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Generation of uniform-hazard earthquake ground motions

Henri P. Gavin, M.ASCE 1 and Bryce W. Dickinson, A.M.ASCE 2

ABSTRACT

This paper presents statistical models for the generation of bi-axial earthquake ground mo-

tion time histories with spectra that match those from samples of ground motion records. The

model parameters define near-field characteristics such as pulse velocity and pulse period, far-fault

characteristics such as velocity amplitude and power spectral density, and envelope characteris-

tics. The samples of ground motions used in this study were previously selected and scaled to be

representative of particular hazard levels in particular geographical regions. A companion paper

presents the fitting of the model to samples of ground motion waveforms. In this paper, the new

concept of a parameter-response correlation spectrum establishes the period-dependence of the

correlation between the response spectrum and ground motion parameters. Parameters that corre-

late to variability of the response spectra are retained as random variables and are then fit to meanand mean-plus-standard deviation of bi-axial response-spectra of the samples of historical records.

Parameter statistics also include correlations between velocity amplitudes and pulse periods.

INTRODUCTIONPeak seismic response statistics of structures that behave with certain nonlinearities including,

but not limited to, discontinuities (e.g., impact and fracture), may be poorly represented by com-

mon distributions (e.g., lognormal). In such cases the computation of peak response distributions

requires many simulations; the sample size required to characterize the peak response distribution

may exceed the of number of available natural ground motions characteristic of the hazard at a site.

Nevertheless, as demonstrated by Dickinson and Gavin (2010) and others, parameters character-

izing a sample of natural ground motionsarewell-characterized by lognormal distributions. Suchparametric ground motions models may then be used to generate any number of additional ground

motions representative of the sample.

The suites of ground motion records developed for the SAC Steel Project were intended to

represent ground motions with an exceedance probability (e.g., 2% in 50 year or 10% in 50 year)

for a representative geographical region (e.g., Los Angeles or Seattle) for a NEHRP site category

SD(firm soil). In addition, a data set representative of near-fault ground motions near Los Angeleswas developed. Ground motions were selected and scaled based upon an approximate deaggrega-

tion for each region and local soil condition, with some preference toward records from nearby or

representative ruptures (Somerville et al. 1997). Each pair of horizontal components was scaled

by a common scaling factor. The scaling factor minimized the weighted sum of the squares of the

differences between the geometric mean spectrum from the records and a 1995 uniform-hazard

spectrum for a NEHRP SD site in the selected region. The spectra were matched at periods of0.3, 1.0, 2.0 and 4.0 s with associated weights of 0.1, 0.3, 0.3, and 0.3. In this sense each pair of

ground motions was scaled to approximate a uniform-hazard spectrum (Somerville et al. 1997).

Each suite provides a sample of ten bi-axial ground acceleration records, with the exception of

1Dept. Civil & Environ. Engg, Duke Univ., Box 90287, Durham NC 277082Rutherford & Chekene, 55 Second Street, Suite 600, San Francisco, CA 94105


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the near-fault set which contains twenty tri-axial records. Within each suite paired horizontal

ground motion records are rotated 45 degrees from their fault-normal and fault-parallel directions

(Somerville et al. 2009). Well over one hundred published studies have made use of this ground

motion database, e.g., (Chang et al. 2002), (Lee and Foutch 2002), (Wen and Song 2003), (Yun

et al. 2002).

A companion study (Dickinson and Gavin 2010) examined the wave form characteristics of fiveof these data sets: Los Angeles 10% in 50 yr (la10in50), Los Angeles 2% in 50 yr (la2in50),

Seattle 10% in 50 yr (se10in50), Seattle 2% in 50 yr (se2in50), and the near-fault (nrfault)

data sets. Each of the twenty or forty records in each of the five data sets were individually and

systematically fit via constrained non-linear least squares methods in order to determine estimates

and standard errors for eleven ground motion parameters. The eleven parameters describe the

power spectral density of the stochastic content of the record (central ground frequency, fg, anddampingg), the envelope of the stochastic part of the record (rise time, 1, constant time,2, anddecay time, 3), the peak velocity of the stochastic part of the record (Vs), and the pulse (peak pulsevelocity, Vp, pulse period, Tp, number of pulse cycles, Nc, phase of the pulse, , and the pulsearrival time,Tpk). Ground motion parameter estimates and their standard errors result in empiricalcumulative distribution functions and distribution statistics (mean and variance) for each parameter

for each data set (Dickinson and Gavin 2010).

