ML20211P661

Rev 0 to Development of Time Histories for 2,000-Year Return Period Design Spectra. with Diskette

ML20211P661
Person / Time
Site:	07200022
Issue date:	08/24/1999
From:	Mok C, Youngs R STONE & WEBSTER ENGINEERING CORP.
To:
Shared Package
ML20211P637	List: ML20211P634 ML20211P645 ML20211P653 ML20211P661 ML20211P666 ML20211P678 ML20211P686 ML20211P700 ML20211P707
References
G(PO18)-3, G(PO18)-3-R00, NUDOCS 9909140064
Download: ML20211P661 (53)

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{{#Wiki_filter:STONE & WEBSTER ENGINEERING CORPORATION CALCULATION TITLE PAGE 'SEE INSTRUCTIONS ON REVERSE SiOr a 501064(FROffT) CLIENT & PRCJECT Private Fuel Storage Facility, LLC PAGE1 OF 34 Private Fuel Storage Facility, Skull Valley, UT CALCULATION TITLE (Indicative of the Objective): QA CATEGORY (/) Development of Time Histories for 2,000-Year Retum Period Design Spectra @ l-NUCLEAR SAFETY RELATED 011 O lll O OTHER CALCULATIONIDENTIFICATION NUMBER OPTIONAL URREE ONONAL WOMAM M. J.O. OR W.O. NO. DMS!ON & GROUP CALC.NO. TASK CODE 05996.02 Geotechnical G(PO18)-3 N/A N/A

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J.OJW.OJCALCULATION NO. REVISION PAGE 2 OF 34 a soio e 05996.02-G(PO18)-3 0 CLIENT / PROJECT Private Fuel Storage Facility, LLC QA CATEGORYlCODE CLASS Private Fuel Storage Facility, Skull Valley, UT I SUBJECT / TITLE Development of Time Histories for 2,000-Year Retum Period Design Spectra OBJECTIVE OF CALCULATION Develop a 3-component set of time histories that are compatible with the 2,000-year retum period design response spectra and satisfy US NRC Standard Review Plan 3.7.1 (P99). CALCULATION METHOD 1 ASSUMPTIONS

1. Select natural recordings consistent with controlling design event.

2. Scale motions to match design :pactra at 5% damping and estimated spectra at 2% and 10% damping

3. Develop target PSD using standard Reves Plan Appendix A and ratio of design spectra to REGGUIDE 1.60

4. Adjust time histories to meet enve8 p requirements for 5% response spectra and PSD of Standard Review Plan.

0 SOURCES OF DATAIEQUATIONS See list of references on Page 10 of 34. Computer Programs:

1. SPECTRA A computer program to compute response spectra Verification in projectfiles CONCLUSIONS A 3-component set of time histories were developed for the 2,000-year Design spectra that satisfy requirements of US NRC Standard Review Plan 3.7.1 (1989).

I REVIEWER (S) COMMENTS PR R DATE g V' \\ R R. Youngs 8/24/99 REVIEWERICHE' K81 DATE C & 71 hin Man Mok 8/24/99 INDEPENDENT REVIEWER DATE Y ^A?L 'fA Chin Man Mok 8/24/99

1 CALCULATION SHEET J.OJw.OJCALCULATION NO. REVISION PAGE 3 OF 34 /) 05996 D2-G(PO18)-3 0 PREPARER /DATE //I /) ] REVIEWERICHECKER/DATEggygNDEPENDENT REVIEWERIDATE Robert R. Youngs / 8/24/99 //dril'V, Chin Man Mok / 8/2d/99 Chin Man Mok / 8/24/99(tCJC,2( SUB.,ECT lTITLE Developmeni of Time Hisfries or 2,000-Year Retum Period Design Spectra QA CATEGORY / CODE CLASS TABLE OF CONTENTS AND HISTORIC DATA (Revisions, Additions, Deletions, Etc.) PAGE REVISION NO. DESCRIPTION NO. DATE REMARKS 1 Calculation Title Page 0 8/24/99 2 Calculation Summary 0 8/24/99 3 Table of Contents 0 8/24/99 4 1.0 Introduction 0 8/24/99 4 2.0 Selected Recording 0 8/24/99 5 3.0 Scaling to Design Spectra 0 8/24/99 7 4.0 Comparison with Regulatory Requirements 0 8/24/99 10 5.0 References 0 8/24/99 Attachment A-0 8/24/99 Abrahamson, N A (199?; Non-Stationary Spectral Matching Program RSFMATCH I i 4 I

