Strong-Motion-Duration-Dependent Power Spectral Density Functions Compatible with Design Response Spectra

Abstract

The development of a suitable set of input ground motions is crucial for dynamic time history analyses. The US Nuclear Regulatory Commission (NRC) requires that these motions generate response spectra closely matching the plant’s design spectrum. Additionally, the NRC recommends verifying the motions’ power spectral densities (PSDs) against a target function to ensure sufficient energy across all frequencies. Current NRC guidelines in Standard Review Plan (SRP) provide a general method for creating target PSDs for any design spectrum. However, this method does not explicitly consider the influence of strong motion duration on the relationship between PSD and response spectrum. This article proposes an improved approach that incorporates the expected strong motion duration into the target PSD generation process. The method first constructs a Fourier amplitude spectrum (FAS) compatible with both the design spectrum and the expected strong motion duration. Subsequently, a large set of synthetic motions based on this FAS is used to construct the target PSD function. It is shown that current target PSD functions tabulated in SRP 3.7.1 implicitly infer an expected strong motion duration of approximately 9 s. The proposed method can be used to construct target PSDs tailored to different strong motion durations.

1. Introduction

Dynamic time-history analyses (DTHAs) are widely used throughout the seismic design and assessment process of nuclear facilities. For example, DTHAs are employed to perform seismic wave transmission analyses to obtain free-field ground motion response spectra from the design response spectra defined for generic rock conditions, e.g., [1]. DTHAs are also utilized in seismic structural and soil–structure interaction analyses to assess the structural members’ seismic design demands, e.g., [2,3], generate floor response spectra, e.g., [4], and investigate seismic response beyond design basis shaking, e.g., [5].

The acceleration time series used as input to perform the DTHAs can significantly influence the analyses’ outcomes and therefore constitute a critical step in the process. Standards and guidelines such as ASCE 4-16 [6], ASCE 43-19 [7], and the United States Nuclear Regulatory Commission (US-NRC) Standard Review Plan (SRP) Revision 4 Section 3.7.1 [8] provide guidance and requirements for developing these input motions. The main requirement in these guidelines is that the response spectra of the input motions closely match the site design response spectrum. Nevertheless, existing research suggests that strict matching of ground motion records to the design spectrum does not guarantee a stable estimate of the mean in-structure response spectrum (ISRS, also called floor response spectrum) due to large power deficiencies in some frequency ranges [9,10]. Notice that the development of the ISRS is crucial as it is used as input for designing and/or assessing the seismic fragility of equipment within the structure. Consequently, it has been suggested that an additional verification of the records’ power spectral densities (PSDs) is necessary [11,12]. In fact, earlier versions of the standards included a PSD check to ensure sufficient energy content across the frequency range of interest and prevent power deficiencies in the input records. This PSD requirement has been de-emphasized in recent revisions in lieu of more stringent criteria to ensure a tight match with the design spectrum. A comprehensive review of the evolution of PSD requirements within relevant codes and standards is provided in [10].

ASCE 4-16 and ASCE 43-19 require a PSD check only if the set mean response spectrum exceeds the design spectrum by more than 30% at any frequency within the range of interest. This scenario is rather unlikely due to advancements in spectral matching techniques that enable modifications of historic records to obtain a very tight match with the target design spectrum, e.g., [13,14,15,16,17]. In contrast, NRC SRP 3.7.1 Revision 4 maintains the requirement to demonstrate adequate power distribution in the input motion. This is typically achieved by comparison of the input motion’s PSD with a target PSD.

Over the years, numerous research studies have proposed analytical and numerical methods for generating PSD functions compatible with a response spectrum. Most of these methodologies are largely rooted in random vibration theory (RVT), which assumes ground motion as a stationary stochastic process. This assumption allows the mean response spectrum to be approximated using the product of the standard deviation of oscillator response and its corresponding peak factor, e.g., [18,19,20,21,22,23]. A challenge with these approaches is the diversity of formulations available in the literature for estimating the peak factor (summarized in [24]). Additionally, [25] has outlined compatibility conditions that a response spectrum must satisfy to ensure that a compatible PSD can be obtained based on RVT approaches. The authors of [25] also point out that some response spectra from design codes may be inadmissible given the way they are constructed and defined, an issue that was also raised in [26]. Moreover, PSDs have proven valuable across various earthquake engineering domains, such as ground motion prediction for hazard analysis (e.g., [27,28]), seismic evaluation (e.g., [29]), and damage detection (e.g., [30,31]).

