A wavelet-based approach for truncating pulse-like records

The paper discusses the fast seismic performance assessment of structures using truncated pulse-like ground motion records. It is shown that it is possible to truncate a pulse-like signal using a novel wavelet-based definition that identifies the duration of the predominant velocity pulse. The truncated time-history is still able to efficiently reproduce the increased seismic demand that near-field records typically produce. Substituting the original ground motion with the truncated signal, can significantly accelerate structural analysis and design. The truncated signal is the part of the original accelerogram that coincides with the duration of the predominant pulse, which is identified using a wavelet-based procedure, previously proposed by the authors. Elastic and inelastic response spectra and nonlinear time-history analyses for SDOF (single-degree-of-freedom) systems are first studied. Subsequently a nine-storey steel frame is examined in order to demonstrate the performance of the proposed approach on a multiple-degree-of-freedom system. The proposed approach is found very efficient for pulse-like ground motions, while it is also sufficient for many records that are not characterized as such.


Introduction
Replacing an acceleration time-history with an equivalent, "truncated", signal has many benefits in terms of accelerating the seismic performance assessment and also for understanding structural response. Although this is a very appealing approach, experience has shown that there is no silver bullet to the problem. It is practically impossible to have a record truncation algorithm that is efficient and accurate for all ground motions and all structural systems possible. However, if the problem is narrowed down to the case of pulse-like ground motions, it is possible to achieve an efficient truncation using a novel wavelet-based definition for the record effective duration. The effective duration is calculated first fitting a wavelet on the ground motion and then truncating the record to the time interval that corresponds to the fitted wavelet.
The problem of record truncation is paired with the definition of "effective" record duration. Once the record effective duration is known, the signal can be truncated to the corresponding time interval. However, existing duration definitions such as "bracketed" and "uniform" duration (e.g. Bolt 1973;Bommer and Martinez-Pereira 1996;2000, Riddell 2007Taflampas et al. 2008), do not permit an efficient signal truncation that can substitute the original signal for time-history simulations. This is due to the fact that the truncated signal does not have zero, or close to zero, acceleration values at their beginning and end. This problem cannot be easily fixed with baseline correction algorithms. Furthermore, the intervals defined using those duration definitions omit important information of the original signal.
One of the early efforts for record truncation was that of Srivastav and Nau (1988)who studied the influence of truncated earthquake records on the response of long-period structures and recommended to truncate the earthquake record at a small value of acceleration in order to reduce the error. Most efforts are based on Arias intensity. For example, Jin et al. (2020) proposed the use of Arias intensity in order to study arch dams, while artificial intelligence approaches have been also proposed, e.g., Khaloo et al. (2016).
The scope of this study is to adopt the wavelet-based duration definition proposed by Repapis et al. (2020) in order to truncate the acceleration time-history and obtain a simpler and shorter signal. The paper shows that this practice can efficiently accelerate the simulation time with minor loss of accuracy. The efficiency of the proposed truncation approach depends on the pulse content of the record, which can be quantified with the aid of a pulse index. The approach proposed is quite efficient in the case of pulse-like records, such as those recorded in the case of near-field ground motions with forward directivity. These signals are characterised by strong, coherent, long period pulses that are found mainly in the strike normal direction. The effect of significant velocity pulse on the structural response, has been highlighted by several studies, i.e. Bertero and Mahin (1978), Chopra and Chintanapakdee (2001), Spyrakos et al. (2008). Figure 1 shows two characteristic pulse-like ground motion records and the corresponding significant pulses. The pulses have been identified using the methodology proposed by Mimoglou et al. (2014), i.e., appropriately fitting a wavelet on the signal. The plot on the right shows the cumulative energy flux and the time limits that correspond to the proposed "wavelet-based duration" (black dashed lines). The plots also show the 5% and the 95% of the record energy flux (dashed grey lines), which was proposed by Trifunac and Bray (1975) as a measure of the record "significant duration". Clearly, the two definitions of record duration differ considerably (Fig. 1a).

Methodology
In order to calculate the record duration, we first extract the predominant pulse. According to Mimoglou et al. (2014), the extraction is based on optimally fitting the Mavroeidis and Papageorgiou wavelet, Mavroeidis and Papageorgiou (2003), on the signal. This is a versatile wavelet, suitable to represent different duration levels, since it includes a parameter that is explicitly associated with the number of pulse cycles. A further advantage of this wavelet is that it is defined as the product of a sinusoidal periodic function and a bellshaped envelope.
