ML20287A295
| ML20287A295 | |
| Person / Time | |
|---|---|
| Issue date: | 07/24/2020 |
| From: | Vladimir Graizer, Jinsuo Nie, Dogan Seber, Jim Xu Office of Nuclear Reactor Regulation |
| To: | |
| J. Nie | |
| References | |
| PVP2020-21132 | |
| Download: ML20287A295 (9) | |
Text
Proceedings of the ASME 2020 Pressure Vessels & Piping Conference PVP2020 July 19-24, 2020, Minneapolis, Minnesota, USA 1
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
Approved for public release; distribution is unlimited.
PVP2020-21132 ESTIMATING STABLE MEAN RESPONSES FOR LINEAR STRUCTURAL SYSTEMS BY USING A LIMITED NUMBER OF ACCELERATION TIME HISTORIES Jinsuo R. Nie, Jim Xu, Vladimir Graizer, and Dogan Seber Division of Engineering Office of Nuclear Regulatory Research U.S. Nuclear Regulatory Commission Washington, DC 20555, USA Jinsuo.nie@nrc.gov ABSTRACT For seismic analyses of linear structural systems including soil-structure systems, the current practice (e.g., the U.S. Nuclear Regulatory Commission (NRC)
Standard Review Plan (SRP) (NUREG-0800 [1], Standard Review Plan for the Review of Safety Analysis Reports for Nuclear Power Plants: LWR Edition) and American Society of Civil Engineers (ASCE)/Structural Engineering Institute (SEI)
Standard 4-16 [2], Seismic Analysis of Safety-Related Nuclear Structures) allows for estimating mean seismic responses by using as few as four or five input time histories.
This paper examined whether this practice can achieve a stable mean response by explicitly considering the uncertainty in the Fourier phase spectra of the input time histories and exploring how this uncertainty can affect the coefficient of variation (CV) of the in-structure response spectra (ISRS). ISRS are the response spectra of the seismic response time history at a location in the structure subjected to an input seismic time history. We found that the maximum CVISRS across all frequencies is around 40% purely due to the uncertainty in the Fourier phase spectra for a typical range of design earthquakes for U.S. nuclear power plants. To estimate a mean ISRS within +/-10% of the true mean ISRS, our analyses showed that this level of CVISRS may require a minimum of 16 input time histories for a confidence level (CL) of 68% and 61 for a CL of 95% for soil-structure systems of low fundamental frequencies. For stiffer systems (for example, with a DISCLAIMER NOTICEThe findings and opinions expressed in this paper are those of the authors, and do not necessarily reflect the view of the U.S. Nuclear Regulatory Commission.
fundamental frequency of 5 hertz (Hz)), the maximum CVISRS is about 30%, and thus, the minimum required number of input time histories may be reduced to 9 for a CL of 68% and 35 for a CL of 95%. In summary, the four or five time histories in the current practice may not be sufficient for estimating stable mean responses, especially for soil-structure systems with very low frequencies.
INTRODUCTION Structural responses due to earthquake ground motions can be considered as stochastic processes and would ideally be captured by probabilistic analyses requiring a sufficient number of samples. However, use of a large number of samples for such analyses would not be practical especially for nuclear systems, structures, and components; instead, analysts often use alternative procedures with as few as four acceleration time histories, such as those provided in the NRC SRP [1] and ASCE/SEI Standard 4-16 [2]. These alternative procedures require the selected acceleration time histories to meet certain response spectrum enveloping criteria, intended to ensure that the analytical results conservatively capture the mean earthquake responses. To demonstrate the adequacy of these procedures, appropriate benchmarking against fully probabilistic analyses could be performed to evaluate whether these procedures can lead to stable and acceptable mean structural responses with a limited number of time histories.
In lieu of a fully probabilistic approach for benchmarking these alternative procedures, which usually requires significant resources, an evaluation approach based on the rule-of-thumb 10% engineering criterion could be used for determining the minimum number of time histories for earthquake response
2 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
Approved for public release; distribution is unlimited.
analysis. This approach uses a relatively large number of acceleration time histories to establish a true mean ISRS, which is then used to determine whether a smaller number of acceleration time histories (e.g., four or five) can lead to a mean estimate of ISRS within the 10% of the true mean ISRS. However, this approach could be sensitive to how the time histories used to establish the true mean ISRS are selected. We propose an approach in this paper to explicitly use the CV of the ISRS from many time history analyses to determine a smaller number of time histories required to estimate a stable mean ISRS estimate. The use of ISRS in determining an adequate number of input time histories is thought to be more appropriate than considering only the response spectrum (RS) enveloping criteria because ISRS represent the structural responses over the entire frequency range of interest, while the input RS enveloping criteria consider only the maximum responses [3, 4].
