ASTUDY OF POWER SPECTRAL DENSITY OiF ...3.15 Equivalent spectra test for 2, 4, 6 and 10 second long...

NSFjCEE-82032

PBB3-1 OlD 14

A STUDY OF POWER SPECTRALDENSITY OiF EARTHQUAKE

ACCELERO,GRAMS

by

PARVIZ MOAYYAD

and

BIJAN MOHRAZ

A Report on a Research Project

Sponsored by the

NATIONAL SCIENCE FOUNDATION

Research Grant No. PFR 8004824

alfPRODUCED BY

NATIONAL TECHNICALINFORMATION SERVICE

u.s. DEPARTMENT Of COMMERCESPRINGfiELD, VA.Ulil

CIVil AND MECHANICAL ENGINEERINGDEPARTMENT

SCHOOL OF ENGINEERING AND APPLIED SCIENCE

SOUTHERN METHODIST UNIVERSITY.DALLAS, TEXAS 75275

June 1982

INFORMATION RESOURCESNATIONAL SCIENCE FOUNDATION

A STUDY OF POWER SPECTRAL DENSITYOF EARTHQUAKE ACCELEROGRAMS

by

PARVIZ MOAYYAD

and

BIJAN MOHRAZ

A Report on a Research ProjectSponsored by the

NATIONAL SCIENCE FOUNDATIONResearch Grant No. PFR 8004824

CIVIL AND MECHANICAL ENGINEERING DEPARTMENT

School of Engineering and Applied ScienceSouthern Methodist University

Dallas, Texas 75275

June 1982

It

ACKNOWLEDGMENTS

This report is based on a doctoral dissertation by Mr. Parviz

Moayyad and was submitted to the faculty of the School of Engineering

and Applied Science of Southern Methodist University in partial fulfill

ment of the requirements for the degree of Doctor of Philosophy in

Engineering Mechanics. The dissertation was prepared under the direc

tion of Dr. Bijan Mohraz, Professor of Civil Engineering.

The authors wish to thank Dr. Henry L. Gray, C.F. Frensley Pro

fessor of Mathematical Science, whose guidance in the early part of

this study played an important role in solidifying the direction of

this investigation.

The study was supported by the National Science Foundation

through Grant PFR 8004824. The opinions, findings, and conclusions

or recommendations expressed here are those of the authors and do

not necessarily reflect the views of the National Science Foundation.

The numerical results were obtained using the CDC 6600 of the SMU

Computer Center.

The authors acknowledge Mrs. Sally Hackett for her diligent

work in preparation of drafts and the final manuscript.

iii

ABSTRACT

An examination of recorded earthquake accelerograms indicates

their nonstationary characteristics, that is, their statistical pro

perties, vary with time. The nonstationary characteristic takes a

special form when the strong motion part of the record is considered.

It is demonstrated in this study that within the strong motion dura

tion the short time mean square value varies with time, whereas the

frequency structure of the record remains time-invarient. This con

c]usion leads to the assumption that the strong motion segments of

accelerograms can be considered to form a locaJly stationary random

process. The power spectral density of such a process is a function

of both time and frequency.

The time-dependent power spectral density for an ensemble of

accelerograms is estimated as the product of a normalized power spec

tral density which is a function of frequency onJy and describes the

frequency structure of the ensemble; a normalized time-dependent scale

factor which is obtained from a short time averaging of mean square

acceleration; and finally the mean square acceleration itself. The

mean square acceleration is obtained from correlations between RMS

vaJue and a variabJe which incorporates four important earthquake

parameters: peak ground acceJeration, earthquake magnitude, epicentraJ

distance, and duration of strong motion.

iv

Time-dependent power spectral densities and correlations between

RMS of the records and the four earthquake parameters are obtained

for horizontaJ and veritical components of accelerograms recorded

on soft, intermediate, and hard sites. The findings are used to esti

mate the power spectraJ density for a given geology, peak ground accele

ration, earthquake magnitude, epicentral distance, and duration of

strong motion. The estimates are then used to predict the response

of a singJe degree of freedom system and to compare the results with

both the reJative dispJacement, relative velocity and absolute accele

ration computed directJy from the record and the mean pJus one standard

deviation response of the ensemble.

v

ACKNOWLEDGMENTS

ABSTRACT

TABLE OF CONTENTS

LIST OF FIGURES

LIST OF TABLES

TABLE OF CONTENTS

iii

iv

vi

viii

xvii

CHAPTER] INTRODUCTION 1

1.1 Background 1].2 Objective and scope 5J.3 NomencJature 7

CHAPTER 2 SOME PRELIMINARY CONCEPTS IN RANDOM VIBRATION ]0

2.J Introductory remarks ]02.2 Random process JO2.3 Stationary random process J22.4 LocaJJy stationary random process ]62.5 Estimating power spectraJ density of stationary

random process ]92.5.J Random error in power spectral density

estimate 222.5.2 Smoothing of power spectra] density estimate 232.5.3 Equivalent power spectral estimate 25

2.6 Input-output reJationship 26

CHAPTER 3 TIME-DEPENDENT POWER SPECTRAL DENSITY OFEIGHT SELECTED RECORDS 35

3.] Introductory remarks 353.2 Duration of strong motion 373.3 Comparisons of proposed duration of strong motion 393.4 Time-dependent power spectra] density 433.5 Comparisons between responses calcuJated from

the time-dependent power spectraJ density andspectra] dispJacement, velocity and acceleration 48

vi

vii

FIGURE

2.1

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

LIST OF FIGURES

Single degree of freedom system.

Comparison of strong motion duration for ImperialValley Earthquake, May 18, 1940, El Centro--componentSOOE.

Comparison of strong motion duration for ImperialValley Earthquake, May 18, 1940, El Centro--componentS90W.

Comparison of strong motion duration for Kern County,California Earthquake, July 21, 1952, Taft--componentN21E.

Comparison of strong motion duration for Kern County,California Earthquake, July 21, 1952, Taft--componentS69E.

Comparison of strong motion duration for Lower California Earthquake, December 30, 1934, El Centro-component SOOW.

Comparison of strong motion duration for Lower California Earthquake, December 30, 1934, El Centro-component S90W.

Comparison of strong motion duration for WesternWashington Earthquake, April 13, 1949, Olympia-component N04W.

Comparison of strong motion duration for WesternWashington Earthquake, April 13, 1949, Olympia-component N86E.

Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (El Centro 1940, SOOE).

Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (El Centro 1940, S90W).

viii

PAGE

34

57

58

59

60

61

62

63

64

65

66

FIGURE PAGE

3.11 Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (Taft 1952, N21E). 67

3.12 Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (Taft 1952, S69E). 68

3.13. Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (El Centro 1934, SOOW) . 69

3.14 Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (El Centro 1934, S90W) . 70

3.15 Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (Olympia 1949, N04W). 71

3.16 Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (Olympia 1949, N86E). 72

3.17 Equivalent spectra test for 1 second long segments(El Centro 1940, SOOE). 73

3.18 Equivalent spectra test for 1 second long segments(El Centro 1940, S90W). 74

3.19 Equivalent spectra test for 1 second long segments(Taft 1952, N21E). 75

3.20 Equivalent spectra test for 1 second long segments(Taft 1952, S69E). 76

3.21 Equivalent spectra test for 1 second long segments(El Centro 1934, SOOW). 77

3.22 Equivalent spectra test for 1 second long segments(El Centro 1934, S90W). 78

3.23 Equivalent spectra test for 1 second long segments(Olympia 1949, N04W). 79

3.24 Equivalent spectra test for 1 second long segments(Olympia 1949, N86E). 80

3.25 Power spectral density for the ensemble of the eightrecords used in the pilot study. 81

3.26 Time variation of normalized mean square value of theensemble of the eight records used in the pilot study. 82

ix

FIGURE

3.27

3.28

3.29

3.30

3.31

3.32

3.33

3.34

3.35

4.1

4.2

4.3

Correlation of RMS with parameter for eighthorizontal components of recorded accelerograms.

Comparison of response for 2 percent of criticaldamping for SOOE component of El Centro, ImperialValley Earthquake of May 18, 1940.




Comparison of response for 2 percent of criticaldamping for N21E component of Taft, Kern County,California Earthquake of July 21, 1952.




Correlation of RMS with peak ground acceleration for367 horizontal and vertical components of recordedaccelerograms.

Correlation of RMS with epicentral distance for 367horizontal and vertical components of recorded accelerograms.

Correlation of RMS with duration of strong motion for367 horizontal and vertical components of recordedacce 1erograms.

x

PAGE

83

84

85

86

87

88

89

90

91

193

194

195

FIGURE PAGE

4.4 Correlation of RMS with earthquake magnitude for367 horizontal and vertical components of recordedaccelerograms. 196

4.5 Correlation of RMS with combined parameter for 367horizontal and vertical components of recordedaccelerograms. 197

4.6 Correlation of RMS with parameter n for 367 horizontaland vertical components of recorded accelerograms. 198

4.7 Correlation of RMS with parameter n for 367 horizontaland vertical components of recorded accelerograms. 199

4.8 Correlation of RMS with parameter n for 161 horizontalcomponents of recorded accelerograms--Soft. 200

4.9 Correlation of RMS with parameter n for 60 horizontalcomponents of recorded accelerograms--Intermediate. 201

4.10 Correlation of RMS with parameter n for 26 horizontalcomponents of recorded accelerograms--Hard. 202

4.11 Correlation of RMS with parameter n for 247 horizontalcomponents of recorded accelerograms--combined Soft,Intermediate, and Hard. 203

4.12 Correlation of RMS with parameter n for 78 verticalcomponents of recorded accelerograms--Soft. 204

4.13 Correlation of RMS with parameter n for 29 verticalcomponents of recorded accelerograms--Intermediate. 205

4.14 Correlation of RMS with parameter n for 13 verticalcomponents of recorded accelerograms--Hard. 206

4.15 Correlation of RMS with parameter n for 120 verticalcomponents of recorded accelerograms--combined Soft,Intermediate and Hard. 207

4.16 Correlation of RMS with parameter n for 239 horizontaland vertical components of recorded accelerograms--Soft. 208

4.17 Correlation of RMS with parameter n for 89 horizontaland vertical components of recorded accelerograms--Intermediate. 209

4.18 Correlation of RMS with parameter n for 39 horizontaland vertical components of recorded accelerograms--Hard. 210

xi

FIGURE PAGE

4.19 Correlation of RMS with parameter n for 367 horizontaland vertical components of recorded accelerograms--combined Soft, Intermediate and Hard. 211

5.1 Power spectral density of the ensemble of 161 horizon-tal components of recorded accelerograms--Soft. 224

5.2 Power spectral density of the ensemble of 60 horizon-tal components of recorded accelerograms--Intermediate. 225

5.3 Power spectral density of the ensemble of 26 horizon-tal components of recorded accelerograms--Hard. 226

5.4 Power spectral density of the ensemble of 78 verticalcomponents of recorded accelerograms--Soft. 227

5.5 Power spectral density of the ensemble of 29 verticalcomponents of recorded accelerograms--Intermediate. 228

5.6 Power spectral density of the ensemble of 13 verticalcomponents of recorded accelerograms--Hard. 229

5.7 Time variation of normalized mean square value of theensemble of 161 horizontal components of recordedaccelerograms--Soft. 230

5.8 Time variation of normalized mean square value of theensemble of 60 horizontal components of recordedaccelerograms--Intermediate. 231

5.9 Time variation of normalized mean square value of theensemble of 26 horizontal components of recordedaccelerograms--Hard. 232

5.10 Time variation of normalized mean square valueensemble of 78 vertical components of recordedaccelerograms--Soft.

5.11 Time variation of normalized mean square valueensemble of 29 vertical components of recordedcelerograms--Intermediate.

5.12 Time variation of normalized mean square valueensemble of 13 vertical components of recordedaccelerograms--Hard.

of the

233

of theac-

234

of the

235

xii

FIGURE PAGE

5.13 Comparison of predicted response (soft) and computedresponse for 2 percent damping, Hollywood StorageP.E. lot, 1952--N90E. 236

5.14 Comparison of mean plus one standard deviationresponse (soft) and computed response for 2 percentdamping, Hollywood Storage P.E. lot, 1952--N90E. 237

5.15 Comparison of predicted response (soft) and computedresponse for 10 percent damping, Hollywood StorageP.E. lot, 1952--N90E. 238

5.16 Comparison of mean plus one standard deviationresponse (soft) and computed response for 10 percentdamping, Hollywood Storage P.E. lot, 1952--N90E. 239

5.17 Comparison of predicted response (soft) and computedresponse for 2 percent damping, El Centro, 1940--S00E. 240

5.18 Comparison of mean plus one standard deviationresponse (soft) and computed response for 2 percentdamping, El Centro, 1940--S00E. 241

5.19 Comparison of predicted response (soft) and computedresponse for 10 percent damping, El Centro, 1940--S00E. 242

5.20 Comparison of mean plus one standard deviationresponse (soft) and computed response for 10 percentdamping, El Centro, 1940--S00E. 243

5.21 Comparison of predicted response (intermediate) andcomputed response for 2 percent damping, Ferndale CityHall, 1952--N44E. 244

5.22 Comparison of mean plus one standard deviationresponse (intermediate) and computed response for 2percent damping, Ferndale City Hall, 1952--N44E. 245

5.23 Comparison of predicted response (intermediate) andcomputed response for 10 percent damping, Ferndale CityHall, 1952--N44E. 246

5.24 Comparison of mean plus one standard deviationresponse (intermediate) and computed response for 10percent damping, Ferndale City Hall, 1952--N44E. 247

xiii

FIGURE PAGE

5.25 Comparison of predicted response (intermediate) and ~

computed response for 2 percent damping, Ferndale CityHall, 1954--N46W. 248

5.26 Comparison of mean plus one standard deviationresponse (intermediate) and computed response for 2percent damping, Ferndale City Hall, 1954--N46W. 249

5.27 Comparison of predicted response (intermediate) andcomputed response for 10 percent damping, Ferndale City _Hall, 1954--N46W. 250

5.28 Comparison of mean plus one standard deviationresponse (intermediate) and computed response for 10percent damping, Ferndale City Hall, 1954--N46W. 251

5.29 Comparison of predicted response (hard) and computedresponse for 2 percent damping, Lake Hughes Station 1,1971--S69E. 252

5.30 Comparison of mean plus one standard deviationresponse (hard) and computed response for 2 percentdamping, Lake Hughes Station 1, 1971--S69E. 253

5.31 Comparison of predicted response (hard) and computedresponse for 10 percent damping, Lake Hughes Station 1,1971--S69E. 254

5.32 Comparison of mean plus one standard deviationresponse (hard) and computed response for 10 percentdamping, Lake Hughes Station 1, 1971--S69E. 255

5.33 Comparison of predicted response (hard) and computedresponse for 2 percent damping, Pacoima Dam, 1971--SI5W. 256

5.34 Comparison of mean plus one standard deviationresponse (hard) and computed response for 2 percentdamping, Pacoima Dam, 1971--S15W. 257

5.35 Comparison of predicted response (hard) and computedresponse for 10 percent damping, Pacoima Dam, 1971--SI5W. 258

5.36 Comparison of mean plus one standard deviationresponse (hard) and computed response for 10 percentdamping, Pacoima Dam, 1971--S15W. 259

xiv

FIGURE

5.37

5.38

5.39

5.40

5.41

5.42

5.43

5.44

5.45

5.46

5.47

5.48

5.49

Comparison of predicted response (soft) and computedresponse for 2 percent damping, Hollywood StorageP.E. lot, 1952--vertical.

Comparison of mean plus one standard deviationresponse (soft) and computed response for 2 percentdamping, Hollywood Storage P.E. lot, 1952--vertical.

Comparison of predicted response (soft) and computedresponse for 10 percent damping, Hollywood StorageP.E. lot, 1952--vertical.

Comparison of mean plus one standard deviationresponse (soft) and computed response for 10 percentdamping, Hollywood Storage P.E. lot, 1952--vertical.

Comparison of predicted response (soft) and computedresponse for 2 percent damping, El Centro, 1940-vertical.

Comparison of mean plus one standard deviationresponse (soft) and computed response for 2 percentdamping, El Centro, 1940--vertical.

Comparison of predicted response (soft) and computedresponse for 10 percent damping, El Centro, 1940-vertical.

Comparison of mean plus one standard deviationresponse (soft) and computed response for 10 percentdamping, El Centro, 1940--vertical.

Comparison of predicted response (intermediate) andcomputed response for 2 percent damping, FerndaleCity Hall, 1952--vertical.

Comparison of mean plus one standard deviationresponse (intermediate) and computed response for 2percent damping, Ferndale City Hall, 1952--vertical.


Comparison of mean plus one standard deviationresponse (intermediate) and computed response for 10percent damping, Ferndale City Hall, 1952--vertical.


xv

PAGE

260

261

262

263

264

265

266

267

268

269

270

271

272

FIGURE PAGE

5.50 Comparison of mean plus one standard deviationresponse (intermediate) and computed response for 2percent damping, Ferndale City Hall, 1954--vertical. 273

5.51 Comparison of predicted response (intermediate) andcomputed response for 10 percent damping, FerndaleCity Hall, 1954--vertical. 274

5.52 Comparison of mean plus one standard deviationresponse (intermediate) and computed response for10 percent damping, Ferndale City Hall, 1954--vertical. 275

5.53 Comparison of predicted response (hard) and computedresponse for 2 percent damping, Lake Hughes Station 1,1971--vertical. 276

5.54 Comparison of mean plus one standard deviationresponse (hard) and computed response for 2 percentdamping, Lake Hughes Station 1, 1971--vertical. 277

5.55 Comparison of predicted response (hard) and computedresponse for 10 percent damping, Lake Hughes Station 1,1971--vertical. 278

5.56 Comparison of mean plus one standard deviationresponse (hard) and computed response for 10 percentdamping, Lake Hughes Station 1, 1971--vertical. 279

5.57 Comparison of predicted response (hard) and computedresponse for 2 percent damping, Pacoima Dam, 1971--vertical. . 280

5.58 Comparison of mean plus one standard deviationresponse (hard) and computed response for 2 percentdamping, Pacoima Dam, 1971--vertical. 281

5.59 Comparison of predicted response (hard) and computedresponse for 10 percent damping, Pacoima Dam, 1971--vertical. 282

5.60 Comparison of mean plus one standard deviationresponse (hard) and computed response for 10 percentdamping, Pacoima Dam, 1971--vertical. 283

xvi

4.8 Parameters used in defining the RMS regression 1i ne.

4.9 Actua1 and predicted RMS for horizontal components--soft.

4.10 Actual and predicted RMS for horizontal components--intermediate.

4.11 Actual and predicted RMS for horizontal components--hard.

TABLE

3.1

3.2

3.3

3.4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

LIST OF TABLES

Properties of the eight records used in the pilotstudy.

Comparison of durations and root mean square valuesfor the eight records.

Comparison of duration and root mean square bydifferent methods.

Actual and predicted RMS for the eight records.

Geological classification of the records.

Earthquake records and data--soft.

Earthquake records and data--intermediate.

Earthquake records and data--hard.

Summary of range and increments used in selectingthe power coefficients P1-P4 in parameter n.

Comparison of correlation coefficients fordifferent n1s.

Comparison of the RMS predicted from different n'sfor the eight horizontal components of records usedin Chapter 3.

xvii

PAGE

52

53

55

56

102

129

144

150

153

154

155

156

157

171

177

TABLE

4.12

4.13

4.14

5.1

5.2

5.3

5.4

5.5

Actual and predicted RMS for vertical components-soft.

Actual and predicted RMS for vertical components-intermediate.

Actual and predicted RMS for vertical components-hard.

Accumulated area as the percentage of total areaunder the power spectral density.

Maximum ordinates of the power spectral densitiesand their corresponding frequencies.

Maximum values of scale factors.

Properties of the six records used in comparingthe predicted and computed response--horizontal.

Properties of the six records used in comparingthe predicted and computed response--vertical.

xviii

PAGE

180

188

191

219

220

221

222

223

CHAPTER 1

INTRODUCTION

1.1 BACKGROUND

Among the approaches used in the seismic analysis and design of

structures and equipments is the response spectrum. The response spec

trum introduced by Biot (1941, 1942) and Housner (1941), provides a

measure of the maximum response of a single degree of freedom system

to an excitation. Subjecting the system to a specific ground motion

of a recorded earthquake and computing its maximum response for a given

frequency and damping gives a point on the response spectrum curve.

For small damping, the relative velocity and absolute acceleration of

the system can be estimated from the relative displacement (Hausner

1970b, Clough 1970). Such quantities are termed pseudo-velocity and

pseudo-acceleration, respectively. For engineering applications the

relative displacement, pseudo-velocity, and pseudo-acceleration are

plotted as a function of the period or frequency of the system on a

tripartite paper (four-way log paper). In seismic analysis such plots

are made for a specific recorded ground motion. To obtain a design

spectrum, the response spectra from individual records are normalized

and averaged at various frequencies or periods. The mean plus one stan

dard deviation response has been generally used as a basis for design

spectrum (Blume et al., 1972; Mohraz et al., 1972; Newmark et al., 1973;

Mohraz 1976).

2

After 1971, studies were carried out to correlate the shape and

magnitude of response spectra to site geology, earthquake magnitude,

duration of strong motion, epicentral distance, and maximum ground ac

celeration. Seed et al. (1976) and Mohraz (1976) independently studied

the influence of site geology on response spectra, and some of their

conclusions were incorporated in a joint report by the Applied Tech

nology Council, the National Bureau of Standards and the National Science

Foundation (1978) for the development of seismic regulations for buil

dings. In a statistical study, McGuire (1974) showed that long epicen

tral distances tend to decrease the response at high frequencies. Later

Trifunac and Brady (1975) correlated the duration of strong motion with

the Modified Mercalli Intensity Scale, earthquake magnitude, epicentral

distance and site geology. Finally, Mohraz (1978a,b) showed that the

earthquake magnitude, peak ground acceleration and duration of strong

motion influence response spectra, and consequently the design spectra.

A different approach in seismic analysis and design of structure

is the use of random vibration theory. The simulation of earthquake

records by a random process has received a great deal of attention.

Both stationary and nonstationary random processes have been used to

model earthquake ground motion. Housner (1947, 1955), Thompson (1959),

Barstein (1960), Bycroft (1960), Tajimi (1960), Rosenblueth and Bustamente

(1962), and Housner and Jennings (1964) used a stationary random process

while Cornell (1960), Bolotin (1960), Bogdanoff et al. (1961), Shinozuka

and Sato (1967), Amin and Ang (1968), Jennings et al. (1969), Iyengar

and Iyengar (1969), Liu (1970), and Trifunac (1971a) employed a nonsta

tionary process. A special form of the nonstationary random process

3

in which an earthquake record is assumed to be a member of a stationary

random process modulated by a time-dependent intensity function has

been widely used by many investigators, for example, Amin and Ang (1968),

Jennings et al. (1968), Goto and Toki (1969), Ruiz and Penzien (1969),

Murakami and Penzien (1975), and Hsu and Bernard (1978). A nonstationary

random process in which the frequency structure of the records is as

sumed to be time-dependent has been used by Trifunac (1971b), Saragoni

and Hart (1972), Kubo and Penzien (1976), and Wong and Trifunac (1979).

The majority of studies in earthquake motion simulation use either

a stationary or nonstationary random process has been to obtain a re

sponse spectrum from artifically generated accelerograms. For example,

Bycroft (1960) used "white noise" to model earthquake ground motion

and related his result to Housner's (1959) standard velocity spectra.