05996.02-G(pol 3)-3 (REV. 0) Page 4 of 34 PRIVATE FUEL STORAGE FACILITY SKULL VALLEY, UTAH DEVELOPMENT OF TIME HISTORIES FOR 2,000-YEAR RETURN PERIOD DESIGN SPECTRA Calculation 05996.02-G(PO18)-3 (Rev. 0)

1.0 INTRODUCTION

In this calculation a three-component set of anificial time histories were generated to match the 2,000-year return-period design response spectra for the Private Fuel Storage Facility located in Skull Valley, Utah. The time histories were generated by selecting an actual ground motion recording from an earthquake and site compatible with the design event defined in Geomatrix (1999). The time histories were then scaled to meet the requirements for a single artificial time history specified in Section 3.7.1.2 of the U.S. Nuclear Regulatory Commission's Standard Review Plan (US Nuclear Regulatory Commission,1989). Scaling was performed using both frequency domain and time domain techniques. 2.0 SELECTED RECORDING The design ground motions for the Skull Valley Private Fuel Storage Facility were developed from a probabilistic seismic hazard analysis. The 2,000-year return period equal-hazard response spectra represents earthquakes with a mean magnitude of M 6.3-6.5 and a mean (log) distance of 4 to 6 km. The controlling fault is the Stansbury fault located 9 km east of the site with a mean maximum magnitude of M 7.0. The site is located in the hanging wall block of the fault and is underlain be shallow stiff soils overlying tertiary semi-consolidated sediments. The design response spectra developed in Geomatrix (1999) include the near-fault effects of directivity and systematic fault-normal to fault-parallel differences at frequencies less than 2 Hz (spectral periods > 0.5 sec). A strong motien recording that provides a good fit to these criteria is the Sturno recording of the November 23,1980 M 6.9 Irpinia, Italy earthquake. The Stumo site is located approximately 11 km from the nonhwest end of the fault rupture in the hanging wall block. The processed time history was obtained from Pacific Engineering and Analysis from a set of time histories processed for the US Geological Survey for the Yucca Mountain project. The Stumo site is indicated to be a rock site (Spudich et al.,1997). However, the spectral scaling will adjust the recording to achieve the desired frequency content. The original time histories were digitized at + t \\DOCJAFN0005W7900nSV 2KTHC DOC 1 ]

05996.02-G(pol 8)-3 (REV. 0) Page 5 of 34 a time step of 0.00244 seconds. For this analysis they were interpolated to a time step of 0.005 seconds. The band width of the processed accelerograms is 0.13 to 30 Hz for the horizontal components and 0.13 to 33 Hz for the vertical component. The horizontal components of the recording are oriented at azimuths of 000 and 270, while the strike of the main rupture was northwest. Therefore the two horizontal recordings were used to compute a fault-normal component at an azimuth of 045 and a fault-parallel component at azimuth 315. The parameters of the recording motions and the interpolated and rotated motions are given in the following table. 5%-75% 5%-95% Energy Energy Component PGA PGV PGD V/A D/A AD/V^2 Duration Duration (g) (cm/s) (cm) (cm/s/g) (cm/g) (sec) (sec) Original Processed Time Histories STUOOO 0.251 37.0 11.8 147.6 46.9 2.1 7.2 15.3 STU270 0.358 52.7 33.1 147.3 92.5 4.2 6.3 15.5 STUVRT 0.260 26.0 10.6 100.2 40.9 4.0 7.3 11.9 1 Interpolated and Rotated Time Histories STU-fn 0.234 43.1 '23.7 184.4 101.4 2.9 7.9 16.9 STU-fp 0.302 46.5 23.4 154.0 77.4 3.2 6.1 12.3 STUvrt ' O.254 26.0 10.6 102.3 41.5 3.9 7.3 11.9 Figure 1 shows the original accelerograms. Figure 2 shows the interpolated and rotated accelerograms, and Figures 3 and 4 show the integrated velocity and displacement time histories. Figure 5 compares the 5% damped response spectra for the interpolated and rotated accelerograms to the design ground motion response spectra from Geomatrix (1999). 3.0 SCALING TO DESIGN SPECTRA The initial scaling of the accelerograms was performed using program R.ASCAL (Silva and Lee, 1987). This program operates in the frequency domain. The stochastic ground motion model (e.g. Boore,1983,1986) is used to generate a Fourier amplitude spectrum for the appropriate magnitude earthquake that is used as an initial estimate. Random vibration theory (RVT) is then used to compute a response spectrum from the Fourier amplitude spectrum. The RVT response spectrum is compared to the target response spectrum and the Fourier amplitude spectrum is 1 adjusted iteratively to minimize the difference between the target response spectrum and the RVT response spectrum. A time history (TH) is then created using the adjusted Fourier amplitude spectrum and the phase from the selected accelerogram. The resulting time history is used to' compute a response spectrum. The TH response spectrum is compared to the target i WDC,,$AFN190 Ol3V-2KTHCDOC b'