As an alternative to using an RVT-based approach, SRP 3.7.1 suggests a numerical procedure that employs sets of synthetic motions generated from an initial PSD function, and this initial function is iteratively modified until reaching convergence with the target response spectrum [32]. However, this approach does not explicitly consider the impact of strong motion duration on the relation between response spectrum and PSD. This article builds upon that work by proposing improvements to the method, specifically by factoring in the expected duration of strong ground motion when developing the target PSDs.

2. Estimation of Power Spectral Density Functions

Given an acceleration time history a(t_j) sampled at N data points with as sampling time Δt: t_j = jΔt (j = 0, 1, … N − 1), a frequency domain representation of the signal can be found by means of the discrete Fourier transform (Equation (1)):

$$F\left[{\omega }_n\right]=\Delta t{\sum }_{j=0}^{N-1}a\left(t_j\right)e^{-2\pi i\left({\displaystyle \frac{nj}{N}}\right)}$$

(1)

The result are complex Fourier coefficients that provide amplitude and phase information for each circular frequency ω_n, where n = 0, 1, … N/2 correspond to the zero and positive frequencies and n = N/2 + 1, … N − 1 correspond to the negative frequencies. In practice, the Fourier transform is implemented using an efficient fast Fourier transform (FFT) algorithm (e.g., [33]). The one-sided Fourier amplitude spectrum (FAS) is computed by taking the absolute values of the Fourier coefficients that correspond to the zero and positive frequencies (Equation (2)):

$$FAS=\left|F\left[{\omega }_n\right]\right|=\Delta t\left|{\sum }_{j=0}^{N-1}a\left(t_j\right)e^{-2\pi i\left({\displaystyle \frac{nj}{N}}\right)}\right|; n=0, 1, \dots N/2$$

(2)

Parseval’s theorem establishes that the total energy in the signal (E_a) can be estimated either in the time domain or in the frequency domain. For a continuous signal a(t) with Fourier transform F(ω):

$$E_a={\int }_{-\infty }^{\infty }{\left|a\left(t\right)\right|}^{2}dt={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\left|F\left(\omega \right)\right|}^{2}d\omega$$

(3)

When dealing with finite length discrete signals like ground accelerations recorded from seismic events, the limits in the integrals of Equation (3) need to be set. That is, the duration of the motion needs to be defined, as well as the frequency range of practical interest. Based on Equation (3), an estimation of the power in the signal and its distribution over the frequencies that make it up can be obtained by dividing the squared FAS (|F(ω)|²) over the duration of the signal. However, this would be appropriate only if the frequency content of the signal does not vary significantly with time, i.e., the signal can be considered stationary. This is certainly not the case for earthquake records, which are rather non-linear and non-stationary signals. To cope with this, NRC SRP 3.7.1 and ASCE 4-16 estimate the one-sided PSD S_o(ω) using only the Fourier coefficients F(ω;T_D) corresponding to the record time lapse (T_D) where the power is near its maximum and the signal can be considered nearly stationary (Equation (4)):

$$S_o\left(\omega \right)={\displaystyle \frac{2{\left|F\left(\omega ;T_D\right)\right|}^2}{2\pi T_D}}$$

(4)

Quantifying the exact duration of near stationarity (T_D) for earthquake records remains challenging due to their inherent complexity. In practice, the strong motion significant duration SD_5–75, defined as the time interval over which 5–75% of the Arias intensity (AI) is accumulated, is often employed as a proxy. The popularity of SD_5–75 stems from its ease of computation and the observation that, for many records, it aligns with a period of roughly linear energy accumulation [34,35]. For the synthetic motions generated here this presumption works well as indicated by the approximately linear behavior of the cumulative AI throughout the SD_5–75 time window. In the case of actual earthquake records the use of SD_5–75 may not be appropriate when the non-stationary features of the signal are dominant and care must be taken when selecting the seed records for spectral matching [32]. Nevertheless, amongst different strong motion duration measures, SD_5–75 is considered the most appropriate for random vibration theory (RVT)-based predictions of ground motion intensity measures when working with observed data [36] and the one showing the strongest correlation with the FAS/PSA relationship [37].

