1 Introduction

In the framework of performance-based earthquake engineering (PBEE), seismic vulnerability is addressed by developing probabilistic seismic demand analyses (PSDA) and fragility curves that depict the conditional probability of the structure exceeding a specific limit state given the Intensity Measure (IM) of the ground motion (Porter 2003). Thus, one of the essential criteria for obtaining reliable fragility curves and PSDA is the selection of appropriate IMs (Giovenale et al. 2004; Padgett et al. 2008). PSDA links structural demand with seismic IM (Cornell et al. 2002; Shafieezadeh et al. 2012; Chen et al. 2022), and it is the foundation of seismic risk assessment and the management of structural decision-making. In PSDA, efficiency and accuracy are primarily determined by selecting suitable IM parameters, and an inappropriate IM may invalidate the probabilistic seismic assessment assumptions. In fact, the IM relates the seismic input to the seismic vulnerability analysis (Cantagallo et al. 2012; Huang et al. 2021). Therefore, selecting the best IM is a critical task that increases confidence in probabilistic seismic demand methods and risk assessments used in decision-making (Wang et al. 2018).

The criteria proposed in the literature for measuring the suitability of an IM are expressed in terms of the information the IM provides to predict the structural response quantities (Jalayer et al. 2012). Luco and Cornell (2007) and Jalayer et al. (2012) have proposed sufficiency as one of the most accurate criteria for measuring the suitability of an IM for representing the dominant features of ground shaking. More specifically, Jalayer et al. (2012) indicate that an IM is sufficient in an absolute sense if the damage measure conditional on this IM depends only on this IM and is entirely independent of other ground motion characteristics. As a result, establishing sufficiency in an absolute sense is nearly impossible unless high-dimensional vector IMs are used. Generally, for a scalar IM, the analysis of the relative sufficiency can be more interesting than the absolute one as it investigates whether one IM is more sufficient and, therefore, more informative than another. The appropriateness of one IM relative to another was initially assessed by Jalayer and Beck (2006) using information theory (Shannon 1948a, 1948b), more specifically with the concept of entropy (Cover and Thomas 1991), which measures the missing information that is required (on average) to specify the value of the uncertain variable. Based on the application of relative entropy, Jalayer et al. (2012) introduced a quantitative measure of relative sufficiency to define how much information one IM gives relative to another about the considered Engineering Demand Parameter (EDP).

In addition to the work by Jalayer et al. (2012), Ebrahimian et al. (2015) and Ebrahimian and Jalayer (2021) applied the relative sufficiency to measure the suitability of alternative IMs in predicting various demand parameters by providing their preliminary ranking. The IMs and the demand parameters are obtained using large sets of natural ground motion records. Due to the high number of ground motions used, the selection was based only on the geophysical characteristics of the recorded events (i.e., magnitude and epicentral distance). Therefore, the record selection did not naturally satisfy the spectrum compatibility code requirements. Current design codes typically specify a maximum misfit between the individual response spectra (ASCE/SEI 7-16: 2017) or the average spectrum of a suite of ground motions and the code design spectrum (UNI EN 1998-1: 2005; Iervolino et al. 2010a, b). This requirement aims to reduce the dispersion in the elastic response spectra of the selected ground motion records and, consequently, the variability in the seismic demand, allowing the use of a limited number of records and analyses while providing the same reliability (Hancock et al. 2008).

Since the spectrum-compatibility criterion required by the seismic codes is very difficult to satisfy, the ground motion records are sometimes modified using different methods in engineering practice. One of the most common modification methods linearly scales the ground motions to the target Peak Ground Acceleration PGA (i.e. to the PGA of the code design spectrum) or to the spectral acceleration corresponding to the first period of vibration of the structure (Nau and Hall 1984; Shome et al. 1998). Even in this case, however, fulfilling the design code requirements is not easy unless high-scale factors are used, thus significantly modifying the energy content of the motion records (Baker and Cornell 2005; Cantagallo et al. 2015). As an alternative to scaling by a constant factor, the spectral matching method (Carballo and Cornell 2000; Hancock et al. 2006) has gained increasing attention among researchers and practitioners because it minimises the computational cost of selecting ground motion records according to the design codes. This method uses wavelets to modify the accelerograms so that their spectra match the target design spectrum, minimising changes to the other ground motion characteristics.