In this paper, we introduce some reasonable assumptions so that simple statistical rules can be applied to derive the minimum required number of input time histories from ISRS, describe how the uncertainties in the input time histories are separately treated, and assess the CV of the input RS and the ISRS and how they affect the minimum number of input time histories. Insights on the required minimum number of input time histories are summarized at the end of this paper.
ASSUMPTIONS AND STATISTICAL BASICS In practice, the input acceleration time histories are often generated using the same procedure or software and meeting the same acceptance criteria for RS enveloping as described in ASCE 4-16 or Approach 2 in SRP Section 3.7.1 and thus generally share some common features. To aid our analysis, we assume that any time histories developed in this way and their RS are statistically independent of all other time histories and their RS. Further, the response time histories and the ISRS are interrelated by means of the same structural model; nonetheless, we assume these quantities are statistically independent of each other as well, relative to the mean responses. In addition, for simplicity, we assume that these quantities have an identical distribution. The assumption of an identical and statistically independent distribution (i.i.d.)
among the sample input motions and responses allows simple statistical rules to be applied in this paper. Nevertheless, these assumptions are considered reasonable because the seeds used to generate these time histories are generally uncorrelated, especially for the seeds in this paper that were generated from random phases uniformly distributed in [0, 2].
Given N input acceleration time histories Ai, their response spectra RSi, and the ISRSi calculated for a location in a structure, the sample mean M and sample CV of the RS (i.e., MRS, MISRS, CVRS, and CVISRS) can be determined from these N samples as functions of frequency (frequencies are omitted from the symbols herein for simplicity but are considered in calculations from 0.1 Hz to 100 Hz, as shown in the figures later in this paper). The CV of MRS and MISRS can then be obtained using the i.i.d. assumption made above:
=
=
(1)
For example, if the sample CV is found to be 30%, the CV of the mean estimate using 500 samples can be calculated using Equation (1) to be 1.3%, so the mean estimate can be claimed as the true mean and the 500 samples can be considered to be the large number of samples, in terms of the approach introduced in the previous section. When using four samples to calculate MRS or MISRS, the CV of these mean estimates would be 15%, thus indicating that four is too few to achieve the 10% rule-of-thumb criterion. Therefore, the essential task in this paper is to estimate a relatively accurate sample CVISRS using a large number of time histories. The smaller number (Ns) of samples to estimate a stable mean ISRS with a CV of 10% can be estimated as:
100 2
, CL=68%
(2)
Equation (2) indicates that a higher sample CVISRS quadratically increases the number of input acceleration time histories to achieve an MISRS estimate with CVMISRS equal to 10%. Table 1 lists the Ns calculated using Equation (2) for some assumed values of the sample CVISRS. As shown in this table, if the sample CVISRS is only 20%, then 4 input time histories would be sufficient, while a sample CVISRS of 70%
would require 49 input time histories. In terms of a +/-10%
confidence interval within which the true mean falls using Ns input time histories, the CL is only about 68%. To achieve a 95% CL for the same +/-10% confidence interval, Ns should be increased by 3.84 (=1.962), or:
384 2
, CL=95%
(3)
TABLE 1 THE NUMBER OF INPUT TIME HISTORIES REQUIRED FOR A GIVEN SAMPLE CVISRS TO ACHIEVE A
+/-10% CONFIDENCE INTERVAL CVISRS (%)
NS (CL=68%)
NS (CL=95%)
20 4
15 22 5
19 26 7
26 30 9
35 40 16 61 50 25 96 70 49 188
3 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
Approved for public release; distribution is unlimited.
Equations (2) and (3) and Table 1 are used later in this paper to determine Ns based on the sample CVISRS that are estimated by analyzing many time histories.