05996.02-G(pol 8)-3 (REV. 0) Page 6 of 34 response. spectrum and the Fourier amplitude spectrum for the TH is adjusted iteratively to minimize the difference between the target response spectrum and the TH response spectrum. Figure 6 shows the developed from the initial frequency domain scaling and Figure 7 compares the response spectra for these time histories to the design response spectra. ] -l The final scaling was performed using the time-domain technique developed by Lilhanand and Tseng (1988). This approach scales the motion to match a target response spectrum by adjusting the time history in small increments in the vicinity of the time for the peak spectral response. Attachment A contains a description of the technique and a computer program to implement it developed by Dr. Norm Abrahamson. One advantage of the approach is that the response spectra for multiple damping levels can be 1 used as the target spectra. Matching the response spectra for multiple damping levels helps 4 prevent development of" holes"in the frequency content of the resulting time history. The design ground motion spectra developed in Geomatrix (1999) are specified for 5% damping only. Abrahamson and Silva (1996) developed adjustment factors to scale 5% damped response . spectra to other damping levels from the analysis of a large number of empirical response spectra. Figure 8 shows the adjustment factors for 2% and 10% damping for horizontal and vertical motions for an M 6.5 eanhquake. These factors were used to create 2% and 10% damping response spectra consistent with the 5% damping design response spectra. The 2% and 10% damping spectra were also used as target spectra to ensure that the resulting time histories are broad-banded. Figure 9 shows the developed from the time domain scaling. These time histories have been baseline corrected to remove displacement drift by subtracting a 6* order polynomial fit to the integrated displacement time history. Figure 10 compares the response spectra for these time histories to the 5% design response spectra and the associated 2% and 10% spectra created using the factors on Figure 8. The final steps in the processing were: shodening the total duration of 30 seconds by removing the long, low amplitude tail of the records; adjusting the PGA value to be more consistent with the design spectrum PGA; and scaling the time histories upward by a small factor to meet the response spectrum envelope criteria specified in Section 3.7.1.2 of the Standard Review Plan. The resulting time histories are shown on Figures 11,12, and 13. The parameters of the design time histories are: IWOC,5AFIM0005w?90OnfV 2KTHCDOC

03996.02-G(PO18)-3 (REV. 0) Page 7 of 34 5%-75% 5%-95% Energy Energy Comrv r.ent PGA PGV PGD V/A D/A ADN^2 Duration Duration e (g) (cm/s) (cm) (cm/s/g) (cm/g) (sec) (sec) fault-normal 0.556 51.8 30.0 93.2 54.0 6.1 10.5 20.3 fault-parallel 0.542 40.4 18.7 74.4 34.4 6.1 8.6 20.4 vertical 0.547 18.9 7.4 34.6 13.5 11.1 8.5 17.2 ( r r Figure 14 shows the time history of normalized cumulative energy [a(i)2 {y(f)2 u.n i.s 4.0 COMPARISON WITII REGULATORY REQUIREMENTS Envelop of Design Response Spectra Section 3,7.1 of the Standard Review Plan specifies that response spectral values computed from a single artificial time history must envelop the target design spectrum such that no more that 5 points of the response spectrum obtained from the time history fall below the design spectrum, with none more than 10% below. Table 3.7.1-1 of the Standard Review Plan provides an acceptable set of frequencies for computation of the response spectrum. This table is reproduced below. Frequency Range Frequency increment (Hz) (Hz) 0.2 - 3.0 0.1 3.0 - 3.6 0.15 3.6 - 5.0 - 0.2 5.0 - 8.0 0.25 8.0 - 15.0 0.5 15.0 - 18.0 1.,0 18.0 - 22.0 2.0 22.0 - 34.0 3.0 The result is a set of 75 frequencies. Table 1 lists these frequencies together with the control points of the design response spectra defined in Geomatrix (1999). The bold entries in Table 1 indicate these control points. The design spectral values between the control points were obtained by linear interpolation oflog(frequency) versus log (spectral acceleration). Also listed in Table 1 are the 5% damped spectral ordinates computed from the time histories. The spectral ordinates were computed using program SPECTRA. Verification of this program is in a separate report. The time histories and computed response spectra are located in directory \\ FINALIZE on the attached disk. The shaded entries in Table 1 indicate the points where the S COCJAFEwo0054790013V.2KTHC DOC