Later Housner and Jennings (1964) used a stationary Gaussian random

process to generate artificial accelerograms from which they computed

response spectra. They demonstrated that the velocity spectra of the

real and artificially generated earthquake motion are similar in shape

and statistical properties. Using Rosenblueth and Bustamente's (1962)

approximate theory based on diffusion analysis and Kanai's (1957) and

Tajami's (1960) semi-empirical equation for power spectral density,

Housner and Jennings presented expressions for damped and undamped ve

locity spectra in terms of power spectral density.

Although the use of random vibration theory in seismic analysis

and design is straightforward (see for example Hurty and Rubinstein,

1964; Lin, 1967; and Penzien, 1970), this theory has not received the

wide attention that the response spectrum has. A random phenomenon,

\ 4

such as an earthquake and its damage potential to a structure, may best

be descriped by probability statements. Once the ground motion is charac

terized through its power spectral density, the mean square response

of the system can be computed and probability statements regarding the

response exceeding a specified value be made by either the use of

Chebychve1s inquality or a normal distribution function, if the process

can be assumed to be normal. Such probability statements provide useful

information regarding the susceptibility of a design to failure as well

as the means for improving it. A designer currently lacks such informa

tion when using a response spectrum approach since it is a collection

of the peak responses and any probability statement based on a statis

tical analysis of response spectra provides statements on peak rather

than on the response.

The difficulty in using the random vibration theory in seismic

analysis and design is the lack of a sufficient number of useful re

corded ground motion and the remote possibility of having them in the

near future. Hence there is a need for a methodology based on the avail

able information which one can use with reasonable confidence. Param

eters such as earthquake magnitude, peak ground motion, epicentral dis

tance, duration of strong motion, energy release, etc., are random vari

ables which characterize an earthquake and therefore should be considered

in the application of the random vibration to seismic design. Motion

recorded at different stations during the same event or at the same

station during different events differ in characteristics; nevertheless,

they should be considered as different realizations of the same random

process with some common underlying features.

5

The first phase in seismic design is to determine the probability

of occurrence of an earthquake at a given site for a specified set of

parameters, such as peak ground acceleration, earthquake magnitude,

epicentral distance, duration of strong motion, etc. The second phase

is to formulate an appropriate statistical description of the ground

motion. The response of a structure to a seismic disturbance can then

be obtained when the above two phases are completed. The determination

of the probability of occurrence of an earthquake has received the at

tention of many investigators (see Esteva, 1976; Burridge and Knopoff,

1976), and is beyond the scope of this investigation. We will, there

fore, restrict our attention to the formulation of statistical descrip

tion of ground motions.

1.2 OBJECTIVE AND SCOPE

Power spectral density, the most useful statistical description

of a random process, is an essential part of seismic design of struc

tures through the use of random vibration theory. As noted previously,

a number of parameters such as site geology, earthquake magnitude, epi

central distance, duration of strong motion, and peak ground accelera

tion influence earthquake ground motion and response spectra. It is

believed that these parameters would also influence the power spectral

density of the recorded ground motion.

The objective of this investigation is to study the power spec

tral densities of a number of recorded earthquake accelerograms. The

study considers the influence of earthquake parameters such as site

geology, earthquake magnitude, epicentral distance, duration of strong

6

motion, and peak ground acceleration on power spectral density. Rela

tionships between the root me~n square acceleration of the records and

an expression characterising the various earthquake parameters are es

tablished. These relationships are used to estimate the power spectral

density of the motion for a set of specified earthquake parameters at

a site. Finally, the estimated power spectral density is used to obtain

the response of a single degree of freedom system and the results are

compared with the response computed from the records directly. The

study includes horizontal as well as vertical ground motion.

Chapter 2 describes some preliminary concepts of random vibration

and the input-output relationship for a single degree of freedom system

subjected to ground motion which is used in this study. Chapter 3,

in a pilot study, shows that with an appropriate selected duration of

strong motion, most earthquake records may be classified as locally

stationary. Procedures for generating a time-dependent power spectral

density from a selected group of records and the feasibility of using

it to compute the response of a single degree of freedom system are

also discussed in that chapter. In Chapter 4 the correlation between

the RMS value of the record and a variable reflecting the effects of

earthquake magnitude, epicentral distance, peak ground acceleration

and duration of strong motion is investigated. Chapter 5 presents the

power spectral densities for the horizontal and vertical components

of earthquake records for three geological classifications. In addition

the findings in Chapters 4 and 5 are used to compute the response of

a single degree of freedom system and the results are compared to those

obtained from the records directly. Summary and conclusions and the

recommendations for further studies are presented in Chapter 6.

7

1.3 NOMENCLATURE

Th~ symbols are defined where they first appear. The majority are sum-

marized below for ease of reference:

a

AA

A, B

E[ J

f

F(t)

G(f)

G(f)

G(f)

h

H(f)

IH(f) I

IHd(f) I

j

k

maximum ground acceleration

absolute acceleration of the mass in a SDOF system[Eq. (3.5)J

constants

resolution bandwidth

constants

epicentral distance

expected value of [ J

cyclic frequency

forcing function

one-sided power spectral density function

raw estimate of power spectral density function

estimate power spectral density function

sampling interval

frequency response function

Transmissibility function, gain factor

Transmissibility function--base accelerations as input,relative displacement of the mass as output

Transmissibility function--base acceleration as input, relativevelocity of the mass as output

Transmissibility function--base acceleration as input,absolute acceleration of the mass as output

;:I, index

index, constant

K

m

M

n

N

NO

PI' P2, P3 , P4

Pr(

r

Rx('r ), R1x(-r )

R(t1, t 2)

RD

RV

Sx(f)

S(t)

t 1, t 2

TI' T2

T

Tru

v

x(t), y(t)

X(f)

8

spring constant, number of records

number of consecutive frequency components

mass of a SDOF system

earthquake magnitude

degrees-of-freedom associated with a random variable

number of data points in a record

number of added zeros to a record

constants

probability that

correlation coefficient

autocorrelation function

nonstationary autocorrelation function

relative displacement of the mass in a SDOF system[Eq. (3.3)]

relative velocity of the mass in a SDOF system[Eq. (3.4)]

two-sided power spectral density function

scale factor

arbitrary times

initial and final times in selecting duration ofstrong motion

observation time

record length

relative displacement of the mass in a SDOF system[Eq. (2.53)]

absolute displacement of the mass in a SDOF system[Eq. (2.54)]

input and output random variable respectively

Fourier Transform of x(t)

z

t,.T

]..l

(J

2(J

T

2X

9

absolute base acceleration of a SDOF system[Eq. (2.53)]

a small probability, level of significance

duration of strong motion

random error

damping ratio

mean value

standard deviation

standard deviation of relative displacement response ofthe mass of a SDOF system

standard deviation of relative velocity response of themass of a SDOF system

standard deviation of absolute acceleration response ofthe mass of a SDOF system

variance

time lag

Chi-Square variable

root mean square

mean square

variable reflecting the combined effects of earthquakeearthquake parameters

CHAPTER 2

SOME PRELIMINARY CONCEPTS IN RANDOM VIBRATION

2.1 INTRODUCTORY REMARKS

This chapter presents a brief review and summary of some of the

concepts in random process which are used in this study. They can be

found in a number of texts, such as Crandall (1963), Lin (1967), Bendat

and Piersol (1971), and Clough and Penzien (1975). The definitions

of ergodic, stationary and locally stationary random processes are given

and means for describing a random process are outlined. The power spec

tral density function, the most important descriptive characteristic

of a stationary random process, is given special attention. The use

of the Fast Fourier Transform (FFT) procedure in computing the power

spectral density is presented, and the errors in estimating it are dis

cussed.

2.2 RANDOM PROCESS

A collection of data representing a physical random phenomenon

cannot be described by an explicit mathematical relationship because

each observation is unique and any observation is only one of the many

possible outcomes. A single time history representing a random phe

nomenon is called a sample record, and a collection of sample records

constitutes a random process.

10

11

The properties of a random process can be estimated at any time

by computing average values over the collection of sample records.

Mean value and the autocorrelation function are usually the quantities

first calculated to study the stationary characteristics of a random

process. For the random process {x(t)}, where { } denotes an ensemble

of sample records'r the mean value ~x(tl) and the autocorrelation func

tion Rx(t1, t 1 + T) at time t 1 are computed as:

(2.1)

(2.2)

where the subscript k indicates the kth sample record of the ensemble

and T is a time lag. The random process {x(t)} is said to be nonsta

tionary if ~x(tl) and Rx(t1, t 1 + T) vary as time t 1 varies. For the

special case where the mean value is constant and the autocorrelation

is only a function of the time lag, that is, ~ (t1) =~ andx ,

Rx(t1, t 1 + T) = RX(T), the' random process {x(t)} is said to be weakly

stationary or stationary in the wide sense. The mean value and the

autocorrelation function are consequences of first and second order

probability distributions. If all possible probability distributions

are independent of time translation, the process is called strongly

stationary. For a Gaussian random process, where all possible distribu-

tions may be derived from the mean value and the autocorrelation func-

tion, stationary in a wide sense implies strong stationary character-

12

istics.

When the time-averaged mean value and the autocorrelation function

of the kth sample record as defined by

fl (k)1 T

= lim f J xk(t) dtx T-+oo 0

R (k,T) = 1im +JT xk(t + T) dtx T-+oo 0

(2.3)

(2.4)

where T is the duration of the record, do not differ when computed over

different sample records, and are equal to those computed over the en

semble, then the stationary random process {x(t)} is said to be ergodic.

Therefore, the statistical properties of an ergodic random process can

be obtained from a single sample record.

Even though the stationary concept applies to a random process,

it is sometimes used to describe a single sample record. A different

interpretation of stationary characteristic is implied when a single

record is being described. In such cases it is generally meant that

the statistical prope~ties computed over a short interval do not vary

significantly from one interval to another.

2.3 STATIONARY RANDOM PROCESS

The following are used to describe the properties of random data:

(a) mean square values, (b) probability density function, (c) autocor

relation function, and (d) power spectral density function. The mean

square value describes the intensity of the data. The probability den-

sity function describes the amplitude properties of the data. The auto-

13

correlation and power spectral density functions, which are Fourier

transform pairs, provide the same information in time and frequency

domains, respectively. The autocorrelation function of a stationary

random process describes the general dependency of the data on each

other at different times and is defined as:

(2.5)

The autocorrelation function is always a real-valued even function with

a maximum at T =0, that is,

(2.6)

(2.7)

The autocorrelation function at time T =0 is equal to the mean square

value, whereas at t = 00 it approaches the square of the mean. In equa-

tion form

= 1/1 2x

R (00) = l.l 2x x

Perhaps the most important single descriptive characteristic of

(2.8)

(2.9)

a stationary data is the power spectral density function, which describes

the frequency composition of the data in terms of its mean square value.

For linear systems with constant parameters (mass. stiffness and damping)

the output power spectrum is equal to the product of the input power

spectrum and the response function of the system. The mean square value

14

of the data in a frequency range of interest is determined by the area

under the power spectrum in that range (note that w2 is the total area

under the power spectrum). The square root of the power spectrum at

zero frequency represents the mean value, ~, of the data. The mean

and mean square values are expressed as:

Wx2

= foo Sx(f) df_00

~x = [f~: \(f) df] 1/2

(2.10)

(2.11)

where the two sided power spectral density function Sx(f), is defined

as the Fourier transform of the autocorrelation function

S (f ) = foo R (T) e- j 27T fT dex _00 x .

(2.12)

From the symmetric property of the autocorrelation function, Eq. (2.6),

it fo llows that

Using Eq. (2.13), Eq. (2.12) can be simplified to

= 2 foo R (T) cos27TfTdTo X

(2.13)

(2.14)

The use of Sx(f) defined over the frequency range of (_00,00) and

the exponentials with imaginary components often simplify mathematical

formulations. The quantity measured in practice is the one sided power

spectral density Gx(f), where the frequency varies over (0,00), and is

defined as

15

(2.15)

Usually a finite upper limit on frequency range is imposed by the sam

pling rate of the data. The quantity Gx(f) can be defined in terms

of the autocorrelation function as

o < f < <Xl (2.16)

An alternative way to define the power spectral density function

is to consider a sample record xk(t) of a stationary random process

in the time interval of 0 < t < T and let

(2.17)

where Xk(f,T) is the finite range Fourier transform of xk(t) defined

by

(2.18)

*and Xk (f,t) is the complex conjugate of Xk(f,t). It should be noted

that an infinite range Fourier transform of xk(t) does not exist. How

ever, by restricting the limits to the range of 0 to T, the finite range

Fourier transform can be obtained.

Defining the power spectral density function of the process as

S (f) = lim E[S (k,f,T)]x T-?<>o X

(2.19)

where, E[Sx(k,f,T)] is the expected value operation over the ensemble

index k, and making use of Eq. (2.17), we obtain

16

= lim t E[!X k(f,T)!2]T-+oo

(2.20)

In terms of the one sided power spectral density function Eq. (2.20)

reduces to

(2.21)

where 0 < f < 00.

2.4 LOCALLY STATIONARY RANDOM PROCESS

Nonstationary data is the class of data whose statistical proper-

ties vary with time. Time-varying mean value, time-varying mean square

value, and time-varying frequency structure of the data indicate the

nonstationary characteristic of the data. Such a conclusion is' a nega-

tive statement denoting the lack of stationary characteristic of the

data. Therefore, nonstationary random processes are defined as those

which do not qualify as a stationary random process, and their time-

dependent statistical properties are determined by time averaging across

the ensemble of records. A particular type of nonstationary random

process, whose frequency structure is time invariant is called a locally

stationary process (Page, 1952; Silverman, 1957; and Bendat and Piersol,

1971) which will be discussed later in this section.

The autocorrelation for a nonstationary process is generally de-

fined as

(2.22)

17

where t 1 and t 2 are two arbitrary times. A further insight in the com

position of the autocorrelation function can be gained by the following

transformation:

t + t 2let -c = t - t and t = 1 (2.23)2 1 2

which results in t 1 = t,

and t 2 = t + I (2.24)- 2"

With these changes of variables, the autocorrelation function becomes

In the above equation t denotes time and, represents a time lag. It

should be noted that Rx(t,,) evaluated at , = 0, gives the time

dependent mean square value function.

(2.26)

For some processes it may be possible to decompose the autocorrelation

Rx(t,,) into a product of two functions,

(2.27)

where R1x(-c) is the stationary autocorrelation function, and R2x (t)

is a slowly varying scale factor defined within a short time interval

as

t. 1 < t < t.1- - 1

(2.28)

18

Nonstationary random processes whose autocorrelation is in the form

of Eq. (2.27) are called locally stationary processes. When R1x (T)

is normalized such that R1x (O) = 1, then

(2.29)

Therefore,

(2.30)

The Fourier transform of Eq. (2.30) gives the two sided time-dependent

power spectral density function

Sx(t,f) = 1JJx2(t) f" R1x (T) e- 2j'rrf TdT

_00

(2.31)

In terms of the one sided time-dependent power spectral density func

tion, Eq. (2.31) reduces to

G (t,f) = 1JJ 2(t) G (f)x x xo < f < 00 (2.32)

where 1JJ~(t) is the time dependent mean square value function and Gx(f)

is the stationary power spectral density function of the process.

The power spectral density function given by Eq. (2.32) can be

estimated by first computing the time dependent mean square value func-

tion, which is averaged across the ensemble of records, and then esti-

mating Gx(f) in the same manner as for a stationary random process.

Since the total area under the power spectral density represents the

19

mean square which is reflected in $x2(t). then one needs to normalize

the total area under the stationary power spectral density function

to unity.

JOO G (f) =1o x

(2.33)

Before estimating the time-dependent power spectral density function.

Gx(t.f). one should demonstrate the validity of the assumption of Eq.

(2.27). To show that the local stationary assumption is a reasonable

one. the power spectral density estimates of different segments of a

record in the ensemble should exhibit similar shapes but have different

scales. In order to have confidence in the assumption. a sufficient

number of records in the ensemble should be tested for similarity of

their segmentally computed power spectral densities.

2.5 ESTIMATING POWER SPECTRAL DENSITY OF STATIONARY RANDOM PROCESS

Equation (2.21) defines the power spectral density of a stationary

random process. A stationary random process contains many sample rec-

ords with infinite duration. whereas the records of physical phenomenon

are few in number and short in duration. Therefore. one can only es

timate the power spectral density function. In order for the power

spectral density function to reveal the characteristic of the data.

the record should be long enough to include all the pertinent frequencies

in the data and. further. the time interval used in the digitization

of the record should be short enough to allow the computation of power

spectral density with a good resolution.

An estimate of the power spectral density function is obtained

20

by first computing the power spectral density function of each of the~

sample records and then averaging the spectral components Gk(f) at each

frequency over the ensemble. This averaging or smoothing operation

is intended to approximate the expected value operation in Eq. (2.21).

Thus, Eq. (2.21) is replaced by the following equations:

Gk(f) =t IXk(f,T)1 2

A 1 k ~G (f) =- \ Gk(f)x k k~l

(2.34)

(2.35)

where Gk(f) represents the raw estimate of the power spectral density

of the kth sample record, and Gx(f) is the estimate of the power spec

tral density of the process {x(t)}.

Assuming that the records contain Ndata points spaced h seconds

apart and letting

xn = x (nh) n = 0, 1, 2 . . . N-1 (2.36)

Eq. (2.18) can be expressed in discrete form as

N-1= h I Xn

e-j2rrfnhn=O

(2.37)

The discrete frequencies at which the Fourier transforms XK(f,t) are

computed are

= f =L.f p T Nh P = 0, 1, 2 ... N-l (2.38)

Substituting Eq. (2.38) into Eq. (2.37), one obtains

21

.21TpnN-1 -J= h L x

ne N

n=Op = 0, 1, 2 . . . N-1 (2.39)

It should be noted that when a continuous record is sampled such

that the time interval between sample values is h seconds, the highest

frequency which can be detected in the data is ~h cps. The cut off

frequency

(2.40)

is called the Nyquist frequency. Therefore, when the N data points

in the record are h seconds apart, the Nyquist frequency occurs at

p =~' Hence the raw estimates of the power spectral density is given

by

k = 1, 2 ... K (2.41)

where the spectral components Xk(fp,T) are computed at the frequencies

NP = 0, 1, 2 ... "2 (2.42)

The smallest frequency increment for which a change in the esti

mate can be detected is called the resolution bandwidth which is defined

as

(2.43)

Equation (2.43) shows that the larger the number of data points in the

sample record, the finer the resolution bandwidth. Because of the na-

22

ture of the Fast Fourier Transforms, one can add zeros to the record

to obtain a finer resolution bandwidth. When padding a record with

zeros, the spacing of the spectral components will be based on the aug

mented rather than the original record length. The resolution bandwidth

is then given by

(2.44)

where NO is the number of zeros added to the beginning or the end of

the record. In such a case the area under the power spectral density

is no longer equal to the mean square value of the original record,

but rather to the mean square value of the augmented record.

2.5.1 RANDOM ERROR IN POWER SPECTRAL DENSITY ESTIMATE

It can be shown (see Bendat and Piersol, 1971) that each compo

nent of Gk(fp) is a chi-square variable with two degrees of freedom.

The random error of the estimate Gk(fp) is the ratio of the standard

deviation to the mean value of the estimate:

(2.45)

The mean and variance of a chi-square variable with n degrees of freedom

are nand 2n, respectively. Hence for two degrees of freedom, the ran-

dom error is

(2.46)

23

which indicates that the standard deviation of the estimate is as large

as the mean value which is obviously n~t desirable. In the next sec

tion techniques for reducing the error are briefly described.

2.5.2 SMOOTHING OF POWER SPECTRAL DENSITY ESTIMATE

The random error associated with Eq. (2.34) in estimating the

power spectral density can be reduced in one of three ways. First is

the frequency smoothing, in which the result of ~ contiguous spectral

components of the estimate of a single sample record are averaged.

Second is the ensemble averaging, which is accomplished by computing

the estimate from K sample records and then averaging the estimates

at each frequency of the spectral components. The third approach uses

a combination of the two.

Frequency Smoothing. When ~ adjacent frequency components are

averaged, the final spectral estimate Gi is given by

(2.47)

There are N/22 such estimates which can be considered as representing

the midpoint of the frequency interval between f i and fi+~-l. By the

x2 (Chi square) addition theorem for independent variables, (see Wagpole

and Myers, 1978) the quantity Gi will be a x2 variable with roughly

n = 22 degrees of freedom. The final effective resolution bandwidth

will approximately be ~/T. Therefore,

24

n = 2£ (2.48a)

Be =i (2.48b)r

s =~ (2.48c)r

Ensemble Smoothing. Assuming that all sample records are of equal

length Tr , the frequencies at which the spectral estimates for each

record are computed by the Fast Fourier Transform procedure will be

identical. Therefore, by averaging across the ensemble of K estimates,

the final spectral estimate is given as

(2.49)

The quantity G(fp) will be a x2 variable with approximately n = 2K

degrees of freedom. The effective resolution bandwidth will still be1y-. Therefore,r

n = 2K (2.50a)

Be1 (2.50b)=Tr

s =~ (2.50c)r

Usually the record lengths in an ensemble are not equal. In such

cases, one needs to pad the records with zeros to achieve equal record

lengths. The effect of adding zeros to records was discussed in section

2.5.

25

Combined Smoothing. For a combination of frequency smoothing

and ensemble averaging, the final effective resolution banqwidth will

be ~ and the resulting estimate will be a x2 variable with n = 2tK

degr~eS of freedom. The random error in this case is given by Er =,IlK.

2.5.3 EQUIVALENT POWER SPECTRAL ESTIMATES

When power spectral density estimates are obtained under different

conditions, e.g. from two parts of the same record or from two indepen-

dent sample records, they may be tested for equivalence. Bendat and

Piersol (1971) give a procedure to determine whether two estimates are

statistically equivalent over the same frequency interval. The test

is based on the statistic

having a chi-square distribution with Nf degrees of freedom.

of acceptance is

(2.51)

The region

n = Nf(2.52)

where the two estimates G1(f) and G2(f) have the same resolution band

width, with n1 and n2 degrees of freedom, respectively; Nf is the number

of bandwidths to cover the frequency range of interest and a is the

level of significance of the test.

It should be noted that this test is valid for the condition when

the power spectra are computed from two statistically independent records

26

or segments of a record. This condition is not strictly satisfied when

overlapping segments of a record are used. However, when the overlap

is small, the segment and the entire record may be considered statis-

tically independent.

2.6 INPUT-OUTPUT RELATIONSHIP

For a single degree of freedom system subjected to a base motion

(Figure 2.1), the governing differential equation of motion is

where

u = relative displacement of mass m,

z = absolute displacement of the base,

wn = natural frequency =~

d · t' cs = amp1ng ra 10 =~

(2.53)

The equation of motion of a single degree of freedom whose mass is sub-

jected to a forcing function is

where,

v = absolute displacement of the mass,

F(t) = forcing function per unit mass.

Since Eq. (2.53) and (2.54) are of the same form, the study of both

(2.54)

types of excitation can be combined into one. The base motion or the

27

forcing function or a combination of both will be referred to as input.

Correspondingly, the induced response of the system, either absolute

or relative displacement, velocity and acceleration of the system will

be referred to as output.