The Mavroeidis and Papageorgiou wavelet depends on four parameters that control the frequency (T p ), the amplitude (A), the number of cycles (γ) and the polarity (φ) of the signal.
The frequency is obtained as the period value that the product of velocity and displacement spectra becomes maximum. An exhaustive search algorithm is then adopted in order to identify the other three parameters; the search is narrowed to two parameters since the amplitude and the number of cycles are related by the expression (Taflampas et al. 2008): where CAD is the cumulative absolute displacement, used in order to define the proposed duration of the strong ground motion, equal to the time integral of the absolute value of ground velocity, v g (t). The wavelet that has the best cross-correlation value with the original signal is considered the most suitable model of the record predominant pulse. Once the wavelet is fitted on the ground motion, e.g., see Fig. 1a, the wavelet-based effective duration of the signal is defined by the time boundaries of the fitted wavelet.
The truncated signal is the part of the original record contained in these time boundaries of the wavelet that represents the predominant pulse ( Fig. 1a black vertical lines). Therefore, the truncated signal contains all the information, including the high frequency information, of the original ground motion. Moreover, according to Fig. 1b, the ends of the proposed duration definition appear at points where the graph of the energy flux shows a horizontal step with zero first gradient. Therefore, there is no significant baseline offset at the beginning and the end of the truncated duration. This is not the case with other duration definitions, e.g. the "significant duration" definition that uses the arbitrary limits of 5-95% of the total energy flux in order to truncate the ground motion. Thanks to the proposed wavelet fitting, the time boundaries identified, ensure that the truncated signal starts and ends at an acceleration value close to zeros, similarly to the case of recorded acceleration time-histories. Therefore, the limits of the envelope can be considered as boundary tapers that attenuate the harmonic function and smooth the baseline of the cut-off instants. This can be better understood looking at Fig. 2, where the acceleration and velocity of six truncated signal are shown. The time-histories approach smoothly the zero acceleration and velocity line which allows to use them for structural response history simulations without any further processing.
Forty-eight (48) pulse-like records (see next section), obtained from the PEER-NGA2-West database, have been adopted throughout this work, unless otherwise specified. Figure 3 shows the proposed truncated duration definition on the energy flux diagrams (Husid 1969;Arias 1970;Sarma 1971) for twelve of the examined pulse-type records. The records shown correspond to ground motions of increasing pulse period, T p . The black line corresponds to the wavelet-based truncated record, shifted in time so that the pulse of the original and of the truncated records coincide. It is seen that in all cases, the proposed duration coincides with the abrupt energy flux release, associated with the near-field effects. Most importantly, for the majority of the examined excitations, the record duration has been significantly reduced, on average by 80%. Figure 4 compares the constant ductility displacement spectra of the ground motions of Fig. 3, using elastoplastic single-degree-of-freedom (SDOF) oscillators with a constant ductility factor equal to 6. Both original and truncated signals present maximum displacement demand at period values close to the predominant period of the pulse. Clearly, high accuracy has been obtained using the truncated records instead of the original acceleration histories. Furthermore, Fig. 5 shows for the set of 48 ground motions, the mean constant ductility displacement spectra for ductility factors equal to μ = 2 and μ = 6 ( Fig. 5a). Figure 5b shows also the corresponding coefficient of variation (CoV) as function of the period of the SDOF system. As expected, the agreement on the mean demand is very good for both ductility levels values considered, i.e. perfect agreement is seen for the two inelastic examined cases (μ = 2, μ = 6) case, while, most importantly, the dispersion is practically not affected when the truncated signal ( Fig. 5b) is adopted instead showing that the proposed wavelet-based truncation approach is robust for elastoplastic SDOF systems.
The agreement between the original and the wavelet-based truncated records is further studied using the constant ductility, μ, acceleration spectra of Fig. 6. In acceleration spectra the maximum demand occurs at period values slightly less than the 0.5T p , as opposed to displacement spectra (Fig. 4) where the maximum occurs at period values close to T p. Figure 6 shows the 5%-damped constant ductility acceleration spectra for four ground motions, with T p = 0.7, 1.23, 2.17 and 4.88 s, respectively, and ductility factors equal to μ = 2 and μ = 6. As in the case of displacement spectra, the elastoplastic constant ductility spectra, of the original and the truncated ground motion perfectly coincide regardless of the period value and the level of inelastic demand.