So far, the scientific literature has focused on establishing whether accelerograms manipulated with spectral matching could introduce significant bias in seismic structural analysis (Hancock et al. 2008; Carballo and Cornell 2000; Bazzurro and Luco 2006; Heo et al. 2011; Huang et al. 2008;  Iervolino et al. 2010a, b; Grant and Diaferia 2013; Reyes et al. 2014) or if this manipulation causes an excessive increase of the energy content of the ground motion records (Romanelli et al. 2023). Several authors found that the spectral-matching method is more accurate and efficient than the other ground motion selection and modification methods (Hancock et al. 2008; Heo et al. 2011; Grant and Diaferia 2013; Reyes et al. 2014). In contrast, others indicate that the method can produce unconservative biased estimates of the average responses (Carballo and Cornell 2000; Bazzurro and Luco 2006; Huang et al. 2021; Iervolino et al. 2010a, b). Currently, there is a lack of work on the suitability of the spectral-matched records in evaluating the IM for PSDAs and fragility curves.

The main scope of the paper is to assess the suitability of different IMs obtained from both natural and spectral-matched records to predict the structural response using the concept of relative sufficiency. The spectral-matched records are widely used in PSDAs and the derivation of fragility curves. For this reason, the analysis of the relative sufficiency of the IMs obtained with this ground motion selection and modification is of increasing interest to the scientific community, as well as the comparison with the results obtained with natural records.

To this purpose, Sects. 2 and 3 present the case-study structures and the setup of the nonlinear numerical models. Three structures are analysed: Structure 1 (Str. 1) to describe the method and obtain the primary results and Structures 2 and 3 (Str. 2 and Str. 3, respectively) to validate the obtained results. In Sect. 4, two ground motion sets of spectrum-compatible records are then selected according to Eurocode 8 (UNI EN 1998-1: 2005), consisting of twenty (unscaled) natural and spectral-matched records, respectively. For each selected record and ground motion component, eight different IMs are then obtained in Sect. 5. Nonlinear Response History Analyses (NRHAs) are used to obtain the Maximum Inter-Story Drift Ratios (MIDRs) for each pair of records. For both sets of ground motion records and each pair of IMs, the relative sufficiency is then calculated to evaluate the IM that is more informative than the other for predicting the MIDR. The results provided in Sect. 6 indicate the impossibility of defining the relative sufficiency for spectral-matching records due to the lack of variability in the IMs and MIDRs using the probabilistic Cloud method in its conventional form. To compute the relative sufficiency of the IMs obtained from the spectral-matched records, Sect. 7 presents an alternative method based on Incremental Dynamic Analyses (IDA) used to enrich the available set of data and, therefore, to improve the response variability, similarly to Zhang et al. (2020). Such an IDA-based procedure is validated by calculating relative sufficiency values of natural ground motion records. In order to validate the obtained results for Str. 1, in Sect. 8 the relative sufficiency of the IMs obtained from a single set of spectral spectral-matched records relative to the MIDRs is calculated for Str. 1, Str. 2 and Str. 3. Finally, the conclusions are drawn, and limitations are provided in Sect. 9.

2 Case study buildings

The case study buildings are existing three- four- and five-storey reinforced concrete (RC) frames designed for gravity loads only using allowable stress design principles according to an old Italian building code (DM 30/05/1974). The building plan layout is rectangular with dimensions of 28.5 × 15.5 m. The three buildings have a constant 3.2 m inter-storey height (Fig. 1). The floor slabs are cast-in-place RC joists with hollow clay bricks between them, topped by a 4 cm thick, lightly reinforced concrete layer. The geometric configuration, mass and stiffness distributions are symmetric with respect to the x and y reference axes, as shown in Fig. 1. There are no internal beams in the y-direction as the load-carrying frames are parallel to the x-direction. The cross-section of all beams is 30 × 60 cm on all floors, while the column sizes decrease along the height of the building. For more details on the case study building, see Cantagallo et al. (2023) and Barbagallo et al. (2023).

