquakes, a standard IDA investigation was carried out by adopting IM = IM el = S a (T el , 5%), where T el = T 1 is the elastic fundamental period of the structure and DM = ID max . The IDA curves obtained are displayed in Figure 7a,b. The plots show a rather high degree of dispersion of DM for a given IM, which indicates that the chosen IM parameter is not effective [19]. Hence, considering the elongated inelastic period instead of the elastic fundamental period might be a better choice, leading to curves with less dispersion.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 23 According to Equation (6), the maximum interstory drift ratio of the building, referred to as , was adopted as the indicator of the maximum demand on the structure (the parameter). All analyses performed herein were referenced to the two-story building displayed in Figure 3 and to the set of spectrum-consistent earthquakes described in Table 2.

Standard IDA Procedure Based on the Elastic Fundamental Period
Referring to the aforementioned building and spectrum-consistent earthquakes, a standard IDA investigation was carried out by adopting = = ( , 5%), where = is the elastic fundamental period of the structure and = . The IDA curves obtained are displayed in Figure 7a,b. The plots show a rather high degree of dispersion of for a given , which indicates that the chosen parameter is not effective [19]. Hence, considering the elongated inelastic period instead of the elastic fundamental period might be a better choice, leading to curves with less dispersion.

IDA Curves Obtained Through Approximate Values of the Elongated Period
The results obtained by calculating the IM parameter in reference to the approximate values of the elongated fundamental period obtained from Equations (1) and (2) [24,33], are presented in this section. Figure 8 shows the IDA curves obtained by considering = 2 , which is the upper value proposed by Kadas et al. [24] (see Equation (1)), while the curves derived through the approximate value given by Di Sarno and Amiri [33] (see Equation (2)) are plotted in Figure 9. Despite referring to the inelastic period, the diagrams in Figures 8 and 9 still show a rather high scatter because the inelastic periods estimated by the methods of Kadas et al. and Di Sarno and Amiri are constants that can greatly differ from the actual values, as also highlighted by the diagrams in Figures 5 and 6.

IDA Curves Obtained Through Approximate Values of the Elongated Period
The results obtained by calculating the IM parameter in reference to the approximate values of the elongated fundamental period obtained from Equations (1) and (2) [24,33], are presented in this section. Figure 8 shows the IDA curves obtained by considering T in = 2T el , which is the upper value proposed by Kadas et al. [24] (see Equation (1)), while the curves derived through the approximate value given by Di Sarno and Amiri [33] (see Equation (2)) are plotted in Figure 9. Despite referring to the inelastic period, the diagrams in Figures 8 and 9 still show a rather high scatter because the inelastic periods estimated by the methods of Kadas et al. and Di Sarno and Amiri are constants that can greatly differ from the actual values, as also highlighted by the diagrams in Figures 5 and 6. Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 23

IDA Curves Obtained through and
References will now be made to the values of the inelastic period proposed in this study, namely, and , which are derived from the Fourier amplitude spectrum and from the smooth Fourier spectrum, respectively. Figure 10 shows the IDA curves obtained when considering the intensity measure given by Equation (3), while the curves in Figure 11 are relevant to the intensity measure provided by Equation (4).

IDA Curves Obtained through and
References will now be made to the values of the inelastic period proposed in this study, namely, and , which are derived from the Fourier amplitude spectrum and from the smooth Fourier spectrum, respectively. Figure 10 shows the IDA curves obtained when considering the intensity measure given by Equation (3), while the curves in Figure 11 are relevant to the intensity measure provided by Equation (4).

IDA Curves Obtained through T inP and T inSP
References will now be made to the values of the inelastic period proposed in this study, namely, T inP and T inSP , which are derived from the Fourier amplitude spectrum and from the smooth Fourier spectrum, respectively. Figure 10 shows the IDA curves obtained when considering the intensity measure given by Equation (3), while the curves in Figure 11 are relevant to the intensity measure provided by Equation (4). Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 23