3. The Relationship between Fourier and Response Spectra

Recall that the objective is to find a PSD function compatible with a given design response spectrum. It is seen in Equation (4) that the PSD can be efficiently estimated based on the FAS corresponding to the nearly stationary part of the record. Therefore, the relationship between PSD and response spectrum would be initially and foremost dictated by the relationship between the FAS and the response spectrum.

In earthquake engineering practice, conversions between response spectrum (typically the 5% pseudo-acceleration response spectrum, PSA) and FAS are often required. For example, when performing site response analyses based on random vibration theory (RVT), the input FAS often needs to be consistent with the design spectrum to comply with code requirements. Nevertheless, the conversion of PSA to FAS presents a significant challenge. The PSA at a specific frequency is not solely influenced by the corresponding FAS amplitude; rather, it reflects the cumulative influence of a broader frequency band within the FAS [37,38]. The conversion can be achieved through iterative schemes based on inverse random vibration theory (IRVT) (e.g., [39,40]). Alternatively, empirical relationships, such as the one presented in Equation (5) [37], can be utilized to approximate the FAS.

$${\displaystyle \frac{FAS}{5\%PSA}}=\left(a_1{S{D}_{5-75}}^{a_2}+a_3\right)F^{\left(b_1{S{D}_{5-75}}^{b_2}+b_3\right)}$$

(5)

In Equation (5), 5%PSA is the pseudo-acceleration response spectrum for 5% damping, F is the frequency (in Hz) at which the ratio is evaluated, a₁ = 0.0512, a₂ = 0.4920, a₃ = 0.1123, b₁ = −0.5869, b₂ = −0.2650, and b₃ = −0.4580. Notice that both approaches, IRVT or empirical equations, require an estimate of ground motion duration, that is, for a given PSA the resulting FAS would vary depending on the expected strong motion duration of the motion.

4. US-NRC SRP 3.7.1 Rev. 4 Methodology

SRP 3.7.1 rev. 4 Appendix B [8] presents an iterative frequency-by-frequency scaling approach to construct target PSD functions consistent with any practical design spectral shape, further details of the methodology are discussed in [32]. The procedure is detailed next, and a flowchart of the algorithm is presented in Figure 1.

• Define the target PSA (PSA_target) and an initial target PSD function. This initial function would be iteratively modified until obtaining a PSD function consistent with the target PSA. In the examples presented the authors used bin average PSDs from the NUREG/CR-6728 time-history database [41] as the initial PSD functions.
• Using the current PSD function and Equation (4), generate a target FAS. Use random numbers (Gaussian noise) to generate a set of synthetic motions that share the target FAS. The authors proposed to generate a set of 10 motions in the 1st iteration, 20 motions in the 2nd iteration, and so on until reaching a set of 100 motions in the 10th iteration. All motions were generated with a sampling frequency of 200 Hz. Here, it is important to notice that the target FAS is obtained using the entire duration of the signals as the instant amplitude is constant (the motions’ amplitude is modulated in the next step).
• Modulate the amplitude of the synthetic motions. The authors used a trapezoidal window with rise time 1.4 s, constant amplitude 10.24 s, and decay time 7 s. This envelope is the same employed in Appendix B of NUREG/CR-5347 [42].
• Calculate the 5% PSA for each of the synthetic time histories in the current set and obtain the set arithmetic average PSA_avg.
• Multiply the PSD function by (PSA_target/PSA_avg)² and go back to step 2 using this adjusted PSD.

Figure 2 shows the results obtained when this procedure is applied to obtain a PSD function compatible with the NUREG/CR-6728 bin representative design response spectra for a moment magnitude M = 7.5 and fault distance R = 150 km event in a rock site in the Central and Eastern United States CEUS (Equation (3) in SRP 3.7.1 Appendix B [8]). The design response spectrum is denoted by the line with circular markers in Figure 2b. The initial PSD function was obtained from the bin average PSDs from the NUREG/CR-6728 time-history database (SRP 3.7.1, Appendix B, Table 3 [8]) and is denoted by the dashed line in Figure 2c.