Synthetic ground motion records are generated from model parameters as follows. A stochastic

acceleration record is generated from power spectral density parameters (fg, g) over a specifiedfrequency range (flo to fhi) by the method of spectral representation (Shinozuka and Jan 1972;Shinozuka and Deodatis 1991). The stochastic acceleration is then time modulated with an en-

velope function (1, 2, 3) and integrated (in the frequency-domain) to give a stochastic velocityrecord. The stochastic velocity record is scaled to the specified amplitude,Vs. A velocity pulse(Tp, Vp, Nc, Tpk, ) is generated and added to the stochastic velocity. The combined velocity isthen differentiated (in the frequency-domain) to give the final synthetic ground motion accelera-

tion. Frequency components above floare used in the frequency-domain integration of accelerationrecords. Frequency components belowfhiare used for the differentiation of velocity records. Ex-pressions used in the waveform synthesis are given in (Dickinson and Gavin 2010).

The parameter estimates developed in the companion study demonstrate that the variability

of wave form characteristics represented in each data set are well-characterized by lognormal

distributions. While these distributions provide insight into the temporal characteristics of each

data set, the effect of these earthquake ground motions on structures is of primary importance for

performance-based design. This study examines the parametrization of five of the SAC data sets

in terms of response spectra.

PARAMETER-RESPONSE CORRELATION SPECTRA

The relationships between ground motion parameters and structural response parameters havebeen investigated recently through a variety of methods (Cordova et al. 2001), (Baker and Cor-

nell 2006), (Baker and Jayaram 2008), (Baker 2007). Correlation analysis between ground motion

parameters and structural response parameters gives a clear picture of the effect that various earth-

quake ground motion characteristics have on structural response. The new concept of a parameter-

response correlation spectrum is introduced here in order to investigate the association between

response spectra and ground motion parameters. Parameter-response correlation spectra give the

statistical correlation between a ground motion waveform parameter and the peak response of a

2 H.P. Gavin and B.W. Dickinson


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structure of a given natural period and damping. These spectra are computed by generating many

(a thousand or more) ground acceleration records using parameters drawn from (uncorrelated) sta-

tistical distributions of ground motion model parameters. For each generated set k of parameters,pk, a corresponding ground motion and its response spectrum, sk, are computed and the covari-

anceVPi,Sj between parameterPiand response spectrum valueSj (at natural periodTj) is updated

recursively,

VPi,Sj ,k =k2

k1VPi,Sj ,k1+

k

(k1)2(pi,k pi,k)(sj,ksj,k), (1)

where the averages of each parameter and response spectrum ordinate are also updated recursively,

pi,k =k1

k pi,k1+

1

kpi,k , (2)

sj,k =k1

k sj,k1+

1

ksj,k. (3)

The Pearson correlation between parameterPi and response spectrum value Sj (at natural period

Tj) isPi,Sj =

VPi,SjVPi,Pi

VSj ,Sj

. (4)

In this study, response spectra and correlation spectra for linear elastic structures with five

percent damping are evaluated. Correlation spectra for acceleration response and displacement

response are identical. This should not be the case for responses of inelastic structures.

Parameters with correlation values within the range of10 percent over the entire range ofnatural periods do not systematically affect response variability, and are treated as constants in

the development of statistical models consistent with the historical earthquake ground motion data

sets.

The correlation spectra in this study are generated from 1000 ground motion simulations foreach of the five SAC data sets considered. In this analysis, the eleven ground motion parameters

were taken as uncorrelated lognormal random variables with means, variances, and offsets obtained

from the fits of the original SAC ground motion waveforms (Dickinson and Gavin 2010). In

order to capture the transient response of long-period structures, the simulated earthquake ground

motions had a minimum duration of sixty seconds. Padding earthquake ground accelerations with

zeroes for the first five seconds reduces errors associated with the frequency-domain integration

method used here (Dickinson 2008), (Boore 2005).