Figures 7-11 show that
is a non-monotonic function of ; the curves often display a twisting pattern with both softening and (counterintuitively) hardening behaviors. Such a trend is also typical for the IDA curves when other choices of and are made, as reported in [19]. Moreover, the curves obtained for different records tend to provide rather dispersed results.
Comparing the graphs in Figures 7-11 almost always reveals more marked scatter in the xdirection with respect to the y-direction regardless of the value of the period (elastic or inelastic, approximated or rigorously calculated). If a reference to the elongated period is made, however, the scatter generally reduces, even if referring to the values of the elongated period does not significantly improve the effectiveness of the IDA procedure.
In contrast, when more rigorous values of the elongated inelastic period are accounted for, namely, and , the IDA curves exhibit less dispersion (see Figures 10 and 11). A remarkable

Figures 7-11 show that
is a non-monotonic function of ; the curves often display a twisting pattern with both softening and (counterintuitively) hardening behaviors. Such a trend is also typical for the IDA curves when other choices of and are made, as reported in [19]. Moreover, the curves obtained for different records tend to provide rather dispersed results.
Comparing the graphs in Figures 7-11 almost always reveals more marked scatter in the xdirection with respect to the y-direction regardless of the value of the period (elastic or inelastic, approximated or rigorously calculated). If a reference to the elongated period is made, however, the scatter generally reduces, even if referring to the values of the elongated period does not significantly improve the effectiveness of the IDA procedure.
In contrast, when more rigorous values of the elongated inelastic period are accounted for, namely, and , the IDA curves exhibit less dispersion (see Figures 10 and 11). A remarkable Figure 11. IDA curves obtained using the elongated smooth peak period T inSP according to Equation (4):

Figures 7-11
show that DM is a non-monotonic function of IM; the curves often display a twisting pattern with both softening and (counterintuitively) hardening behaviors. Such a trend is also typical for the IDA curves when other choices of DM and IM are made, as reported in [19]. Moreover, the curves obtained for different records tend to provide rather dispersed results.
Comparing the graphs in Figures 7-11 almost always reveals more marked scatter in the x-direction with respect to the y-direction regardless of the value of the period (elastic or inelastic, approximated or rigorously calculated). If a reference to the elongated period is made, however, the scatter generally reduces, even if referring to the values of the elongated period does not significantly improve the effectiveness of the IDA procedure.
In contrast, when more rigorous values of the elongated inelastic period are accounted for, namely, T inP and T inSP , the IDA curves exhibit less dispersion (see Figures 10 and 11). A remarkable reduction in dispersion in both the x-direction and the y-direction is found when the IM parameter is calculated through the smooth peak period T inSP . A comparison between the graphs in Figure 11 and those in Figures 7-10 shows that the IM parameter provided by Equation (4) is, in fact, the best choice among those considered in the present investigation, at least in reference to the building in Figure 3. To assess whether and to what extent Equation (4) may provide a good IM indicator, the effectiveness of the different IDA procedures will be evaluated in terms of the dispersion of the DM results at given IM values for the considered building.
According to the conventional procedure presented in [19], the collapse is assumed to be reached at the point from which the slope of the curve falls to a value less than 20% of the elastic slope. The values of IM and DM at the collapse point are referred to as Sa collapse and ID collapse . The collapse point enables to determine the response reduction factors [7] and to assess the seismic performance of existing structures [6]. To this purpose, the median of the collapse points of the IDA curves is used as global collapse criterion.
The 16th, 50th, and 84th percentiles of Sa collapse and ID collapse achieved in this study are provided in Figures 12 and 13, for the two analysis directions and relevant to the five values of period considered. The histograms in Figure 12 show that, for any percentile, the values of Sa collapse relevant to the elastic period are the highest, reaching almost 6 g for the 84th percentile in both directions. Still, rather high values are reached for the approximate elongated period given by Di Sarno and Amiri [33]. On the contrary, the Sa collapse values obtained with the approximate period of Kadas et al. [24] and with the periods T inP and T inSP increase slightly from the 16th to the 84th percentile, reaching in general values lower than 3 g, which are compatible with the expected capacity of the type of structure considered.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 23 reduction in dispersion in both the x-direction and the y-direction is found when the parameter is calculated through the smooth peak period . A comparison between the graphs in Figure 11 and those in Figures 7-10 shows that the parameter provided by Equation (4) is, in fact, the best choice among those considered in the present investigation, at least in reference to the building in Figure 3. To assess whether and to what extent Equation (4) may provide a good indicator, the effectiveness of the different IDA procedures will be evaluated in terms of the dispersion of the results at given values for the considered building. According to the conventional procedure presented in [19], the collapse is assumed to be reached at the point from which the slope of the curve falls to a value less than 20% of the elastic slope. The values of and at the collapse point are referred to as and . The collapse point enables to determine the response reduction factors [7] and to assess the seismic performance of existing structures [6]. To this purpose, the median of the collapse points of the IDA curves is used as global collapse criterion.
The 16th, 50th, and 84th percentiles of and achieved in this study are provided in Figures 12 and 13, for the two analysis directions and relevant to the five values of period considered. The histograms in Figure 12 show that, for any percentile, the values of relevant to the elastic period are the highest, reaching almost 6 g for the 84th percentile in both directions. Still, rather high values are reached for the approximate elongated period given by Di Sarno and Amiri [33]. On the contrary, the values obtained with the approximate period of Kadas et al. [24] and with the periods and increase slightly from the 16th to the 84th percentile, reaching in general values lower than 3 g, which are compatible with the expected capacity of the type of structure considered.  increase, reaching values below 4%. It is important to highlight that the values obtained with the period are very close to those calculated for the same building through a Pushover analysis in [2]. Furthermore, such values of collapse interstory drift are compatible with those relevant to this type of structures [40]. This makes the results obtained through the smooth period more reliable.