Let us consider a single degree of freedom system with a constant

mass, stiffness, and damping subjected to a Gaussian stationary input

x(t) with zero mean. Since the system is linear, the output y(t) will

also be Gaussian stationary with zero mean. The relationship between

the input and output power spectral density is given by

(2.55)

where Gx(f) and Gy(f) are the one sided power spectral density of the

input and output, respectively, and the function IH(f)1 2 is the trans-

missibility function or the gain factor, which prescribes the portion

of the energy to be transmitted through the system at various frequen

cies. It follows that the mean square value of the output is given

by

1/J 2 = t' G (f)df = (' IH(f)12 G (f)dfy 0 Y 0 x

(2.56)

The integration of Eq. (2.56) for obtaining the mean square re-

sponse can be carried out in a closed form if a mathematical expression

for Gx(f) is available; however, if Gx(f) is given in a tabular form,

a numerical integration is necessary. When Gx(f) is a smooth function

with no sharp peaks, a good approximation of equation (2.56) can be

obtained for small damping as follows: For small damping ratio ~, the

28

transmissibility function IH(f)!2 is sharply peaked around the natural

frequency f n and it reduces considerably for small changes in frequency.

Therefore, the major contribution to the integral in equation (2.56)

comes from the region around the natural frequency f n. In addition

if the power spectral density varies slowly in the vicinity of the natu

ral frequency, then the contribution of Gx(f) to the integral outside

that vicinity is minimal. For such cases Gx(f) in Eq. (2.56) can be

taken outside the integral. Thus,

(2.57)

Considering the relationship between the mean square, variance, and

the mean value

and noting that the mean of the output is zero

II = 0Y

the variance of the output can be written as

cr22 = G (f) f~ IH(f)!2 df

x 0

(2.58)

(2.59)

(2.60)

The mean and the variance of a stationary Gaussian process are the only

quantities needed to describe the probability density function of the

process.

Since the excitation x(t) is random, the response y(t) is also

random, and it is conceivable that it may exceed a specified level Ymax'

29

Consequently, the output is described by making probability statements

regarding the response exceeding Ymax' The probability of y(t) exceed

ing Ymax is

fCXl 1

Pr(y > Ymax) = IZITcr eYmax Y

or

where the error function is defined as

2erf(x) =~ JX e-P dp

I7r 0

Since ~y = 0, we can write equation (2.62) as

dy (2.61)

(2.62)

(2.63)

(2.64)

In design the sign of y(t) is unimportant. The probability that

the absolute value of y(t) exceeds Ymax is

Pr( IYI > Ym•x) = 1 - erf (;m~;) (2.65)

by allowing Ymax = kcry, where k = 1, 2, 3, ... , we obtain the following

30

probability statements:

Pr(lyl > cry) = 1 - erf (~) = 31.74%

Pr(!YI > 2cry) = 1 - erf (~) = 4.56%

Pr(lyl > 3cry) = 1 - erf (A-) = 0.26%

(2.66)

It should be noted that the above probability statements result when

the mean and the variance of the output are known.

So far our discussion of the input-output relation has been gen

eral. If we now consider support acceleration as input and the relative

displacement of the mass as output, we can formulate the transmissibility

function as

(2.67)

Upon substitution of Eq. (2.67) into Eq. (2.60) and integrating, we

obtain the variance of the relative displacement as

(2.68)

where the subscript x indicates that the power spectral density G"(f)xis obtained from the input acceleration. The transmissibility function

for support acceleration as input and the relative velocity as output

is

31

(2.69)

Similarly from Eq. (2.60), the variance of the relative velocity is

(2.70)

Finally the transmissibility function for support acceleration as input

and the absolute acceleration as output is

the variance of the absolute acceleration is

7ff (1 + 4~2) G··(f )2 n x n

(J a = 4~

(2.71)

(2.72)

Knowing the mean and the variance of the output, it is a simple matter

to make probability statements such as those given by Eq. (2.66).

For a locally stationary random process the input power spectral

density is time-dependent (Eq. 2.32); therefore, the variance of the

output will also be time-dependent. In this case the time dependent

mean square value is

(2.73)

32

and the relative displacement, velocity and the absolute acceleration

of a single degree of freedom system subjected to a base motion are

given by

2 G·· (t, f )a (t) + X 3 n3y 64TI sf

n

G·· (t f )2 x' n

a (t)y = 16TIsfn

(2.74)

(2.75)

(2.76)

Probability statements for output are obtained from expressions similar

to those of Eq. (2.66). Thus,

k = 1, 2, 3, . .. (2.77)

It is interesting to note that y and o(t)y on the left side of Eq. (2.74

2.76) depend on time whereas the probability statements are not time

dependent.

Equations (2.74-2.76) have been obtained under the assumption

of smooth or slow varying power spectral density (ideal white noise).

For cases where the power spectral density is not flat Eq. (2.56) must

be utilized in which case the response takes the following form:

(2.78)

(2.79)

33

(2.80)

where the functions IH(f)1 are given in Eq. (2.67), (2.69), and (2.71),

respectively.

The materials presented in this chapter are employed in the sub

sequent chapters to obtain a time-dependent power spectral density,

to predict the response of a single degree of freedom system to a set

of base excitations, and finally, to compare the predicted response

to the response of the system computed directly from the records.

34

v

u

---,I m ---i

~I I I~! I I

~,I c I

/ K /~I

I I

~II I

I II

I

Iz

m = massc = damping

K = spring constant

u = rel~tive displacement of mass mv = absolute displacement of mass mz = absolute motion of base

FIG. 2.1. Single degree of freedom system.

CHAPTER 3

TIME-DEPENDENT POWER SPECTRAL DENSITY OFEIGHT SELECTED RECORDS


An examination of earthquake records reveals some of their general

characteristics. First, due to a finite energy release at the source

the resulting motion is transient. Second, this transitory phenomenon

manifests itself in three distinct zones in an accelerogram: an initial

build-up zone, an intermediate zone of strong motion, and a decaying

zone. Third, the acceleration oscillates around a zero line. These

general observations lead one to believe that earthquake records are

nonstationary. In fact, Amin and Ang (1968) confirm the nonstationary

characteristics of the eight earthquake records used originally by Housner

(1959) to establish a standard velocity reponse spectrum.

As mentioned in Section 2.4, the nonstationary characteristic

could manifest itself in the time-varying mean value, time-varying mean

square value, and time-varying frequency structure of the data. For

earthquake records one can readily eliminate the time-varying mean value

as a contributing factor to nonstationary characteristic. Time aver-

aging results show insignificant changes in the mean value as a function

of time, whereas significant changes are observed in the mean square

value and the frequency structure of the records.

Realizing that the low amplitude impulses which usually appear

35

36

toward the beginning and the end of an earthquake accelerogram have

little effect on the energy content within the strong motion duration

of the earthquake, we will consider only that part of an earthquake

record which contains the strong motion. We further assume that no

significant variation in the spectral composition occurs during the

strong motion segment of the record. Under such conditions it will

be justified to assume that the nonstationary earthquake random process

is of locally stationary form (Page 1952, Silverman 1957, Priestley

1965, Bendat and Piersol 1971).

In this chapter, using the eight selected records considered by

Housner (1959), we will show that it is possible to select a strong

motion segment of earthquake records during which its frequency struc

ture remains reasonably constant. Using such selected segments for

the eight records a time-dependent power spectral density is formulated

whose frequency structure remains time-invariant, whereas its magnitude

(area under the power spectral density) becomes a function of time.

This time-dependent magnitude is the ensemble short time mean square

value of the eight records. Finally~ we will show that a good correla

tion exists between the RMS value of the selected duration, and a param

eter reflecting peak ground acceleration, duration of strong motion,

earthquake magnitude and epicentral distance; thereby permitting one

to estimate the average magnitude of the time-dependent power spectral

density from the knowledge of the mentioned earthquake parameters.

37

3.2 DURATION OF STRONG MOTION

The duration of strong motion is widely recognized as an important

characteristic of ground motion. Studies by Bolt (1974), Trifunac and

Brady (1975b), and McCann and Shah (1979) suggest that the duration

of strong motion depends on the purpose for which it is used. The in-

tention in this study is to determine the duration of strong motion

during which the frequency structure of the record remains nearly the

same.

The definition proposed by Bolt (1974), which is known as the

bracketed duration, is useful to find the duration of strong motion

during which the structure will be subjected to a level of accelera

tion equal to or greater than a specified limit. Another definition,

which is related to the structural response, is that of Trifunac and

Brady (1975b) where they define the duration as the time interval during

which a significant contribution to the integral ft a2dt takes place.a

The first and last 5 percent contributions to this integral is omitted

and the remaining 90 percent is defined as the significant or the strong

motion contribution to the integral. The time interval between the

low and the high 5 percent cut-off points (5 and 95 percent, respec

tively) is defined as the duration of strong motion.

The definition of strong motion suggested by McCann and Shah (1979)

is related to the average energy arrival rate and is obtained by consid-

ering the cumulative root-mean square function of the record. A search

is performed on the derivative of this function to identify the cut-

off times. The final cut-off time, T2, is taken as the last time at

which the derivative of cumulative root-mean square function is positive.

38

To obtain the initial time T1, the same procedure is repeated except

the record is now considered from the "tail-end."

The method proposed here is similar to McCann and Shah's (1979),

with one slight difference in the manner that the cut-off points are

determined. In this case the cut-off time is selected as the last time

at which the slope of the cumulative root mean square function is equal

to or greater than one cm/sec2/sec. For accelerogram spacing of .02

seconds, this corresponds to a change of .02 cm/sec2 in the cumulative

RMS function. The derivative of cumulative root mean square function,

in addition to sharp peaks and valleys, exhibits flat regions. The

selection of unity or any other appreciable slope instead of any posi

tive slope (which could be extremely small) ensures that the cut-off

points are determined where the contribution to the cumulative RMS func

tion is no longer significant. Two other slopes, 0.5 and 2.0 cm/sec2/sec,

were also examined and it was determined that in general they did not

result in satisfactory durations of strong motion. As will be shown

later, the criteria used in this study provides durations of strong

motion for which the frequency structure of data remains time-invariant

more often than the methods proposed by either Trifunac and Brady (1975b)

or McCann and Shah (1979).

Another procedure for determining the duration of strong motion

such that the frequency structure of the data would be time-invariant

is to apply the equivalent power spectral test (see Section 2.5.3) to

the normalized power spectra of two different segments of the accelero

grams. If the two normalized power spectral densities are equivalent,

the frequency structure of the two accelerogram segments are the same;

and therefore, time-invariant.

39

Establishing equivalency between the

normalized power spectral densities of consecutive segments of an ac

celerogram would give a duration for which the frequency structure re-

mains time-invariant. However, such a procedure is extremely time con-

suming and not economical when large number of records are to be ana-

lyzed. In addition, this method mayor may not yield consecutive seg

ments with consistent frequency structure. Therefore, this procedure

was not used.

3.3 COMPARISONS OF PROPOSED DURATIONS OF STRONG MOTION

In order to determine the suitability of the proposed method of

computing the duration of strong motion, the eight strong motion records

used originally by Housner (1959) were used to compute the durations

and compare them with those obtained using the procedures given by Trifunac

and Brady (1975b) and McCann and Shah (1979). Table 3.1 lists these

records and some of their properties. In Table 3.2 the initial time

T1, the final time T2, the duration of strong motion ~T, the root-mean

square RMS, and the percent contribution to the integral f a2dt for

different procedures are presented and compared.

Figures 3.1 through 3.8 compare the duration of strong motion

for the three methods. The method proposed here consistently gives

shorter duration than either of the two other methods. As suspected,

for a given accelerogram, a shorter duration of strong motion results

in a larger RMS value. This is attributed not only to the insignificant

contribution of smaller pulses at the latter portion of the accelerogram

to the total RMS value, but also to the fact that fewer number of pulses

40

are used in the computation as well.

It should be noted that a recomputation of the duration and the

RMS values by the procedure proposed by McCann and Shah (1979) did not

reproduce their reported values exactly. Their study includes six of

the eight components of the records used in this chapter. The dura

tions and RMS values as reported by them and recomputed in this study

are presented in Table 3.3. Also shown are the values reported later

by McCann (1980). The values for El Centro 1940 are extremely close

to each other; however, large discrepencies are noted for the other

two records. It should be noted that the results reported by McCann

and Shah (1979) and by McCann (1980) are also slightly different from

each other. Although Trifunac and Brady (1975b) did not report dura

tions and RMS values in their study, the values computed by McCann and

Shah (1979) using their procedure and those computed in this study for

the six records are in close agreement. The discrepency observed in

Table 3.3 may be attributed to the different methods of calculating

the derivatives of the cumulative RMS function. A centeral 3-point

difference formula was used in this study to obtain the derivative.

To check the durations computed by the three procedures for their

consistency of frequency structure, the equivalent power spectra test

of Section 2.5.3 was used. The test compares the normalized power spec

tral density of consecutive segments of the selected duration with the

normalized power spectral density of the total selected duration. It

is necessary to normalize the power spectral densities, since we com

pare their shapes (frequency structure) rather than their magnitudes.

The normalization is accomplished by setting the area under power spectral

41

densities to unity. A brief description of the test procedure follows:

First, the power spectral density of the selected duration is

estimated using the Fast Fourier Transform procedure (Bendat and Piersol

1971) and then it is smoothed and normalized. Next, a segment from

the selected duration is padded with zeros to make its length equal

to the length of the selected duration. The padding is performed in

order to obtain spectral estimates at the same frequencies as those

for the selected duration (see Section 2.5). The power spectral density

for this augmented segment is then estimated, smoothed and normalized.

The same degree of smoothing is performed on the power spectral density

of the augmented segment and the power spectral density of the selected

duration. The normalized power spectral density of the augmented seg

ment is then compared with the normalized power spectral density of

the selected duration in the frequency range of a to 25 Hertz using

a Chi-square test with a 5 percent level of significance. Other con

secutive segments are chosen and the procedure is repeated until the

selected duration is exhausted. With the exception of the last segment,

all segments are equal in length.

The above procedure was used to test the duration of the strong

motion for the eight records computed by the three methods. The results

of such comparisons for 2, 4, 6 and 10 second long segments are presented

in Figs. 3.9 through 3.16. The plots show the ratio of computed Chi

square to the theoretical one. The two power spectral densities (for

the selected duration and the segment) are accepted as being equivalent

when this ratio is less than or equal to one (see Section 2.5). Two

observations can be made from the result presented in Figs. 3.9 through

42

3.16. First, the Chi-square ratio is less than one for most of the

records, regardless of the method employed in determining the duration

of strong motion. Second, the Chi-square ratios for longer segments

(6 and 10 seconds) are closer to zero than the ratios for shorter seg

ments (2 and 4 seconds) indicating that for longer segments the fre

quency structure of the segment is closer to that of the selected dura

tion. This is to be expected, since for longer segments more charac

teristics of data are taken into account in the comparison. In the

limit, when the length of the segment is equal to that of the selected

duration, no difference in the frequency structure can be detected and

the Chi-square ratio would be zero (see Eq. 2.52) indicating identical

data sets.

These figures indicate that in the majority of cases even a two

second segment gives acceptable Chi-square ratios for three procedures.

As mentioned previously (Section 2.5.3) the equivalent spectra test

is more reliable when the segment length is short as compared to the

entire duration of the record. For this reason Chi-square ratios were

also computed for a one second segment of the selected duration as de

termined by the three procedures as well as for the entire record length.

The results are shown in Figs. 3.17 through 3.24. The Chi-square ratios

for the entire record length of the eight records (Figs. 3.17 to 3.24)

clearly indicate that the frequency structure of the record changes

with time. The change is more pronounced towards the latter portion

of the records which correspond to the region of decaying activity.

The figures show that in a majority of cases the ratios computed using

the method proposed herein is within the acceptable limit more often

43

than those computed by the other two procedures. Therefore, it was

decided to adopt the proposed procedure for determining the duration

of strong motion in this study. Since the spectrum of one second long

segments is equivalent to the spectrum of the selected duration of strong

motion, the locally stationary assumption for earthquake records is

justified. Consequently, a one second segment is used in the computa

tion of the magnitude of the time-dependent power spectral density.

3.4 TIME-DEPENDENT POWER SPECTRAL DENSITY

It was shown in Section 3.3 that the power spectral density com

puted from one second long segments of the records remains the same

for the selected duration for the eight records. This implies that

the normalized power spectral density for the selected duration is a

good representation of the local (one second long segment) normalized

power spectral densities. Since the frequency structure of the data

(the shape of power spectral density) remains time-independent, then

the .time dependency of the power spectral density must manifest itself

in its magnitude. It should be noted that the term magnitude here is

referred to the area under the power spectral density curve, which is

also equal to the variance (mean square value when the mean of the rec

ord is zero) of the record (see Section 2.3).

An inspection of accelerograms indicates that the short time mean

square value of the records changes with time. If one estimates the

ensemble power spectral density of the eight records and normalizes

its area to unity, then its time-dependent magnitude should be computed

by performing a short time mean square operation on the ensemble of

44

the records. Therefore, we may consider the time-dependent power spec

tral density to be composed of the product of two functions--the nor

malized power spectral density representing the frequency structure

of the data, and the time-dependent magnitude representing the area

under the power spectral density. The normalized power spectral density

is estimated from the ensemble of the records using the selected dura

tions as if they were stationary records, and the magnitude is computed

using a one-second long mean square averaging on the ensemble of the

records.

When the magnitude is computed within the duration of the shor

test record in the ensemble, it includes all records in the ensemble.

Beyond the shortest duration, fewer records are included in the computa

tion resulting in a larger variance as the end of the longest record

is approached. To compute the time-dependent magnitude in a consistent

manner for the duration of longest record, the other records are padded

with zeros to make their length equal to that of the longest record

in the ensemble before the mean square averaging is carried out. How

ever, with padding the average mean square value is no longer correct

and in fact, will be smaller than the average of the individual mean

square values of the records in the ensemble. This problem is overcome

by reducing the average value to one, and using the normalized time

dependent magnitude as a scale factor which represents the variation

of the local magnitude (one second long mean square value) to an average

magnitude of unity. The scale factor can now be adjusted to reflect

the average mean square value of any record by multiplying the scale

factor by the record's mean square value which is computed from the

45

selected duration of the record.

Therefore, one may consjder the time-dependent power spectral

density to be composed of three parts. First a normalized power spec

tral density which describes the frequency structure of the ensemble

and remains the same for the segments of the records considered; second,

a time-dependent scale factor, obtained by performing a short time av

eraging on the square of the acceleration and normalizing the mean of

the resulting function to one, thereby describing the normalized varia

tion of the localized mean square acceleration; and finally the mean

square acceleration. Since the mean square acceleration can be computed

for each record, we can obtain a time-dependent power spectral density

corresponding to each of the records in the ensemble. Correlations

between the mean square acceleration and earthquake parameters would

enable one to estimate the time-dependent power spectral density for

a given set of earthquake parameters.

The most important ground motion parameter which is widely used

in design is the peak ground acceleration. As it will be shown later

in Chapter 4, a good correlation exists between the peak ground accelera

tion and the RMS value. Nevertheless, a better correlation is obtained

when a combination of peak ground acceleration, earthquake magnitude,

duration of strong motion and epicentral distance is considered.

The procedure for computing the normalized power spectral density

is as follows: First the durations of strong motion are determined

based on the method proposed in this chapter (modified McCann and Shah1s

method). Then enough zeros are added to the end of each record to make

their length equal to 120 seconds or 6000 data points (6000 data points

46

would result in a fine resolution in the spectral estimates). Power

spectral density~of each augmented record is then estimated and norma

lized (area is set equal to one). The ensemble smoothing procedure

similar to one presented in Section 2.5.2 is used to smooth the power

spectral density. The only difference is using a duration weighted

average to take into account the unequal durations of the unpadded rec

ords. Figure 3.25(a) shows the normalized power spectral density of

the ensemble of the eight records in the frequency range of 0-25 Hz.

Because of a spacing of 0.02 seconds in the accelerograms, the power

spectral density is estimated in the frequency range of 0-25 Hz (see

Eq. 2.40); however, the dominant frequencies appear in the frequency

range of 0 to 10 Hz. Since we used a small number of records in this

'chapter, the random error associated with the estimated power spectral

density shown in Fig. 3.25(a) is quite high (sr =0.35). To reduce

this random error, the frequency smoothing technique of Section 2.5.2

is applied to this estimate, where every 100 neighboring spectral or

dinates are averaged. Figure 3.25(b) shows the power spectral density

after combined smoothing, where the random error is reduced to 0.035.

The ordinates in Fig. 3.25(b) are joined by straight lines where in

Fig. 3.25(c) third degree polynomial segments join them. The power

spectral density in Fig. 3.25(c) was obtained by using a cubic-spline

interpolation (DeBoor 1978).

The time-dependent scale factor is determined as follows: The

longest duration was found to be almost 25 seconds. Therefore, the

other seven records are padded with zero up to 25 seconds in order for

all records to have the same length. The accelerations are then squared

47

and averaged across the ensemble at every 0.02 second interval. The

average value of this IIbiased ll mean square functio l1 is reduced to one.

The normalized mean square function is shown in Fig. 3.26(a). Since

we showed that the spectrum remained time-independent for one second

long segments, we average the normalized mean square function at one

second intervals. The result of this short time averaging which are

joined by straight lines is shown in Fig. 3.26(b). Finally the scale

factor after using a cubic spline interpolation to join the ordinates

is presented in Fig. 3.26(c).

We have now determined the normalized power spectral density and

the scale factor which are shown in Figs. 3.25(c) and 3.26(c), respec-

tively. What remains is an estimate of the average magnitude of the

power spectral density (mean square value) for a set of earthquake pa-

rameters generally specified in design. As will be discussed later

in Chapter 4, relationships between RMS values and earthquake parameters

such as peak ground acceleration, earthquake magnitude, duration of

strong motion, and epicentral distance can be ~stablished which enables

one to estimate the RMS value or the mean square acceleration for a

given set of earthquake parameters. Such a relationship was established

for the eight earthquake records used in this chapter and the equation

of the regression line is given below:

1jJ(a,M,T,O) [

41 ].5510= 101. 9468 a ( M1. 3 ).

0. 066 r 31 (3.1)

where 1jJ is the predicted RMS value in cm/sec2, a is the peak ground

acceleration in g, Mis the earthquake magnitude, T is the duration

48

of strong motion in seconds, and D is the epicentral distance in kilo-

meters.

Table 3.4 lists the properties of the eight records used in this

pilot study as well as their actual and predicted RMS values using Eq.

(3.1). The actual and predicted RMS values are also shown in Fig. 3.27.

The correlation coefficient for the fit is 0.9278 which indicates that

nearly 86 percent of the variation in RMS values can be accounted for-

by the above relationship. In the next chapter we will discuss in de-

tail the correlation between RMS values and the ground motion parameters

for several larger groups of records with common site geology.

The time dependent power spectral density is formulated as:

2G(a,M,T,D,t,f) = ~ (a,M,T,D) S(t) G(f) (3.2)

where ~2(a,M,T,D) is the average magnitude of the power spectral density

which can be obtained from Eq. (3.1) for a given set of ground motion

parameters; S(t) is the time-dependent scale factor representing the

variation of localized mean square to an average mean square of one,

Fig. 3.26(c); and G(f) is the normalized power spectral density, Fig.

3.25(c).