05996.02-G(pol 8)-3 (REV. 0) Page 8 of 34 time history response spectrum falls below the design response spectrum. For the fault-parallel time history, only one point falls below the 5% damped design spectrum because the fit at other damping levels controlled the time history scaling. The time histories meet the requirements for enveloping the design response spectra. Envelop of Target Power Spectral Density Section 3.7.1 of the Standard Review Plan specifies that if a single time history is to be used, then it must have a power spectral density (PSD) that exceeds 80% of a target PSD. The one-sided PSD, So(,>, is related to the Fouder amplitude spectrum, lF( 3 l by the relationship: 2lF,l' S.) = 24 oc (l) where To is the duration of near maximum and near stationary power of an acceleration time history. Equation (2)in Appendix A of the Standard Review Plan Section 3.7.1 provides a target PSD for the Regulatory Guide 1.60 horizontal response spectrum anchored to 1.0 g PGA. The elationship is: For f < 2.5 Hz S,3 = 650 in / sec '(f / 2.5)"2 2 og For 2.5 5 f < 9 Hz S,(,3 = 650 in ' / sec '(2.5 / f)" II For 9 s f < 16 Hz S,(,3 = 64.8 in / sec '(9 / f)' 2 For f 216 Hz 2 8 S,3 = 11.5 in / sec (16 / f)' og Appendix A of the Standard Review Plan Section 3.7.1 indicates that a target PSD for the Regulatory Guide 1.60 hodzontal response spectrum anchored to an other PGA value can be obtained by multiplying Equation (2) by the square of the design PGA. A target PSD for the Skull Valley 2,000 year design response spectra was obtained by extending this approach. At each frequencyf, the target PSD for the Skull Valley design response spectrum, (So(,3}sy, is obtained by the expression: l e moc.smwxxamonsv.2mc oac j

05996.02-G(PO18)-3 (REV. 0) Page 9 of 34 [)$V ([)RGl.60 h) ~ oM sv " OM Rai 6o where S4(/) is the acceleration response spectral ordinate at frequencyf, and the subscripts SV and RG1.60 refer to the Skull Valley design spectrum and the Regulatory Guide 1.60 horizontal response spectrum anchored to 1.0 g PGA, respectively. A small computer code, TPSD, located in directory \\TPSD on the attached disk was used to scale Equation (2) using Equation (3). The PSD for the time histories shown on Figures 11,12, and 13 were computed using a small computer code, PSD, located in directory \\PSD on the attached disk. The program uses as input a Fourier spectrum of the time hi. story. Each frequency in the Fourier spectrum, Sw is computed using Equation (1). The appropriate duration for nearly constant power, To, was I estimated from the 5%-75% cumulative energy duration of the time histories listed above. The average value for the three time histories is 9.2 seconds, somewhat longer than the average value of 7.1 seconds obtained fcr the original time histories. The Fourier spectra were computed using ) program FT located in directory \\FT on the attached disk. The program outputs columns containing the frequency, real, imaginary, and absolute amplitude Fourier components. Testing of the program indicates that the Fourier amplitude spectrum must be scaled by At to obtain units of g-seconds when the input time history is in g's. The Fourier amplitudes output from program 2 FT were scaled by a factor of 1.9311 = 0.005*386.22 in/sec /g with program PSD. Program . PSD then smoothes the PSD by computing the average value over a frequency window ofi20% off, following the procedure described in Appendix A to Section 3.7.1 of the Standard Review Plan. Figures 19,20, and 21 compare the PSD's computed for the time histories to 80% of the target PSDs shown on Figure 18. All of the target PSDs are enveloped at the 80% level. Component-to-Component Cross-Correlation ASCE Standard 4-86 (American Society of Civil Engineers,1986) recommends that the cross-correlation between the three components of a time history set used in nuclear plant analysis be less that 0.3. EXCEL @ spreadsheet CC.XLS in directory \\ FINALIZE on the attached disk contains columns with the three time histories. The CORREL function was used to compute the zero-lag cross-correlation between the three time histories. The values are: Fault-normal to fault-parallel -0.07 Fault-normal to vertical -0.04 Fault-parallel to vertical 0.02 IWOC,$AFEwo00Sw7e00h5V 2KTHC, DOC

05996.02-G(pol 8)-3 (REV. 0) Page 10 of 34 These values indicate that the time histories are uncorrelated.