Dispersion of the IDA Curves' Results
To quantify the dispersion in the IDA curves, the coefficient of variation (COV) relevant to the collapse values of the and parameters will be considered. Since COV = 0 reflect a lack of statistical variation, the COV decrease as the degree of dispersion decreases and as the efficiency of the indicator increases. According to Vamvatsikos and Cornell [19], the collapse threshold of each IDA curve was determined by evaluating the interstory drift ratio value at which the curve's slope ( / ) drops under 20% of the initial slope. This value typically varies from curve to curve, although a smaller variation indicates less dispersion and, thus, higher effectiveness. The values of = and = ( , 5%), both of which are taken at the collapse point, will be referred to as the collapse interstory drift ratio and collapse spectral acceleration, respectively. The histograms in Figure 13 show that ID collapse corresponding to the 16th percentile has moderate values for any kind of period considered in this study. When the 50th and 84th percentile are concerned, the ID collapse values calculated with the elastic period and with the Di Sarno and Amiri [33] elongated period, increase notably instead, particularly in the y-direction, where 12% is exceeded. In contrast, the values obtained with the T inP and T inSP periods show a moderate increase, reaching values below 4%. It is important to highlight that the ID collapse values obtained with the T inSP period are very close to those calculated for the same building through a Pushover analysis in [2]. Furthermore, such values of collapse interstory drift are compatible with those relevant to this type of structures [40]. This makes the results obtained through the smooth period T inSP more reliable.

Dispersion of the IDA Curves' Results
To quantify the dispersion in the IDA curves, the coefficient of variation (COV) relevant to the collapse values of the DM and IM parameters will be considered. Since COV = 0 reflect a lack of statistical variation, the COV decrease as the degree of dispersion decreases and as the efficiency of the IM indicator increases. According to Vamvatsikos and Cornell [19], the collapse threshold of each IDA curve was determined by evaluating the interstory drift ratio value at which the curve's slope (IM/DM) drops under 20% of the initial slope. This value typically varies from curve to curve, although a smaller variation indicates less dispersion and, thus, higher effectiveness. The values of DM = ID max and IM = S a (T 1 , 5%), both of which are taken at the collapse point, will be referred to as the collapse interstory drift ratio and collapse spectral acceleration, respectively.
With reference to the IDA curves in Figures 7-11, the statistical results relevant to the collapse interstory drift ratio and to the collapse spectrum acceleration, for the x and y directions, are given in Tables 4 and 5, and in Tables 6 and 7, respectively. The second and third columns of these tables provide the mean and standard deviation, respectively, while the fourth column lists the COV, calculated as the ratio between the standard deviation and the mean.  Table 6. Statistical data for the collapse spectral acceleration (x-direction).  Table 7. Statistical data for the collapse spectral acceleration (y-direction). The results given in Tables 4 and 5 show that, on average, the standard procedure (referring to the elastic fundamental period) and the approximate procedures (referring to the constant values of the elongated period proposed by Kadas [24] and by Di Sarno and Amiri [33]) both lead to higher collapse interstory drift ratio values with respect to the alternative procedure proposed in this investigation based on either the inelastic peak period T inP or the inelastic smooth peak period T inSP , that is, based on the IM indicators given by Equations (3) and (4).