The results obtained are denoted by the violet lines in Figure 2. From Figure 2a it is seen that the strong motion durations SD_5–75 for all 550 synthetic motions generated during the 10 iterations follow mostly a normal distribution with an average of 9.1 s. Notice that the average response spectrum obtained from the synthetic records constructed from the PSD bin average (i.e., the average response spectrum from the 1st iteration of the iterative procedure, marked in Figure 2b as PSD bin avg. based spectrum) falls below both the target design spectrum and the bin average response spectrum (dashed line in Figure 2b). However, from the same plot can be seen that by the last iteration, the procedure successfully matched the motions’ average response spectrum to the design spectrum. Moreover, the final PSD function (violet line in Figure 2c) is in close agreement with the values tabulated in SRP 3.7.1 Appendix B Table 1 [8]. Notice that the SRP 3.7.1 values may have been smoothed using splines and manually adjusted for inconsistencies at extreme frequencies, while the PSD functions calculated here were only smoothed by taking a moving average over a frequency window width of ±20% the subject frequency, as required for PSD checks of input motions in SRP 3.7.1.

5. Issues with Current US-NRC SRP 3.7.1 Rev. 4 Methodology

When studying and implementing the algorithm described in SRP 3.7.1 two main issues became apparent. One is that the effect of strong motion duration is not directly accounted for, the target PSDs for all different settings of magnitude and distance were generated using the same total duration for the motions. However, it has been shown that duration plays a key role in the relation between PSD, FAS, and PSA (Equations (3)–(5)). The second issue is that the target FAS is constructed using the whole duration of the unmodulated signal. However, the motions’ response spectra, and consequently the correction factors used to update the FAS, are estimated based on the modulated signal. The effect of each of these issues on the resultant target PSD is discussed next.

5.1. Duration

To explore the influence of duration on the PSD function predicted by SRP 3.7.1, two additional scenarios were considered. These scenarios employed motions where the durations of the original constant amplitude portion of the trapezoidal window used in SRP 3.7.1 were halved and doubled. The rise and decay times were held constant.

The results for these additional durations are presented by the blue and salmon lines in Figure 2. The SD_5–75 average values for the short- and long-duration scenarios were 5.56 s and 16.27 s, respectively. As for the original scenario, the PSD bin-average-based response spectrum falls below the target design spectrum; however, the average response spectra from the last iterations closely match the design spectrum (Figure 2b). Nevertheless, significant variations are observed in the resulting target PSD functions (Figure 2c). The short-duration scenario’s PSD significantly exceeds the original scenario’s amplitudes, while the long-duration scenario’s PSD falls considerably below. To quantify these differences, the ratio between the obtained PSDs within the typical frequency range of interest (0.2–50 Hz) is presented in Figure 2d. It is seen that the short- and long-duration scenarios deviate from the original scenario’s target PSD by up to 50% above and 40% below, respectively.

5.2. Inconsistencies in the Calculation of the FAS/PSD and PSA

This issue becomes evident when a different envelope function is used to modulate the amplitude of the synthetic motions. In addition to the trapezoidal function, target PSDs were developed for the same design spectrum using a flat window (i.e., the signal is not modulated) and the window function proposed by Saragoni and Hart [43]. The Saragoni and Hart (S&H) window was developed based on the study of historic earthquake records and is a more realistic representation of the amplitude envelope. Following [44], the window can be defined using Equations (6) and (7):

$$w\left(t\right)=a{\left(t/t_n\right)}^b{e}^{-c\left(t/t_n\right)}$$

(6)

$$a={\left(e/\epsilon \right)}^b;b=-\epsilon ln\left(\eta \right)/\left(1+\epsilon \left(ln\left(\epsilon \right)-1\right)\right);c=b/\epsilon$$

(7)

In these equations e is Euler’s number and ln() the natural logarithmic function. The parameters a, b, and c in Equation (6) were determined using Equation (7), which ensures the generation of a window function that reaches a peak amplitude of w(t) = 1 at t = ε × t_n × t_f and an amplitude w(t) = η at t = t_n × t_f, where t_f is total duration of the signal. Values of ε = 0.2, η = 0.2, and t_n = 0.6 were used for all the simulations in this work.