3.5 COMPARISON BETWEEN RESPONSES CALCULATED FROM THE TIME-DEPENDENTPOWER SPECTRAL DENSITY AND SPECTRAL DISPLACEMENT, VELOCITY ANDACCELERATION

In Section 2.6, the mean and variance of relative displacement,

relative velocity and absolute acceleration for a Gaussian stationary

and a nonstationary random process were formulated. The input-output

relationship presented in that section are applicable to a system which

49

is subjected to an earthquake ground motion under the following two

assumptions: First, the individual earthquake records form a locally

stationary random process (see Section 2.4); and second, the process

is normal. Since we showed in Section 3.3 that the spectrum remained

time invariant even for as short as a one second long segment, it is

reasonable to assume that by selecting strong motion duration the earth-

quake records constitute a locally stationary random process. The as

sumption of normality is accepted since it is generally recognized that

the motion recorded at a station results from the arrival of multiple

waves after many reflections and refractions of the initial disturbance

at the source of the earthquake.

With the above two assumptions, we can rewrite Eqs. 2.78-2.80

to describe the relative displacement RD, relative velocity RV, and

absolute acceleration AA of a single degree of freedom system for a

given probability that the response may exceed a specified limit. Thus,

k [~2(a,M,T,D) S(t) t' f/2RDkcr = kcry = IHd(f)1 2 G(f) df0

k [~2(a,M,T,D) t' . ] 1/2RV kcr = kcr' = S(t) IHv(f) 1

2 G(f) dfY 0

k [~2(a,M,T,D) S(t) t' f/2AAkcr = kcra = IHa(f)1 2 G(f) df0

(3.3)

(3.4)

(3.5)

where y and yrefer to relative displacement and velocity, respectively,

and a refers to absolute acceleration. It should be noted that Goo inx

Eqs. 2.78-2.80 has been replaced by its equivalent from Eq. 3.2.

Equations 3.3-3.5 were used to predict the response of a single

50

degree of freedom system subjected to a base motion. The results for

two records (El Centro 1940, SOOE Component and Taft 1952, ~21E Compo

nent) at 3cr level (k =3) and for damping coefficients of 2, 5, 10 and

20 percent of critical are presented in Figs. 3.28-3.35. With the knowl

edge of earthquake parameters (peak ground acceleration a, earthquake

magnitude M, duration of strong motion T, and epicentral distance D)

for each of the two records, the appropriate value of mean square ac

celeration w2(a,M,T,D) is estimated from either Eq. 3.1 or Fig. 3.27.

Also shown in the figures are the corresponding spectral relative dis

placement, spectral relative velocity, and spectral absolute accelera

tion as reported by Trifunac et al., 72-75.

It should be noted that spectral response obtained from an earth

quake record at a given frequency and damping is the absolute maximum

value of the response regardless of the time at which it occurs. The

probability that the response equals the maximum during the duration

of the record is very small (one over the number of points at which

the response is computed). In order to compare the results of this

study with the spectral values one should select a low probability for

exceeding the response and account for the maximum response in the com

putation. To achieve the maximum response, one should use the maximum

scale factor S(t). To reduce the probability for exceeding the response

one needs to compute the response at a high cr level (k = 3 or 4). The

maximum scale factor obtained from Fig. 3.26 is 3.49. Selecting a 3cr

level (k = 3), the probability that the maximum response will be exceeded

is .0026.

(1970).

It should be noted a 3cr level is also suggested by Penzien

51

In general the responses computed from the power spectral density

follow the shape of the spectral curves and for the most part envelope

the curves even at higher damping ratios. In spite of the small sample

size of eight, the results obtained compare well with the spectral values

both in shape and magnitude. As mentioned previously, in the following

chapters similar results from a large number of records with various

geological classifications will be presented and discussed.

TABL

E3.

1

PROP

ERTI

ESOF

THE

EIGH

TRE

CORD

SUS

EDIN

THE

PILO

TST

UDY

Stro

ngM

otio

nPe

akR

ecor

dE

pice

ntra

lR

ecor

dE

arth

quak

eD

ate

Sit

eCo

mpo

Mag

.D

urat

ion*

Acc

.L

engt

hD

ista

nce

(sec

)(g

)(s

ec)

(km

)

Impe

rial

SOOE

.348

53.7

4AO

OlVa

11ey

5/18

/40

E1C

entr

oS9m~

6.7

25-3

0.2

1453

.46

11.5

Ker

nN2

1E.1

5654

.36

A00

4C

ount

y7/

21/5

2T

aft

S69E

7.7

14-1

7.1

7954

.38

41.4

Low

erSO

OW.1

6090

.28

B024

Cal

ifor

nia

12/3

0/34

ElC

entr

oS9

0W6.

517

-25

.183

90.2

266

.3

Wes

tern

N04W

.165

89.0

6B0

29W

ashi

ngto

n4/

13/4

9O

lym

pia

N86E

7.1

21-2

5.2

8089

.04

16.9

*H

ausn

eran

dJe

nnin

gs(1

964)

(Jl

N

53

TABLE 3.2

COMPARISON OF DURATIONS AND ROOT MEAN SQUARE VALUESFOR THE EIGHT RECORDS

T1 T2 ~T RMSfa 2dtRecord Camp. Method* (sec) (sec) (sec) (cm/sec2)

a 0.00 53.74 53.74 46.01 100

b 1.68 26.10 24.42 64.75 90SOOE

c 0.88 26.32 25.44 65.60 96

d 1.38 26.30 24.92 65.88 95E1 Centro

1940a 0.00 53.46 53.46 38.85 100

b 1.66 26.20 24.54 54.39 90S90W

c 0.80 26.62 25.82 54.73 96

d 1.32 26.42 24.92 55.14 94

a 0.00 54.34 54.34 25.03 100

b 3.70 34 0 24 30.54 31. 70 90N21E

c 2.14 36.46 34.32 30.85 96

d 3.46 20.66 17.20 40.19 82Taft

1952a 0.00 54.38 54.38 26.10 100

b 3.66 32.52 28.86 33.96 90S69E

c 2.34 35.30 32.96 32.71 95

d 3.18 17.34 14.16 46.20 82

54

TABLE 3.2 - continued

-Tl T2 L1T RMS

Record Compo Method* (sec) (sec) (sec) (cm/sec2) fa 2dt

a 0.00 90.28 90.28 19048 100

b 2.82 23.92 21.10 38.27 90SOOW -

c 1.92 23.88 21.96 38.38 94

d 1.96 14.98 13.02 46.83 83£1 Centro

1934a 0.00 90.22 90.22 20.76 100

b 2.86 23.14 20.28 41.57 90S90W

c 1.62 20.10 18.48 44.26 93

d 2.00 17.78 15.78 46.80 89

a 0.00 89.06 89.06 22.98 100

b 1. 78 27.58 25.80 40.51 90N04W

c 0.08 23.02 22.94 43.73 93

d 1.06 20.18 19.12 46.59 88Olympia1949

a 0.00 89.02 89.02 28.10 100

b 4.34 22.42 18.08 59.22 90N86E

c 0.28 21.80 21.52 55.48 94

d 4.34 20.46 16.12 61.50 87

* a - Entire Recordb - Trifunac and Brady's Methodc - McCann and Shah's Methodd - This Study

55

TABLE 3.3

COMPARISON OF DURATION AND ROOT MEAN SQUAREBY DIFFERENT METHODS

Tl T2 liT RMSRecord Compo Method* (sec) (sec) (sec) (cm/sec2)

a 1.16 26.36 25.20 65.76SOOE b 1.36 26.20 24.84 65.63

E1 Centro c 0.88 26.32 25.44 65.60

1940 0.88 26.28 25.40 54.95aS90W b 1.08 26.20 25.12 54.89

c 0.80 26.62 25.82 54.73

a 3.18 14.38 11.20 46.96

N21E b 3.38 14.40 11.02 47.15

Taftc 2.14 36.46 34.32 30.85

1952 a 3.20 15.80 12.60 48.24S69E b 3.40 15.80 12.40 48.16

c 2.34 35.30 32.96 32.71

a 2.00 15.00 13.00 46.80SOOW b 2.10 15.00 12.90 46.95

E1 Centro c 1.92 23.88 21.96 38.38

1934 a 2.00 17.60 15.60 46.72S90W b 2.24 17.60 15.36 46.59

c 1.62 20.10 18.48 44.26

* a - Reported by McCann (1980)b - Reported by McCann and Shah (1979)c - Computed in this study by McCann and Shah's (1979) method

TABL

E3,

4

ACTU

ALAN

DPR

EDIC

TED

RMS

FOR

THE

EIGH

TRE

CORD

S

RMS

Val

ueD

urat

ion

ofE

pice

ntra

1(c

m/s

ec2 )

Rec

ord

Com

pone

ntPe

akA

ce.

Stro

ngM

otio

nD

ista

nce

Mag

.

(g)

(sec

)(k

m)

Pre

dict

edA

ctua

l

E1C

entr

oSO

OE.3

4824

.92

66.5

165

.88

1940

U.5

6.7

S90W

.214

24.9

250

.88

55.1

4

Taf

tN2

1E.1

5617

.20

44.8

440

.20

1952

41.4

7.7

S69E

.179

14.1

649

.03

46.2

0

E1C

entr

oSOO~J

.160

13.0

243

.80

46.8

319

3466

.36.

5S9

0W.1

8315

.78

46.5

446

.80

Oly

mpi

aN0

4W.1

6519

.12

45.4

246

.59

16.9

7.1

1949

N86E

.280

16.1

261

.52

61. 5

1

01

0'1

0'1 -....J

41]

353D

Val

ley

Ear

thqu

ake,

May

18.

1940

,

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.3.

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.3.

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.3.

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21,

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.3.

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1934

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.3.

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Was

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ton

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lil]

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* 2 sec2

r + 4 sec Strong motion durationTrifunac &Brady (1975b)

~ 6 sec

X 10 sec

*+ * x+

*+**+ *+* +

* * ~ ~~* ~

0 +~

.....J 0 4 8 12 16 2D 24 28 32c::eu......I-LJ.J0::: 2aLJ.J Strong motion duration::r::

McCann &Shah (1979)I-

aI-

ClLJ.Jl-:=)

0-

*::E +a +.f *u~

* *+ ~a * *+t......

*I- * +* ~ +c::e0:::

+~LJ.J 0 ! X0:::c::e

'1 8 i2 i6 2IJ ~u 28 32~ 00 c.C/)

I......::r::u :!...

Strong motion durationThis study

*+

* * * *+~ JoE

* * + <!>+0 +,<!> X

0 '1 B i2 i6 20TIME, sec

28 32

FIG. 3.9. Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (E1 Centro 1940, SOOE).

~~+

~+~~~ ~ ~ +~

~~ * +~~ z

+ ~+

~

lJ z

lJ'"'

8 12 16 20 24 28 32TIME, sec

FIG. 3.10. Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (El Centro 1940, S90W).

oo

~

~ + ~ ~ + ~

~ ~ ~ ~+ + x ~

~ v'

3 6 9 12 15 18 21 24TIME, sec

FIG. 3.11. Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (Taft 1952, N21E).

l1

o

~

* ~

*~ ~ ~ + ::&: ~

+ ~+

3 6 9 12 is 18 21 2'1TIME, sec

FIG. 3.12. Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (Taft 1952, S69E).

69

* 2 sec2

+ 4 sec Strong motion duration

~ 6 secTrifunac & Brady (1975b)

X 10 sec

* * + ** x ~¢

* * x + ~x + + x¢ +

0¢

...J 0 3 6 9 i2 Ie i8 21 2l.Jo::t: ,--u......I-WJ

a:: -o C

Il.LJ Strong motion duration:cI- McCann & Shah (1979)0l- IClWJ

I

l- X;::).

~I0 x :+ xu

0 X......

I* x + ¢ ¢

l- X+

X + X X lito::t: *a:: + ¢ ¢

l.LJ 0 Xa::o::t:;::)

0 :t 5 :3 i2 is i8 21 2l..J0-(/)

I......:cu 2


o

o

~

x x+ * x XxX +

+ ¢ ¢X

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TIME, sec

FIG. 3.13. Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (El Centro 1934, SOOW).

* * +~ :¥

* * * * * *+ + * + + it~ ~0 x

- 0 3 Ei 9 i2 is 18 21 2Y...Jc:(U......I-UJ

20::::0

Strong motion durationUJ::I:

McCann &Shah (1979)I-

0l-

eUJI-:::>0-::E:0u

*0

* ~ ~......Xl- X + * + X X X X + ~c:(

~ +0::: ~ XUJ [j ,X0:::c:(

!] 3 6 9 i2 is i8 21 2Y:::>0"V)

I......::I:U 2


o

o

XX

*:>E * :>E

* + + + f~~ ~x

3 6 9 i2 is i8 2! 24TIME, sec

FIG. 3.14. Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (El Centro 1934, S90W).

~ 2 sec 712

r+ 4 sec Strong motion duration

~ 6Trifunac &Brady (1975b)

secI is: 10 *I sec

* ~ '*++

* * * is:?IE * * *+ ~

+ ~ +~ +

~

0 X

-l f] Y a i2 15 2U 2Y 28 320::(u......I-l.J.J

§52

f

l.J.J Strong motion duration::cI- McCann &Shah (1979)0I-

Cll.J.Jl-=:> .~ ,.;:EI

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0 +* *

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I......::cu .~

r::

rStrong motion durationThis study

* ++ * * ¥

* ~ + * * *~ * + z~~ +

0 f

0 3 5 9 12 i5 18 21 24TIME, sec

FIG. 3.15. Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (Olympia 1949, N04W).

72

* 2 sec2

+ 4 sec Strong motion duration

~ 6 sec Trifunac &Brady (1975b)

x 10 sec

* ** + ~ +* * + * )( + * ~

~ ~ x0 X

-l 0 ~ 6 9 i2 is is 21 211-'c:t:UI-l

I-LJ.J

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Strong motion durationLJ.J:J::I- McCann &Shah (1979)0I-

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0 3 5 9 i2 is is 2i 211:=>0'(/')

II-l

:J::U 2


{]

{]

** +

* ** * + * + * ~

~ ~ X,x

3 5 9 12 'J: i8 2i 24I ...

TIME", sec

FIG. 3.16. Equivalent spectra test for 2, 4, 6 and 10 secondlong segments (Olympia 1949, N86E).

3h

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19.

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FIG

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20.

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S69E

).

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92

69Lt

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FIG

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22.

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for

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,S9

0W).

3E

ntir

ere

cord

3S

tron

gm

otio

ndu

rati

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nac

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rady

(197

5b)

Str

ong

mot

ion

dura

tion

McC

ann

&Sh

ah(1

979)

\/\

J

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ong

mot

ion

dura

tion

Thi

sst

udy

'-J

'-0

28

/I.

,1

21IY

"1l]

2 3 2l]

9269

'-Hi

23

2.....

...-l

<::

(u ~I

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a lJ.J :::c

I- 01

]I-

Cl

I]lJ

.JI- :::>

0-

::?:

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"

83

I-

<::

(n:

:LJ

.J n::~2

CY

(/) I ........ :::c u

.....-~..

.L

--------

-.I]

,--T

IIJ

L16

1218

2YI]

5III

152U

TIM

E,se

c

FIG

.3.

23.

Equ

ival

ent

spec

tra

test

for

1se

cond

long

segm

ents

(Oly

mpi

a19

49,

N04

W).

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ntir

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tron

gm

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5b)

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McC

ann

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126

~.

,-....

\J~~

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ong

mot

ion

dura

tion

Thi

sst

udy

Ll

23IJ2

9269

Y623

2

........

-l

<:( u ~I

w ~ 0 w :c l- oU

I-

Cl

IIw I-

:::J

0-

::E

0 U ........ 83

I-

<:(~ w ~ :3

20

'U

') II-t

:c uI U

,-~~-,

Ll

U6

1218

2YLl

G12

182Y

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FIG

.3.

24.

Equ

ival

ent

spec

tra

test

for

1se

cond

long

segm

ents

(Oly

mpi

a19

49,

N86

E).

81

5(a) Raw

2

..-l0.-l

X 0N:c 0 5 II] is 20 25-N

NUOJ(/)-yE (b) Smoothedu

..-..4-.........0

>-I- 2~

C/)

ZLJ.JC

.....Ie:t:0:::I-uLJ.J 00..C/)

0::: 0 5 10 is 20 2SLJ.J:;::00..

ClLJ.J YN...... (c) Sp1ined.....I

~0:::0z

2 f~I \I

I0

0 5 10 is 20 2SFREQUENCY, Hz

FIG. 3.25. Power spectral density for the ensemble ofthe eight records used in the pilot study.

82

12(a) Raw

5

oo 5 iO is 2]] 25

'1(b) One second average

..........+-l

V1

~

02I-

Uc:(LL

I..LJ....Jc:(UV1

0

0 5 ilJ is 20 25

(c) Splined

2

o

o 5 15 20 2STIME, sec

FIG. 3.26. Time variation of normalized mean squarevalue of the ensemble of the eight records used in the pilotstudy.

83

100

90 r = 0.9278A = 1. 9468

80 B = 0.5510P = 0.41

70

NU(])

60Vl'-Eu~

?-

U'l 50:::E:c::

40

30.2 .3 .4 .5 .6 .7

FIG. 3.27. Correlation of RMS withparameter n for eight horizontal componentsof recorded accelerograms.

84465

Eu

· 31[1(/)......Cl

LW>.....

iSSl-e::( a-lLW0:::

I)

U

a - Response at 3cr level (this study)

b - Computed response (Trifunac et al., 1972)

o

SB5

·uue::(

LWI::::l-lo(/)coe::(

285uQ)V'l-Eu i9rI

·-lLW>LW 95>.....l-e::(-lLW0:::

0

ilS5 aN

uQ)V'l-B illlJ

IJ S ilJ isFREQUENCY, Hz

21J 25

FIG. 3.28. Comparison of response for 2 percent ofcritical damping for SOOE component of El Centro, ImperialValley Earthquake of May 18, 1940.

85210

Eu

• iYO(/)......Cl



!.L.J:>

~ 10c::x::wj a0:::

o

o

310

i3S

.uuc::x::!.L.Jf:::::....Jo(/)

coc::x::

u<J)

Vl..........Eu gil.

....J!.L.J:>

!.L.J .,5:>......~c::x::....J!.L.J0:::

0

jjiD aN

u<J)

Vl..........E

1YOu

o 5 j[] isFREQUENCY, Hz

20 25


86

a - Response at 30 level (this study)


o

·uuc::eL.LJ 210I-:::>....JaU)

coer::

i35

Eu

· 9IJU).....Cl

L.LJ>..... 'isI-er:: a-lL.LJ0::

[f

9IluOJV1

........Eu

GEl·-l

l.LJ>l.LJ

3IJ:>.....I-er::-ll.LJex:

0

8lO aN

uOJV1

........E 5YOu

o 5 lO isFREQUENCY, Hz

20 25


87gO

a - Response at 3cr level (this study)Eu

b - Computed response (Trifunac et al., 1972)60

(/)......Cl

u.J::::-......

30f-c::e: a-lu.J0:::

0

l5u<llV1

........Eu 50.

-lu.J::::-u.J 25

I::::-......f-c::e:-lu.J0:::

11

GOD aN

u<llV1........E

YOOu

.uuc::e:w 200f-:;:)-l0(/)

coc::e:

0

0 5 iO is 20 25

FREQUENCY, Hz

FIG. 3.31. Comparison of response for 20 percent ofcritical damping for SOGE component of El Centro, ImperialValley Earthquake of May 18, 1940.

88



o

395

3;S

Eu

. 2mV1..-.Cl

WJ:>..-. iDSl-e:(-l aWJc:::

U

iSSu(JJVl-Eu

i3tJ.-IWJ:>

WJ:> 65..-.l-e:(-IWJc:::

l]

iiBS a

uue:(

WJI:::l-IoV1coe:(

NU(JJVl-G 19tJ

[] 5 10 isFREQUENCY, Hz

20 25

FIG. 3.32. Comparison of response for 2 percent ofcritical damping for N21E component of Taft, Kern County,California Earthquake of July 21, 1952.

89



o

.uu~

l.JJ 250l-:=>--JaVlco~

lsil

Eu

~ iDilVl......Cl

l.JJ>...... sIJI-~--Jl.JJ a0::::

[J

9£IuQ)In-Eu

60.--Jl.JJ>l.JJ> 30......I-~--Jl.JJ0::::

0

150 a

NuQ)In-5 500

!J S ilJ is 20 25FREQUENCY, Hz


909lJ

Eu

.VJ.....Cl

L.LJ::>.....I<t:-IL.LJe:::

5fJ

3lJa



o

-IL.LJ::>

L.LJ::>......I<t:-IL.LJe:::

o i!LD-==:::::::~"""" -J- ~ .....l. ......I

50

20

o

o

555N

UOJVl

........5 311].

uu<t:

L.LJ iSSI-;::)

-IoVJco<t:

lJ

a

5 iO is 20 25

FREQUENCY, Hz


91HI r a - Response at 30- level (this study)

Eu

'iIJb - Computed response (Trifunac et al., 1972)

U)......Cl

I.W::>......

2IJI-c:t a--JI.W0::

IJ

50uQ)Vl

........Eu 'i0.

--JI.W::>

I.W 2IJ::>......I-c:t--JI.W0::

0

'iDS aN Iu

Q)Vl........E 2lOu

.uuc:tw i3SI-=:l--J0U)

coc:t

0

0 S iIJ is 20 25FREQUENCY, Hz


CHAPTER 4

CORRELATION BETWEEN RMS VALUEAND EARTHQUAKE PARAMETERS


In Chapter 3 a correlation between the RMS values computed for

the selected duration and the combination of peak ground acceleration,

earthquake magnitude, epicentral distance and the duration of strong

motion was established. Even though the sample size of eight was small,

the correlation was excellent. In this chapter, we will compute the

duration of strong motion for a large number of records (see Section

3.2) and examine the correlation between the RMS values and the same

parameters for horizontal, vertical and combined components of records

with different geological classifications.

4.2 RECORD SELECTION AND CLASSIFICATION

The 987 components of the recorded earthquake accelerograms com-

piled by the Earthquake Engineering Research Laboratory of California

Institute of Technology (Hudson, et al.~ 1971-1975) was used in this

study. Neither the records which were identified as after shocks~ nor

those that were obtained from accelerographs mounted at the mid-heights

or upper stories of building were considered. Among the remaining ac-

celerograms those with at least one of the horizontal components having

a peak ground acceleration equal to or greater than .05 g were chosen.

92

93

It should be noted that when one of the horizontal components had a

peak, ground acceleration equal to or greather_ than .05 g the complete

set (all three components) were selected. Using the above criteria,

a total of 371 components (one of the horizontal components for Parkfield

California Earthquake of June 27, 1966, Cholame, Shandon Array No.2

is not available) were selected.

It i~ generally recognized that the geological condition of the

area near the ground surface has an important influence on the nature

of the ground motion recorded there. Seed, Ugas, and Lysmer (1976)

and Mohraz (1976) show that the site geology influences the response

spectra to a significant degree. Therefore, it seems reasonable to

suspect that the geology of the recording station would also influence

the shape and the magnitude of power spectral density. For this reason,

both the horizontal and vertical components of the 371 selected records

were grouped according to the estimated geological condition of their

sites. Table 4.1 lists the geological descriptions and the locations

of the recording stations. The table has been arranged according to

station number in ascending order. The descriptions were obtained from

four different sources when available, namely: Trifunac and Brady (1975a),

Hudson (1971), Seed, Ugas, and Lysmer (1976) and Mohraz (1976). It

should be noted that if no description is listed by the author's name,

his description is identical to the one given by Hudson (1971).