5.0 REFERENCES

American Society of Civil Engineers (1986), Seismic analysis of safety-related nuclear structures . and commentary on standard for seismic analysis of safety-related nuclear structures: 1 ASCE Standard 4-86. l Abrahamson, N.A., and Silva, W.J.,1996, Empirical ground motion models: in Silva, W.C., Abrahamson, N., Toro, G., and Costantino, C.,1998, Description and validation of the stochastic ground motion model: Report submitted to Brookhaven National Laboratory, 3 Associated Universities, Inc., New Yor'c. ) Boore, D.M.,1983, Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra: Bulletin of the Seismological Society of America, v. 73, p.1865-1894. Boore, D.M.,1986, Short period P-and S-wave radiation from large earthquakes: Implications i for spectral scaling relations: Bulletin of the Seismological Society of America, v. 76, p. 43-64. Geomatrix Consultants, Inc.,1999, Development of design ground motions for the Private Fuel Storage Facility, Skull Valley, Utah: report prepared for Stone & Webster Engineering Corporation, March,6 p. Lilhanand, K., and Tseng, W.S,1988, Development and application of realistic earthquake time histories compatible with multiple-damping response spectra: Ninth World Conference on Earthquake Engineering, Tokyo, Japan, v. II, 819-824. Silva, W.J., and Lee, K.,1987, WES RASCAL code for synthesizing earthquake ground motions: State-of-the-art for assessing earthquake hazards in the United States: U.S. Army Waterways Experiment Station, Report 24, Miscellaneous Paper S-73-1,120 p. Spudich, P., Fletcher, J.B., Hellweg, M., Boatwright, J., Sullivan, C., Joyner, W.B., Hanks, T.C., Boore, D.M., McGarr, A., Baker, L.M., and Lindh, A.G.,1997, SEA 96 - a new predictive relation for earthquake ground motions in extensional tectonic regimes: Seismological Research Letters, v. 68, p.190-198. U.S. Nuclear Regulatory Commission,1989, Standard Review Plan; Office of Nuclear Reactor Regulation, Revision 2, Document NUREG-0800. 1 I WOC,$AFEM000lN7900hSV 2KTHC. DOC m '

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05996.02-G(pol 8)-3 (REV. 0) Page 28 of 34 3 i i 2 's 1 s n ~ s v s d o .5 'O \\ g N 'e. ~ O O \\ O 2 q 's N 0 \\ M U .1 O 9. CQ .05 5?.' Damping ~ Design ~ Spectrum ' Design Time History .02 .02 - .05 .1 .2 .5 1 2 5 Period (sec) Figure 15 Comparison of Design and Time History Spectro for Fault-normal Component I WOC,$AFEN790 Ol4V-2KTHC DOC

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o 05996.2 G(pol 8-3 (Rev.0) Attachment A Page 1 of19 Non-Stationary Spectral Matching Program RSPMATCH N. A. Abrahamson July 16,1993 Introduction - Various methods have been developed to modify a reference time history so that its response spectrum is compatible with a specified target spectrum. A review of spectral matching methods is given by Preumont (1984). A commonly used method adjusts the Fourier amplitude spectrum based on the ratio of the target response spectrum to the time history response spectrum while keeping the Fourier phase of the reference time history fixed. While this approach is straight-forward, it has two drawbacks. First, it generally does not have good convergence properties, particularly for multiple damping spectra. Second, it can alter the non-stationary character of the time history if the shape of the Fourier amplitude spectrum is changed significantly. f An alternative approach for spectral matching adjusts the time history in the time domain by adding wavelets to the reference time history. A formal 4 optimization procedure for this type of time domain spectral matching was first proposed by Kaul (1978) and was extended to simultaneously match spectra at I multiple damping values by Lilhanand and Tseng (1987,1988). While this l procedure is more complicated than the frequency domain approach, it has good convergence properties and in most cases preserves the non-stationary character of the reference time history. k

05996.2-G(pol 8-3 (Rev.0) Att:chment A Pige 2 of 19 The accompanying program, RSPMATCH, implements the Lilhanand and Tseng algorithm with some modifications that help preserve the non-stationary character of the reference ground motion for a wider range of time histories. Methodology Usually, response spectra studies are only concerned with the maximum response of the oscillator; however, for the time domain methods, the time and polarity of the peak response must also be considered. I will refer to the response to indicate the oscillator time history, not just its maximum value. Let a(t) be the reference time history and Qi e the target spectral value for b frequency mi and damping pi. Also let R be the absolute value of the peak i response, t be the time of the peak response, and P be the polarity of the peak i i response (P = 1 if the maximum response is positive and P = -1 if the maximum i i response is negative). The difference between the target spectrum and the computed spectrum is given by AR = (Oi - R ) Pi (1) i i .where AR is the spectral misfit whichincludes the polarity of the response. i Assuming that the time of the peak response of a(t) will not be perturbed by adding a small adjustment to a(t), the basic method is to determine an adjustment time history,6a(t), such that the response of 6a(t) at time ti s equal to i AR for alli. Let i