For both new cases, flat and S&H windows, the total duration of the synthetic motions was manually adjusted so that the matched motion’s SD_5–75 approximately matches the mean SD_5–75 of 9.1 s obtained for the motions generated using a trapezoidal window. The results obtained are presented in Figure 3. It is seen that independent of the envelope used, the algorithm was able to generate motions that successfully match the design spectrum (Figure 3a). Moreover, the target PSD obtained with the flat envelope is very close to the one obtained previously using the original trapezoidal window (Figure 3b). However, the target PSD obtained with the S&H envelope is significantly different to the PSDs obtained with the other two windows (Figure 3b), surpassing their amplitudes by approximately 25% (Figure 3c).

The discrepancy and similarities between the target PSD functions obtained using the three types of window functions can be explained by looking at the characteristics of the synthetic motions generated by each of them. Figure 4 presents the time histories and normalized Arias intensities for the 100 motions in the sets from the algorithm’s last iteration, with the vertical lines denoting the sets’ average SD_5–75 interval. It is seen that for the trapezoidal envelope, the SD_5–75 interval falls within the flat portion of the envelope, therefore the incongruency generated by using the whole unmodulated signal for FAS/PSD calculation is avoided, and the obtained target PSD ended up being practically the same as the one obtained using motions with a flat envelope. However, if the SD_5–75 interval does not fall on a time lapse where the instant amplitude of the motion is constant, or a more realistic non-linear envelope window is used (such as the S&H window), estimating the FAS and PSD based on the unmodulated signal would trigger inconsistencies in the procedure and the resultant target PSD would not be reliable.

To address the duration issue, the proposed methodology incorporates the expected strong motion duration as an input, directly influencing the records’ amplitude envelope. To mitigate inconsistencies in the correction factors, instead of directly converting between PSA and PSD, a PSA-compatible FAS is initially constructed, which is subsequently used to generate the target PSD function.

6. Proposed Methodology

The proposed methodology accounts for the marked effect that strong motion durations have on the relationship between FAS/PSD and PSA. Therefore, the required input is not only the target spectrum but also the expected strong motion duration. Moreover, recognizing that the relation between a motion PSD and PSA is mostly controlled by the motion FAS (e.g., PSD can be constructed from the FAS of the quasi-stationary part of the motion), the proposed algorithm first focuses on developing a FAS compatible with the target design spectrum. Once the compatible FAS is obtained, it is used to produce a large set of synthetic FAS-compatible signals (e.g., [45]) from which the target PSD is estimated. The procedure is detailed next and a flowchart outlining the algorithm is presented in Figure 5.