UNCERTAINTIES IN INPUT TIME HISTORIES To address the uncertainties leading to the CV of the input RS and ISRS, we separate the uncertainty in Fourier amplitude spectra and the uncertainty in Fourier phase spectra of the acceleration time histories. This paper will address only the uncertainty in Fourier phase spectra, which are commonly observed to be distributed uniformly in [0, 2] and thus are considered irreducible. Consideration of the uncertainty in the Fourier amplitude spectra should increase CVRS and CVISRS; however, the extent of such an increase requires further study.
In the frequency domain, a time history can be represented as a Fourier amplitude spectrum and a Fourier phase spectrum, which are generally uncorrelated. This can be illustrated by plotting the Fourier amplitude spectrum versus the Fourier phase spectrum, as shown in Figure 1 for one of the classic El Centro records for the Imperial Valley earthquake that occurred on October 15, 1979. In Figure 1, the two inner circles represent the two horizontal directions, and the outer circle represents the vertical direction. To generate an acceleration time history, we prescribe the Fourier amplitude spectrum deterministically and randomly generate a Fourier phase spectrum over [0, 2] so that the acceleration time history can be developed by an inverse transform of the Fourier spectra (determinisistc amplitudes and random phases). Using many time histories generated in this way, the CVRS or CVISRS is ensured to be solely the effect of the randomness in the Fourier phase spectra. The Fourier amplitude spectrum is assumed to be smooth and can be expressed equivalently as a power spectral density (PSD) function.
It should be noted that these sample time histories are not generated to meet the RS enveloping criteria in ASCE 4-16 or the SRP Section 3.7.1, Approach 2, guidance. Also note that these criteria mostly concern the Fourier amplitudes but not so much the Fourier phases
- because, for
- example, a
freqeuncy-domain generation method usually maintains the phase spectrum very well during the time history synthesis process. The uncertainty in Fourier phase spectrum affects CVRS and CVISRS but not so much the MRS and MISRS because they are determined by the prescribed deterministic input PSD function, except in practice for the variations in estimated mean properties when a small number of sample time histories are used.
THE METHOD TO CALCULATE CVISRS The study for this paper explored 21 cases, each represented by a PSD function taken from the 20 tabulated target PSD functions in SRP Section 3.7.1 [1] and a unit PSD function. For each case, 500 sample acceleration time histories were generated, each with a time step of 0.005 seconds (s), 4,096 acceleration points, and thus a total duration of 20.48 s.
An intensity function from NUREG/CR-5347 [5], Recommendations for Resolution of Public Comments on USI A-40, Seismic Design Criteria, issued May 1989, was applied to create the ramp and decay durations in the time histories so that a time-domain method could be used to compute the RS and ISRS. All RS are calculated using a damping ratio of 5%. The frequency series used to develop the RS and ISRS are those described in ASCE 4-16 and SRP Section 3.7.1, Approach 2 (i.e., 100 frequency points per frequency decade spanning from 0.1 Hz to 100 Hz).
FIGURE 1 FOURIER AMPLITUDES PLOTTED OVER FOURIER PHASE ANGLES (TWO INNER CIRCLES FOR THE TWO HORIZONTAL COMPONENTS AND THE OUTER CIRCLE FOR THE VERTICAL COMPONENT)
For each time history, a series of single-degree-of-freedom oscillators with a damping ratio of 5% were used to generate the response time histories at a frequency range of 0.1 Hz to 100 Hz. These oscillators are used to represent various modes of a complex structural model such as a realistic soil-structure system. However, note that this paper does not explicitly model any specific soil-structure system.
The fundamental frequencies of these oscillators were selected based on a recommendation in NRC Research Information Letter RIL-1901 [4], Assessment of Artificial Acceleration Time History Guidance in Standard Review Plan Section 3.7.1, Seismic Design Parameters, issued December 2019:
log = 3
1
1,
= 0, 1,, 1 (4) where Fi is the frequency point, and n=151 and p=0.6. The oscillator fundamental frequencies were selected to provide good coverage of the frequency range and are somewhat different from the frequencies used to compute RS to avoid confusion but bearing no other specific benefits. The response
4 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
Approved for public release; distribution is unlimited.
time histories were developed in the frequency domain through the convolution of the oscillator transfer function and the input Fourier spectra. For each oscillator (conceptually, a single-degree-of-freedom structure or a structural mode), the RS of the response time histories (i.e., the ISRS) resulted from the 500 input time histories were used to estimate the sample CVISRS for this oscillator.