Fig. 1
figure 1

Plan layout of Str.1, Str. 2 and Str.3 (a) and 3D model (b) of Str.1 (five story building)

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3 The structural model

The nonlinear analyses are carried out in the computational platform OpenSees (McKenna et al. 2010) using the pre/post-processor STKO (Petracca et al. 2017). The nonlinear elements are modelled using Beam-with-Hinges elements (Scott and Fenves 2006), with a predefined plastic hinge length at the two ends, while the remaining central element is linear elastic. The end hinges are modelled using a fibre-based discretisation (Spacone et al. 1996). The hinge length is equal to the cross-section height for the beams and to the average of the two section sizes for the columns. The inertia of the element central elastic part is reduced to account for cracking (50% EcIg for beams and 80% EcIg for columns, where Ec is the concrete elastic modulus and Ig is the gross section inertia) following the suggestions of the current design codes (DM 17/01/2018; ACI 318-19: 2019). The constitutive laws Concrete01 (Kent and Park 1971) and Steel01 (Mazzoni et al. 2006) are used for the concrete and steel fibres, respectively. For Concrete 01, the mean compressive strength is fpc = 28 MPa, the strain at maximum strength is εc0 = 2.5‰, the strain at crushing strength is εcu = 3.5‰ and the crushing strength is fpcu = 5.6 MPa. These data are used both for the core and cover fibres because the buildings designed according to DM 30/05/1974 have few stirrups with 90° rather than 135° hooks that are thus considered ineffective in providing confinement. For Steel 01, the elastic modulus is Es = 206,000 MPa, the yield strength is fy = 400 MPa and the strain-hardening ratio is b = 0.0049. Floor diaphragm constraints are applied to all floors. This constraint prevents relative (in-plan) displacements between the floor nodes, thus generating fictitious axial forces in the beam elements. To overcome this problem, an axial release was introduced at the end of each floor beam (Barbagallo et al. 2020). The diaphragm is assumed to be fully effective although the lack of beams in one direction may result in a construction system in which not all floor constraints are fully effective.

The gravity loads are computed using the nominal values found in DM 16/01/1996. They are applied statically on the structural model before the ground motion records are applied dynamically at the structure base. The first three structural periods of Str. 1 (five stories) are T1 = 2.12 s, T2 = 1.23 s and T3 = 1.01 s, and the mass participation ratios in the structural directions Ux, Yy, and Rz are m1,Ux = 0, m1,Uy = 78.7%, m1,Rz = 0, m2,Ux = 0, m2,Uy = 0, m2,Rz = 80.6%, m3,Ux = 79.0%, m3,Uy = 0, m3,Rz = 0, respectively, indicating that the first and the third mode are mainly translational and the second one is torsional. A similar behaviour is observed in Str. 2 (four stories) and Str. 3 (three stories). Specifically, the first three vibration periods for Str. 2 are T1 = 1.69 s, T2 = 1.01 s, and T3 = 0.85 s. The corresponding participating masses are m1,Ux = 0%, m1,Uy = 80.2%, m1,Rz = 0%; m2,Ux = 0%, m2,Uy = 0%, m2,Rz = 83.2%; m3,Ux = 81.7%, m3,Uy = 0%, m3,Rz = 0%. Similarly, for Str. 3, the periods are T1 = 1.25 s, T2 = 0.76 s, and T3 = 0.67 s, with participating masses equal to m1,Ux = 0%, m1,Uy = 81.3%, m1,Rz = 0%; m2,Ux = 0%, m2,Uy = 0%, m2,Rz = 84.6%; m3,Ux = 85.1%, m3,Uy = 0%, m3,Rz = 0%.