It is clear from Table 4.1 that it would be impossible to describe

the site geology precisely. For this reason the classification was

accomplished by considering the firmness of the underlying material

at the recording station. The three geological groups which were selected

94

are soft, intermediate and hard. The geological descriptions given

in Table 4.1 were examined and the underlying material at each station

was identified as either soft, intermediate or hard. It should be men

tioned that the boundaries between the three geological classifications

are not precisely defined and there is some overlap between the soft

and intermediate and the intermediate and hard classifications. Tables

4.2-4.4 list the records with their pertinent properties in chronological

order of the earthquakes for the three classifications. Also listed

in the tables are the Cal Tech identification number, epicentral dis

tance, station number, peak ground acceleration and the record length.

The method of establishing the duration of strong motion discussed

in Section 3.2 was applied to the 371 records selected. The method

successfully determined the initial and final times and, therefore,

the duration of the strong motion for 367 of the 371 records. It failed

to determine the duration of strong motion for the following four rec

ords: El Alamo, Baja California Earthquake of Feb. 9, 1956 (A011);

Western Washington Earthquake of April 13, 1949 (8028); ~orthern Cali

fornia Earthquake of June 5, 1960 (V308); and Torrence-Gardena Earth

quake of November 14, 1941 (V316), where the initial cut-off time was

greater than the final cut-off time. A possible explanation for the

failure could be the stringent condition that was specified for the

derivative of the cumulative RMS function (see Section 3.2). These

four records were not considered in the study.

95

4.3 PREDICTION OF RMS VALUE

Seed, et al., (197~) and Mohraz (1976) in their statistical study

showed that the site geology influences the shape and magnitude of re

sponse spectra to a significant degree. Using a regression analysis

McGuire (1974) studied the effect of earthquake magnitude and epicentral

distance on response spectra. The influence of earthquake magnitude,

peak ground acceleration and the duration of strong motion on response

spectra have also been studied by Mohraz (1978a, 1978b). The effect

of earthquake magnitude, epicentral distance and site geology on Fourier

amplitude spectra was first studied by Trifunac (1976). Later McGuire

(1978) presented an empirical model for estimating of Fourier amplitude

spectra and confirmed Trifunac·s finding that site geology, earthquake

magnitude and epicentral distance influence the Fourier spectra.

The above studies indicate that parameters such as earthquake

magnitude, duration of strong motion, epicentral distance, site geology

and peak ground acceleration are important and should be considered

in design. Since these parameters influence response spectra in general

and the Fourier amplitude which is directly proportional to power spec

tral density in particular, it is conceivable that these parameters

will also influence the power spectral density of the recorded accelero

grams. In addition to the five parameters mentioned above, McGuire

(1974) suggests that other parameters such as stress drop, seismic mo

ment, direction of propagation and length of rupture, etc., may be im

portant and could be included in the study. However, at present there

is not enough documented information on these parameters to include

them in the analysis. The influence of site geology on the power spectral

96

density will be accounted for by studying the three geological groups

separately. The effect of the other four parameters will be examined

by considering their correlation to the RMS value. Although the rela

tionship between the RMS value and the peak ground acceleration, dura

tion of strong motion, epicentral distance and earthquake magnitude

can individually be investigated, it will be extremely difficult to

combine the individual influences. Therefore, it is desirable to cor

relate the RMS value with a combination of the four parameters, and

to ascertain the validity of predicting the RMS value from them.

In order to gain an insight as how to combine the four parameters

into one, the RMS value was correlated to each of the parameters using

all 371 selected records without regard to components and site geology.

Figures 4.1-4.4 show the correlation between the RMS value and each

of the four parameters. The figures show the scatter of the data and

regresssion line fitted to them. Also given in the figures is the cor

relation coefficient r. The strongest correlation is obtained with

the peak ground acceleration where the correlation coefficient is .9404.

For the other parameters the correlation is very weak as indicated by

their correlation coefficients. Nevertheless from the sign of the cor

relation coefficients, one can note that the RMS value is directly pro

portional to peak ground acceleration and earthquake magnitude, and

inversely proportional to duration of strong motion and epicentral dis

tance. This observation led to the study of correlation between the

RMS value and the parameter aM/DT. The result of this correlation is

presented in Fig. 4.5. The correlation coefficient for the regression

line is 0.6629 which indicates that there is a better correlation between

97

the RMS value and the parameter aM/DT than any of the individual param

eters with the exception of the peak ground acceleration.

Each of the four parameters (a~ M~ 0 and T) is related linearly

to the combined parameter aM/DT~ which may not be the case for the best

possible correlation. In order to obtain a correlation as good as or

better than the one with acceleration~ a nonlinear combination of the

parameters was considered. Consequently each of the four parameters

was raised to a different power as indicated in Eq. 4.1

PI P2n = a M (4.1)

P3 P40 T

A correlation between the RMS value and n can be established for

a given set of power coefficients P1-P4. For a given set of power co

efficients the RMS values were correlated with n and the correlation

coefficient computed. All possible combinations of P1-P4 were tried

and the set of power coefficients which resulted in the best correlation

was identified. The above procedure was repeated with different range

and increments for P1-P4. The range, the increment and the selected

value of each power coefficient as well as the best correlation coef

ficient are given in Table 4.5. It is noted from the table that all

correlation coefficients for the combination of power coefficients shown

are better than the one obtained when correlating with the peak ground

acceleration alone. The results indicate that the best correlation

is obtained for PI = 1.53~ P2 = 1.30, P3 = .066~ and P4 = .31. The

correlation between the RMS value and n computed using the above powers

is shown in Fig. 4.6.

98

Different sets of coefficients P1-P4 can be obtained for various

geological classifications as well as ·horizontal and vertical components

of the records. Since it would be difficult to compare four different

coefficients (P1-P4) for each category, it was decided to correlate

the RMS value with a new parameter n which is obtained by rewriting

Eq. 4.1 as

[

P ] PP M 2n=Cn) =a

P3 P4D T

(4.2)

where P =fr- Retaining the same P2-P4 of Eq. 4.1, the best correla1

tion between the RMS value and n (Eq. 4.2) was established for P = .65

with a correlation coefficient of r = .96322 which is identical to the

correlation coefficient presented in Table 4.5. The correlation with

the new parameter is shown in Fig. 4.7. Since previously the best cor

relation between the RMS value and the individual parameters was ob-

tained for the peak ground acceleration (r = 0.9404), the improvement

(r = 0.9632) can be attributed to the quantity

(4.3)

It should be noted that the expression for n presented in Eq.

4.2 is by no means the only expression that can be correlated with the

RMS value. The RMS value was also correlated with the following two

parameters:

99

n" = (aM)P(D+200)·45P T· 4{2-P)

(4.4)

(4.5)

The results are compared in Table 4.6. The table presents the power

coefficient P and the correlation coefficient r for each of the three

parameters n, n l, n" for horizontal, vertical, and combined components

of the three geological classifications. The table indicates that in

general (Eq. 4.2) gives slightly better correlation with RMS than n'

and n". It is interesting to note that would also give a better cor

relation coefficient than n l and n" for the eight records considered

in Chapter 2. The results of this comparison are given in Table 4.7.

The result of correlation between the RMS value and parameter

n (Eq. 4.2) for the twelve categories (soft, intermediate, hard and

combined geological classifications; horizontal, vertical and combined

components are presented in Table 4.8. Also shown are the number of

components of records used in the correlation N, the coefficient P,

the slope A, the intercept B and the percent variation of the RMS value

accounted for by n. The equation for the regression line can be ex

pressed in arithmatic scale as ~ = lOA (n)B. Table 4.8 indicates that

best correlations are obtained for the hard sites followed by the group

containing all 367 records. Figures 4.8-4.11 show the correlation for

the horizontal components of the records for soft, intermediate, hard

and combined geological classification. Shown in Figs. 4.12-4.15 are

100

similar correlations for vertical components. The correlation for the

combination of horizontal and vertical components for the three geo

logical classifications and for all 367 records are presented in Figs~

4.16-4.19. In Figs. 4.8-4.19 in addition to the regression line the

95% interval on the future observation (see Walpole and Myers, 1978)

are also presented. The RMS values predicted from the appropriate re

gression line as well as the actual RMS (computed using the strong mo

tion duration--see Chapter 3) for the horizontal components of the three

geological classifications are given in Tables 4.9-4.11. Also presented

in the tables are the initial and final times, the duration of strong

motion, the peak ground acceleration, the epicentral distance and the

earthquake magnitude. Similar information for the vertical components

are presented in Tables 4.12-4.14.

Comparisons of the slope A and the intercept B of the regression

line given in Table 4.8 indicate that for large values of n, the pre

dicted RMS value for the vertical components are greater than those

for the horizontal components for each geological classification. The

correlation coefficient given in Table 4.8 and repeated in Figs. 4.8

4.19 indicate that the RMS values can reliably be predicted from the

parameter n. The correlation coefficient r indicates that lOOr 2 percent

(see Walpole and Myers, 1978) of the variation of the RMS value is ac

counted for by the combined seismic parameter n. The values of r listed

in this table are extremely close to unity (between .9453 to .9880)

indicating that between 89 to 96 percent of the variation of RMS is

accounted for by the relationship with parameter n. It is also observed

from Table 4.8 that in general better correlations are obtained for

101

the "hard" geological classification than the other two geological clas

sifications. The RMS value for any of the 12 categories listed in Table

4.8 can be predicted using the corresponding regression line for that

category. However, the last category which includes all 367 records

considered in the study results in a correlation coefficient which is

just as good or better than most other categories and it is recommended

for predicting RMS values as it covers the largest range of data. It

should be noted that the plots presented in Figs. 4.8-4.19 should be

used for the range of n presented.

TABL

E4.

1

GEOL

OGIC

ALCL

ASS

IFIC

ATI

ON

OFTH

ERE

CORD

S

Sta

tion

Num

ber

Loc

atio

nR

efer

ence

Gro

up*

Geo

logi

cal

Des

crip

tion

0S

eatt

le,

Was

h.D

istr

ict

Tri

funa

caTO

Sand

,sil

t,an

dgr

avel

over

blue

clay

Eng

inee

rsO

ffic

eat

Arm

yha

rdpa

nB

ase

13C

hola

me,

Shan

don,

Cal

i-H

udso

nbA

lluvi

umfo

rnia

Arr

ayN

o.2

Tri

funa

cTO

cS2

Sti

ffso

il15

0ft

.de

epSe

ed

14C

hola

me,

Shan

don,

Cal

i-H

udso

nA

lluvi

umfo

rnia

Arr

ayN

o.5

Tri

funa

cTO

Unc

onso

lida

ted

shal

low

soil

and

allu

vium

,ov

erly

ing

plio

-ple

isto

cene

loos

lyco

n-so

lida

ted

sand

,gr

avel

,sil

tan

dcl

ay

Seed

S2S

tiff

soil

100

ft.

deep

Moh

razd

MO

-

--'

a I'\.

)

TABL

E4.

1-

cont

inue

d

Sta

tion

Geo

logi

cal

Des

crip

tion

Num

ber

Loc

atio

nR

efer

ence

Gro

up*

15C

hola

me,

Shan

don,

Cal

i-H

udso

nA

lluvi

umfo

rnia

Arr

ayN

o.6

Tri

funa

cTO

Moh

raz

MO

16C

hola

me,

Shan

don,

Cal

i-H

udso

nA

lluvi

umfo

rnia

Arr

ayN

o.12

Tri

funa

cTO

Unc

onso

lidat

edsh

allo

wso

ilan

dal

luvi

um,

over

lyin

gpl

io-p

leis

toce

nelo

osly

con-

soli

date

dsa

nd,

grav

elsil

t,an

dcl

ay

Moh

raz

MO

22E

urek

aF

eder

alB

uild

ing

Seed

51D

eep

cohe

sion

less

soil

250

ft.

deep

23F

ernd

ale

Cit

yH

all

Tri

funa

cT1

1500

ft.

ofpl

io-p

leis

toce

nelo

osly

con-

soli

date

dm

assi

veco

nglo

mer

ate,

sand

ston

e,an

dcl

ayst

one

Seed

SID

eep

cohe

sion

less

soil

500

ft.

deep

o w

TABL

E4.

1-

cont

inue

d

Sta

tio

n~erence

IG

roup

'I

Geo

logi

cal

Des

crip

tion

Num

ber

IL

ocat

ion

Moh

raz

IM2

I40

-80

ft.

of

allu

vium

over

1000

ft.

of

sand

ston

eov

ersi

ltst

on

e

28H

oll

iste

rC

ity

Hal

lT

rifu

nac

TOR

ecen

tun

cons

olid

ated

allu

vium

over

par

tly

cons

olid

ated

gra

vel

s.an

dw

ell

cons

olid

ated

mar

ine

sand

ston

ean

dsh

ale.

Wat

erta

ble

from

85-9

5ft

.I

0 .j::>

Moh

raz

IMO

I500

ft.

of

allu

vium

over

ceno

zoic

rock

.W

ater

tab

leat

50ft

.

52OS

OPu

mpi

ngP

lan

t,H

udso

nA

lluv

iurn

Gor

man

,C

alif

orni

aT

rifu

nac

T1

77Sa

nF

ranc

isco

Gol

den

Gat

eT

rifu

nac

T1O

utcr

oppi

ngo

fF

ranc

isca

nch

ert

and

thin

Park

inte

rbed

ded

shal

e

Seed

S3R

ock

Moh

raz

M3S

ilic

eous

sand

ston

e

Sta

tion

Num

ber

80

Loc

atio

n

San

Fra

ncis

coS

tate

Bld

g.B

asem

ent

TABL

E4.

1-

cont

inue

d

Ref

eren

ceIG

roup

*

Tri

funa

cI

Tl

Geo

logi

cal

Des

crip

tion

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d

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n

Num

ber

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ence

Gro

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Geo

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Des

crip

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283

San

taB

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udso

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Tri

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Moh

raz

M2

288

Ver

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ble

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ater

than

300

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1-

cont

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d

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tion

Num

ber

Loc

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ence

Gro

up*

Geo

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289

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d

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tion

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ber

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ence

Gro

up*

Geo

logi

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325

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sil

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420

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413

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1800

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428

5900

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616

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orm

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1150

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1760

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449

2500

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3354

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6562

.56

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62.5

4

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TABL

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2-

cont

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._-_.

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Rec

ord

Rec

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eM

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Len

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(km

)N

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.107

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8647

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8277

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.271

54.4

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8832

.612

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54.5

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8942

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4259

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66.5

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8566

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..... W -.....J

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8857

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40.3

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9640

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TABL

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Rec

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45.6

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8145

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2835

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9327

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27.3

8

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48.2

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48.2

8

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3437

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6449

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Rec

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.118

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4288

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5741

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3241

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5541

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OFRA

NGE

AND

INCR

EMEN

TSUS

EDIN

SELE

CTIN

GTH

EPO

WER

COEF

FICI

ENTS

PI-P

4IN

PARA

MET

ERn

PIP2

P 3P4

No.

of

rit

erat

ion

sR

ange

Inc.

Sel

ect.

Ran

geIn

c.S

elec

t.R

ange

Inc.

Sel

ect.

Ran

geIn

c.S

elec

t.

0.1

0.5

.1.5

0.1

0.5

.1.5

00.

10.

5.1

.10.

10.

5.1

.1.9

5689

625

0.4

1..1

1.0.

41.

.1.1

0-0

.20

.2.1

0.0

-0.2

0.2

.1.2

.962

1912

25

0.80

1.5

.11.

1.4

1..1

.10

-0.2

0.2

.10

.00.

10

.5.1

.2.9

6279

1400

1.1

1.2

.11.

0.50

0.90

.1.7

0-0

.20.

2.1

0.0

0.1

0.5

.1.2

.962

7925

0

1.3

1.5

.11.

51.

11

.5.1

1.4

-0.2

0.2

.1.1

0.0

0.4

.1.3

.963

0931

5

1.3

1.1

.11

.51.

31

.7.1

1.4

-0.2

0.2

.1.1

0.0

0.4

.1.3

.963

0962

5

1.40

1.60

.05

1.50

1.30

1.50

.05

1.30

0.06

0.14

.02

.06

0.20

0.40

.05

.30

.963

2062

5

1.48

1.52

.01

1.52

1.28

1.32

.01

1.29

0.05

80.

062

.001

.062

0.28

0.32

.01

.31

.963

2762

5,

1.52

1.55

.01

1.5

31.

281

.32

.01

1.3

0.06

20.

061

.001

.066

0.28

0.32

.01

,.3

1.9

6322

600

01

W

TABL

E4.

6

COM

PARI

SON

OFCO

RREL

ATIO

NCO

EFFI

CIEN

TSFO

RDI

FFER

ENT

nBS

Ini

ln

nC

ompo

nent

Geo

109Y

Type

Pr

Pr

Pr

Sof

t0.

860

.995

01.

57.9

543

1.20

.956

1

Inte

rmed

iate

1.32

0.9

475

1.66

.941

31.

14.9

438

Hor

izon

tal

Har

d0.

890

.988

01.

60.9

879

1.16

.989

2

All

Thr

ee0.

840

.956

61.

57.9

560

1.22

.957

2

Sof

t0.

620

.945

31.

51.9

453

1.32

.944

4In

term

edia

te0.

750

.961

91.

56.9

618

1.28

.961

9V

erti

cal

Har

d0.

440

.980

11.

48.9

799

1.64

.985

3A

llT

hree

0.62

0.9

527

1.50

.952

91.

34.9

525

Sof

t0.

630

.960

21.

53.9

601

1.30

.959

5H

oriz

onta

lIn

term

edia

te0.

970

.962

41.

60.9

608

1.22

.961

8an

dH

ard

0.78

0.9

815

1.55

.981

21.

24.9

818

Ver

tica

l,

All

Thr

ee0.

650

.963

21.

53.9

632

1.30

.961

3

01 +:>

155

TABLE 4.7

COMPARISON OF THE RMS PREDICTED FROM DIFFERENT n'S FOR.THE EIGHT HORIZONTAL COMPONENTS OF RECORDS USED IN CHAPTER 3

Predicted RMS FromActual

Record Componentn l II

RMS n n

r = .9278 r = .9098 r = .8907

E1 Centro SOOE 65.88 66.51 64.77 65.55

1940 S90W 55.14 50.88 51.80 52.29

Taft N2 E 40.20 44.84 45.32 44.461952 S69E 46.20 49.03 49.58 47.30

E1 Centro SOOH 46.83 43.80 43.00 42.29

1934 S90W 46.80 46.54 44.55 45.22

Olympia N04W 46.59 45.42 47.41 47.07

1949 N86E 61.51 61.52 61.89 59.95

TABL

E4.

8

PARA

MET

ERS

USED

INDE

FINI

NGTH

ERM

SRE

GRES

SION

LIN

ES

RMS

Com

pone

ntG

eolo

gyTy

peN

PA

Br

Var

iati

on*

Per

cent

Sof

t16

10.

861.

8514

0.83

920.

9550

91

Hor

izon

tal

Inte

rmed

iate

601.

321

.557

30.

7890

0.94

7590

Har

d26

0.89

1.80

500.

8728

0.98

8098

All

Thr

ee24

70.

841.

8521

0.84

130.

9566

92

Sof

t78

0.62

1.97

210.

9067

0.94

5389

Ver

tica

lIn

term

edia

te29

0.75

1.90

300.

8837

0.96

1993

Har

d13

0.44

2.04

400.

9097

0.98

0196

All

Thr

ee12

00.

621.

9508

0.87

760.

9527

91

Sof

t23

90.

632.

0025

0.91

700.

9602

92H

oriz

onta

lIn

term

edia

te89

0.97

1.75

970.

8557

0.96

2493

and

Har

d39

0.78

1.86

750.

9202

0.98

1596

Ver

tica

lA1

1T

hree

367

0.65

1.97

400.

8984

0.96

3293

*RM

Sv

aria

tio

nac

coun

ted

for

byth

epa

ram

eter

n;co

mpu

ted

from

lOO

r2(W

alpo

lean

dM

yers

,19

78)

..... U'1

0'\

TABL

E4.

9

ACTU

ALAN

DPR

EDIC

TED

RMS

FOR

HORI

ZONT

ALCO

MPO

NENT

S-

SOFT

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Tim

eTi

me

Ace

.D

ista

nce

Rec

ord

Com

poM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

edT 1

T 2T

a0

MljIa

ljIp

AOOl

SOOE

1.3

826

.30

24.9

2.3

4811

.56.

765

.88

75.6

4

A00

1S9

0W1.

3226

.24

24.9

2.2

1411

.56.

755

.14

50.3

0

A00

3SO

OE14

.64

30..1

615

.52

.047

126.

97.

712

.92

15.9

3

A00

3S9

0W13

.66

26.6

813

.02

.053

126.

97.

719

.40

18.3

2

A00

4N2

1E3.

4620

.66

17.2

0.1

5641

.47.

740

.20

44.9

3

A00

4S6

9E3.

1817

.34

14.1

6.1

7941

.47.

746

.20

52.6

7

A00

5N4

2E6.

7221

.36

14.6

4.0

9088

.47.

728

.57

28.3

2

A00

5S4

8E6.

5220

.40

13.8

8.1

3188

.47.

731

.77

39.2

7

A00

6SO

OW12

.18

26.1

413

.96

.055

120.

37.

715

.97

18.6

5

A00

6N9

0E12

.90

24.4

011

.50

.044

120.

37.

716

.19

16.1

5

-'-

--

---_

._--

--"-

_.

..... 01

-.....J

TABL

E4.

9-

cont

inue

d

-In

itia

lF

inal

Peak

Epi

cent

ral

RMS

Val

ueD

urat

ion

Ric

hter

(cm

/sec

2 )Ti

me

Tim

eA

ce.

Dis

tanc

eR

ecor

dCa

mp.

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

redi

cted

T 1T 2

Ta

DM

1J!a

1J!p

A00

7SO

OW11

.88

26.1

814

.30

.059

120.

37.

716

.40

19.6

8

A00

7N9

0E12

.92

24.4

611

.54

.042

120.

37.

716

.53

15.5

2

A00

8N

llW2.

966.

683.

72.1

6824

.06.

567

.85

58.9

6

A00

8N7

9E3.

067.

003.

94.2

5824

.06.

598

.30

83.4

4

AO

I0N3

1W.2

61.

701.

44.1

029.

65.

846

.30

45.0

3

AO

I0N5

9E.3

65.

104.

74.1

089.

65.

823

.40

36.1

8

A01

1SO

OW9.

8815

.58

5.70

.033

121.

26.

812

.85

13.2

1

AO

llS9

0W4.

3823

.82

19.4

4.0

5112

1.2

6.8

14.8

714

.46

A01

8SO

lW.9

411

.90

10.9

6.0

6522

.25.

724

.34

18.5

2

A01

8N8

9W.8

29.

989.

16.1

7922

.25.

737

.67

45.1

1

A01

9SO

OW7.

1221

.12

14.0

0.1

3067

.36.

426

.20

33.1

7

A01

9S9

0W13

.56

29.4

615

.90

.057

67.3

6.4

17.6

816

.14

-

--I

U1

00

TABL

E4.

9-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Ace

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

redi

cted

T 1T 2

Ta

DM

¢$

pa

B021

S08W

.96

12.3

811

.42

.133

50.4

6.3

32.1

035

.35

B021

N82W

1.26

11.4

210

.16

.154

50.4

6.3

29.6

841

.04

B024

SOOW

1.96

14.9

813

.02

.160

66.3

6.5

46.8

340

.74

B024

S90W

2.00

17.7

815

.78

.183

66.3

6.5

46.8

043

.68

B028

S02W

10.4

231

.58

21.1

6.0

6857

.77.