FIGURE 2 INPUT RS FOR CEUS M65D10 As an example, Figure 2 shows the RS of 500 sample time histories as the thin gray curves and their MRS as the thick red curve and the green and blue curves indicating MRS x (1 +/- CVRS), for the input PSD function in SRP Section 3.7.1 for the Central and Eastern U.S. (CEUS) bin M65D10 (magnitude (M) 6.5 and source-to-site distance (D) 0-10 kilometers) [1]. The generally decreasing curve shows the frequency-dependent CVRS (also abbreviated as COV in the figures in this paper), which has a maximum value of 0.26 in the low-frequency range and a minimum value of 0.08 in the high-frequency range. It can be seen that CVRS is independent of the magnitudes of the RS, which are governed by the input PSD function.
Figure 3 shows two plots for the ISRS of a 0.49-Hz oscillator using the 500 sample time histories shown in Figure
- 2. The top plot shows the 500 ISRS, MISRS, and MISRS x (1 +/- CVISRS), and the bottom plot shows the frequency-dependent CVISRS in blue, together with the CVRS in red (also shown in Figure 2) and their ratio as the gray dashed line. It is obvious that this oscillator amplifies the CV of the input motions at its fundamental frequency, below which it slightly reduces the CV. CVISRS maintains a higher level than CVRS beyond the zero-period acceleration (ZPA) frequency (i.e., the ZPA effect). This higher level of CVISRS in frequencies higher than the ZPA frequency roughly reflects the CVRS at the oscillator frequency, which is higher than the CVRS in higher frequencies. This makes sense because the ZPA of the ISRS is simply the RS of the input motions at the oscillator frequency [3].
FIGURE 3 ISRS AND CVISRS FOR OSCILLATOR AT 0.49 HZ FOR CEUS M65D10 Figure 4 is similar to Figure 3 but pertains to an oscillator at a fundamental frequency of 19.74 Hz. Although this oscillator is at a frequency close to the peak frequency of the input motions and the ISRS are much higher than those shown in Figure 3 as expected, the CVISRS is generally much lower than the CVISRS for the 0.49-Hz oscillator. The CVISRS shows some amplification at 19.74 Hz, but the amplification is much less pronounced than that for the other oscillator at 0.49 Hz.
At frequencies lower than the oscillator frequency, the CVISRS is essentially the same as the CVRS of the input motions.
Coefficient of Variation CV 500 Sample RS, Overlaid by MRS and MRS x (1 +/- CVRS) 500 Sample ISRS, Overlaid by MISRS and MISRS x (1 +/- CVISRS)
5 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
Approved for public release; distribution is unlimited.
FIGURE 4 ISRS AND CVISRS FOR OSCILLATOR AT 19.74 HZ FOR CEUS M65D10 EFFECT OF PHASE UNCERTAINTY ON CV AND THE NUMBER OF INPUT TIME HISTORIES To determine the minimum required number of input time histories, CVISRS should be considered for the entire structure and the entire frequency range of interest to the seismic analysis and design.
Using 21 cases that consist of the 20 target PSD functions as documented in SRP Section 3.7.1 and a unit PSD function (i.e., a phase wave), this section calculates CVRS, CVISRS, and the minimum number of input time histories required to achieve a CVMISRS that is small enough so that the true mean MISRS lies within +/-10% of the sample mean ISRS.
CVRS of the Input Acceleration Time Histories Figure 5 shows the CVRS for the 20 target PSD functions from SRP Section 3.7.1 in thin grey lines and their mean in a thick red line and the bounds showing the maximum and minimum CVRS among these 20 cases. Except for very low frequencies (below 0.2 Hz), the variations in the CVRS are generally very small among different PSD functions. The few cases of very low CVRS at the very low frequencies may be related to those PSD functions that were manually adjusted at this frequency range so that the target PSD functions may not show drastic changes during their development [1]. This issue deserves further exploration; however, it does not affect this study because the upper bound appears to follow the general trend very well and governs the determination of the minimum number of input time histories. The spread in the CVRS from 0.6 Hz to 30 Hz appears to be constant regardless of frequencies and the magnitude of the CVRS and the PSD functions, which have vastly different shapes. Discarding those cases with very low CVRS at the very low frequencies, the cause for the slightly larger spread of the CVRS below 0.6 Hz or above 30 Hz is not immediately known; however, these slightly larger spreads do not affect the assessment of the overall behavior of the CVRS. In summary, the largest CVRS is about 30% (at 0.1 Hz) and the smallest CVRS is 7% (around 60~70 Hz), and all CVRS curves are generally decreasing functions of frequency.