Correlation spectra for each of the eleven parameters are shown in Figure 1. In Figure 1, |R|maxrefers to the spectral displacement from a single-component ground motion record. Results from

each data set for each parameter are plotted together. The ground motion parameters that correlate

most significantly to peak response statistics (in order of decreasing correlation) are: the peak ve-

locity of the stochastic part of the ground motion,Vs, the peak pulse velocity,Vp, the pulse period,Tp, the central frequency of the stochastic ground motion, fg, and the number of cycles in the pulse,Nc. Correlation between the stochastic ground velocity,Vs, and peak response characteristics aresignificant for structures of all periods (0.1s< Tn 0.1), nonetheless. The pulse velocityamplitude,Vp, is significantly correlated to the responses of long-period structures (Tn >1 s) and



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is significantly correlated to the response of structures of all periods for the nrfault data set. The

pulse period,Tp, is negatively correlated to peak responses at periods from 1 to 3 seconds and ispositively correlated at periods from 3 to 10 seconds, especially for the nrfault, la2in50, and

la10in50 suites. The central frequency of the stochastic content,fg, is somewhat correlated tothe peak responses of short-period structures (Tn < 1 s), and the number of cycles in the pulse,

Nc, is only slightly correlated to peak responses of long-period structures (Tn > 1.5 s) for thela2in50 and the la10in50 suites.

Correlations of peak responses to the other ground motion parameters are within10percent.The importance of the pulse phase, , has previously been found to be insignificant (Mavroeidisand Papageorgiou 2003), (Mavroeidis et al. 2004). This result is confirmed here.

Based on these correlation spectra, a ground motion model involving a reduced number of ran-

dom variables is proposed. The ground motion variables that are important to the response of linear

elastic structures (with five percent critical damping) are Vs,Vp,Tp, andfg. The variability in peakresponses due to variability in the remaining ground motion parameters is small in comparison.

CORRELATIONS BETWEEN HORIZONTAL VELOCITY COMPONENTS AND PULSE

PERIODThe transient response analysis of asymmetric structures (with lateral-torsional modes) requires

the specification of ground motions as paired horizontal components. This study adopts the des-

ignations X and Y for the paired horizontal components. The X and Y components of the

pulse velocity,VpXand VpY, and the peak stochastic velocity,VsXand VsY, are taken as identically-distributed and partially-correlated parameters. For physically-motivated reasons, the pulse period,

Tp, and the ground frequency, fg, for ground motions in the X and Y directions are considered to beidentical. Further, the ground frequency is assumed uncorrelated with the other five random vari-

ables. The bi-axial ground motion model therefore involves six random variables, five of which

are correlated.

Correlations amongVpX,VpY,Tp, VsX, andVsY, are estimated from previously-identified pa-rameter values (Dickinson and Gavin 2010). For a data set containing m X records and m Yrecords, themparameter estimatesVpXi,VpYi,TpXi,TpYi,VsXi, andVsYi, are arranged in a2m-by-5 matrixP,

P=

VpX1 VpY1 TpX1 VsX1 VsY1VpX1 VpY1 TpY1 VsX1 VsY1VpX2 VpY2 TpX2 VsX2 VsY2VpX2 VpY2 TpY2 VsX2 VsY2

... ...

... ...

...

VpXm VpYm TpXm VsXm VsYmVpXm VpYm TpYm VsXm VsYm

(5)

Values for the X and Y components of the pulse velocity and stochastic velocity parameters are

repeated for each pulse period value. Since offset lognormal distributions represent the variability

of these parameters, their standardized values are given by the 2m-by-5 matrixZ,

Z= 1

p

ln [P 1po] 1

lnp

1

2ln

p

p+ 1

, (6)

where po is row-vector of parameter offsets given in Table 3 of (Dickinson and Gavin 2010), 1

is a2m-by-1 vector of 1s, pis a row-vector containing the average of the columns of the matrix



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TABLE 1. Correlation matrices for the bi-axial model