Ground motion records
A set of 48 pulse-like ground motion records, included in the PEER-NGA2-West database, PEER (2013), have been selected for our work. The properties of the ground motions are summarized in the Appendix. The excitations were recorded on different soil types and distances from the rupture plane and have different values of predominant pulse periods. Moreover, the selected ground motion sample contains records with strong directivity, mainly in the fault-normal direction (e.g., Loma Prieta), or records where both the fault-normal and fault-parallel components show prominent directivity effects (e.g., Erzincan-Turkey).

Correlation with pulse index
The proposed wavelet-based truncation, in principle, applies to pulse-like ground motions. When pulse-type records are considered, the seismic energy is suddenly released from the fault producing few strong ground motion cycles. In this case the seismic demand obtained with the truncated and the original ground motions is expected to be close. However, often the proposed approach gives sufficiently accurate results also for records that have been a priori classified as non pulse-like. It is, therefore, very useful to have a metric that can be used to determine, prior any analysis, if the proposed truncation can be used to substitute the original signal. An obvious metric suitable for this purpose is a "pulse index" such as the one proposed by Baker (2007) and/or Kardoutsou et al. (2017). Kardoutsou et al. (2017), proposed as a pulse index (PI) the cross-correlation of the original signal and the extracted wavelet. This work was also based on fitting the Mavroeidis and Papageorgiou wavelet, as discussed in Mimoglou et al. (2014). The authors recommend that records with PI less than 0.55 are non-pulse like, while when PI > 0.65 the record is definitely pulse-like. Records with intermediate values, i.e., 0.55 < PI < 0.65, are characterized as ambiguous. Figure 7 compares the ratio of peak displacement demand of the original and the wavelet-based truncated time-history versus the Pulse Index (PI). The comparison is based on the record set of FEMA P-695 (2009). A different and wellknown record database is adopted for this comparison in order to use ground motions that are completely different from those used in their previous studies of the authors (Kardoutsou et al. 2017;Mimoglou et al. 2014). Moreover, the ground motion set of FEMA P-695 (2009) consists of 44 far-field records, 28 near-field records characterized as "pulsetype" and 26 near-field records characterized as "non pulse-type". The far-field set includes earthquakes of large magnitude (M w > 6.5), recorded on soil types C and D according to the NEHRP classification. The pulse-type set contains records, with varying pulse periods, corresponding to events of magnitude between 6.5 and 7.6. These records present strong directivity effects in the fault-normal/and or the fault parallel direction.
The ratio of displacements shown in Fig. 7 is computed for elastoplastic SDOF systems with natural periods T 1 equal to 0.2 and 2 s and constant R y factors equal to 1 and 4. R y is the strength reduction factor, equal to the ratio of the ζ%-damped spectral acceleration demand Sa (T 1 ,ζ) over the corresponding yield acceleration. The horizontal axis in Fig. 7 shows the PI value of every record and the vertical axis shows the ratio of SDOF displacement demand of the truncated over the original signal u truncated /u original . For the "pulse-like" ground motions, it is seen that all records have a PI value above 0.60, while nearly perfect agreement between the truncated and the original record has been achieved, i.e. almost all points lie on a horizontal line with u truncated /u original equal to one. However, for "non-pulse like" records, the index clearly receives small values, as low as PI < 0.3, while the majority of the records have PIs below 0.65; thus for non-pulse records there is loss of accuracy that may reach 50%. Nevertheless, even in this case, there are simulations where the prediction of the truncated record is close to that of the original signal. Finally, for the "far-field" ground motions the PI index varies between 0.3 and 0.9, with a median PI value about equal to 0.6. Relatively good accuracy is also achieved for this set, despite the large inelastic demand due to the R y = 4 assumption.
Overall, for the T 1 = 0.2 s and T 1 = 2 s cases, the predictions are considered better in the former case. Another point of interest is that there seems to be a bias towards underestimating the demand when the truncated motion is adopted. This is expected since the truncated record is a "simplified" version of the original record and hence contains less energy. However, there are some simulations where the demand is, slightly, overestimated. This unexpected behaviour can be attributed to a distortion in the baseline which in turn affects the displacements estimate.
Based on the discussion above, a PI index, e.g., that of Kardoutsou et al. (2017), can be used to identify if the record is pulse-like and hence if the proposed wavelet-based truncation is applicable. Large PI values, above 0.6, indicate that the substitution of the original ground motion will be successful, while smaller values imply that attention is required when truncating the signal.