FIGURE 5 MEAN, MAXIMUM, AND MINIMUM CVRS FOR THE TARGET PSD FUNCTIONS IN SRP SECTION 3.7.1 Figure 6 shows the RS and the statistics for 500 sample time histories randomly generated from a unit PSD function (i.e., a phase wave). The generally decreasing CVRS is very similar to those shown in Figure 2 and Figure 5 for some realistic PSD functions of nonunit amplitudes. The range of the CVRS for the phase wave is [0.08, 0.30], also consistent with those shown in Figure 5. These 21 cases with prescribed PSD functions that are vastly different from each other clearly confirm that CVRS is independent of the input PSD functions.
For a unit PSD function that extends to all frequencies, the RS of the sample time histories should not show the tendency of leveling at the high frequencies in Figure 6. This leveling is probably caused by the time-domain RS calculation 500 Sample ISRS, Overlaid by MISRS and MISRS x (1 +/- CVISRS)
Coefficient of Variation 0.6 Hz 30 Hz Cases with very low CVRS.
6 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
Approved for public release; distribution is unlimited.
procedure that is affected by frequency bounds (e.g., the Nyquist frequency of the sample time histories with a time step of 0.005 s is only 100 Hz). To verify this aspect, Figure 7 shows results using sample time histories with a time step of 0.0025 s and also using a frequency-domain RS calculation procedure. The tendency of leveling at the high frequencies is greatly minimized. The ringing effect at the low-frequency range is caused by the lack of frequency resolution in that range, because the total duration of the time histories is 20.48 Hz. Nevertheless, the overall trend and the extreme values of the CVRS are similar to those in Figure 6.
In summary, CVRS is a direct result of the irreducible uncertainty in the phase spectra and does not show any correlation with the prescribed deterministic PSD functions (or equivalently the Fourier amplitude spectra). Except for a few cases that merit further exploration at the very low frequencies
(< 0.2 Hz), CVRS is generally a monotonically decreasing function of frequency. The variation in CVRS among the 21 cases is small. The largest and smallest CVRS are 30% and 7%, respectively. For the low-frequency range (0.1-1 Hz),
midfrequency range (1-10 Hz), and high-frequency range (10-100 Hz), the ranges of the CVRS are found in Figure 5 to be roughly [0.3, 0.19], [0.19, 0.11], and [0.11, 0.9],
respectively.
FIGURE 6 MEAN, MAXIMUM, AND MINIMUM CVRS FOR THE UNIT PSD FUNCTION (RS CALCULATED IN THE TIME DOMAIN)
FIGURE 7 MEAN, MAXIMUM, AND MINIMUM CVRS FOR THE UNIT PSD FUNCTION (RS CALCULATED IN THE FREQUENCY DOMAIN)
Assessment of CVISRS and the Number of Minimum Input Time Histories For each of the 20 target PSD functions in SRP Section 3.7.1 and the unit PSD function, CVISRS were calculated for the 151 oscillators as prescribed by Equation (4). For each PSD function, the 151 CVISRS are plotted together to assess how a complex structure of many modes would respond to the input motion, represented by the PSD function with random phases, at various locations in the structure across the entire frequency from 0.1 Hz to 100 Hz.
Of course, the modal combination for complex structures will affect CVISRS, most likely by increasing CVISRS. However, these plots would still offer useful insights because many structural modes dominate structural responses without a significant effect of combination with other modes.
Figure 8 through Figure 11 show the CVISRS for the cases of CEUS M65D10, CEUS M75D10, Western U.S. (WUS)
M65D10, and WUS M75D10, as designated in SRP Section 3.7.1 reflecting the region, magnitude, and the site-to-source distance. Figure 12 shows the CVISRS for the case of the unit PSD function. Despite some minor differences, these figures show very similar results.