i 05996.2-G(pol 8-3 (Rev.0) Attachment A Page 3 of 19 N 6a(t) = { bj fj(t) (2) j i=1 where fj(t)is a set of adjustment functions, bj is the set of coefficients to be determined, and N is the number of spectral points (frequency and damping pairs) to match. A restriction on fj(t)is that fj(t) = 0 for t<0. The acceleration response of 6a(t) for frequency mi and damping pi at time tiis given by r= 6R = ba(r) h (t -T) dr (3) i ii J0 where h (t) is the acceleration impulse response of the oscillator for frequency mi i and damping pi. Substituting Eq. (2)into Eq. (3) gives N f* 6R={b; fj(t) h (t T) dx. (4) i ii j=1 JO The acceleration impulse response is given by exp(cipi)[(2pf-1) sin (wi)-2 VI- [ cos(dt)) (5)

• i h;(t) =

t t i i V1-pf where d = wi V 1-pf (6) i 1 and hi(t) =0 for t<0.

05996.2-G(pol 8-3 (Rev.0) Attachment A Page 4 of19 Next let cij be the response at time t for the ith frequency and damping i resulting from the motion fj(t): t f, fj(t) h (t;.r) dr (7) c;; = i J0 (the upper limit of the integral is t because h (t) =0 for t<0). Substituting Eq. (7) i i into Eq. (4) gives N 6R; = { bj cij (8) j=1 If the response of the adjustment time history,6Ri, is equal to the spectral misfit, ARi, then I N AR = bj cij (9) i al Given the spectral misfits, AR, and the response coefficients cij, Eq. 9 can be i solved for the coefficients bj. In matrix notation, the solution is simply b = C-16R (10) { Given, the bj, the adjustment time history,6a(t), can be computed by Eq. 2. The adjusted time history for the first iteration is given by i 4(t) = a (t) + y 6a(t) (11) ,l o

1 059%.2-G(pol 8-3 (Rev.0) i Attachment A Pcge 5 of 19 where y is a relaxation parameter (between 0 and 1) to damp the adjustments. The algorithm is repeated using the adjusted time history until the desired spectral match is achieved. Selection of the adjustment function - The key to the non-stationarity of the method is the selection of the adjustment function fj(t). The selection of the form of the adjustment function, fj(t), is where the seismological considerations enter the problem. What sort of adjustments can be made and yet still yield realistic seismograms? In considering the forms of fj(t), we need to also consider the numerical stability of ' the algorithm. For the method to work efficiently, the timing of fj(t) should be such that the response of fj(t) is in phase with the peak response of a(t). For numerical speed, fj(t) should be chosen so that the elements of C given by the integral in Eq. 7 can be computed analytically. For numerical stability, the off-diagonal terms of C should be as'small as possible. As mentioned earlier, in the frequency domain approach, only the Fourier amplitude spectrum is modified. This is equivalent to using fj(t) = cos( mjt + 0;) (12) where ej is the Fourier phase of the reference time history (Figure 1). The adjustments are computed from the ratio of the computed response to the target response which is equivalent to using b=6R. The poor convergence of this method results from the simplified estimation of b which ignores the cross-terms

05996.2-G(pol 8-3 (Rev.0) Attachment A Page 6 of 19 in the C matrix ( e.g. it is equivalent to assuming that C is the identity matrix). In addition, the timing of fj may not be in phase with the, teak response of a(t). In the Lilhanand and Tseng algorithm, fj(t) is given by fj(t) = hj(tj - t) (13) which is just the oscillator impulse response in reverse time order. Since h(t)=0 for t<0, this form is one-sided. as shown in Figure 2. There are several numerical aspects that make this function attractive. First, it leads to a symmetric C matrix. Second, the abrupt stop insures that the response will peak at time t and not i resonate to larger values at greater values of t for all damping values. At high frequencies, the f has a short duration, but at short frequencies f has a long i i duration. From a seismological point of view, this form of the adjustment function has some undesirable features, particularly at long periods. The adjustment is emergent and stops abruptly. This is contrary to the behavior of strong motion time histories that generally have a sharp initiation and gradual decay. Forcing the long period adjustment early in the time history can lead to unrealistic ground motions if the long period response peaks early in the record as can be the case for near-fault recordings (dist < 5 km). As an alternative, a tapered cosine wave can be used for the adjustment function: fj(t) = cos{coj(t tj+Atj)} exp{-lt-tj+Atjl aj} (14)