VpX VpY Tp VsX VsY

nrfault

VpX 1 0.35 -0.51 0.20 0.10

VpY 0.35 1 -0.08 -0.03 -0.04

Tp -0.51 -0.08 1 -0.11 -0.09

VsX 0.20 -0.03 -0.11 1 0.68

VsY 0.10 -0.04 -0.09 0.68 1

la2in50

VpX 1 0.75 -0.58 -0.08 -0.25

VpY 0.75 1 -0.62 -0.65 -0.68

Tp -0.58 -0.62 1 0.32 0.43

VsX -0.08 -0.65 0.32 1 0.84

VsY -0.25 -0.68 0.43 0.84 1

la10in50

VpX 1 0.52 -0.26 -0.17 0.03

VpY 0.52 1 -0.51 0.01 0.33Tp -0.26 -0.51 1 -0.05 -0.39

VsX -0.17 0.01 -0.05 1 0.70

VsY 0.03 0.33 -0.39 0.70 1

se2in50

VpX 1 0.94 -0.60 -0.43 -0.23

VpY 0.94 1 -0.62 -0.48 -0.16

Tp -0.60 -0.62 1 -0.18 -0.18

VsX -0.43 -0.48 -0.18 1 0.55

VsY -0.23 -0.16 -0.18 0.55 1

se10in50

VpX 1 0.82 0.08 -0.46 -0.61

VpY 0.82 1 -0.05 -0.56 -0.38Tp 0.08 -0.05 1 0.01 -0.27

VsX -0.46 -0.56 0.01 1 0.44

VsY -0.61 -0.38 -0.27 0.44 1

[P1po], andp is a row-vector containing the unbiased standard deviation estimate of the columnsof the matrix [P1po]. The correlation matrix ofZ is the correlation matrix for the five parameters.This method captures correlations between the individual measurements of pulse period and each

of the X and Y components of the pulse amplitude and the peak stochastic velocity.

The resulting 5-by-5 correlation matrices for each data set are given in Table 1. As would be ex-

pected, pulse amplitudes (VpXand VpY) and peak stochastic velocities (VsXand VsY) are positivelycorrelated. Standardized pulse periods are negatively correlated with standardized pulse velocities.

The correlations between peak stochastic velocities and pulse parameters depend substantially on

the data set from which they were estimated.

GENERATING SAMPLES OF GROUND MOTION PARAMETERS AND WAVEFORMSGiven means, variances, offsets, and correlations for lognormal waveform parameters corre-

sponding to a particular ground motion suite, a sample of waveform parameters may be generated.

The lower Cholesky factor, L, of the standardized parameter correlation matrix is used to generate



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correlated standard normal samples, Lz, from which correlated lognormal samples for VpX,VpY,Tp,VsX, andVsY are computed.

p= po

+p exp

Lz

ln

p

p+ 1

1

2ln

p

p+ 1

(7)

The offset values, po

, listed for Vsand Tpin Table 2 enforce a non-zero minimum value for theseparameters. Extremely large values ofVp,Tp, andVs are avoided by specifying that the generatedparameter fall below an allowable limit,

Vp < (Vp) + n(Vp), (8)

Tp < (Tp) + n(Tp), (9)

Vs < (Vs) + n(Vs), (10)

where the number of standard deviations,n, should be between 3 and 4. Additionally, the arrival

time of the pulse is restricted such that its arrival is neither too early nor too late to meet physicalexpectations about impulsive behavior of ground motions. The pulse arrival time should be later

than the start of the strong ground motion and within the first quarter of the duration of the record.

(1+ TpNc)/2< Tpk< (1+ 2+ 3)/4. (11)

If a set of random variables does not meet these requirements, the set is discarded and another set

of parameters are generated. Given a sample of ground motion parameters, a sample of associated

waveforms may be generated using the method described briefly in the Introduction and completely

in (Dickinson and Gavin 2010).

RESPONSE SPECTRA COMPATIBLE MODEL FOR BI-DIRECTIONAL GROUNDMOTIONSome adjustment to the parameter statistics, informed by correlation spectra, is required to fit

the ground motion model to response spectra. In the fitting of the ground motion model to response

spectra of the original data sets, parametersNc,Tpk,,1,2, and3were fixed to the mean valuesshown in Table 2. The means and variances ofVp,Tp,Vs, andfg, and deterministic values ofg,flo, andfhiwere adjusted (by trial and error) in order to match the mean and variance of the SACresponse spectra.