Degrading single-degree-of-freedom (SDOF) oscillators
Three degrading single-degree-of-freedom (SDOF) oscillators are studied in order to further investigate the accuracy and efficiency of the proposed truncation approach. The oscillators have in-cycle degradation (i.e., degradation of the monotonic envelope) as shown in Fig. 8. Compared to the elastoplastic systems already examined, these are simple SDOF systems but more realistic since real-world systems are not elastoplastic with infinite capacity. The importance of studying degrading systems in the case of pulse-like ground motions is further discussed in Dimakopoulou et al. (2013).
The oscillator of Fig. 8a is a system considered representative of typical non-ductile systems commonly found in Southern Europe; it will be used as our reference oscillator. Using the parameterization of Fig. 8a, the properties of the backbone oscillator are a h = 5%, a c = -50%, μ c = 2 and r = 20%. A "brittle" and a "ductile" oscillator are also considered. The backbone curves of all three oscillators are shown in Fig. 8b and their properties are summarized in Table 1. The cyclic response of all examined systems is based on the "hysteretic material" available in the material library of the OpenSees software (McKenna and Fenves 2001). The force and displacement pinching parameters of the model are set equal to a moderate value.
In order to investigate the response of the SDOF systems, plots of the oscillator period versus the inelastic displacement ratio C R are shown. The ratio C R is calculated as C R = u m /u el = μ/R y . In C R , the subscript "R" denotes that C R has been calculated keeping constant the strength reduction factor R y . For a constant R y value, C R is linearly related to   Dimakopoulou et al. (2013), the plots have been normalized as function of the predominant pulse period, T p . This is better explained in Fig. 9; in Fig. 9a the curves are plotted against the period T 1 as opposed to Fig. 9b where the period axis has been normalized with the pulse period. Plotting T 1 /T p results to shifting the spectral values towards the vicinity of T p . Therefore, normalizing with T p , reduces the dispersion, which indicates that the ratio T 1 /T p is better correlated with the demand. Figures 10, 11 and 12 compare the C R demand as function of the ratio T 1 /T p for systems with strength reduction factor equal to R y = 2 and 6. The figures show the reference, the brittle and the ductile oscillator, respectively. The figures on the left (Figs. 10a, 11a  and 12a) show the median C R demand, while on those on the right , show the dispersion, calculated as = (Prc 84% − Prc 16% )∕2 . For the reference oscillator (Fig. 10), the inelastic displacement ratio decreases exponentially as the normalized period T 1 /T p increases, while, for both R y values shown, the response predicted by the truncated signal is Fig. 9 Reference oscillator (R y = 6) under original ground motions: C R ratio values versus (a) oscillator period T 1 , (b) oscillator period normalized with the pulse period T 1 /T p Fig. 10 (a) Median C R of the reference SDOF systems subjected to the original ground motion and the corresponding truncated for R y = 2 and R y = 6, (b) dispersion of C R , calculated for R y = 2 and R y = 6 1 3 practically identical to that of the original record. Good accuracy is also observed for the brittle (Fig. 11) and the ductile oscillator (Fig. 12), although the median response is different for each of the oscillators considered. It is also shown that the R y value controls the slope of the descending branch as the median curves approach the horizontal line at C R = 1, which corresponds to the equal displacement rule, Newmark and Hall (1982). Close agreement is also seen in the dispersion plots. This means that despite the record-to-record variability, the truncated signal can adequately substitute the original ground motion throughout the period range and regardless the level of R y and the oscillator properties. Figure 13 shows force-displacement hysteretic plots for the reference oscillator. The plots refer to record NGA #1063 for period values T = 0.2 and 2 s and R y equal to 2. This is a record with strong directivity in the fault-normal direction and a moderate pulse period, T p = 1.11 s. As shown already (e.g., Fig. 10) the maximum demand due to the Fig. 11 (a) Median C R of the brittle oscillator subjected to the original ground motion and the corresponding truncated for R y = 2 and R y = 6, (b) dispersion of C R , calculated for R y = 2 and R y = 6 Fig. 12 (a) Median C R of the ductile oscillator subjected to the original ground motion and the corresponding truncated for R y = 2 and R y = 6, (b) dispersion of C R, calculated for R y = 2 and R y = 6 original and the truncated record is almost the same, and therefore the hysteretic curves of Fig. 13 appear practically identical, but there are still small amplitude cycles, outside the proposed effective duration, that cannot be easily seen.