121

. 21

19.4

9

B028

N88W

10.7

627

.64

16.8

8.0

6757

.77.

118

.50

20.2

5

B02

9N0

4W1.

0620

.18

19.1

2.1

6516

.97.

146

.59

44.4

8

B02

9N8

6E4.

3420

.46

16.1

2.2

8016

.97.

161

.51

72.0

3

B031

N21E

5.36

11.2

25.

86.0

6542

.85.

917

.16

21.3

3

B031

S69E

5.52

9.94

4.42

.068

42.8

5.9

19.5

023

.59

B032

S04E

5.78

12.9

67.

18.1

3760

.96,

544

.07

41.0

3

B032

S86W

5.24

12.4

87.

24.1

9860

.96.

553

.52

55.7

8

-'

U1

\0

TABL

E4.

9-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Ace

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

redi

cted

T 1T 2

Ta

DM

1JIa

1JIp

B033

N65E

2.66

4.84

2.18

.489

58.5

5.6

207.

4413

5.75

B034

N05W

5.56

12.5

46.

98.3

5556

.15.

671

.02

80.1

3

B034

N85E

4.92

11.1

46.

22.4

3456

.15.

687

.80

97.3

3

B035

N50E

1.58

9.40

7.82

.237

25.4

5.6

45.5

457

.80

B035

N40E

1.68

7.00

5.32

.275

25.4

5.6

59.3

671

.37

B036

N50E

2.32

16.0

213

.70

.053

53.5

5.6

12.8

714

.00

B036

N40W

2.42

17.5

215

.10

.064

53.5

5.6

13.3

216

.05

C048

NOOW

3.36

14.8

011

.44

.255

21.1

6.4

77.4

564

.54

C04

8S9

0W2.

8020

.56

17.7

6.1

3421

.16.

445

.92

34.0

9

C051

N36E

.18

10.3

610

.18

.100

41.1

6.4

33.8

529

.25

C051

N54W

.26

7.64

7.38

.125

41.4

6.4

37.9

937

.90

-

-'

O"l o

TABL

E4.

9-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Ace

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(I<m

)A

ctua

lP

redi

cted

T 1T 2

Ta

DM

1JJa

lJI p

0057

SOOW

1.50

12.6

411

.14

.106

35.6

6.4

32.9

930

.32

0057

N90E

1.56

9.26

7.70

.151

35.6

6.4

50.4

444

.31

0058

SOOW

1.52

7.54

6.02

.171

35.6

6.4

58.4

951

.97

0058

N90E

1.56

9.16

7.60

.211

35.6

6.4

67.6

858

.85

0059

N46W

.30

7.56

7.26

.136

38.5

6.4

35.9

140

.97

0059

S44W

.38

7.66

7.28

.150

38.5

6.4

43.4

944

.46

0062

N38W

4.82

14.8

09.

98.1

2041

.36.

435

.16

34.2

4

0062

S52W

4.50

12.4

07.

90.1

3341

.36.

449

.87

39.3

3

0065

SOOW

.06

5.84

5.78

.150

38.5

6.4

40.3

046

.81

D06

5S9

0W0.

006.

166.

16.1

5938

.56.

453

.18

48.4

6

0068

NOOE

0.00

7.46

7.46

.083

33.5

6.4

27.9

127

.09

0068

N90E

.08

7.28

7.20

.100

33.5

6.4

30.6

831

.92

-

-'

O'b

TABL

E4.

9-

cont

inue

d

-In

itia

lF

ina1

Peak

Epi

cent

ral

RMS

Val

ueD

urat

ion

Ric

hte

r(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Ace

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(kill

)A

ctua

lP

red

icte

d

T 1T

Ta

0M

ljJa

IjJp

2

EO72

N75W

0.00

11.0

811

.08

.084

38.1

6.4

27.7

324

.89

E072

N15E

.02

12.0

011

.98

.117

38.1

6.4

32.5

232

.30

E075

NOOE

1.76

14.2

412

.48

.136

38.7

6.4

38.8

636

.29

E075

S90W

1.64

12.1

810

.54

.114

38.7

6.4

34.7

632

.50

E083

SOOW

1.44

8.64

7.20

.161

38.6

6.4

57.4

147

.29

E083

N90E

1.34

10.0

08.

66.1

6538

.66.

446

.60

46.3

2

F086

N83W

2.02

11.3

69.

34.1

0747

.96

.430

.55

31.3

4

F086

S07W

2.20

13.4

211

.22

.082

47.9

6.4

24.4

724

.06

F088

S70E

1.30

7.06

5.76

.271

32.6

6.4

101.

7577

.57

F088

S20W

1.28

9.94

8.66

.213

32.6

6.4

69.3

457

.85

F089

S53E

4.82

11.3

46.

52.1

3442

.66.

449

.51

41. 2

5

F089

S37W

4.96

14.8

69.

90.1

4242

.66.

433

.77

39.4

5-----'-

--

-

m N

TABL

E4

.9-

cont

inue

d

Init

ial

Fina

lPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Ace

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

redi

cted

T 1T 2

Ta

DM

I/J aI/J

p

F095

S88E

2.92

10.1

67.

24.0

9836

.16

.432

.37

31.2

4

F095

S02W

3.00

11. 7

68.

76.0

8536

.16.

431

.29

26.5

6

F098

S53E

5.08

11.5

86.

50.2

4141

.36

.456

.75

67.6

6

F098

S37W

5.02

11. 7

06.

68.1

9641

.36

.453

.43

56.5

4

Fl03

NOOE

.48

10.2

69.

78.0

9345

.06

.427

.64

27.6

6

F103

N90W

.42

iO.6

610

.24

.123

45.0

6.4

35.2

034

.61

Fl04

NOOE

0.00

8.86

8.86

.087

53.5

6.4

20.4

726

.52

Fl04

N90W

.12

2.88

2.76

.105

53.5

6.4

26.9

240

.31

G10

7NO

OE5.

1614

.28

9.12

.095

38.4

6.4

27.4

028

.82

G10

7N9

0E5.

3416

.86

11.5

2.1

0938

.46.

432

.42

30.6

9

G10

8NO

OE4.

5214

..82

10.3

0.2

0238

.46.

445

.09

52.8

1

G10

8N9

0E4.

6815

.40

10.7

2.1

8538

.46

.441

.63

48.6

2-

......

O'l

W

TABL

E4.

9-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Rec

ord

Cam

p.Ti

me

Tim

eA

cc.

Dis

tanc

eM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

ed

T 1T 2

Ta

DM

1/Ja

1/Jp

G11

2N3

8E2.

5810

.88

8.30

.104

41.1

6.4

30.7

531

.65

G1l

2N5

2W2.

7013

.44

10.7

4.0

8041

.16.

424

.87

23.9

7

G1l

4S6

0E1.

1815

.68

14.5

0.1

1332

.66.

435

.67

30.2

8

G1l

4S3

0W1.

0217

.74

16.7

2.1

3932

.66.

428

.96

34.9

0

H1l

5N

llE4.

3621

.02

16.6

6.2

2528

.16.

455

.77

52.7

0

H1l

5N7

9W4.

1422

.26

18.1

2.1

4928

.16.

440

.81

36.5

9

H12

1S9

0W6.

4216

.62

10.2

0.1

2241

.76.

436

.33

34.5

3

H12

1SO

OW6.

2614

.62

8.36

.114

41.7

6.4

36.9

034

.11

1128

NOOE

5.50

15.7

610

.26

.062

35.8

6.4

21.9

219

.68

1128

S90W

5.28

12.8

67.

58.0

9335

.86.

427

.40

29.6

0

1131

N50E

6.72

14.3

07.

58.1

8836

.86.

452

.19

53.3

6

1131

N40W

6.42

15.7

09.

28.1

6436

.86.

439

.29

45.4

8

O'l

+::0

TABL

E4.

9-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Rec

ord

Com

poTi

me

Tim

eA

cc.

Dis

tanc

eM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

edT 1

T 2T

a0

M1J!

a1J!

p

1134

N54E

6.56

14.5

47.

98.1

0037

.56.

434

.17

31.0

3

1134

S36E

6.66

14.1

07.

44.0

8437

.56.

427

.68

27.2

3

1137

S81E

5.44

24.9

219

.48

.143

27.8

6.4

31.4

334

.80

1137

S09W

5.52

21.2

215

.70

.131

27.8

6.4

41.4

833

.93

J145

SOOW

2.82

18.1

415

.32

.116

33.4

6.4

41.2

830

.54

J145

S90W

1.86

18.7

416

.88

.105

33.4

6.4

39.8

827

.49

M17

6N3

7E3.

4813

.34

9.86

.085

41.4

6.4

30.7

825

.70

M17

6S5

3E2.

6815

.34

12.6

6.1

1841

.46.

428

.68

32.0

1

N18

6S3

7E.6

812

.12

11.4

4.0

9852

.66.

423

.54

27.7

0

N18

6S5

3W1.

0810

.04

8.96

.099

52.6

6.4

29.9

129

.50

0199

N28E

7.08

17.0

29.

94.1

4140

.56.

438

.06

39.2

7

0199

N62W

6.64

14.6

27.

98.2

4340

.56.

460

.51

65.1

3-

--

--'

0)

<.n

TABL

E4.

9-

cont

inue

d

-In

itia

lF

inal

Peak

Epi

cent

ral

RMS

Val

ueD

urat

ion

Ric

hter

(cm

/sec

2 )R

ecor

dCa

mp.

Tim

eTi

me

Ace

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

redi

cted

T 1T 2

Ta

DM

1/Ja

1/Jp

P217

SOOW

1.68

14.0

612

.38

.110

38.6

6.4

30.2

930

.43

P217

N90E

1.48

11.8

810

.40

.090

38.6

6.4

27.7

026

.73

Q23

3S1

2W5.

1216

.10

10.9

8.2

4828

.16.

474

.12

62.7

7

Q23

3N7

8W4.

8420

.06

15.2

2.2

0128

.16.

450

.72

48.9

2

Q23

6SO

UT4.

9414

.44

9.50

.171

33.4

6.4

44.9

147

.07

Q23

6EA

ST5.

0416

.34

11.3

0.1

2533

.46.

429

.47

34.8

1

Q23

9SO

UT5.

1616

.86

11.7

0.1

2137

.06.

438

.02

33.4

5

Q23

9EA

ST5.

2013

.18

7.98

.165

37.0

6.4

43.9

047

.27

R244

N53W

5.70

13.8

68.

16.1

5240

.46.

438

.92

43.7

2

R244

S37W

5.86

15.3

89.

52.1

2940

.46.

437

.14

36.8

1

R246

SOUT

5.48

15.9

010

.42

.118

34.1

6.4

33.2

533

.74

R246

EAST

5.32

16.0

210

.70

.109

34.1

6.4

37.3

531

.38

--'-

-.

CTI

CTI

TABL

E4.

9-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

ich

ter

(cm

/sec

2)

Rec

ord

Com

poTi

me

Tim

eA

cc.

Dis

tanc

eM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

ed

T 1T 2

Ta

0M

1J!a

1J!p

R248

SOUT

5.38

15.1

29.

74.1

8834

.26.

450

.73

50.6

3

R248

EAST

5.48

16.1

410

.66

.178

34.2

6.4

49.5

847

.39

R249

N44E

5.48

13.7

48.

26.0

8137

.96.

428

.05

25.7

9

R249

S46E

5.38

16.0

610

.68

.086

37.9

6.4

26.2

425

.60

R251

N37E

3.56

13.1

09.

54.1

9940

.36.

455

.08

52.9

4

R251

S53E

3.38

10.4

47.

06.1

9240

.36.

449

.84

54.9

5

R253

N30W

6.24

13.2

47.

00.2

4740

.66.

460

.22

67.9

9

R253

S60W

6.08

17.9

211

.84

.225

40.6

6.4

45.7

555

.89

S255

N08E

1.20

11.6

410

.44

.126

37.6

6.4

38.0

035

.47

S255

N82W

1.30

14.3

413

.04

.131

37.6

6.4

30.4

034

.87

S258

N29E

6.96

16.4

69.

50.0

5743

.26.

423

.14

18.4

9

S258

S61E

7.60

19.1

011

.50

.085

43.2

6.4

24.1

024

.78

---

......

0'\

""-J

TABL

E4.

9-

cont

inue

d

Init

ia1

Fin

alPe

akE

pic

entr

alRM

SV

alue

Dur

atio

nRi

chte

r(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Ace

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

red

icte

d

TT 2

Ta

0M

I/Ja

I/Jp

1

5261

N59

E0.

007.

367.

36.1

0038

.26.

434

.33

31. 5

7

S261

N31W

0.00

7.32

7.32

.110

38.2

6.4

30.3

634

.24

S262

N83W

2.58

14.7

612

.18

.070

37.7

6.4

26.7

020

.92

S262

,S07

W3.

4217

.02

13.6

0.0

9637

.76.

427

.29

26.6

1

S266

NORT

5.88

18.5

212

.64

.157

38.6

6.4

36.7

940

.82

S266

WES

T5.

9417

.30

11.3

6.1

3238

.6' 6

.437

.34

36.1

5

5267

NORT

6.66

18.1

811

.52

.057

50.6

6.4

17.1

117

.58

S267

EAST

7.70

21.5

813

.88

.063

50.6

6.4

15.7

218

.34

T274

NORT

0.00

4.02

4.02

.029

12.6

3.0

8.75

6.62

T274

EAST

.02

2.50

2.48

.050

12.6

3.0

12.1

311

.66

T286

NORT

2.68

16.8

014

.12

.060

46.2

6.5

15.2

417

.87

T286

EAST

2.90

17.6

014

.70

.047

46.2

6.5

12.5

914

.43

------

--'

O'l

CP

TABL

E4.

9-

cont

inue

d

Init

ia1

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2

)Ti

me

Tim

eA

ce.

Dis

tanc

eR

ecot

'dCo

mpo

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

redi

cted

T 1T

Ta

DM

W a~,

2P

,....- T2

92NO

RT0.

008.

568.

56.0

6423

.25.

414

.94

18.3

2

T292

EAST

0.00

8.54

8.54

.072

23.2

5.4

14.4

520

.24

U29

9N

45E

.04

1.74

1.70

.238

36.4

5.9

97.5

284

.24

U29

9S4

5E0.

001.

721.

72.1

7636

.45.

982

.03

65.2

2

U301

N89W

2.76

8.70

5.94

0197

1909

503

40.3

650

.57

U301

S01W

2.56

7.56

5.00

.122

19.9

5.3

42.7

935

015

U30

5N8

9W2.

3813

.58

11'.2

000

5326

.85.

313

.88

14.3

7

U30

5.

SOIW

1.76

13.2

211

.46

.050

26.8

5.3

14.6

113

.62

U30

7N8

9W.3

67.

387.

02.0

578.

05.

016

.68

17.0

1

U30

7S0

1W.3

84.

824.

44.0

368.

05.

013

.56

12.8

2

V31

4N3

9E3.

0418

.98

15.9

4.0

6457

.46.

321

.08

17.6

5

V31

4N5

1W2.

9815

.08

12.1

0.0

9757

.46.

330

.34

26.6

1-

_.

-..

0)

1.0

TABL

E4.

9-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2

)R

ecor

dC

amp.

Tim

eTi

me

Ace

.D

ista

nce

r~ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

ed

T 1T 2

Ta

DM

ljIa

ljip

V31

5SO

UT.8

66.

946.

08.1

9628

.46.

365

.54

57.9

2

V31

5W

EST

.22

7.06

6.84

.159

28.4

6.3

56.6

947

.33

V31

6NO

RT5.

7210

.00

4.28

.040

3.4

5.4

14.9

915

.80

V31

6EA

ST5.

229.

564.

34.0

553.

45.

420

.10

20.5

8

V32

9SO

UT0.

004.

264.

26.1

676.

34.

736

.36

44.7

5

V32

9W

EST

0.00

.50

.50

.089

6.3

4.7

39.1

742

.62

W33

8NO

RT.0

81.

621.

54.1

1629

.85.

444

.44

43.7

8

W33

8EA

ST.0

82.

362.

28.0

5929

.85.

427

.20

22.7

4

"'-

----

-----

'--.

--

'-J o

TABL

E4.

10

ACTU

ALAN

DPR

EDIC

TED

RMS

FOR

HORI

ZONT

ALCO

MPO

NENT

S-

INTE

RMED

IATE

"-J

alF

inal

Peak

Epic

entt

'a1

RMS

Val

ueD

urat

ion

Ric

hter

(cm

/sec

2 )Ti

me

Ace

.D

ista

nce

)(s

ec)

Mag

.(s

ec)

(g)

(km

)A

ctua

lP

redi

cted

T 2I

aD

MW a

~Jp

-

/l9.

367.

62.1

0456

.25.

824

.02

25.7

2

B9.

467.

48.1

1256

.25.

828

.07

27.4

3

216

.72

10.5

0.1

5940

.06.

552

.02

38.7

2

014

.60

8.50

.201

40.0

6.5

47.9

249

.88

63.

642.

88.0

8514

.25.

330

.01

29.2

2

64.

964.

10.0

5614

.25.

319

.81

18.7

6

06.

544.

74.1

4455

.25.

531

.89

36.1

1

/l7.

145.

10.0

8955

.25.

526

.44

24.1

3

08.

786.

08.0

6210

3.7

6.4

16.8

520

.15

B5.

625.

54.0

3910

3.7

6.4

14.7

014

.40

-----

--'-

.

2.1. 1. 2.1. 6. 6.

Init

1

Tim

e

(sec II

-_

1--

_

Rec

ord

ICa

mp.

A00

2S4

4W

A00

2N4

6W

A00

9N4

4E

A00

9N4

6W

A01

6S0

9E

A01

6S8

1W

B026

N45E

B026

S45E

B027

N45E

B027

S45E

L_

TABL

E4.

10-

cont

inue

d

Init

ial

Fin

alPe

akRM

SV

alue

,

Epi

cent

ral

Dur

atio

nR

icht

er(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Ace

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

,(k

m)

Act

ual

Pre

dic

ted

T 1T 2

Ta

0M

t/Ja

t/Jp

i-_

B03

0N

44E

4.66

9.70

5.04

.054

43.1

5.5

22.1

716

.61

B03

0S4

6E4.

789.

785.

00.0

7643

.15.

521

.39

21.8

1

C05

4N5

2W2.

168.

866.

70.1

5040

.66.

441

.54

41.8

2

C05

4S3

8W2.

0412

.06

10.0

2.1

1940

.66.

433

.21

30.6

0

0056

N21E

1.02

8.02

7.00

.315

29.5

6.4

70.9

075

.69

0056

N69W

1.02

14.9

613

.94

.271

29.5

6.4

62.5

853

.82

E078

N50W

1.62

9.08

7.46

.129

41.1

6.4

36.6

035

.84

E078

S40W

1.72

10.0

48.

32.1

7241

.16.

436

.76

43.4

1

E081

S08E

0.00

8.52

8.52

.217

33.3

6.4

41.1

652

.50

E081

S82W

.02

1.98

1.96

.202

33.3

6.4

76.3

879

.74

F092

S62E

.20

7.84

7.64

.065

41.6

6.4

23.3

620

.69

F092

S28W

.02

9.50

9.48

.081

41.6

6.4

24.6

722

.96

--'

'-J

N

TABL

E4.

10-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Rec

ord

Com

poTi

me

Tim

eA

ce.

Dis

tanc

eM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

edT 1

T 2T

aD

MW a

W p

FI05

SOOW

2.82

8.50

5.68

.085

37.4

6.4

29.9

028

.34

FlO

SN

90E

2.80

10.6

07.

80.0

7937

.46.

424

.38

24.1

5

G11

0S8

2E2.

127.

665.

54.2

1230

.16.

458

.13

59.6

4

G11

0S0

8W2.

3210

.40

8.08

.142

30.1

6.4

34.9

338

.49

J144

N21E

.84

3.92

3.08

.353

'24

.36.

411

4.37

109.

39

J144

N69W

.30

3.76

3.46

.283

24.3

6.4

102.

0988

.50

J148

NOOE

6.10

18.4

612

.36

.110

38.5

6.4

32.6

326

.97

J148

S90W

5.92

16.2

410

.32

.114

38.5

6.4

38.5

729

.40

L166

NOOE

2.02

9.38

7.36

.167

29.3

6.4

36.9

945

.16

L166

S90W

1.80

7.36

5.56

.151

29.3

6.4

51. 7

945

.67

M18

3N6

5W6.

7814

.68

7.90

.043

70.0

6.4

12.9

814

.25

M18

3N2

5E7.

0415

.38

8.34

.057

70.0

6.4

15.3

317

.50

-

.......

w

TABL

E4.

10-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nRi

chte

r(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Ace

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

red

icte

d

T 1T 2

Ta

0M

~Il/J

pa

M18

4S6

5E5.

1812

.94

7.76

.044

70.0

6.4

13.1

814

.60

M18

4S2

5W4.

4812

.70

8.22

.058

70.0

6.4

15.6

517

.82

N18

5S5

0E1.

6012

.98

11.3

8.0

6974

.26.

420

.61

18.3

3

N18

5S4

0W1.

8010

.78

8.98

.069

.74

.26.

424

.47

19.7

8

N18

7N1

5E1.

749.

567.

82.0

5771

.06.

417

.83

17.8

5

N18

7N7

5W2.

149.

747.

60.o

n71

.06.

426

.08

22.8

3

N19

2N2

9E4.

5215

.86

11.3

4.0

9939

.36.

427

.83

25.4

8

N19

2N6

1W4.

5411

.80

7.26

.101

39.3

6.4

36.3

629

.90

P214

S89W

1.3

28.

086.

76.1

5734

.86.

455

.86

43.6

9

P214

SOlE

1.36

7.18

5.82

.159

34.8

6.4

61.3

246

.32

Q24

1N3

7E6.

4816

.78

10.3

0.0

8840

.36.

433

.85

23.9

1

Q24

1N5

3W6.

3814

.98

8.60

.141

40.3

6.4

35.4

636

.n

......

'-J

.p.

TABL

E4.

10-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Acc

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

redi

cted

T 1T 2

Ta

0M

1/Ja

1/Jp

S265

SOUT

6.02

18.6

012

.58

.106

38.5

6.4

29.6

026

.04

5265

WES

T6.

0416

.44

10.4

0.1

2838

.560

432

.24

32.1

4

U30

0N4

5W6.

5011

.64

5.14

.121

64.9

6.4

29.8

637

.24

U30

0S4

5W6.

049.

803.

76.1

1664

.96.

435

.02

39.8

4

U30

8N4

6W2.

688.

065.

38.0

5960

.25.

714

.77

17.8

9

U30

8S4

4W6.

7413

.46

6.72

.075

60.2

5.7

14.4

420

.12

U31

2N4

6W6.

348.

181,

84.1

0530

.55.

837

.90

42.7

6

U31

2S4

4W6.

3811

.60

5.22

.237

30.5

5.8

31.4

458

.05

V31

9N3

6W1.

129.

568.

44.0

5476

.06.

012

.29

15.2

2

V31

9S5

4W1.

585.

684.

10.0

3676

.06.

014

.16

13.9

5

W33

4S6

5E.0

62.

362.

30.1

4213

.25.

455

.65

48.5

5

W33

4S2

5W.0

42.

302.

26.1

9813

.25.

459

.67

63.4

7-

"-l U1

TABL

E4.

10-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Rec

ord

Com

poTi

me

Tim

eA

ce.