As discussed earlier for oscillators, the structures amplify the CV in the input motion and result in a larger CV in the ISRS at the modal frequencies. This effect can be observed only in the low frequencies (< 0.3 Hz), above which this amplification effect is shadowed by the ZPA effect discussed earlier. In other words, although the CVRS generally decreases CV 500 Sample RS, Overlaid by MRS and MRS x (1 +/- CVRS)
CV 500 Sample RS, Overlaid by MRS and MRS x (1 +/- CVRS)
7 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
Approved for public release; distribution is unlimited.
with frequency, the CVISRS always show plateaus after some higher frequency, which would depend on the lowest modal frequency of the structure that affects a particular location in the structure. For a soil-structure system with a low soil column frequency around 0.5 Hz, the CVISRS can be retained at a relatively higher level due to the ZPA effect.
Without looking into specific structures, this study seems to indicate that the maximum CVISRS is generally at a level of 0.4. For the 20 PSD functions, the smallest maximum CVISRS was found to be 0.35 for CEUS M75D200, while the largest maximum CVISRS was found to be 0.56 for CEUS M55D100.
The maximum CVISRS is around 0.4 for most cases except for the four low-magnitude cases (M55) that all have a maximum CVISRS above 0.53.
Neglecting the earthquakes smaller than magnitude 6 and considering only those larger ones as typical design earthquakes for U.S. nuclear power plants, using Equations (2) and (3) or Table 1, a CVISRS of 40% would indicate that the minimum required number of input acceleration time histories should be 16 for a CL of 68% and 61 for a CL of 95%. Note that this finding assumes there is no uncertainty in the Fourier amplitude spectrum of the input motion. All the variation in the input motion and the ISRS is due to the effect of random phase spectrum uniformly distributed in [0, 2].
For stiffer structural systems (for example, with a fundamental frequency of 5 Hz), the maximum CVISRS is about 30%, the same as that of the input motion. As shown in Figure 4, a higher structural mode would basically carry the CV of the input motion at lower frequencies to ISRS. A CVISRS of 30% would indicate that the minimum required number of input acceleration time histories should be 9 for a CL of 68% and 35 for a CL of 95%.
FIGURE 8 CVISRS FOR THE 151 OSCILLATORS FOR CEUS M65D10 FIGURE 9 CVISRS FOR THE 151 OSCILLATORS FOR CEUS M75D10 input input
8 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
Approved for public release; distribution is unlimited.
FIGURE 10 CVISRS FOR THE 151 OSCILLATORS FOR WUS M65D10 FIGURE 11 CVISRS FOR THE 151 OSCILLATORS FOR WUS M75D10 FIGURE 12 CVISRS FOR THE 151 OSCILLATORS FOR THE UNIT PSD FUNCTION A BRIEF DISCUSSION OF UNCERTAINTY IN FOURIER AMPLITUDE SPECTRA When a design acceleration time history is generated, the common practice is to ensure its RS envelops the design RS following a set of RS enveloping criteria. This practice was commonly applied until very recently when studies found that a PSD check is a necessary addition to the RS enveloping criteria because the RS by itself is not a sufficient descriptor of the power content of the input motion implicitly assumed in the specification of the design RS [3, 4]. A PSD check rules out those time histories with a power below 80% or 70% of the target PSD function in a wide frequency window (per Appendix A or Appendix B to SRP Section 3.7.1 [1]). For design time histories meeting both the RS enveloping criteria and passing the PSD check, the uncertainty in the PSD functions of the input time histories would generally be smaller than without a PSD check because a PSD check basically restricts the large variations in the resultant time histories. The reduced uncertainty in the Fourier amplitude spectra of the PSD-checked time histories certainly still increases the CVISRS on top of those determined from the random phase spectra. However, since the PSD check ensures the input time histories have sufficient power over the entire frequency range of interest, those increased CVISRS may not be a safety concern as they represent those variations occurred on the conservative side.