4 Selection and modification of ground motion records

4.1 Natural ground motion records

Natural ground motion records are selected according to the provisions of Eurocode 8 (UNI EN 1998-1: 2005) and NTC2018 (DM 17/01/2018). The reference site is located on rock soil in L’Aquila, Italy (latitude = 42.350° and longitude = 13.399°). The seismic hazard for a 10% probability of exceedance in 50 years is considered. The intervals of magnitude Mw and epicentral distances R of the earthquakes that contribute the most to the seismic hazard are obtained from the disaggregation of the probabilistic seismic hazard analysis (Stucchi et al. 2011; Spallarossa and Barani 2007). Based on these intervals (5.5 ≤ Mw ≤ 7.6 and 0 ≤ R ≤ 50 km), a large set of 89 pairs of ground motion records is pre-selected from the European Strong-motion Database ESD (Ambraseys et al. 2004) and the Engineering Strong-Motion database ESM (Luzi et al. 2016). As recommended by Beyer and Bommer (2006), for each pre-selected pair of ground motion components, a single response spectrum is obtained by calculating the geometric mean Sa(T) of the two corresponding horizontal spectral components Sax(T) and Say(T). Twenty ground motion records are then extracted according to the spectrum compatibility criterion so that in the 0.2T1–2T1 period range (where T1 is the structure first vibration period of Str. 1), the average spectrum from all the records is larger than 90% of the corresponding target design spectrum. No upper bound is prescribed in Eurocode 8 (UNI EN 1998-1: 2005). In the current study, a 110% upper bound is added to the 90% lower bound. Str. 1 is subjected to ground motion records not altered by the application of scaling factors (Baker and Cornell 2005). This allows the determination of IMs that are effectively representative of the considered seismic inputs and the prediction of an EDP that represents the structural demand corresponding to a real record. Figure 2a and b compare the average spectrum of the selected twenty ground motion records with the normalised target spectrum in normal-normal and log–log plots, respectively. Table 1 reports the twenty records belonging to the spectrum-compatible combination together with their seismological features (Database, Code of the Station, Earthquake Name, Earthquake Date and Time, Moment Magnitude Mw, Epicentral Distance R and Site Class).

Fig. 2
figure 2

Selection of 20 natural records for Str. 1: check of the spectrum-compatibility criterion in normal-normal (a) and log–log (b) plots

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Table 1 Characteristics of the twenty selected ground motion records selected for Str. 1
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4.2 Spectral-matched ground motion records

The twenty natural ground motion record pairs of Sect. 4.1 are matched to the target spectrum with wavelet transforms. The wavelet transform consists of modulating functions, selectively located in time to modify the spectrum of the signal, where and when it is needed, to match the target spectrum (Hancock et al. 2006). The matching of the records is carried out with the software SeismoMatch (Seismosoft 2022) using the wavelets algorithm proposed by Al Atik and Abrahamson (2010) and a spectrum matching range 0.42–4.24 s, corresponding to 0.2T1 − 2T1. Figure 3a and b show the spectrum compatibility of the twenty spectral-matched records in normal-normal and log–log plots. Each plot reports the geometric mean of the response spectra of each record (in grey), the target spectrum (in solid black), the average spectrum (in red), the upper and lower limits (in dashed black) and the spectrum compatibility range (in dotted black).

Fig. 3
figure 3

Selection of 20 spectral-matched records: spectrum-compatibility criterion in normal-normal (a) and log–log (b) plots

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5 Intensity measures

Eight different IMs are considered in this study. Following Mollaioli et al. (2013), these IMs are classified into two groups: (1) non-structure-specific IMs, calculated directly from the ground motion time histories and (2) structure-specific IMs, obtained from response spectra of the ground motion time histories. Non-structure-specific IMs are further divided into (i) acceleration-related IMs and (ii) velocity-related IMs. Structure-specific IMs are divided into those obtained from spectral ordinates at specific periods and those obtained from integrating the response spectra.

The IM values are determined using the software SeismoSignal (Seismosoft 2018). Each selected IM is derived for the horizontal components of the selected ground motion records. For each ground motion record, a single IM is obtained as the geometric mean of the two IMs calculated for the two horizontal components (Beyer and Bommer 2006). In this study, the effect of the vertical component is not considered.

5.1 Non-structure-specific intensity measures

5.1.1 Acceleration-related intensity measures

The most commonly used measure of the amplitude of a particular ground motion is the Peak Ground Acceleration (PGA). The PGA is the largest (absolute) value of acceleration over time a(t).

$${\text{PGA}} = \max \left| {a\left( t \right)} \right|$$
(1)

The Arias Intensity (AI) is the integral of the square of the absolute acceleration time history (Arias 1970). AI is an energy-based parameter that considers the amplitude and duration of the ground motion, but it is unable to capture the frequency characteristics of ground motions (Ghimire et al. 2021) and is not sensitive to long acceleration pulses in the excitation (Sucuoǧlu and Nurtuǧ 1995).

$${\text{AI}} = \frac{\pi }{2g}\int_{0}^{{t_{\max } }} {\left| {a\left( t \right)} \right|^{2} dt}$$
(2)

The Cumulative Absolute Velocity (CAV) is the area under the absolute accelerogram (EPRI 1988):

$${\text{CAV}} = \int_{0}^{{t_{\max } }} {\left| {a\left( t \right)} \right|dt}$$
(3)