The area of the hysteretic plots has been calculated in order to measure the total energy dissipated. Therefore, if E trunc is the energy dissipated by the system subjected to the truncated record and E org is energy dissipated by the original ground motion, the loss of energy between the two signals is defined as 1−E trunc /E org . For the hysteretic plots of Fig. 13, the energy loss is 2.8 and 11%, respectively. Table 2 summarizes the energy loss for four fundamental period values and for the reference oscillator for the 48 pulse-type records considered. In an average sense, the loss of energy dissipation between the original and the truncated ground motion does not exceed the 25%. As already discussed above, this difference corresponds to cycles that do not fall within the proposed effective duration and hence they are not responsible for the maximum demand values. They would have been important if other response quantities were of interest, e.g., residual displacements, cumulative damage, etc.

Nine-storey steel frame
The efficiency of the proposed record truncation method is also studied using a testbed multidegree-of-freedom (MDOF) structure. The building adopted is known as LA9 building and is a nine-storey steel moment-resisting frame. The building was designed during the SAC/ FEMA project (FEMA 355C 2000) according to 1997 NEHRP guidelines for a Los Angeles site. It is a building with peripheral frames which are designed to resist the seismic actions. The gravity loads and the mass of the internal gravity-resisting frames are placed on a leaning column, which does not contributes to the lateral stiffness. The frame examined is shown in Fig. 14 and it consists of five bays and a hinge-storey basement. The fundamental period of the frame was found equal to T 1 = 2.35 s and the mass modal participation of the first mode  amounts to 84% of the total mass. Thus, the frame is essentially dominated by the first mode, while higher modes may also contribute to the response. More details about the LA9 building can be found in Gupta andKrawinkler (1999) andFEMA 355C (2000). A centreline model is adopted using the OpenSees platform (McKenna and Fenves, 2001). The model is able to explicitly account for the geometric nonlinearities in the form of P − Δ effects. The columns are assumed linear-elastic, while a quadrilinear model is adopted for the beam-column connections. The backbone of the moment-rotation relationship is based on a model similar to the force-deformation relationship of the degrading SDOF systems, e.g., see Fig. 8. More specifically, the moment-rotation relationship has properties are equal to a h = 10%, a c = −50%, μ c = 3, r = 50%. These values are assumed similar for every beam-column connection. Figure 15 shows a comparison similar to that of Fig. 7, where the ratio of peak interstorey drift is plotted against the pulse indicator (PI) for the three record sets of FEMA P-695. The difference in the maximum peak interstorey drift demand between original and truncated records is small for ground motions with a pulse indicator PI above the 0.65 Fig. 14 The nine-story steel moment-resisting frame threshold. This observation holds even for records that belong to the far-field set. The largest differences are again found for the far-field records (red dots), but even for PI < 0.55 the differences do not seem to exceed on average 18%. Overall, for the building examined, the scatter in the computed drift values is significantly decreased as PI increases, regardless of the classification of the ground motion record. This indicates the efficiency of the proposed wavelet-based approach for truncating a ground motion record and also the potential on the proposed pulse index PI metric to quickly assess using the efficiency of the proposed wavelet-based truncation.
The LA9 steel frame was subjected to the 48 pulse-like records. Figure 16a presents the median peak interstorey drift demand and the corresponding 16% and 84% percentile curves of the interstorey drift demand. Excellent agreement for all the stories has been achieved along the height of the frame. Furthermore, the effect of record-to-record variability on the maximum interstorey drift demand is studied in Fig. 16b. Although minor differences are seen for individual records, for the majority of ground motions the drift estimates of the original and the truncated signal practically coincide. Figure 16b deserves further attention. A different marker colour has been adopted depending on the storey that the maximum interstorey drift occurred. Overall, the seismic response computed for the truncated signals, follows the same damage pattern observed for the Fig. 15 Ratio of the maximum interstorey drift for the 9-storey steel moment frame versus the calculated cross correlation Fig. 16 (a) Profile of the median and the 16 and 84% percentiles drift demand for the 48 pulse-type ground motions; (b) Maximum interstorey drift obtained using the original and the truncated records versus the T 1 /T p ratio original records while for most cases the maximum drift demand occurs at the same level. Since the structure studied is sensitive to the first mode, the larger demand is due to records with T 1 /T p ≤ 1. Furthermore, for these records, the peak drift appears at the middle stories (stories 4, 5 and 6).