The average and standard deviation of bi-axial response spectra (with 5% of critical damping)

were calculated for each of the original SAC data sets using the paired horizontal ground motion

records. In this study the bi-axial spectral response,max |R(Tn)|, is the peak magnitude of the

response componentsrx(t; Tn)andry(t; Tn)(in the X and Y directions),

max |R(Tn)|= maxt

r2x(t, Tn) + r

2y(t; Tn)

1/2, (12)

where the responses rx(t; Tn)and ry(t; Tn)are the responses in the X and Y directions of a bi-axialoscillator of natural periodTn (and 5% critical damping) to ground accelerations applied simulta-neously in the X and Y directions. Similarly, bi-axial response spectra were calculated for 1000

pairs of synthetic ground motions drawing ground motion parameters from the sample statistics



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for each SAC data set. Within each bi-axial time history analysis total acceleration responses,

max |A(Tn)|, and relative displacement responses, max |D(Tn)|, were computed independently.Means and standard deviations of the random ground motion parameters were adjusted to match

the response spectra for each ground motion data set, in terms of its mean and variance of accel-

eration and displacement response. In an effort to realistically model potentially large motions,

priority was given to matching the mean and the mean plus one standard deviation responsespectra. Figure 2 qualitatively describes the sensitivity of acceleration and displacement response

spectra to variation in ground motion parameter statistics. The pulse velocity was adjusted to match

the response at long periods and the peak stochastic velocity was adjusted to match the response at

short periods. Increasingfg increases acceleration response and decreases displacement response.Increasing Vpincreases the responses of structures with natural periods around Tp. IncreasingTpincreases the range of natural periods affected by the pulse, and increasing Vpand Vsincreasesthe standard deviation of the response spectrum.

Spectral responses of the SAC records at short and long periods indicate that the frequency

limits for the stochastic part of the ground motion, floand fhi, should be included as parameters inorder to match response spectra. To match the Los Angeles data sets, the frequency range should

be 0.1 Hz to 9 Hz and to match the Seattle data sets, the frequency range should be 0.1 Hz to 20 Hz.

The resulting adjusted parameter means and standard deviations for the bi-axial response spectra-

compatible model are shown in Table 2. As compared to parameter values fit to waveforms, the

pulse velocity decreased and the peak stochastic velocity increased (with the exception of the nr-

fault data set). On the other hand, the variability in the pulse velocity increased (except for the

nrfault data set). For the nrfault suite, the mean pulse velocity, mean ground damping, and the

variance of the ground frequency fit to waveforms are significantly lower than those fit to response

spectra. For the nrfault records, the variance in the pulse velocity fit to waveforms is larger than

that fit to response spectra. The adjustments to pulse period statistics were minimal and showed

no uniform trends. The mean and variance of the ground frequency increased for the Seattle data

sets. These results indicate that, except for near-fault scenarios, fitting ground motion parametersto wave forms tends to overestimate pulse amplitudes and underestimate the variability in pulse

amplitudes, as compared to parameters fit to response spectra.

The bi-axial response spectra for the SAC data sets are compared to spectra from the param-

eterized model in Figures 3 through 7. In most cases, the mean minus one model spectra also

match the mean minus one spectra from the original data sets. Although not shown here, re-

sponse spectra for 2% and 20% damping show that the model is response-spectrum-compatible at

multiple levels of damping (Dickinson 2008).

The sequence of generating synthetic earthquake ground motions for two of the five ground

motion suites is illustrated in Figures 8 and 9. Figure 8 illustrates the synthesis process for the

nrfault parameters and Figure 9 illustrates the process for the se10in50 parameters: the limiting

cases of the most and least significant pulses. The power spectral density of the synthetic groundmotion is compared to the target power spectral density as defined by the ground frequency and

ground damping parameters. Note that the significant variability among power spectra computed

from a set of synthetic records is due to their short and transient nature. In many cases, long period

and high amplitude peaks occur in the stochastic velocity record. Many of these peaks are similar

to peaks in the original data set and would have been identified as a velocity pulse in fitting the

model to waveforms. Such occurrences directly support the conclusion that fitting ground motion

parameters to waveforms can overestimate pulse amplitudes and underestimate the amplitude of



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TABLE 2. Model parameters for 2D response-spectrum-compatible records.