5.1.2 Velocity-related intensity measures

The Specific Energy Density (SED) is the integral of the square of the absolute velocity time history v(t). It measures the overall energy of the record (the larger the SED, the more significant the energy and expected damage).

$${\text{SED}} = \int_{0}^{{t_{\max } }} {\left| {v\left( t \right)} \right|^{2} dt}$$
(4)

5.2 Structure–specific intensity measures

5.2.1 Spectral intensity measures

Two different spectral intensity measures are used in this work: the spectral acceleration corresponding to the first vibration period of the structure Sa(T1) (Shome et al. 1998) and the spectral acceleration corresponding to the structural period T*, Sa(T*) (Cantagallo et al. 2012). T* is the period of the initial branch of the bilinear capacity curve obtained from the nonlinear static (pushover) analysis, according to Eurocode 8 (UNI EN 1998-1: 2005: Annex B) and accounts for the elongation of the fundamental period due to nonlinear effects. Cantagallo et al. (2012) reported that Sa(T*) is well correlated with the deformation demand and produces the lowest variability in structural demand among several intensity measures. T* depends on the distribution of lateral loads; in this study, T* is computed for an applied triangular load pattern.

The case study structure has two fundamental periods of vibration: T1x in the x- and T1y in the y-direction. Given the spectral accelerations Sa(T1x) and Sa(T1y) for the two horizontal components, Sa(T1) is the geometric mean of Sa(T1x) and Sa(T1y) (Beyer and Bommer 2006); thus a single spectral acceleration value is associated to each record. The same procedure is followed for Sa(T*). T*x and T*y are obtained from nonlinear static analyses with loads applied in the x and y structural directions, respectively.

5.2.2 Integral intensity measures

Von Thun et al. (1988) introduced the Acceleration Spectrum Intensity (ASI), defined as the area under the pseudo-acceleration response spectrum in the 0.1–0.5 s period range.

$${\text{ASI}} = \int_{0.1}^{0.5} {S_{a} \left( {\xi = 0.05,T} \right)dT}$$
(5)

Since many structures have fundamental periods between 0.1 and 2.5 s, the ordinates of the pseudo velocity spectrum in this period range should provide an indication of the potential response of these structures. For this reason, Housner (1952) defines the Hausner Intensity HI as the area under the pseudo-velocity spectrum Sv in the 0.1–2.5 s period range.

$${\text{HI}} = \int_{0.1}^{2.5} {S_{v} \left( {\xi = 0.05,T} \right)dT}$$
(6)

6 Nonlinear response history analyses

NRHAs are carried out by applying the selected combinations of ground motion records to the nonlinear structural model. A Rayleigh damping model (Rayleigh 1945) is used with 2% damping at the first and third vibration frequencies of the model before application of the ground motions.

The NRHAs results are reported and discussed here using the Maximum Inter-storey Drift Ratio (MIDR) as the Engineering Demand Parameter (EDP). For each record, the inter-storey drifts IDRx(t) and IDRy(t) are computed from the centres of mass of two subsequent floors in the structural directions X and Y, respectively. The IDR at an instant t is then computed as the SRSS combination of the inter-storey drifts IDRx(t) and IDRx(t): \({\text{IDR}} \left( t \right) = \sqrt {{\text{IDR}}_{\text{X}} \left( t \right)^{2} + {\text{IDR}}_{Y} \left( t \right)^{2} }\). MIDR is then calculated as the maximum IDR over each record duration; that is MIDR = max|IDR(t)|.

A single MIDR is finally obtained for each story and ground motion record as the maximum MIDR of the five storeys of the case study building.

6.1 Correlation between ground motion intensity measures and maximum inter-storey drift ratio

Figure 4 shows the correlations between the IMs calculated from the two selected ground motion sets and the corresponding MIDRs using the Cloud method (Jalayer et al. 2015). Each plot reports the results for the twenty natural ground motion records (yellow points) and the twenty spectral matched records (cyan squares) in the log–log format. The continuous black lines represent the regression lines fitted through the data of the natural ground motion records; the dashed lines show the same values plus and minus the lognormal standard deviation. As expected, the cyan points of the spectral-matched records are less scattered and are close to each other for all IMs. The matching process clearly reduces the scatter of the seismic demand and does not allow for estimating a statistically significant relationship between IM and EDP.