Dis

tanc

eM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

edT 1

T 2T

aD

Mljia

ljip

W33

6S5

4E.0

42.

182.

14.0

5722

.25.

424

.05

23.3

4

W33

6S3

6W0.

002.

082.

08.0

7122

.25.

424

.81

28.0

1-...

..J m

TABL

E4.

11

ACTU

ALAN

DPR

EDIC

TED

RMS

FOR

HORI

ZONT

ALCO

MPO

NENT

S-

HARD

Init

ia1

Fin

alPe

akE

pice

ntra

lRM

SVa

'lue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Tim

eTi

me

Ace

.D

ista

nce

Rec

ord

Cam

p.M

ag.

(sec

)(s

ec)

(sec

)(9

)(k

m)

Act

ual

Pre

dict

ed

T 1T 2

Ta

DM

1/Ja

W p

A01

5N1

0E1.

082.

261.

18.0

8311

.55.

333

.94

33.2

0

A01

5S8

0E1.

162.

761.

60.1

0511

.55.

340

.65

37.8

9

B025

SOOW

1.58

3.30

1.72

.146

6.3

6.0

47.9

858

.03

B025

S90W

1.64

3.24

1.60

.145

6.3

6.0

63.2

958

.70

B037

N65W

2.38

6.06

3.68

.269

59.6

5.6

67.4

068

.47

B037

S25W

2.76

4.50

1.74

.347

59.6

5.6

117.

4110

2.40

C041

Sl5W

2.12

9.86

7.74

1.17

07.

26.

4.2

53.6

026

3.36

C041

S74W

1.18

10.5

09.

321.

075

7.2

6.4

222.

7223

3.90

G10

6SO

OW1.

908.

766.

86.0

8934

.76.

428

.61

26.4

1

G10

6S9

0W3.

169.

246.

08.1

9234

.76.

456

.15

53.1

8

'---

-

'-J

'-J

TABL

E4.

11-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nRi

chte

r(c

m/s

ec2 )

Rec

ord

Com

poTi

me

Tim

eA

ce.

.Di

stan

ceM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

ed

T 1T 2

Ta

DM

1/Ja

1/Jp

J141

N21E

2.14

15.2

413

.10

.148

30.8

6.4

33.1

835

.44

J141

S69E

2.26

15.8

013

.54

.111

30.8

6.4

26.8

327

.35

J142

S69E

2.06

7.94

5.88

.172

28.0

6.4

43.6

849

.24

J142

S21W

2.04

6.30

4.26

.146

28.0

6.4

45.5

146

.12

J143

N21E

.22

4.84

4.62

.122

27.7

6.4

36.0

338

.69

J143

N69W

.08

3.52

3.44

.112

27.7

6.4

35.4

338

.55

0198

SOOW

3.74

9.82

6.08

.180

32.5

6;4

55.0

050

.44

0198

S90W

3.56

13.1

29.

56.1

7132

.56.

455

.02

43.2

5

0207

N56E

.16

4.70

4.54

.066

34.0

6.4

21.2

322

.49

0207

N34W

.12

2.52

2.40

.099

34.0

6.4

32.8

737

.35

P221

N03E

.64

9.72

9.08

.140

42.0

6.4

38.8

536

.29

P221

N87W

.26

8.62

8.36

.169

42.0

6.4

42.4

443

.63

'..J

(X)

TABL

E4.

11-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Ace

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

redi

cted

T 1T 2

Ta

DM

1/Ja

1Pp

P223

N55E

0.00

8.88

8.88

.071

63.7

6.4

17.0

519

.75

P223

N35W

.66

7.44

6.78

.054

63.7

6.4

16.7

116

.60

W33

5S8

5E.0

21.

141.

12.0

7118

.95.

435

.12

29.1

5

W33

5S0

5W0.

001.

601.

60.0

5618

.95.

421

.73

21.7

4

,

-_

_l.

--.

......

1.0

TABL

E4.

12

ACTU

ALAN

DPR

EDIC

TED

RMS

FOR

VERT

ICAL

COM

PONE

NTS

-SO

FT

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Tim

eTi

me

Ace

.D

ista

nce

Rec

ord

Com

poM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

ed

T 1T 2

Ta

fJM

1JJa

1/J p

--

A00

1VE

RT.1

811

.50

11.3

2.2

1011

.56.

747

.40

54.7

4

A00

3VE

RT19

.30

19.3

4.0

4.0

3012

6.9

7.7

28.0

225

.40

A00

4VE

RT1.

0828

.00

26.9

2.1

0541

.47.

722

.85

26.5

0

A00

5VE

RT8.

1219

.36

11.2

4.0

4488

.47.

713

.56

13.6

4

A00

6VE

RT21

.96

22.0

2.0

6.0

2312

0.3

7.7

21. 9

418

.63

A00

7VE

RT18

.04

19.4

21.

38.0

2112

0.3

7.7

9.07

9.94

A00

8VE

RT0.

007.

747.

74.0

8324

.06.

520

.73

23.9

9.

A01

0VE

RT.5

42.

902.

38.0

459.

65.

814

.00

16.1

2

A01

8VE

RT1.

2614

.02

12.7

6.0

5022

.25.

710

.50

12.6

5

A01

9VE

RT7.

4612

.20

4.74

.030

67.3

6.4

9.21

9.88

.__

__

__

__

L-.

..... ex:> o

TABL

E4.

12-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Acc

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

redi

cted

T 1T 2

Ta

DM

$a$p

B021

DOWN

.20

3.88

3.68

.152

50.4

6.3

30.0

444

.95

B02

4VE

RT.8

615

.34

14.4

8.0

6966

.36.

518

.40

17.5

2

B029

DOWN

0.00

21.8

621

.86

.092

16.9

7.1

21.1

023

.75

B031

VERT

2.16

13.6

611

.50

.036

42.8

5.9

8.72

9.57

B032

VERT

.14

10.5

210

.38

.061

60.9

6.5

17.3

516

.66

B033

DOWN

1.48

6.52

5.04

.206

58.5

5.6

64.8

751

.15

B034

DOWN

.52

9.54

9.02

.119

56.1

5.6

31.9

728

.14

B035

DOWN

.98

6.22

5.24

.079

25.4

5.6

26.7

621

. 97

B036

DOWN

.94

8.14

7.20

.046

53.5

5.6

16.5

912

.39

C04

8DO

WN0.

0021

.12

21.1

2.1

7121

.16.

442

.24

38.5

4

C051

DOWN

2.42

9.42

7.00

.049

41.4

6.4

13.4

114

.67

------

-------

-

--0

OJ

--0

TABL

E4.

12-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Rec

ord

Com

poTi

me

Tim

eA

cc.

Dis

tanc

eM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

ed

TT 2

Ta

DM

1jJa

1jJp

1

0057

UP.0

69.

889.

82.0

5135

.66

.417

.47

14.4

2

0058

UP.1

09.

849.

74.0

8935

.66.

421

.73

23.9

3

0059

DOWN

.54

6.80

6.26

.068

38.5

6.4

18.2

020

.19

0062

DOWN

3.42

15.9

412

.52

.076

41.3

6.4

15.3

819

.74

D06

5DO

WN.0

411

.08

11.0

4.0

7438

.56.

422

.20

19.7

4

D06

8DO

WN0.

008.

268.

26.0

5833

.56.

418

.64

16.7

4

E072

DOWN

0.00

15.0

215

.02

.066

38.1

6.4

14.5

216

.88

E075

DOWN

1.66

13.0

811

.42

.048

38.7

6.4

15.3

213

.25

E083

DOWN

0.00

13.4

613

.46

.057

38.6

6.4

15.3

115

.05

F086

UP1.

0212

.44

11.4

2.0

4347

.96.

412

.93

11.9

0

F088

DOWN

.72

9.78

9.06

.134

32.6

6.4

45.5

935

.23

"---.

__

__

__

1-..

._

__

__

__

__

_L

..-.

--l

co N

TABL

E4.

12-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Rec

ord

Com

poTi

me

Tim

eA

ce.

Dis

tanc

eM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

edT 1

T 2T

aD

MljJ

aIjJ

p

F089

DOWN

3.30

12.6

29.

32.0

7742

.66.

423

.87

21.0

0

F095

DOWN

5.66

15.4

89.

82.0

2736

.16.

48.

358.

10

F098

DOWN

3.58

12.1

28.

54.0

7141

.36.

421

.44

19.8

4

Fl03

DOWN

.64

11.6

811

.04

.048

45.0

6.4

11.1

113

.26

Fl04

DOWN

.14

8.00

7.86

.036

53.5

6.4

9.55

10.7

7

G10

7DO

WN3.

2611

.84

8.58

.095

38.4

6.4

20.0

625

.88

G10

8DO

WN4.

2811

.54

7.26

.093

38.4

,6.4

30.9

126

.13

G1l

2DO

WN3.

2412

.02

8.78

.054

41.1

6.4

15.3

915

.40

G11

4DO

WN.7

412

.36

11.6

2.0

8832

.66.

425

.50

23.0

4

H11

5DO

WN.8

814

.22

13.3

4.0

9628

.16.

425

.90

24,4

7

H12

1DO

WN1.

2016

.20

15.0

0.0

8141

.76.

420

.82

20.2

6

-'

00 w

TABL

E4.

12-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nRi

chte

r(c

m/s

ec2 )

Rec

ord

Cam

p.Ti

me

Tim

eA

ce.

Dis

tanc

eM

ag.

(sec

)(s

ec)

(see

)(g

)(k

m)

Act

ual

Pre

dict

ed

T 1T 2

Ta

DM

1Jia

1Jip

1128

DOWN

3.78

16.7

813

.00

.037

35.8

6.4

10.3

710

.26

1131

DOWN

7.14

14.6

87.

54.0

3836

.86.

413

.92

11. 5

5

1134

DOWN

6.52

14.4

07.

88.0

6437

.56.

418

.70

18.3

7

1137

DOWN

1.42

16.9

215

.50

.102

27.8

6.4

25.0

925

.20

J145

DOWN

1.70

24.3

622

.66

.108

33.4

6.4

29.2

224

.67

M17

6DO

WN.9

611

.70

10.7

4.0

4241

.46.

414

.13

11.8

4

N18

6DO

WN0.

007.

107.

10.0

6052

.66.

49.

6117

.43

0199

DOWN

5.32

13.3

28.

00.1

5140

.56.

436

.00

39.8

0

P217

DOWN

.86

13.5

812

.72

.061

38.6

6.4

14.7

416

.17

Q23

3UP

1.14

14.4

613

.32

.099

28.1

6.4

26.9

425

.17

Q23

6UP

4.96

13.7

68.

80.0

7533

.46.

424

.58

20.9

0

00

.f::>

TABL

E4.

12-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Rec

ord

Cam

p.Ti

me

Tim

eA

ce.

Dis

tanc

eM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

.Act

ual

Pre

dict

ed

T 1T 2

Ta

0M

1/Ja

1/Jp

Q23

9UP

4.52

12.5

07.

98.0

4137

.06.

413

.99

12.2

5

R244

UP4.

3012

.58

8.28

.044

40.4

6.4

15.4

012

.93

R246

UP2.

1815

.80

13.6

2.0

7634

.16.

419

.81

19.5

9

R248

UP2.

3414

.40

12.0

6.0

9134

.26.

424

.00

23.5

6

R249

UP5.

5413

.02

7.48

.058

37.9

6.4

16.7

516

.95

R251

UP1.

3812

.44

11.0

6.0

6940

.36.

416

.18

18.4

9

R253

UP3.

8013

.54

9.74

.083

40.6

6.4

21.3

422

.35

5255

UP.8

610

.50

9.64

.048

37.6

6.4

13.1

913

.67

5258

UP3.

6016

.18

12.5

8.0

5643

.26.

414

.54

14.9

3

5261

UP.2

65.

184.

92.0

6638

.26.

423

.59

20.5

0

5262

UP1.

828.

947.

12.0

3437

.76.

413

.42

10.5

4

-" co <J"l

TABL

E4.

12-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

5V

alue

,D

urat

ion

Rich

ter

(cm

/sec

2 )R

ecor

dCo

mpo

Tim

eTi

me

Ace

.D

ista

nce

,M

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

edT

1T 2

Ta

DM

ljialjip

5266

UP.4

419

.02

18.5

8.0

5538

.66.

416

.56

13.7

8

5267

UP10

.06

14.9

84.

92.0

2650

.66.

47.

158.

72

T274

UP.0

23.

563.

54.0

2212

.63.

04.

294.

80

T286

UP4.

767.

122.

36.0

2646

.26.

57.

7310

.05

T292

UP0.

001.

921.

92.0

5823

.25.

412

.71

19.3

3

U29

9UP

.14

6.78

6.64

.070

36.4

5.9

15.0

719

.37

U301

UP0.

007.

247.

24.0

7119

.95.

318

.16

18.2

7

U30

5UP

0.00

9.28

9.28

.024

26.8

5.3

6.89

6.47

U30

7UP

1.28

5.66

4.38

.024

8.0

5.0

8.98

7.40

V31

4UP

1.46

11.7

810

.32

.065

57.4

6.3

18.6

417

.30

V31

5UP

.10

5.92

5.82

.285

28.4

6.3

82.3

674

.95

,---.

--'-,

---

co O'l

TABL

E4.

12-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nRi

chte

r(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Acc

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

redi

cted

T 1T 2

Ta

D.M

1/Ja

1/Jp

V32

9UP

.74

2.92

2.18

.025

6.3

4.7

8.32

8.36

-W

338

DOWN

.04

1.46

1.42

.054

29.8

5.4

22.3

518

.91

. -

......

(Xl

........

Rec

AD AO AO BO 80 BO CO DO EO EO

TABL

E4.

13

ACTU

ALAN

DPR

EDIC

TED

RMS

FOR

VERT

ICAL

COM

PONE

NTS

-IN

TERM

EDIA

TE

Init

ia1

Fina

lPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er2

Tim

eTi

me

Ace

.D

ista

nce

(em

/sec

)j

Comp

oM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ua1

Pre

dict

edT 1

T 2T

aD

MW a

W p

VERT

3.00

9.58

6.58

.027

56.2

5.8

9.98

8.51

VERT

7.96

16.5

68.

60.0

4340

.06

.5,

15.4

213

.61

VERT

1.30

3.00

1.70

.044

14.2

5.3

18.3

417

.00

DOWN

3.64

8.12

4.48

.032

55.2

5.5

7.68

10.2

3

DOWN

5.96

6.92

.96

.020

103.

76

.411

.03

10.2

7

DOWN

5.52

8.50

2.98

.030

43.1

5.5

11.7

1.1

0.62

DOWN

1.10

10.4

29.

32.0

5340

.66

.416

.33

15.8

7

DOWN

.84

8.80

7.96

.156

29.5

6.4

41.5

743

.17

DOWN

.26

10.5

810

.32

.068

41.1

6.4

18.3

619

.37

DOWN

.16

11.4

211

. 26

.065

33.3

6.4

14.7

618

.45

------'-

-------

---

co co

TABL

E4.

13-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2

)R

ecor

dCo

mpo

Tim

eTi

me

Ace

.D

ista

nce

Mag

.(s

ec)

(sec

)(s

ec)

(g)

(km

)A

ctua

lP

redi

cted

T 1T

2T

a0

M1J

i alJl p

F092

DOWN

.38

7.36

6.98

.050

41.6

6.4

15.8

915

.98

FI05

UP.8

29.

108.

28.0

6837

.46.

424

.53

20.3

5

GllO

DOWN

2.46

10.2

27.

76.1

2930

.16.

433

.12

36.6

6

J144

DOWN

0.00

3.54

3.54

.107

24.3

6.4

41.8

136

.85

J148

DOWN

.46

15.0

414

.58

.053

38.5

6.4

15.1

614

.51

L166

DOWN

1.9

89.

767.

78.0

7129

.36.

423

.81

21.6

4

M18

3DO

WN4.

3811

.28

6.90

.023

70.0

6.4

7.89

7.88

M18

4DO

WN2.

168.

666.

50.0

2570

.06.

47.

788.

59.'

N18

5DO

WN1.

8210

.90

9.08

.042

74.2

6.4

13.0

512

.66

.N

187

DOWN

2.08

7.12

5.04

.029

71.0

6.4

9.90

10.3

2

N19

2DO

WN3.

6215

.64

12.0

2.0

4339

.36.

414

.67

12.5

4

---~._-

--

..... co lD

TABL

E4.

13-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Rec

ord

Com

poTi

me

Tim

e.A

ce.

Dis

tanc

eM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

edT 1

T 2T

aD

M1/Ja

1/Jp

P214

DOWN

.30

8.78

8.48

.118

34.8

6.4

28.4

833

.06

Q24

1UP

4.60

13.8

49.

24.0

6240

.36.

419

.49

18.2

7.,

5265

UP.6

413

.12

12.4

8.0

5538

.56.

415

.26

15.4

8

U30

0UP

7.20

11.4

44.

24.0

3864

.96.

415

.36

13.6

3

U31

2UP

7.84

9.98

2.14

.033

30.5

5.8

11.3

913

.15

V31

9UP

1.98

4.40

2.42

.027

76.0

6.0

8.94

10.6

2

W33

4DO

WN0.

003.

043.

04.0

5413

.25.

420

.95

18.4

3

W33

6DO

WN0.

001.

861.

86.0

3822

.25.

412

.17

14.6

1

\

-'

1.0

a

TABL

E4.

14

ACTU

ALAN

DPR

EDIC

TED

RMS

FOR

VERT

ICAL

COM

PONE

NTS

-HA

RD

,---

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Tim

eTi

me

Ace

.D

ista

nce

Rec

ord

Com

poM

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

edT 1

T 2T

a0

MW a

tP p

A01

5VE

RT.8

02.

301

.50

.038

11.5

5.3

11.6

012

.00

8025

DOWN

.82

3.24

2.42

.089

6.3

6.0

30.1

426

.58

8037

DOWN

.10

5.92

5.82

.132

59.6

5.6

22.1

331

.01

C041

DOWN

.52

9.44

8.92

.709

7.2

6.4

165.

4915

3.87

G10

6DO

WN2.

128.

246.

12.0

8534

.76.

421

.18

22.4

6

J141

DOWN

0.00

7.66

7.66

.095

30.8

6.4

30.1

924

.24

J142

DOWN

.22

5.92

5.70

.154

28.0

6.4

38.6

939

.12

.

J143

DOWN

.16

5.06

4.90

.073

27.7

6.4

21.3

320

.22

0198

DOWN

.90

10.4

89.

,58

.123

32.5

6.4

28.6

229

.78

0207

UP.8

85.

684.

80.0

34I

34.0

6.4

9.42

10.0

6

---

-,

..... \0

TABL

E4.

14-

cont

inue

d

Init

ial

Fin

alPe

akE

pice

ntra

lRM

SV

alue

Dur

atio

nR

icht

er(c

m/s

ec2 )

Tim

eTi

me

Acc

.D

ista

nce

Rec

ord

Cam

p.M

ag.

(sec

)(s

ec)

(sec

)(g

)(k

m)

Act

ual

Pre

dict

edT 1

T 2T

aD

M1Jia

1Jip

P221

DOWN

0.00

12.4

812

.48

.049

42.0

6.4

14.3

712

.39

P223

DOWN

.80

5.34

4.54

.039

63.7

6.4

12.3

011

.29

W33

5DO

WN.0

21.

081.

08.0

6018

.95.

416

.12

18.9

1

.

--------

-

I.D

N

193

500

l! ! I

jI ii.. !

r = 0.9404 "

100. J~

C'JUQ) •Vl M'-Eu M

~

V1::E

K~ 10. K)C

1..1

PEAK ACCELERATION a, g

1 I I f { I r r I I.J-__.:..----.:._.l--.:..-J.--.:....~.L.___.l..___:~..:.._..:.._:._.:....:....:....!___ _.I

.01

FIG. 4.1. Correlation of RMS with peak groundacceleration for 367 horizontal and vertical componentsof recorded accelerograms.

194

500. I I I I I ! ! I ! !

Ie

r = -0.2234 " ""

M M

100. f "MM

"M ~,,= iii

"N MMUOJVl

M- "E ... tCU

,.;..tC tC

-i3- IIIIe

(/) III::E: M Ie ,.c::

"I(Ie

10. ~

" M

'"M)IE

'"

100.10.

EPICENTRAL DISTANCE D, km

1. :...1 :...-_:.--...;,_1;......;.1......:..'-"1.....,...;1 ...:.-_...:.-......:..' ' ' ....1_1.......' ...1__--'

1.

FIG. 4.2. Correlation of RMS with epicentraldistance for 367 horizontal and vertical componentsof recorded accelerograms.

500.

195

r = -0.0227

100.

NUC!)Vl

"Eu.. tS

?

"en::E:e:::

10.

K tS

M: M~ "illJti

,rIlC ~",* IIIIII N Ie Ie

iii IG Ie

",Ie iii~

iii IeIe ~IC

Ifs

--* IeIe

IIIJC

III 1ll*,?lJC

.1 1. 10.

DURATION OF STRONG MOTION T, sec.

FIG. 4.3. Correlation of RMS with duration ofstrong motion for 367 horizontal and vertical componentsof recorded acce1erograms.

196

500.

r = 0.1725

100.

,N ,

u f-OJUl i- 1

E ru!i

..;3- r(/'):E:c:::

10. ......,r-r-riii

"I

,

Ii"it~

,

" ,

.-

1. -.-------------------------~-1. 2. 3. 4. 5. 10.

EARTHQUAKE MAGNITUDE M

FIG. 4.4. Correlation of RMS with earthquakemagnitude for 367 horizontal and vertical componentsof recorded accelerograms.

5000

197

IIIIIIIIIIIIIIIIII

10 .

I I II II I

1.

, I I II I II

.1

I I I I 1 II ,

199

r = 0.9632

FIG. 4.7. Correlation of RMS with parameter nfor 367 horizontal and vertical components of recordedaccelerograms.

L.01

1000.

100.

NUa.JVI

.......Eu..

?

Vl:E:a:::

10.

•

10.1..1.01

r = 0.9632

198

FIG. 4.6. Correlation of RMS with parameter ~

for 367 horizontal and vertical components of recordedaccelerograms.

1.1...--.l...-l.....l-.:...L.LlJ..l.--.l...--l-...l....l..J....:..J...I.l..----I.-J..-"-l...:..w..l..l-.----l.---i......l-'-...1..U.J.j

.001

100.

NU(l)Vl

.........Eu

n

?

V')

::E.:0::

10.

II-IIIIIIIIIIIIIIII•

200

· 1000.

r = 0.9549A = 1.8514B = 0.8392

"p = 0.86

100.N

UOJU1

.........Eu..

?

VI:E:c:r::

10.

l. I I , I , ! II I , I II III I , 1 I II'

.01 .1 1. 10.

[ ~11.3 ] Pn =

a O.066 r 31

FIG. 4.8. Correlation of RMS with parameter nfor 161 horizontal components of recorded accelerograms-Soft.

201

r = 0.9475A = 1.5573B = 0.7890P = 1.32

C'JU(])VI

'Eu

100.

10.

10 .1 1'111111 I IlIllll I I ifill.:-_~--"---:""':""~~__':""--':"'''':''''''':-':'~..:..l-_---:'_''':'-~'-'-~

.01 .1 1.

FIG. 4.9. Correlation of RMS with parameter nfor 60 horizontal components of recorded accelerograms-Intermediate.

202

r = 0.9880A = 1.8050B = 0.8728P = 0.89

NUIVVlEu

10 .