For a time history without a PSD check, the uncertainty of the Fourier amplitude spectrum would be larger because the RS enveloping criteria check only the ZPA on the entire ISRS curve of the resultant response time history. For a structure input input input
9 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
Approved for public release; distribution is unlimited.
that consists of multiple modes, the ZPA effect would still exist because the responses of lower frequency modes would affect not only the ZPA of the resultant response but also the periodic, amplified responses at higher frequency modes. In addition, the larger uncertainty in ISRS is not necessarily on the conservative side because the power in frequencies below the ZPA frequency on the ISRS curve could be much lower than that implicitly required by the design RS [3]. For these design time histories, the CVISRS is expected to be higher than those that passed the PSD check, and the number of required time histories for a linear analysis would be expected to be greater than 16 for a CL of 68% and 61 for a CL of 95%. For example, using realistic soil-structure models and hundreds of input time histories, Reference [6] determined that CVISRS can be as large as 70%, which effectively leads to a minimum required input time histories of 49 for a CL of 68% and 188 for a CL of 95%.
Of course, similar to this study of Fourier phase spectra, a systematic assessment of the effect of the Fourier amplitude spectra on CVISRS would be a very beneficial future study.
SUMMARY
We proposed an approach to explicitly use CVISRS to determine the minimum number of input acceleration time histories to estimate a stable mean response for linear structural systems. We also proposed a separate treatment of the uncertainty in the Fourier amplitude spectra and the uncertainty in the Fourier phase spectra as they are deemed uncorrelated. This paper covers only the effect of the uncertainty in the Fourier phase spectra.
We found that the CVRS of the input motion using 500 sample time histories are generally a monotonically decreasing function of frequency, and they do not vary much for significantly different PSD functions. The maximum CVRS is about 30% and the minimum CVRS is about 7%.
Neglecting the earthquakes smaller than magnitude 6 and considering only those larger ones as typical design earthquakes for U.S. nuclear power plants, we found the maximum CVISRS to be around 40%, which is purely due to the randomness in Fourier phase spectra. Our analyses indicated that this level of CVISRS would require a minimum of 16 input acceleration time histories for a CL of 68% and 61 for a CL of 95%. For stiffer structural systems, the maximum CVISRS is about 30%, the same as that of the input time histories; thus, the minimum number of input time histories can be reduced to 9 for a CL of 68% and 35 for a CL of 95%.
The minimum required number of input time histories is defined as those that can reduce the CV of the mean ISRS estimate to a level at which the true mean ISRS falls within
+/-10% of the mean ISRS estimate with a predetermined CL.
To assess the uncertainty in the Fourier amplitude spectra, further study is needed. However, a study in Reference [6]
using realistic soil-structure models and hundreds of input time histories determined that the CVISRS can be as large as 70%, which translates to a minimum required number of input time histories of 49 for a CL of 68% and 188 for a CL of 95%.
In summary, the four or five time histories in the current practice may not be sufficient for estimating stable mean responses, especially for soil-structure systems with very low frequencies.
REFERENCES
- 1. U.S. Nuclear Regulatory Commission (NRC), Standard Review Plan for the Review of Safety Analysis Reports for Nuclear Power Plants: LWR Edition, NUREG-0800, Washington, DC.
- 2. American Society of Civil Engineers (ASCE) )/Structural Engineering Institute (SEI) 4-16, Seismic Analysis of Safety-Related Nuclear Structures, Reston, VA.
- 3. Nie, J.R., J. Pires, and D. Seber (2019). Understanding the Assumptions in Design Response Spectra for Seismic Time History Analyses, Transactions, SMiRT-25, Charlotte, NC.
- 4. NRC, Research Information Letter RIL-1901 (2019).
Assessment of Artificial Acceleration Time History Guidance in Standard Review Plan Section 3.7.1, Seismic Design Parameters, Washington, DC.
(Agencywide Documents Access and Management System (ADAMS) Accession No. ML19308A045).
- 5. Philippacopoulos, A.J. (1989). Recommendations for Resolution of Public Comments on USI A-40, Seismic Design
- Criteria, NUREG/CR-5347, Prepared by Brookhaven National Laboratory for the
- NRC, Washington, DC.
(ADAMS Accession No.
- 6. Houston, T.W., G.E. Mertz, M.C. Costantino, and C.J. Costantino (2010). Investigation of the Impact of Seed Record Selection on Structural
- Response, American Society of Mechanical Engineers Pressure Vessels and Piping Conference (PVP2010-25919),
Bellevue, WA.