Fig. 4
figure 4

Correlations between IMs and MIDRs for natural and spectral matched records and linear fit of natural ground motion records

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The linear regressions of the natural records data (Fig. 4) indicate that HI, Sa(T1) and Sa(T*) have the lowest values of standard deviation (and the lowest variability of the data). The dashed lines (the ± logarithmic standard deviation interval) are very close to the continuous lines. This indicates that HI, Sa(T1) and Sa(T*) are the most efficient IMs in correlating with the selected EDP for natural ground motion records. These results are in line with those obtained by Akkar and Özen (2005), Riddell (2007), Yakut and Yilmaz (2008), Cantagallo et al. (2012), Jayaram et al. (2010), and Mollaioli et al. (2011).

6.2 Relative sufficiency of the ground motion intensity measures

For a given EDP θ, if θmax = max|θ (t)|, for a set of acceleration time histories \(\underline {\ddot{x}}_{g}\), the relative sufficiency of IM1 relative to IM2 is computed according to the formulation by Jalayer et al. (2012) as:

$$\begin{aligned} I\left( {\theta_{\max } |{\text{IM}}_{2} |{\text{IM}}_{1} } \right) & = E\left[ {D\left( {\theta_{\max } |{\text{IM}}_{1} |\underline {\ddot{x}}_{g} } \right) - D\left( {\theta_{\max } |{\text{IM}}_{2} |\underline {\ddot{x}}_{g} } \right)} \right] \\ & = \int {\log_{2} \frac{{p\left( {\theta_{\max } |{\text{IM}}_{2} |\underline {\ddot{x}}_{g} } \right)}}{{p\left( {\theta_{\max } |{\text{IM}}_{1} |\underline {\ddot{x}}_{g} } \right)}} \cdot p\left( {\underline {\ddot{x}}_{g} } \right)} \cdot d\left( {\underline {\ddot{x}}_{g} } \right) \\ \end{aligned}$$
(7)

where \(D\left( {\theta_{\max } |{\text{IM}}_{i} |\underline {\ddot{x}}_{g} } \right)\) is the relative entropy, equal to \(\int {p\left( {\theta_{\max } |\underline {\ddot{x}}_{g} } \right)\log_{2} \frac{{p\left( {\theta_{\max } |\underline {\ddot{x}}_{g} } \right)}}{{p\left[ {\theta_{\max } |{\text{IM}} \left( {\underline {\ddot{x}}_{g} } \right)} \right]}} \, } d\theta_{\max }\).

The relative sufficiency I(θmax|IM2|IM1) can be interpreted as a measure of how much more information, on average, is gained about the uncertain structural response parameter θmax by knowing IM2 instead of IM1.

  • If the relative sufficiency measure is zero, the two IMs provide the same amount of information about θmax on average. In other words, they are equally sufficient.

  • If the relative sufficiency measure is positive, on average, IM2 provides more information than IM1 about θmax, so IM2 is more sufficient than IM1. Similarly, if the relative sufficiency measure is negative, IM2 provides, on average, less information than IM1, so IM2 is less sufficient than IM1.

Tables 2 and 3 show the values of relative sufficiency calculated for Str. 1 for each pair of IMs obtained for the natural and the spectral-matched records sets, respectively. The left column (grey background) indicates the IM2, and the top header row (grey background) indicates the IM1; thus, positive cell values (shown by the green background) suggest that the IM2 of the given cell is more sufficient (and therefore gives more information) than the corresponding IM1). For example, the value of 0.2061 (fourth row, second column) in Table 2 indicates that SED provides more information about the MIDR than PGA. Conversely, negative values (red background) indicate that the IM is less sufficient than the other (e.g., the value − 0.2976 Table 2 indicates that ASI provides less information than SED about the MIDR). The quantities shown in Tables 2 and 3, on the other hand, reveal the measure of relative sufficiency by quantitatively indicating how much one IM provides more (or less) information than the other. For example, the values of the relative sufficiency of HI relative to AI and SED in Table 2 are 0.6694 and 0.3612, indicating that AI should contain more information than SED to represent the EDP uncertainty (i.e. the sufficiency of HI relative to AI is greater than the sufficiency of HI relative to SED).