suite parameter Vp Tp Nc Tpk Vs 1 2 3 fg g flo fhi

cm/s s - s rad cm/s s s s Hz - Hz Hz

nrfault mean 115 1.5 1.4 5.0 0 60 5.0 0.5 5.1 0.7 2.0 0.10 8

std.dev. 40 1.2 0 0 0 35 0 0 0 0.2 0 0 0

la10in50 mean 20 1.5 1.3 5.0 0 60 4.7 0.2 5.5 1.0 1.8 0.07 9

std.dev. 35 1.0 0 0 0 25 0 0 0 0.1 0 0 0

la2in50 mean 50 1.7 1.2 4.8 0 120 5.5 0.5 4.8 0.7 1.2 0.07 9

std.dev. 60 1.0 0 0 0 40 0 0 0 0.1 0 0 0

se10in50 mean 10 2.1 1.3 6.1 0 25 13.2 0.2 11.4 1.8 0.9 0.10 20

std.dev. 20 0.8 0 0 0 15 0 0 0 0.2 0 0 0

se2in50 mean 40 1.0 1.7 6.6 0 60 16.7 0.2 11.6 1.7 1.0 0.10 20

std.dev. 12 1.0 0 0 0 35 0 0 0 0.2 0 0 0

offset 0 0.8 0 0 0 10 0 0 0 0 0 0 0

the stochastic velocity content.

The adjusted ground motion parameter statistics provide a bi-axial ground motion model cal-

ibrated to the ground motions of the SAC data sets. This model contains the means, standard

deviations, correlations, and offsets for ground motion waveform parameters. It also specifies the

frequency range required for generation of stochastic ground motion content. The model captures

the mean and variability of response spectra through six random variables and is shown to be ac-

curate in terms of displacement and acceleration responses for a range of natural periods from 0.1

sec to 10 sec, and for damping ratios from 2 percent to 20 percent. This model is implemented in

Matlab code available at: http://www.duke.edu/hpgavin/groundmotions/.

SUMMARY AND CONCLUSIONSSynthetic ground motions are generated via the superposition of a long-period pulse with a

stochastic acceleration record generated from a power spectral density function and modulated

by an envelope function. In this study, parameter statistics have been calibrated to the mean and

variance of response spectra from a number of ground motion data sets, but could be fit to otherspectra as well. Synthetic ground motion records generated using this model qualitatively match

the wave form characteristics of the original records as well as their response spectra.

Parameter-response correlation spectra, as introduced in this paper, illustrate correlations be-

tween the parameters of synthetic ground motion models to peak structural responses as a function

of natural period. Such spectra indicate the association between peak structural responses and

parameters of synthetic ground motion models. In this study, a ground motion model having thir-

teen parameters (pulse velocity, pulse period, number of pulse cycles, pulse phase, arrival time

of the pulse, peak stochastic velocity, central frequency of the stochastic content, bandwidth of

the stochastic content, envelope rise time, envelope constant time, envelope decay time, low fre-

quency limit, and high frequency limit) was fit to spectra from five sets of earthquake ground

motion records. The pulse velocity, the pulse period, the peak stochastic velocity, and the groundfrequency correlate significantly (greater than ten percent) to peak structural responses. These pa-

rameters are random variables in the ground motion model while the remaining parameters take

constant values.

The resulting bi-axial ground-motion model includes correlations between horizontal compo-

nents of the pulse velocity and random velocity amplitude, and the direction-independent pulse

period. The bi-axial model was used to simulate 1000 synthetic accelerations for each data set and

the average response spectra for these ground motions were calculated and fit to corresponding



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spectra of the ground motion data sets. The mean pulse amplitudes were less than those fit to the

waveforms, except for the near-fault data set. The stochastic velocity amplitudes have a coefficient

of variation (c.o.v.) of approximately 50%. The c.o.v. of the pulse velocity amplitude for the Los

Angeles and the Seattle 10% in 50 years sets was approximately 100%. The c.o.v. for the ground

frequency for all data sets is approximately 10%.

The methods described in this paper should be extended to confirm the ground motion modelfor inelastic response spectra before application to hysteretic or collapse-sensitive structures. It

is anticipated that duration parameters such as number of cycles of a pulse and envelope param-

eters will have a more significant impact on the response of these structures and would need to

be included in the set of random variables of the model. The correlation spectra for acceleration

response and displacement response were nearly identical for the 5% damped, elastic case. It is

expected that this would not be the case for inelastic structures, and correlation spectra for accel-

eration and displacement response of inelastic structures would need to be evaluated separately.