Mean (g) Standard Deviation COV
Regarding the COV of the collapse interstory drift ratio, however, different levels of effectiveness are exhibited by the two new IM indicators given by Equations (3) and (4). In fact, although the COV in the y-direction is more than halved (see Table 4), the IM indicator based on T inP does not improve the results in the x-direction (see Table 3). On the other hand, when the IDA curves are calculated with the elongated smooth peak period T inSP , the COV of the collapse interstory drift ratio is drastically reduced with respect to the other standard methods in both directions (almost halved in the x-direction and reduced by more than 60% in the y-direction).
On the other hand, Table 6 shows that lower COV values of the collapse spectrum acceleration are found in the x-direction for the inelastic period, regardless of whether the period is conventionally estimated or rigorously calculated. In contrast, a much more remarkable reduction in the COV is attained when the smoothed peak period T inSP is considered in place of the peak period T inP . A reduction in the COV of approximately two-thirds of the COV in the x-direction with respect to the elastic period is attained with T inSP , in contrast to a reduction of less than one-half with T inSP .
In the y-direction, the dispersion detected when considering the Kadas or Di Sarno and Amiri values of the inelastic period (second and third rows of Table 7) is even higher than that considering the elastic period (first row of Table 6). Therefore, the use of the conventional inelastic periods given by Equations (1) and (2) does not lead to a reduction in the dispersion of the collapse spectral acceleration results in the y-direction. The COV of the collapse spectral acceleration decreases only when T inP and T inSP are considered (fourth and fifth rows in Table 7). Likewise, the value of COV relevant to T inSP is markedly lower than that obtained for T inP , which means that Equation (4) provides a better choice than Equation (3) for the IM indicator.