Table 2 Relative sufficiency measures of Str. 1 for IMs obtained from natural ground motion records
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Table 3 Relative sufficiency measuresof Str. 1 for IMs obtained from spectral-matched ground motion records
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Figure 5a and b summarise the numerical information in Tables 2 and 3, respectively. The numerical positive and negative values are mapped into a coloured grid with different red and green shades. The black cells correspond to relative sufficiency values equal to or near zero. Table 2 and Fig. 5a indicate that Sa(T1), Sa(T*) and HI are the most suitable intensity measures for natural, unscaled ground-motion records. The relative sufficiency values shown in Table 3 and Fig. 5b are all close to zero, leading to the impossibility of evaluating the most suitable IM to use in PSDM for spectral-matched records. The spectral-matching process significantly reduces the variability of both IMs and EDPs, leading to non-significant statistical linear regression. The spectra of the matched time histories are made similar to each other by the wavelet transforms. Consequently, the IMs and the corresponding MIDRs have minimum differences, and the relative sufficiency value becomes close to zero.

Fig. 5
figure 5

Relative sufficiency of Str. 1 for IMs obtained from natural (a) and spectral-matched (b) ground motion records

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7 Incremental dynamic analyses

The relative sufficiency of the spectral-matched records is evaluated in this section using a version of IDA (Vamvatsikos and Cornell 2002) aimed at increasing the variability of the samples, similar to Zhang et al. (2020). In a conventional IDA, a set of ground motions compatible with a design scenario is scaled to multiple intensity levels, and the structure response is evaluated at each intensity level. In this study, the set of twenty spectral-matched records obtained from Str. 1 is scaled so that the geometric mean of the spectral components has Sa(T*) varying from 0.1 to 4.0 m/s2, with a step of 0.1 m/s2. In general, the use of IDAs is conditioned by the use of significantly high scale factors that can generate unrealistic time series. In this work, the scaling process of the spectral-matched records is however an approximation due to the need to calculate the relative sufficiency of this type of ground motion records. IDA is used only as a scaling approach to restoring the ground motion variability of spectrally matched records, and it is not finalised to the definition of fragility curves. IDAs are also applied using the set of natural accelerograms to complete the work and validate the procedure.

The eight IMs are computed for each of the 400 × 2 scaled records, and the selected EDP is calculated from the NTHA. Figures 6 and 7 show the log–log plots of the correlations between the MIDRs and the IMs for natural and spectral-matched records, respectively. For low IMs (and low-scale factors), the MIDRs correlate well with each considered IM. Conversely, very high and irrealistic MIDRs are obtained for high-scale factors, indicating the non-convergence of the NTHAs. To solve the non-convergence issue, only the NTHAs carried out with accelerograms with Sa(T*) lower or equal to 2.0 m/s2 are considered for the evaluation of the relative sufficiency.

Fig. 6
figure 6

Natural ground motion records–correlation between the MIDRs computed from IDAs carried out on Str. 1 and the corresponding IMs

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Fig. 7
figure 7

Spectral-matched ground motion records–correlation between the MIDRs computed from IDAs carried out on Str. 1 and the corresponding IMs

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Figure 8a and b show the relative sufficiency plots obtained from the IDAs for natural and spectral-matched ground motion records, respectively. As for Fig. 5, red cells correspond to negative relative sufficiency values, green cells to positive values and black cells to null values. The results obtained for natural ground motion records indicate that, coherently with the results obtained by applying a single set of unscaled accelerograms, the IM with the highest relative sufficiency values is Sa(T1). The other two suitable IMs are ASI and Sa(T*). However, in this case, HI does not have high relative sufficiency values and cannot be considered a sufficient predictive IM. HI is, in fact, defined as the area under the pseudo-velocity spectrum in the range of periods 0.1–2.5 s. The first period of vibration T1 of the analysed structure is equal to 2.12 s; for ground motion records characterised by high intensities and scale factors as those used in the IDAs, there is a significative elongation of T1 and consequently, because of the range of periods considered HI loses its validity.

Fig. 8
figure 8

Relative sufficiency of Str. 1 for IMs obtained from IDAs carried out with natural (a) and spectral-matched (b) ground motion records

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The relative sufficiency values for spectral-matched records confirm, in general, that the most suitable IMs are Sa(T1), ASI and Sa(T*). Similar to natural records, the IM with the highest relative sufficiency values is Sa(T*).