• Define the target design spectrum and the expected strong motion duration SD_5–75 for which the target PSD function would be developed.
• Use the target PSA and target SD_5–75 to construct an initial target FAS. In this work, Equation (5) [37] was employed. Another option is to use the IRVT approach [40], while not as simple as the empirical equation, the procedure has been already implemented and made publicly available as a Python library [46].
• Determine the total duration (t_f) of the synthetic motions and construct the window function that will be used to modulate the amplitude of the motions. The S&H window (Equation (6)) is used in this work. The total duration of the motions shall be defined so that the FAS-compatible motions (constructed in step 4) exhibit an SD_5–75 close to the target duration defined in step 1. Through a numerical study it was found that for the S&H window with parameters ε = 0.2, η = 0.2, and t_n = 0.6, the total duration can be estimated as t_f ~ 3.54 × SD_5–75.
• Use the FAS from the previous step as a target to generate a set of synthetic motions. First, generate Gaussian noise signals of duration t_f and modulate the amplitude using the window function. Then, construct the FAS for each signal and take the frequency-by-frequency ratios between the target FAS and the signal’s FAS. Use these ratios to modify the signals Fourier coefficients and go back to the time domain using the inverse Fourier transform. The resulting signals are FAS-compatible motions, that is, all motions would share the exact same FAS. The number of motions in the set is defined using the same incremental approach suggested in [32], 10 motions are generated in the 1st iteration, and the number of motions is increased by 10 in each subsequent iteration.
• Calculate the PSA for each of the motions in the set and average them to obtain the set mean PSA.
• Calculate the mean relative true error between the set mean PSA and the design spectrum within the frequency range of interest (the range used in this work was 0.2–50 Hz). If the error is below the allowable error (2.5% was used for the examples presented), stop iterating and go to step 8, otherwise go to step 7. In most of the examples performed the 2.5% allowable error was satisfied within the first three to six iterations.
• Take the frequency-by-frequency ratios between the design spectrum and set mean PSA. Use these ratios to update the current target FAS and go back to step 4.
• Generate a larger set of motions compatible with the most recent target FAS using the same procedure described in step 4 (100 motions were used for the examples presented here). For each motion calculate their PSD using Equation (4) and average the amplitudes over a frequency range ±20% the subject frequency, as recommended in SRP 3.7.1. Average all PSDs obtained to define the target PSD function.
• As a final check, verify that the mean SD_5–75 from the FAS-compatible motions used to build the target PSD function is consistent with the expected strong motion duration, otherwise go back to step 3 and modify the total duration of the synthetic motions accordingly. When using the window parameters and total durations suggested here, the set mean SD_5–75 was found to be in close agreement with expected strong motion duration for all the scenarios evaluated.

Figure 6 and Figure 7 present the results obtained using the proposed approach to obtain a PSD function compatible with the same SRP 3.7.1 CEUS rock spectrum used in the previous examples (Figure 2 and Figure 3). The target SD_5–75 was specified at 9.1 s, which is the same strong motion duration implicitly used in the development of the SRP 3.7.1 target PSD. Accordingly, the synthetic motions are developed with a total duration of 3.54 × 9.1 s = 32.22 s and an amplitude envelope defined by an S&H window constructed using the parameters recommended in step 3. Figure 6 shows the development of the compatible FAS. The dashed lines in the figures denote the results obtained during the first iteration, i.e., using the FAS estimated from Equation (5) to generate the motions. It is seen that the amplitudes of the mean PSA obtained during the first iteration are significantly below the design target spectrum. Nevertheless, it took only three iterations to obtain a compatible FAS that generates motions with a mean PSA matching the target with an error below 2.5%.

The compatible FAS is used to generate a set of 100 motions from which the target PSD is constructed. As shown in Figure 7, these motions exhibit a strong motion duration very close to the target SD_5–75 (Figure 7a,b) and their average response spectrum tightly matches the target design spectrum (Figure 7c). Furthermore, the resultant target PSD function, defined as the mean of the motions’ PSDs, is in close agreement with the tabulated SRP 3.7.1 PSD function (Figure 7d). Recall here that attempting to use the algorithm proposed in SRP 3.7.1 with an S&H window and motions’ total duration adjusted to obtain an SD_5–75 close to 9.1s resulted in a target PSD function that largely deviates from the ones tabulated in SRP 3.7.1 (Figure 2 and Figure 3).

Additional results obtained for different combinations of target design spectra and strong motion durations are presented in Figure 8 and Figure 9. The target design spectra were constructed using the equations in SRP 3.7.1 Appendix B [8] for a peak ground acceleration (PGA) of 1 g, moment magnitude of 7.5, and two different fault distances of 150 km and 30 km. The results in Figure 8 correspond to rock sites in CEUS and Figure 9 shows the results for rock sites in the Western United States (WUS). For each case, target PSD functions were developed using expected strong motion durations of 6, 9, 12, and 15 s. The results obtained confirm the high dependency of the target PSD on the expected strong motion duration. The cases with the lowest SD_5–75 generated target PSD functions with peak amplitudes about 50% larger than the peak amplitudes exhibited by the target PSD functions with the largest SD_5–75. Figure 8 and Figure 9 also show the target PSD function tabulated in SRP 3.7.1 (lines with the circular markers). By comparison with the duration-dependent target PSD functions obtained with the proposed approach, it is confirmed that the SRP 3.7.1 values correspond to an SD_5–75 of around 9 s.