05996.2-G(PO 8-3 (Rev.0) Attachment A Page 7 of 19 where At is the time delay between the maximum of fj(t) and a peak in the response of fj(t). This function is shown in Figure 3. The a term controls the time duration of fj(t). The frequency dependence of a can be estimated from the reference time history which helps to preserve the non-stationary character of the reference time history. That is,if the reference time history has a short duration at a particular frequency, the a should be selected such that the adjustment function at that frequency will also have a short duration. The model for a(f) used here is given by r ai for f < fi f"') a(f) = l ai + (a2-at)(f -f ) for f < f < f2 2 t a2 for f > f: An example of a(f)is shown in Figure 4. Numerical Aspects Conceptually, the time domain spectral matching method appears straight-forward; however, there are some numerical difficulties. The C matrix in Eq. (10) is singular or near-singular for a large number of closely spaced frequencies and multiple damping values. The numerical problem is to find a way to handle this near-singular matrix. Lilhanand and Tseng subdivided the target spectrum into several smaller subsets that each have about 20 to 30 frequency and damping pairs (Lilhanand, personal communication). Each subset should sample the entire frequency range rather than using low frequency, moderate frequency, high frequency,

05996.2-G(pol 8 3 (Rev.0) Attachment A hge 8 of 19 subgroups. For example,if 4 subsets are used, the first subset contains the 1st. 5th,9th,... frequencies. For each subset, the C matrix can be inverted numerically. The inverse of C is computed using singular value decomposition. For numerical stability, the eigenvectors corresponding to the small eigenvalues are removed. Since the elements of C are computed analytically, the time step of the reference time history should be small enough so that the numerical calculation of the response is close to the analytical calculation. For most cases, this condition can be met using 200 samples /sec. Specifying the Target Spectrum The method makes narrow band modifications to the time series. Therefore, it is important to use a fine enough frequency sampling to ensure that the response spectmm at frequencies not matched will remain smooth. Based on the bandwidth of the oscillator response, about 30 frequencies per decade (equally spaced on the log frequency axis) is sufficient. If multiple damping are specified, then the relative levels of the spectra at the different damping values must be realistic for the program to converge. Acknowledgments I thank Kiat Lilhanand for providing explanations of his algorithm. References Kaul, M. K. (1978). Spectrum-consistent time-history generation (1978). ASCE J. Eng. Mech., EM4, 781-788. ~. i

05996.2-G(pol 8-3 (Rev.0) Attachment A Page 9 of 19 Lilhanand, K. and W. S. Tseng (1987). Generation of synthetic time histories compatible with multiple. damping response spectra, SMiRT-9, Lausanne, K2/10. Lilhanand, K. and W. S. Tseng (1988). Development and application of realistic earthquake time histories compatible with multiple damping response spectra, Ninth World Conf. Earth. Engin., Tokyo, Japan, Vol II,819-824. 1 Preumont, A. (1984). The generation of spectrum compatible accelerograms for ) { the design of nuclear power plants, Earth. Engin. Struct. Dyn.,12,481-497. 1

~ 05996.2-G(PO!8-3 (Rev.0) Attachment A P:ge 10 of 19 Figure Captions Figure 1. Adjustment time history (top) and corresponding oscillator response (bottom) used in frequency domain methods. This makes stationary adjustments and can significantly change the character of the time history. (A) 5 Hz (B) 0.2 Hz Figure 2. Adjustment time history (top) and corresponding oscillator response (bottom) for Eq. (13). For this example, the peak time is at t =10.0 i seconds. This model produces a symmetric oscillator response. For high frequencies, the adjustment time history has a short duration. For low frequencies, the adjustment time history has a long duration. If the long period response peaks early in the time history, then this model can build up unrealistic long period energy at the start of the time history. (A) 5 Hz (B) 0.2 Hz Figure 3. Adjustment time history (top) and corresponding oscillator response (bottom) for Eq. (14). For this example, the peak time is at t =10.0 i seconds. This model uses a symmetric time history which produces an asymmetric oscillator response. Compared with Figure 2, this model makes a long period adjustment over a longer time interval. (A) 5 Hz (B) 0.2 Hz Figure 4. Functional form for a(f) used in model 6 to taper the cosine wave. ..,4, 4

lL"lL29"lh";TN rnodel 0. 5 Mr fy = 10 Sec Adjustment Time History 5 Hi 2. .i....i....i....i.... O. ,,,,r,,,,i.,,,i,,,,i ...i.... .0 5 10 15 20 25 30 Time.(sec) Oscillator Response ygpgg 0. -12. ~ O 5 10 15 20 25 30 Time (sec)