8 Validation of the results

To validate the results obtained for Str. 1 and extend the obtained results to other types of structures, the Cloud analysis was carried out on three different types of structures (Str. 1, Str. 2, and Str. 3), using the same set of spectral-matched records. By using the same seismic input, it is possible to observe the variability in the results based only on the type of structure. Consistent with the procedure applied in previous sections, the set of spectral-matched records is derived using the spectrum-compatibility criterion specified in Eurocode 8 (UNI EN 1998-1: 2005). However, in this case, a spectrum matching range of 0.2T1,min–2T1,max is used, where T1,min and T1,max are the minimum and maximum vibration periods obtained from the three considered structures, respectively. Figure 9a–c show the relative sufficiency measures for IMs and MIDRs calculated for Str. 1, Str. 2 and Str. 3, respectively. In line with with the results shown in Fig. 5b, the results obtained for the three analysed structures confirm that IMs and MIDRs obtained from spectral-matched records have minimum differences, and the relative sufficiency values are therefore close to zero. This shows that, regardless of the type of structure, when using spectral-matched records it is not possible to define one IM more efficient than another for predicting EDP uncertainty.

Fig. 9
figure 9

Relative sufficiency of Str. 1 (a), Str. 2 (b) and Str. 3 (c) for IMs obtained from the same set of spectral-matched ground motion records

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9 Summary and conclusions

This paper applies the concept of relative sufficiency to calculate the most suitable IM for PSDAs when natural and spectral-matched ground motion records are employed. For this purpose, three reinforced concrete building designed for gravity loads following an old 1974 Italian building were selected as a case studies. The three buildings have the same plan layout but different heights of five, four and three stories, respectively. In order to obtain the main results, the five-story building was analysed with NTHAs using both natural and spectral-matched ground motion records. Both sets of records consist of 20 pairs of horizontal components. Eight different IMs were computed for each record pair. For each NTHA, the value of a single EDP (the MIDR) was calculated to assess the relative sufficiency of the eight IMs. For spectral-matched ground motion records, the relative sufficiency values are almost zero due to the wavelet transforms that generate a lack of variability in seismic input and consequently demand. To overcome this problem, IDAs are applied to the case-study building to restore this variability using scaled spectral-matching records and compute the corresponding relative sufficiency values. The IDAs are then used to evaluate the relative sufficiency of the selected IMs for both natural and scaled ground motion records. The obtained results are finally validated and extended to different types of structures calculating the relative sufficiency measures of Str. 1, Str. 2 and Str. 3 using the same set of spectral-matched records.

The main outcomes of the study are summarised hereafter:

  1. (a)

    For unscaled natural ground motion records, the most suitable IMs to use for a PSDA or, in general, for predicting the seismic demand are Sa(T1), Sa(T*) and HI;

  2. (b)

    When spectral-matched ground motion records are used, the (IM, EDP) points are very similar to each other. The wavelet transform matches each spectrum to the target one, significantly reducing the variability of IMs and EDPs. In this case, the relative sufficiency values are almost insignificant, and the estimation of the most suitable IM for spectral-matched time histories is not possible;

  3. (c)

    For spectral-matched records, the relative sufficiency values of the IMs are then computed using IDAs. In this case, the scaling process guarantees the variability of the ground motion records and the possibility of applying the relative sufficiency concept. For spectral-matched records, the most sufficient and informative IMs to use in PSDA are Sa(T1), Sa(T*) and ASI;

  4. (d)

    IDAs from natural ground motion records confirm that Sa(T1) is the most sufficient and informative IM, validating the results obtained for the set of matched records. Other suitable IMs for a PSDA are Sa(T*) and ASI.

  5. (e)

    The comparison of the relative sufficiency calculated for different types of structures and using the same set of spectral-matched records confirms that, regardless of the type of building, the spectral-matched records drastically reduce the variability of IMs and EDPs, making difficult to define the most appropriate IM to use in PSDAs.

Future research is required to evaluate the effects of the lack of variability in spectral-matched records, which could further influence PSDA and PBEE applications. As for example, other structures with different plan irregularity levels could be considered. Also, nonlinear IMs and IMs calculated on mainshock-aftershock sequences should be considered (Iervolino et al. 2012; Di Sarno and Pugliese 2021), together with the formulation of a more detailed approach for using IDAs. Finally, future studies could specifically explore how the epicentral distance affects the identification of efficient IMs, thereby enhancing the understanding of how the nature of ground motion records impacts PSDAs.