ACKNOWLEDGMENTSThis material is based upon work supported by the the Civilian Research and Development

Foundation for the Independent States of the Former Soviet Union (CRDF) under Award No.

MG1-2319-CH-02 and by the National Science Foundation under Grant No. NSF-CMMI-0704959

(NEES Research), and Grant No. NSF-CMS-0402490 (NEES Operations). Any opinions, find-

ings, and conclusions or recommendations expressed in this material are those of the authors and

do not necessarily reflect the views of the National Science Foundatioon.

The authors express sincere thanks to the reviewers, whose perceptive comments were instru-

mental in revising this manuscript.

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-1

-0.5

0

0.5

1

0.1 1 10

Vp-|R|m

ax

correlation

natural period, sec

(a)

nrfaultla10in50la2in50

se10in50se2in50

-1

-0.5

0

0.5

1

0.1 1 10

Tp-|R|m

ax

correlation

natural period, sec

(b)


se10in50se2in50

-1

-0.5

0

0.5

1

0.1 1 10

Nc-|R|max

correlation

natural period, sec

(c)


se10in50se2in50

-1

-0.5

0

0.5

1

0.1 1 10

Tpk-|R|max

correlation

natural period, sec

(d)


se10in50se2in50

-1

-0.5

0

0.5

1

0.1 1 10

-

|R|max

correlation

natural period, sec

(e)


se10in50se2in50

-1

-0.5

0

0.5

1

0.1 1 10

Vs-|R|max

correlation

natural period, sec

(f)


se10in50se2in50

-1

-0.5

0

0.5

1

0.1 1 10

1-|R|max

correlation

natural period, sec

(g)


se10in50se2in50

-1

-0.5

0

0.5

1

0.1 1 10

2-|R|max

correlation

natural period, sec

(h)


se10in50se2in50

-1

-0.5

0

0.5

1

0.1 1 10

3-|R|max

correlation

natural period, sec

(i)


se10in50se2in50

-1

-0.5

0

0.5

1

0.1 1 10

fg-|R|max

correlation

natural period, sec

(j)

nrfault

la10in50la2in50se10in50se2in50

-1

-0.5

0

0.5

1

0.1 1 10

g-|R|max

correlation

natural period, sec

(k)

nrfault

la10in50la2in50se10in50se2in50

FIG. 1. Correlation spectra between ground motion parameters and peak structuralresponse statistics for five of the SAC ground motion data sets. Ground motion

parameters in order of decreasing correlation are: Vs,Vp,Tp,fg, andNc.



12/17

peak response displacement

peakresponseacceleration

averag

eresponsesp

ectrum

avera

ge+

1std

.dev.

responsesp

ectrum

responsesp

e

ctrum

average

1std

.dev.

fg

2

V

V

Tp

Tp

2

V pVp

s

s

FIG. 2. Response spectra sensitivity to input parameters



13/17

0.5

1

1.5

2

2.5

3

3.5

4

0.1 1 10

max|A|,g

natural period, s

50

100

150

200

250

0.1 1 10

max|D|,cm

natural period, s

0

0.5

1

1.5

2

2.5

3

3.5

4

0 50 100 150 200 250

max|A|,g

max|D|, cm

nrfault

0.20.5s 1.0s

2.0s

5.0s

FIG. 3. nrfault bi-axial linear elastic response spectra, 5% damping, mean,mean+std.dev, mean-std.dev,=SAC records, =model.

0.5

1

1.5

2

2.5

3

0.1 1 10

max|A|,g

natural period, s

20

40

60

80

100

120

0.1 1 10

max|D|,cm

natural period, s

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100 120

max|A|,g

max|D|, cm

la10in50

0.2 0.5s 1.0s

2.0s

5.0s

FIG. 4. la10in50 bi-axial linear elastic response spectra, 5% damping, mean,

mean+std.dev, mean-std.dev,=SAC records, =model.