Conclusions
A new IDA procedure is presented in this paper with the aim of reducing the high record-to-record variability of the curves representing the maximum demand on a given building at different scaled seismic intensities. Here, the seismic intensity is measured through the ξ = 5% damped spectral acceleration at the structure's first-mode period, where the first-mode period is assumed to be the elongated inelastic period exhibited by the system when it deforms in the plastic range. Two different methods are adopted to obtain the value of the inelastic fundamental period: the first one obtains the inelastic peak period T inP from the fundamental frequency taken at the maximum peak of the Fourier amplitude spectrum of the acceleration of a building's reference point; the second method calculates the inelastic smooth peak period T inSP from a smooth Fourier spectrum derived from the original spectrum by means of a suitable filtering technique. The efficacy of adopting a smooth elongated period is expected to still be valid regardless of the approach adopted to smooth the Fourier spectrum in the range where the elongated period of the considered structure falls.
An investigation is presented in the paper to assess the effectiveness of the proposed new IDA procedure. By referring to a R/C building and to a set of 10 earthquakes, consistent with the design response spectrum relevant to a high-seismicity zone of Italy, multi-record IDA is carried out by considering a 3D finite element model of the building with non-linear behavior (distributed plasticity model).
The first part of the present study analyzed the variation in the inelastic period as a function of the seismic intensity, the latter being evaluated in terms of PGA. The following results were found in this preliminary analysis.
(i) The value of the elongated "inelastic" period calculated as the peak period T inP or as the smooth peak period T inSP can be much higher than (even more than twice) the elastic fundamental period of the structure and increases as the earthquake intensity level increases. (ii) Some approximation methods proposed in the literature to estimate the elongated inelastic period are found to often deviate from the rigorous values (given by T inP or by T inSP ). (iii) The values of the elongated period obtained through time-history analysis (T inP or T inSP ) are found to depend on the PGA of the given accelerogram. As expected, however, this dependence is far from monotonic and predictable since the PGA is not able to unequivocally define the violence of an earthquake. Of course, higher PGA values typically imply that the structure reaches higher levels of damage, which directly affects the value of the elongated period. However, further studies on the variation in the elongated period with respect to the characteristics of the structure and of the earthquakes are still required.
(iv) Much more gradual growth of T inSP with increasing PGA values is found with respect to T inP . In fact, T inSP is obtained from the smooth Fourier spectrum, which avoids abrupt changes in the period values while guaranteeing that the predominant frequency is found in the frequency band containing the highest Fourier amplitudes. This generally leads to a more suitable relationship between the elongated structural period and the intensity measure of the earthquake.
The second part of the study investigated the effectiveness of the proposed IDA procedure based on calculating the earthquake IM as the spectral acceleration taken at either the peak period T inP or the smooth peak period T inSP . A comparison is performed among the IDA curves derived through the new procedure, those obtained with the standard procedure that refers to the elastic fundamental period, and those obtained with the procedure that refers to the approximate values of the inelastic period proposed in the literature. The following conclusions can be drawn from the results.
(i) Among the five different IDA procedures compared in this paper, the standard procedure that refers to the elastic fundamental period produces IDA curves with the most dispersion. The coefficient of the collapse interstory drift ratio is COV = 0.76 in both the xand the y-directions, while COV = 0.98 and COV = 0.73 are found for the collapse spectral acceleration in the xand y-directions, respectively. (ii) On the other hand, referring to approximate values of the inelastic period may not lead to significant improvements in the effectiveness of the IDA procedure. In the present study, the methods proposed by Kadas et al. and Di Sarno and Amiri are unable to reduce the COVs of the collapse interstory drift ratio in either of the two directions, while they reduced the COV of the collapse spectral acceleration only in the x-direction. (iii) When the IDA procedure is applied by considering the calculated values of the inelastic period T inP and T inSP , the dispersion of the results can be significantly reduced, although different impacts of the two values of the inelastic period are detected. (iv) The most remarkable improvement is achieved in reference to the inelastic smooth peak period T inSP . Calculating the IDA curves through this value of the elongated period is found to always reduce the COV of both the DM indicator and the IM parameter in any direction with respect to all the other methods. The COV values are halved or even reduced to one-third of those obtained with respect to the standard procedure that refers to the elastic period.
It is not surprising that smoothing the spectra decreases the coefficient of variation of the results. However, the adoption of a smooth Fourier spectrum allows to fix issues regarding the dispersion of the results and the inconsistencies in the values of the "inelastic" period, which may occur when applying the IDA procedure. On one hand, it is widely recognized that referring to the elastic period usually leads to highly dispersed IDA curves. Much lower dispersed IDA curves are obtained when a reference to the inelastic smooth peak period T inSP is made instead. On the other hand, it is found in this paper that inconsistent values of period can be taken from the unsmoothed Fourier spectrum. In fact, the higher the scale factor of the seismic record, the more the Fourier amplitude spectrum can exhibit peaks in the range where the predominant frequency falls, makes it rather hard to determine its correct value with confidence. This may lead to obtaining lower inelastic periods for higher scale factors of the input record (that is for higher levels of damage in the structure). This inconsistency can be avoided when reference to the inelastic smooth peak period T inSP is made.
Of course, the fact that the IDA curves are less dispersed and that the values of period are more consistent may be considered merely as an index of more reliable results. Further investigation, however, is needed to assert this fact with certainty. It should also be stressed that the above findings are relevant to the building and to the set of earthquakes considered in this study. Therefore, the effectiveness of the proposed IDA procedure should be assessed in future studies by also considering other types of structures and different sets of earthquakes.
Future studies can be also directed to determine the intensity measure (IM) by using "inelastic" response spectra relevant to equivalent single degree of freedom systems with elastic-plastic behavior. This would allow a more moderate growth of the intensity measurement.
Finally, it is to note that, although most of the works that have sought to reduce the variability of IDA curves have focused almost exclusively on intensity measurements, it is of special importance to reduce the scatter in the DM parameter since determining the collapse value of the building is crucial to carry out the performance-based design.