05996.2-G(pol 8-3 (Rev.0) Attachment A Pese 12 of19 model 0,0.2 H2 Adjustment Time History o, gg 2. ....i....i....i....i....i.... O. 'i'i'i'i'i' -2. O 5 10 15 20 25 30 Time (sec) Oscillator Response 10. ....i....i....i....i.... O. \\ 4 k k I k f l f i 0 h 0 5 10 15 20 25 30 Time (sec)

05996.2-G(pol 8-3 (Rev.0) Attachment A Page 13 of 19 mose:1, sa: bc = lO se.c Adjustment Time History r na 29. >><>l ><>>l >>g ill! ^::.',,'lll 0. -29. O 5 10 15 20 25 30 Time (sec) Oscilator Response 158. g >>.l l l. > >. s... .. i' llli,,,. O. s, - -158. O 5 1"O 15 20 25 30 Time (sec) 7A

05996.2-G(pol 8-3 (Rev.0) Attachment A Page 14 of 19 rresel 1. 0.1 Hz Adjustment Time History 02 g3 2. ....i....i....i....i....i.... O. -2. ' ' ' ' ' ' ' ' ' i '~' ' ' ' ' ' ' ' ' ' '.' ' 'f O 5 10 15 20 25 30 Time (sec) Oscilator Response s. i....i....i....i....i.... l 0. -6. 0 5 10 15 20 25 30 Time (sec) w

05996.2-G(PO13-3 (Rev.0) Attachment A Page 15 of 19 .moel s s Hz /o.sec. ft = Adjustment Time History sHs 2. ....,....i....i....,....i.... il j .llllllI'l 0. -2. O 5 10 15 20 25 30 Time (sec) { ) Oscillator Response 10. ll l!... 4 O. p.- -10. 0 5 ~10 15 20 25 .30 Time (sec) w

05996.2-G(pol 8-3 (Rev.0) Attachment A Page 16 of 19 model s. o.2 Hz Adjustment Time History 0.1 % 1. \\ s> g n O. ,,,,I I I I 0 5 10 15 20 25 30 Time (sec) Oscillator Res3onse 4. s... ... s > s -- g l >l l .. s. l 0. -4. O 5 10 15 20 25 .30 Time (sec) t

059%.2-G(PO18-3 (Rev.0) Attachment A Page 17 of 19 i l gg j i I 6 a l l 0 4 8 6 9 9 ai l 8 0 0 4 0 g I e l 3 e i f1 f2 Frequency r0 q k

g ge6 fPT5 f kia i 3 58095721 81 1 7983530 %2846275323572761 80 0 9.8.8 7.7.6 6 6.6.6 6 6.7. 7.8.9.9.0 2 000000000000000001 543500039202666960 %3060731 9891 8280 5 9.9.8 8.7.7.7 6 6 6.7.3 7 17.7.8 8 9 9.0 1 000000000000000001 ~ 971 539765781 0621 20 %420641 099902572590 )ec 0 9 9 9 8 8 8 8 7 7 7 8 8 8 8.9.9.9.0 1 n 000000000000000001 se ed un l e ap 838098398940359470 Ve %76431 0099901 235790 aD 7 9 9.9 9.9.9.9.8 8 8.9.9.9.9.9.9.9.0 l 000000000000000001 r e tc c en pa t S s 9659409801 68991 570 i dD %468024456654207400 e 3 0.0011 11 1 11 11 1 1 0000 pr 11 1 1 1 11 111 1 1 11 1 1 1 1 o m e a d Du 9790552071 63044660 t %i %81 491 468898740381 0 n 5 g 2 01 1 1.2.2.2 2.2 2.2.2.2 2 1 0 0 0 oa 111111111111111 111 tsM ot it u ao 61 1 790330584436790 Rh %494040379009485630 i 1 2.3.3 4 4 4 4.5 5.4 4.3 2. 1 0 0 t 11 W 111 111 11 11 11 1 1 1 111 ( %241 820687925990280

5. 1 2 3.3 4.5.i",6 6 6 7 7 6.5.3 2.0 0 841 952 2581 0469670 0

1 1 1 11 1 11 11 1 1 1 1 1 1 11 5 5 d 5 5 75432 o 5 7 5 4.3.2 1 1 0 0.0.0.0 ir54321 1 000000000000 ep

05996.2-G(PO13-3 (Rev.0) Attachment A Page 19 of19 Spectral Ratios Factors for Horizontal Ground Motions 0.5%, 1%, 2%, 3%, 7%, 10%, 15%, 20% 2 m 1 1.8 - 1.6 / x 1.4 - / E 2B / m\\ 7 8.1.2 N E E 1 N e $0.8 k 5 0.6 i G 0.4 W e 0.2 e 0 0.01 0.1 1 10 Period (sec)

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