14/17

0.5

1

1.5

2

2.5

3

3.5

4

0.1 1 10

max|A|,g

natural period, s

20

40

60

80

100

120

140

160

0.1 1 10

max|D|,cm

natural period, s

0

0.5

1

1.5

2

2.5

3

3.5

4

0 20 40 60 80 100 120 140 160

max|A|,g

max|D|, cm

la2in50

0.2 0.5s 1.0s

2.0s

5.0s

FIG. 5. la2in50 bi-axial linear elastic response spectra, 5% damping, mean,mean+std.dev, mean-std.dev,=SAC records, =model.

0.5

1

1.5

2

2.5

0.1 1 10

max|A|,g

natural period, s

5

10

15

20

25

30

35

40

45

0.1 1 10

max|D|,cm

natural period, s

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30 35 40 45

max|A|,g

max|D|, cm

se10in50

0.2 0.5s

1.0s

2.0s

5.0s

FIG. 6. se10in50 bi-axial linear elastic response spectra, 5% damping, mean,

mean+std.dev, mean-std.dev,=SAC records, =model.



15/17

1

2

3

4

5

6

0.1 1 10

max|A

|,g

natural period, s

10

20

30

40

50

60

70

80

90

100

0.1 1 10

max|D|,cm

natural period, s

0

1

2

3

4

5

6

0 20 40 60 80 100

max|A|,g

max|D|, cm

se2in50

0.2 0.5s

1.0s

2.0s

5.0s

FIG. 7. se2in50 bi-axial linear elastic response spectra, 5% damping, mean,mean+std.dev, mean-std.dev,=SAC records, =model.



16/17

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

FREQUENCY, Hz

NORMALIZEDPOWERSPECTRA

A

0 5 10 15 20 25 30 35 40

10

8

6

4

2

0

2

4

6

8

10

TIME, sec

ACCEL,cm/s/s

B

0 5 10 15 20 25 30 35 40

5

4

3

2

1

0

1

2

3

4

5

TIME, sec

ACCEL,cm/s/s

C

0 5 10 15 20 25 30 35 40

150

100

50

0

50

100

150

TIME, sec

VELOC,cm/s

D

0 5 10 15 20 25 30 35 40

200

150

100

50

0

50

100

150

200

TIME, sec

VELOC,cm/s

E

0 5 10 15 20 25 30 35 40

200

150

100

50

0

50

100

150

200

TIME, sec

VELOC,cm/s

F

0 5 10 15 20 25 30 35 40

1000

800

600

400

200

0

200

400

600

800

1000

TIME, sec

ACCEL,cm/s/s

G

0 5 10 15 20 25 30 35 40

80

60

40

20

0

20

40

60

80

TIME, sec

DISPL,cm

H

FIG. 8. nrfault simulation: (a) power spectral density, (b) stochastic acceleration

content, (c) time modulated stochastic acceleration content, (d) normalized veloc-ity content, (e) velocity pulse, (f) superimposed pulse and stochastic velocity, (g)

synthetic acceleration, (h) synthetic displacement.



17/17

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

FREQUENCY, Hz

NORMALIZEDPOWERSPECTRA

A

0 10 20 30 40 50 60

15

10

5

0

5

10

15

TIME, sec

ACCEL,cm/s/s

B

0 10 20 30 40 50 60

10

8

6

4

2

0

2

4

6

8

10

TIME, sec

ACCEL,cm/s/s

C

0 10 20 30 40 50 60

30

20

10

0

10

20

30

TIME, sec

VELOC,cm/s

D

0 10 20 30 40 50 60

6

4

2

0

2

4

6

TIME, sec

VELOC,cm/s

E

0 10 20 30 40 50 60

30

20

10

0

10

20

30

TIME, sec

VELOC,cm/s

F

0 10 20 30 40 50 60

500

400

300

200

100

0

100

200

300

400

500

TIME, sec

ACCEL,cm/s/s

G

0 10 20 30 40 50 60

8

6

4

2

0

2

4

6

8

TIME, sec

DISPL,cm

H

FIG. 9. se10in50 simulation: (a) power spectral density, (b) stochastic acceler-ation content, (c) time modulated stochastic acceleration content, (d) normalizedvelocity content, (e) velocity pulse, (f) superimposed pulse and stochastic velocity,

(g) synthetic acceleration, (h) synthetic displacement.

17 H P Gavin and B W Dickinson

gavin+dickinson_jse

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