Our findings are in agreement with Baker and Cornel (2008) who highlight the significant effect of the pulse period on the seismic behaviour of MDOF systems. They investigated the effect of the pulse on the higher modes of excitation through MDOF structures sensitive to second mode excitation and suggested that the ratio T 1 /T p is indicative of the level at which the peak displacement occurs; short period records excite higher modes, while for records with long period pulses the maximum response is expected at the low stories, indicating that first mode response governs the peak displacements of the building. According to Fig. 16b, for records with short period pulses, T 1 /T p > 2 the higher modes of vibration are excited and the maximum interstorey drift is located at the upper stories (stories 7, 8 and 9). On the other hand, for records with T 1 /T p < 1, the maximum drift demand is observed at lower stories, indicating that the response is first mode dominated. Figure 17 shows the maximum drift demand for the 2nd, the 5th, the 7th storey and the roof drift. The agreement is practically perfect for all stories with the exception of the top storey where some minor errors appear for ground motions with T 1 /T p > 2. The maximum interstorey drifts occur mainly for records with T 1 /T p between 0.6 and 2, and in some cases for T 1 /T p > 2, while for the structure considered, the peak values appear at the middle stories of the building. Figure 18a shows the profiles of the median shear forces along the height of the building and the corresponding 16 and 84% percentiles. Very good agreement is again obtained, since the damage patterns are very close. Some minor differences are observed with respect to the demand in stories 3, 4 and 5. Furthermore, Fig. 18b compares the base shear demand for the original and the truncated signal. It should be pointed out that the maximum values of the base shear are computed in the period range 0.5 < T 1 /T p < 1, while (although not shown) most differences appear at the 1st and 5th storey, as also applies for the interstorey drifts (Fig. 17). Figure 19 compares pairs of the maximum roof drift versus the corresponding base shear for the truncated and for the original ground motions. For comparison purposes, the capacity curve obtained with pushover analysis based on a first-mode lateral load pattern is also provided. For both record types considered, the agreement is sufficient and practically the same ultimate building capacity and elastic stiffness is obtained. Note that, due to the fracturing of the beam-column connections, the pushover capacity curve has a negative stiffness branch after a roof drift value of 0.03; this degradation cannot be captured by the dynamic simulations that always provide the maximum response quantities. Overall, the Pushover capacity curve plotted against pairs of maximum roof drift and the corresponding base shear for the pulse-like records considered response obtained for the majority of pulse-type motions follows the shape of the capacity curve and good agreement is observed for original and truncated signals.
Similar to the hysteretic plots of Figs. 13, 20 shows the roof drift vs the shear force plots for a corner column of the first storey of the building. Two pulse-type ground motions with different pulse periods are considered, i.e., NGA#179-1979 (Imperial Valley/El Centro Array #4) with T p = 4.32 s, and NGA#1009-1994 (Northridge-01/LA-Wadsworth VA Hospital North) with T p = 2.35 s. The reduction in the record duration is 80% for the NGA #179 and 85% for the NGA#1009 record. The loss of energy dissipation measured as 1−E trunc /E org is 0.23 and 0.14, respectively. For both records of Fig. 20, although the original ground motion has a considerably longer duration the maximum force and displacement demand is practically the same. Therefore, although the original record has a larger duration and more hysteretic cycles, in the case of pulse-type ground motions, this does not affect the maximum displacement or force demands and the shape of the hysteretic plots is very close. Figure 21 compares the duration of the truncated signal with that of the original ground motion, proving that the proposed wavelet-based truncation can be used to efficiently accelerate seismic performance assessment studies. Furthermore, Fig. 21b shows the improved CPU run times, which also leads to reduced storage demand of the analyses output. It is therefore evident that the proposed record truncation can reduce considerably the simulation time with a minor loss of accuracy, also in the case of multi-degree-of-freedom systems.

Conclusions
The proposed record truncation approach can be used to considerably simplify the input signal and accelerate the time required for seismic performance assessment, especially in the case of pulse-like ground motions. More specifically, the truncation procedure is efficient for records with a pulse index that exceeds 0.65 which is in agreement with the findings of Kardoutsou et al. (2017). A lower pulse index (PI < 0.65) may also give accurate results, although this is not a priori guaranteed. Furthermore, the truncation does not require baseline correction and thus it can be integrated to give truncated realistic velocity and displacement time-histories. The proposed approach has been validated through the study of different structural systems. Elastoplastic systems are first studied in order derive inelastic displacement and acceleration spectra. Subsequently, three single-degree-offreedom oscillators with in-cycle and cyclic degradation are investigated. The effect of the proposed truncation approach is finally applied on a nine-storey steel building, the wellknow LA9 building. All the examined systems follow a similar damage pattern, for both the original and the truncated pulse-type ground motion. Almost in all cases, the seismic demand, in terms of displacement, drift or shear forces, obtained using the truncated signal will produce close estimates within a fraction of the CPU time required when the original complete record is used.   Table 3 (continued)