7. Error and Convergence Analysis

As mentioned in the previous section, the iterations in the proposed algorithm are controlled by the mean relative true error between the current set average response spectrum and the target design spectrum. For all examples in this study, the iterative process was stopped when this error fell below 2.5%.

This section investigates the relationship between the accuracy in the matching of the target design spectrum and the precision of the predicted target FAS and target PSD functions. While a direct comparison to the target design spectrum allows for calculation of a true relative error for PSA, error estimation for FAS and PSD relies on comparing the outputs from consecutive iterations.

Figure 10 and Figure 11 depict the error evolution for PSA, FAS, and PSD during the development of target PSD functions for the CEUS M7.5 R150 and WUS M7.5 R30 spectra, respectively. Although the algorithm normally constructs a target FAS in each iteration and a target PSD only once based on the final FAS, it was modified to generate target FAS functions at every iteration for this analysis. Additionally, the algorithm was allowed to run for 10 iterations regardless of reaching the target error.

Both cases exhibited similar error reduction trends, eventually plateauing at comparable levels. However, developing the target PSD for the WUS spectrum proved more challenging, requiring roughly double the iterations to achieve the 2.5% target error and displaying a bumpier decay in FAS and PSD errors.

Similar results were obtained for the other target spectrum cases. Considering all 16 target functions developed for this work (Figure 8 and Figure 9), the average true error in the matching of the target response spectrum was 2%, and the average relative errors in the predicted target FAS and target PSD functions were 3% and 4.5%, respectively.

8. Conclusions

• Strong motion duration has a significant influence on the relationship between earthquake record response spectra, Fourier amplitude spectra (FAS), and power spectral density (PSD) functions. This finding is consistent with previous research and is further confirmed in this study.
• The current SRP 3.7.1 methodology for developing response-spectrum-compatible target PSD functions does not explicitly consider strong motion duration. Upon closer examination, it becomes apparent that the target PSD functions tabulated in SRP 3.7.1 implicitly assume an expected strong motion duration of approximately 9 s for all earthquake magnitude and rupture distance scenarios.
• A fundamental inconsistency in the current SRP 3.7.1 methodology arises from calculating record FAS and PSD based on the unmodulated signal while constructing the response spectrum from the modulated signal. SRP 3.7.1 avoids this inconsistency because the characteristics of the trapezoidal window used for signal modulation ensures that the strong motion duration interval (SD_5–75) used for estimating the PSD falls within the flat portion of the envelope. However, it was shown that this inconsistency would be a trigger and cause unreliable results if non-linear functions were used to modulate the signals or the strong motion duration part of the signals falls on a time lapse where the instant amplitude is not constant.
• To address these issues, an alternative approach has been proposed that generates spectrum-compatible target PSD functions consistent with a predefined strong motion duration.
• The sample results presented employed design spectra obtained from SRP 3.7.1, and the target PSD functions obtained with the proposed procedure are compared with the PSD functions tabulated in SRP 3.7.1. The results show the large impact that the specified strong motion duration has on the target PSD amplitudes and further corroborates that SRP 3.7.1 tabulated target PSD values correspond to a strong motion duration of around 9 s.
• A critical challenge in implementing the proposed procedure is specifying the expected strong motion duration. While seismic codes do not explicitly define strong motion duration values, ASCE 4-16 and SRP 3.7.1 stipulate that input motions should have durations consistent with the characteristic magnitude, distance, and site conditions of the controlling seismic events. Predictive equations from the literature (e.g., [47,48,49]) can be used to estimate these durations.
• Further studies are needed to assess the implications of using strong-motion-duration-dependent target PSD functions in the generation of the input motions for seismic structural design and safety assessments.

As a final disclaimer, note that the minimum PSD requirement is included in NRC guidelines to ensure reliable results from elastic analyses, such as the analyses used to construct in-structure response spectra. If non-linear analyses are to be performed, as in beyond design basis seismic assessments, the duration of strong ground motion continues to be significant, albeit for reasons distinct from its impact on the relationship between FAS, PSD, and PSA (e.g., [50,51]).