1 I 11 11,,1 I 11 11,,1 I 11 II,]• L-_....l-.-...:.-l.--l-.:....:..~__..:..--.:...---"-...:....:~~_-..:_...:.-..l-:.....:...~

.01 .1 1. 10.

FIG. 4.10. Correlation of RMS with parameter nfor 26 horizontal components of recorded accelerograms-Hard.

203

r = 0.9566A = 1.8521B = 0.8413P = 0.84

10.

100.

1000. ....I=--,.-~.,..-,r-r-j ""jM!j"Tj---r--r--r-"T-rj""T!"T!""'j."---,.---r-"T'I....,.-jT""jjiT"l-W

NUOJVl

au

10 .1 ' 'Ilflfl I I 11111 I IIlfif.=--_...:..-......:-........:.....:..~:..:.-_-'---:......:.. ........~............----_.........------........

. 01 .1 1.

FIG. 4.11. Correlation of RMS with parameter nfor 247 horizontal components of recorded accelerograms-combined Soft, Intermediate, and Hard.

1000.

204

r = 0.9453A = 1.9721B = 0.9067P = 0.62

100.N

U<llVI-Eu

?

tf)

~e::::

10.

1..01

, I I I " II.1

I I I ,,"I1.

, , I

l,J

10.

FIG. 4.12. Correlation of RMS with parameter nfor 78 vertical components of recorded accelerograms-Soft.

205

Jr = 0.9619A = 1.9030B = 0.8337P = 0.75

10.

100.N

UQ)VI

E-u

1. I I I " III I I I I I Ii I I I I I 11 I

. 01 .1 1. 10 .

= [ M1.3 rn

a 0. 066 r. 31

FIG. 4.13. Correlation of RMS with parameter nfor 29 vertical components of recorded accelerograms-Intermediate.

206

1. 1.-1 _--.;_..;.I---:.I~I...:.,1...:.,'..:,.1..:.,.11.-1 _--.;_..;.1---:.1--..:..1...:.,1..,:.1..:,.1..:.,.1:...1_~:...-..:.I--:.I~I...:.,1..,:.1..:,.1..:..:'

.01 .1 1. 10.

FIG. 4.14. Correlation of RMS with parameter nfor 13 vertical components of recorded accelerograms-Hard.

207

r = 0.9527A = 1. 9508B = 0.8776p = 0.62

100.N

U(l)Vl'-Eu..?

Vl:E:0:::

10.

1. I I I II " , I 1 , I , I rl I , I I I II

.01 .1 1. 10.

ljJ = [ r~I. 3 JPa 0. 066 r 31

FIG. 4.15. Correlation of RMS with parameter nfor 120 vertical components of recorded accelerograms-combined Soft, Intermediate and Hard.

208

1000.

r = 0.9602A = 2.0025B = 0.9170p = 0.63

100.N

U(1)(/)-Eu

ft

-,3-

(/)::::0::

10.

1..01

, I I 11111

. 1

I , I , 11 , I1.

I I I I II 1

10 .

FIG. 4.16. Correlation of RMS with parameter nfor 239 horizontal and vertical components of recordedaccelerograms--Soft.

209

1000.

r = 0.9624 ~A = 1. 7597B = 0.8557 ]P = 0.97

100.N

UOJVI

.........Eu

ft

?

(/)

::E:0:::

10.

1. '-- .......'___.'...'........' .;..'';....;1....' ,___.;'___..'.-.1..;"....1....11 ...;..-.--:..,---.'..;'........' ...,1'......1I.01 .1 1. 10.

FIG. 4.17. Correlation of RMS with parameter nfor 89 horizontal and vertical components of recordedaccelerograms--Intermediate.

210

r = 0.9815A = 1. 8675B = 0.9202P = 0.78

100.N

UOJVl

........Eu

ft

?

V"l~0:::

10. K

l..01

I I I If III

.1

I 1 I II III

1.

I 1 I II I I

10.

r M1. 3 JPn = a lO.066 T. 31

FIG. 4.18. Correlation of RMS with parameter nfor 39 horizontal and vertical components of recordedaccelerograms--Hard.

211

r = 0.9632A = 1. 9740B = 0.8984p = 0.65

10.

100.

FIG. 4.19. Correlation of RMS with parameter nfor 367 horizontal and vertical components of recordedaccelerograms--combined Soft, Intermediate and Hard.

CHAPTER 5

POWER SPECTRAL DENSITIES AND SCALE FACTORSFOR DIFFERENT GEOLOGICAL CONDITIONS


In a pilot study in Chapter 3, a power spectral density and a

scale factor for an ensemble of eight strong motion accelerograms were

estimated and used to compute the response of a single degree of freedom

system. Using the procedure outlined in that chapter, power spectral

densities and scale factors for the three geological classifications

(soft, intermediate, and hard) and for both the horizontal and vertical

components of a number of records are computed and presented in this

chapter. In Chapter 4, relationships between the RMS of the records

and a variable n reflecting the earthquake parameters were obtained

for different groupings of the records including the six used in this

chapter. The information presented in Chapter 4 and this chapter is

used to predict the response of a single degree of freedom system and

the results are compared with spectral relative displacement, relative

velocity and absolute acceleration computed directly from several records.

5.2 POWER SPECTRAL DENSITIES AND SCALE FACTORS

Using the procedure outlined in Section 3.4, power spectral den

sities were estimated for six different classifications of records.

Normalized power spectral densities for the ensemble of the horizontal

212

213

components of accelerograms recorded on soft, intermediate, and hard

sites are shown in Figs. 5.1-5.3 respectively. Similar plots for the

vertical components are presented in Figs. 5.4-5.6. The accumulated

area under the power spectral density as a percentage of the total area

for the three site classifications and for the horizontal and vertical

components are given in Table 5.1. The peak ordinate for each of the

six power spectral densities and their corresponding frequency is shown

in Table 5.2.

From the shape of the power spectral densities in Figs. 5.1-5.6

and the rate of accumulation of areas in Table 5.1, one may make the

following observations: First, power spectral densities reach their

maximum values at low frequencies and practically vanish at frequencies

greater than 10-Hz indicating that the dominant frequencies in earth

quake accelerograms are within 0-10 Hz. Second, the power spectral

densities for accelerograms recorded on softer geology display fewer

peaks, whereas those recorded on harder sites show several peaks, in

dicating that the geology of the recording station influences the fre

quency structure of the data. Third, a comparison of the power spectral

densities for horizontal and vertical components shows a wider range

of dominant frequencies in the vertical components, indicating that

the energy contained in the horizontal motion is concentrated in a nar

rower band than that in the vertical motion. This can also be seen

from Table 5.2, where the maximum ordinates of the power spectral den

sities for the horizontal motion are consistently greater than those

corresponding to the vertical motion. Fourth, the rate of energy ac

cumulation (see Table 5.1) is faster for the accelerograms recorded

214

on softer geology and is also faster for the horizontal components than

their corresponding vertical components.

The scale factors (the normalized mean square acceleration) for

the six categories are computed using the procedure outlined in Section

3.4. The variation of the scale factor with time for the horizontal

components of acce1erograms recorded on soft, intermediate, and hard

geology is presented in Figs. 5.7-5.9, respectively. Similar plots

for the vertical components are presented in Figs. 5.10-5.12. Table

5.3 lists the durations of the scale factors, their maximum ordinates

and the corresponding time at which they occur. Some general observa

tions regarding the scale factors can be made. The stiffer geological

classifications have shorter durations. This is true for both the hori

zontal and the vertical components. In transient response of a single

degree of freedom, the duration of the scale factor could play an impor

tant role. For a given mean square value, the shorter duration imparts

energy into the system in a shorter time than a longer one. The maximum

scale factors for the soft site are generally greater than those for

the stiffer sites and they occur at an earlier time.

5.3 RECORDS SELECTED FOR COMPARING THE RESPONSE

To study the application of the power spectral densities and scale

factors presented in Figs. 5.1-5.2 in predicting the response of a single

degree of freedom system for a given set of earthquake parameters, a

total of twelve records--two records for each of the six classifications-

were selected. The records and some of their properties are listed

in Tables 5.4 and 5.5. The records were selected from five seismic

215

events (Imperial Valley 1940, Kern County 1952, Eureka 1954, Northern

California 1952, and San Fernando 1971) with an earthquake magnitude

between 5.5 and 7.7. The records are from six different stations with

an epicentral distance ranging from 11 to 120 kilometers. For each

geological classification two records were selected, one with a high

peak horizontal acceleration and the other with a low peak horizontal

acceleration but a high peak velocity to peak acceleration ratio. As

seen from Table 5.4, the properties for the six horizontal records cover

a wide range; i.e. peak ground accelerations between .05 to 1.172 g,

peak ground velocities between 2.74 to 44.49 in./sec., peak ground dis

placements between .80 to 14.84 in., durations of strong motion between

5.04 to 24.92 sec., and actual RMS values between 16.53 to 253.6 cm/sec2.

A wide range of properties is also observed from Table 5.5 for the verti

cal components.

5.4 COMPARISON OF PREDICTED AND COMPUTED RESPONSE

The expressions for computing the relative displacement, relative

velocity, and absolute acceleration of a single degree of freedom are

presented in Eqs. 3.3-3.5. In addition, the relationships between the

RMS of the records and a variable n reflecting the earthquake parameters

(peak acceleration, earthquake magnitude, epicentral distance and the

duration of strong motion) for different geological groupings were de

veloped and presented in Chapter 4. Using the normalized power spectral

densities and scale factors presented in this chapter and the informa

tion given in Chapters 3 and 4, the response of a single degree of free

dom for several specified sets of earthquake parameters is predicted

216

and compared with that computed directly from the records. The mean

plus one standard deviation response (normal distribution) of the en

semble, which is presently used as a basis in developing design spectrum,

is also compared with the response computed directly from the records

(Trifunac et al., 1972-1975).

Since a response spectrum represents the maximum response of a

system, the maximum scale factor and a low probability of exceeding

the maximum response (3cr level) were used in the comparisons. The re

sults were compared for 2, 5 and 10 percent of critical damping; however

only 2 and 10 percent are presented here.

A comparison between the responses predicted using the power spec

tral density and those computed directly from the six horizontal compo

nents of the records listed in Table 5.4 is presented in Figs. 5.13

5.36. Also shown (in separate figures) is the mean plus one standard

deviation response of the appropriate ensemble. The plots are arranged

in three sets. Figures 5.13-5.20 show the comparison of the response

for soft sites, Figs. 5.21-5.28 for the intermediate sites, and Figs.

5.29-5.36 for the hard sites. Figures 5.37-5.60 show similar comparisons

for the vertical motion. In Figs. 5.13-5.60, each comparison between

the predicted and the computed response is immediately followed by a

comparison of the mean plus one standard deviation and the computed

response. The reason for separating the two comparisons is to provide

an easier examination of the figures.

Figures 5.13-5.60 indicate that in general there is a close agree

ment between the shape of the predicted and the mean plus one standard

deviation response. In many instances this shape ciosely resembles

217

the shape of the computed response (see for example Fig. 5.29 and 5.35).

The response predicted from the power spectral density in the majority

of cases envelopes the computed response over the entire frequency range

of .06 to 25 Hz. This envelope seems to be closer to the computed re

sponse in the higher frequency region and is particularly evident for

the horizontal components on hard geology. In the lower frequency region

however, the predicted response seems to generally overestimate the

computed response. This is not true for all the comparisons. For example,

Fig. 5.35 shows a very good agreement between the predicted and computed

response for both high and low frequency regions. The agreement is

equally good for relative displacement, relative velocity and absolute

acceleration. Comparison of Figs. 5.35 and 5.36 indicates that the

computed response is closer to the predicted than to the mean plus one

standard deviation response. Figures 5.29-5.30 show a similar comparison.

Figures 5.27-5.28 and 5.49-5.50 show the comparison of the horizontal

and vertical components of Ferndale City Hall 1954 in which the relative

velocity predicted from PSD is much closer to the computed response

than the mean plus one standard deviation. The agreement between the

predicted and computed absolute acceleration is seen in Figs. 5.21-5.22

and 5.23-5.24. In general the predicted response for both horizontal

and vertical components of the records for all geological classifica-

tions compare well with the computed response. The difference noted

between the predicted and computed response at low frequencies, espe

cially for low damping, indicates that 3a level is too high for such

cases. Penzien (1970) has confirmed this result. As the probability

for exceeding the predicted response at 3a level (.26%) indicates, the

218

computed response should seldom exceed the predicted one. This can

easily be seen from the comparisons presented in Figs. 5.13-5.60. On

the other hand, the mean plus one standard deviation response .which

represents the 84.1 percentile level is exceeded more often as expected.

The eight records used in the pilot study in Chapter 3 were clas

sified in Chapter 4 in the soft category. It is interesting to compare

Figs. 3.28 and 5.17, and Figs. 3.30 and 5.19, where the computed re

sponse for 2 and 10 percent of critical damping for El Centro 1940 NS

component is compared with the predicted response. These figures in

dicate that not only the predicted response in Chapter 5 is higher than

those of Chapter 3, but the shapes are different as well. The higher

values of response are attributed to higher predicted RMS value for

that record (75.64 vs. 51.56) and higher value of maximum scale factor

(4.29 vs. 3.49) used in Chapter 5. Further the shape of the predicted

responses are different because the power spectral densities (Fig. 3.25

and Fig. 5.1) are not the same.

219

TABLE 5.1

ACCUMULATED AREA AS THE PERCENTAGE OF TOTAL AREAUNDER THE POWER SPECTRAL DENSITY

Frequency Horizontal VerticalRange(Hz) Soft Inter. Hard Soft Inter. Hard

o - 1 17.94 10.75 7.22 11. 75 10.10 6.58

o - 2 41. 62 30.25 26.96 26.14 24.59 15.58

o - 3 60.32 48.81 39.49 39.89 37.59 22.53

o - 4 73.28 64.65 52.18 52.13 49.31 32.70

o - 5 81.85 75.91 65.93 62.29 59.29 43.73

o - 6 87.80 83.64 76.25 70.06 66.51 54.70

o - 7 91.57 88.10 82.09 76.15 72.27 63.35

o - 8 94.32 91.82 87.91 82.08 76.90 70.03

o - 9 96.15 94.67 91.83 86.09 80.82 76.58o - 10 97.32 96.89 94.22 89.13 85.38 82.45

o - 11 98.13 97.94 95.59 91.74 89.63 87.34

o - 12 98.69 98.65 96.76 93.53 92.30 90.41o - 13 99.09 99.09 97.68 95.01 94.31 92.79

o - 14 99.33 99.36 98.45 96.16 95.82 94.68

o - 15 99.49 99.52 98.84 97.11 96.84 95.91

o - 16 99.60 99.66 99.15 97.77 97.73 97.01

o - 17 99.69 99.74 99.39 98.33 98.38 97.80

o - 18 99.75 99.81 99.57 98.75 98.88 98.52o - 19 99.81 99.86 99.69 99.03 99.18 98.93

o - 20 99.84 99.89 99.78 99.25 99.40 99.24

o - 21 99.88 99.92 99.85 99.46 99.56 99.52

o - 22 99.91 99.94 99.90 99.61 99.68 99.67

o - 23 99.94 99.96 99.93 99.74 99.80 99.76

o - 24 99.97 99.98 99.97 99.86 99.90 99.89

o - 25 100.00 100.00 100.00 100.00 100.00 100.00

220

TABLE 5.2

MAXIMUM ORDINATES OF THE POWER SPECTRAL DENSITIESAND THEIR CORRESPONDING FREQUENCIES

Frequency Peak ValueClassification Geology Type

(cm/sec2)2/Hz(Hz)

Soft 0.98 .2548

Horizontal Intermediate 1.55 .1995

Hard 1.34 .2173

Soft 0.90 .1587

Vertical Intermediate 1.14 .1547

Hard 4.76 .1118

221

TABLE 5.3

MAXIMUM VALUES OF SCALE FACTORS

Classification Geology Type Duration Time Peak Value(sec) (sec)

Soft 25 0.98 4.29

Horizontal Intermediate 14 0.84 4.07

Hard 14 5.57 2.21

Soft 27 0.86 3.91

Vertical Intermediate 15 0.88 3.11

Hard 13 5.49 3.61

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2SDu

uue::(

UJ iYSI-:::::>-l0(/)

co

"" 0

Q iQ !5 20

FREQUENCY, Hz

FIG. 5.13. Comparison of predicted response (soft)and computed response for 2 percent damping, HollywoodStorage P.E. lot, 1952--N90E.

237

a - Mean plus one standard deviation responseEu

• HiDV')......aI.JJ:>......!;: BD...JI.JJ0:::


aD

i:fSuOJ1/1

...........Eu

90·....J

I.JJ::>

I.JJ::> "IS......l-e:(...JI.JJ0:::

D

1135N

uOJ1/1-Eu 29tJ·u

uc:(

I.LJi Lf 5I-

:::>...J0V')coc:(

0

0 5 II] it: 2D 25.~

FREQUENCY, Hz

FIG. 5.14. Comparison of mean plus one standarddeviation response (soft) and computed response for 2percent damping, Hollywood Storage P.E. lot, 1952--N90E.

lS239

a - Mean plus one standard deviation response

b - Computed response (Trifunac et al., 1972)SI]

25

Eu

·(/)....aUJ::::.-~...JUJ~

IJ

Y5u(1JVl

"E-u 3D..·...J

UJ::::-UJ

IS::::-.-!;(...JUJ~

lJ

2lDN

u(1JVl........Eu It; 0

·uu0:::(

UJ lOI-:=l...J0(/)

a::l0:::(

I]

0 "- ID is 20 2S...I

FREQUENCY, Hz

FIG. 5.16. Comparison of mean plus one standarddeviation response (soft) and computed response for10 percent damping, Hollywood Storage P.E. lot, 1952--N90E.

240ii25

a - Response at 30 level (this study)Eu b - Computed response (Trifunac et a1., 1972)· 150

U1.....c:l

l.J.J:::-..... 315l-oa:-Jl.J.J ae:::

Ib,IJ

EIJDu(1)Vl

.........Eu YDlJ

·-ll.J.J:::-l.J.J

a:::- 20D.....l-oa:-ll.J.J0:::

!J

2D1DN

u(1)Vl

.........E

1380u

·uuoa:l.J.J 530I-::)-l0U1cooa:

0

lJ 5 iD 'r 20..,,...

I:; C:l

FREQUENCY, Hz

FIG. 5.17. Comparison of predicted response (soft)and computed response for 2 percent damping, E1 Centro,1940--S00E.

2411125

a - Mean plus one standard deviation responses

et al.~ 1972)u b - Computed response (Trifunac· lSn

(/)......0

L.W>...... 315~..JL.Wex:

lJ

EDDu(lJIII

........Eu

4lJtI·..J

L.W>L.W

2IJD>......l-e::(..J

~lJ.J0::::

IJ

2ITlfjN

u(l)III........ asu 1380

·uue::(

LJ..l EiSDI-::::>..J0(/)

coe::(

[I

0 5 l[J IS 2[1 25

FREQUENCY, Hz

FIG. 5.18. Comparison of mean plus one standarddeviation response (soft) and computed response for2 percent damping, E1 Centro, 1940--500E.

2423lJD

a - Response at 3cr level (this study)Eu

b - Computed (Trifunac et al., 1972).. response· 2lJIJ

(/)......Q

UJ>...... lDDl-e:::...JUJ a0::

lJ

lSDuQ)(/).......Eu

i2D·...J

UJ>UJ> ED......l-e:::...JUJ0::

Q

SEiDN

uQ)(/)

.......Eu EYO·u

ue::: (UJ 320

WI-::>...J0(/)CQ=::(

!J J

0 5 i!J 15 20 25

FREQUENCY, Hz

FIG. 5.19. Comparison of predicted response (soft)and computed response for 10 percent damping, El Centro,1940--500E.

243

b - Computed response (Trifunac et al.~ 1972)


o

·wwex:LU 320~:;:,-JoV')coc:e

3lJIJ

Eu.. 2DD·V').....

Cl

LU:> IOD.....~c:e-ILU0:::

a!J

IBDuQ)l/l--E i2Du

·-JLU::::-LU 50::::-.....~ex:-J'-W0:::

IJ

9EON

uCIJl/l--E EY[}u

1] lD is 20 25

FREQUENCY, Hz

FIG. 5.20. Comparison of mean plus one standarddeviation response (soft) and computed response for 10percent damping, El Centro, 1940--500E.

244iElJ

a - Response at 30' level (this study)Eu

b - Computed response (Trifunac et al., 1972)~ i2lJ·V')

.......Q

LLJ:>....... EDl-e:::::-l aLLJc:.::

D

IDSuOJIn.......Eu 10·-..l

LLJ:>

LLJ35:>

>-0

l-e:::::-..lLLJc:.::

IJ

yeo rNuQ) IV1.......

lEu 320

·uue:::::LLJ i6Dt:;-..l0V')

coe:::::

D

D 5 ID 'c; 2D 25L ..

FREQUENCY, Hz

FIG. 5.21. Comparison of predicted response(intermediate) and computed response for 2 percent damping,Ferndale City Hall, 1952--N44E.

245


b - Computed response (Trifunac et al. 9 1972)

IBO

Eu.. 12D·(/)

.......C!

LiJ::>....... EDI-c:e...JI.IJ~

aIJ

lJJ5uCLlVl

........Eu 10·...J

LiJ:>

w 35::>.......I-c:e....JLiJc:::

0

YEO rNuQ)Vl

........Eu 320

·uuc:eLiJ iEiDI-:::>-l0(/)

co<l:

IJ

D 5 iD is 20 2S

FREQUENCY, Hz

FIG. 5.22. Comparison of mean plus one standarddeviation response (intermediate) and computed responsefor 2 percent damping, Ferndale City Hall, 1952--N44E.

ED247


YO

Eu

·V)ow::>.-. 2Dle:(

-'W0:::

1]


a

2D

3D

·-'w::>

uOJ11"I

.........Eu

IJ

225

[N

UOJ11"I

.........Eu iSD

· luue:(

w 15I-::J-'0V)

coe:(

D

0 5 1IJ is 20 25

FREQUENCY, Hz

FIG. 5.24. Comparison of mean plus one standarddeviation response (intermediate) and computed response for10 percent damping, Ferndale City Hall, 1952--N44E.

Eu

·(/').....Q

LJ.J>.....le:(-.oJLJ.Jc:::

33D

lEiS

IJ

a

248



285uOJVl.......Eu ISD·-.oJ

LJ.J>l.I..J 95>.....~-.oJl.I..Jc:::

D

tY25N

uOJVl.......E

350u

·uue:(

l.I..J YlSI-~

-.oJ0(/')

c:le:(

0

D 5 15

FREQUENCY, Hz

20 25

FIG. 5.25. Comparison of predicted response(intermediate) and computed response for 2 percent damping,Ferndale City Hall, 1954--N46W.

249

Eu...

enc::l

LJ.J::>-~-'WJex:

330

iES

a



285u<1.ltil"-Eu

1900

.....IWJ::>WJ

9S::>-!;;i:....JWJ~

lJ

iY25

rNu<1.ltil"-Eu gSa.

uu0:::(

WJ i.llSI-=:l.....I0enco0:::(

U

U 5 iD is 20 25

FREQUENCY, Hz

FIG. 5.26. Comparison of mean plus one standarddeviation response (intermediate) and computed responsefor 2 percent damping, Ferndale City Hall, 1954--N46W.

250'::J t;l .... ~

a - Response at 30 level (this study)Eu

b - Computed (Trifunac et a1. , 1972)response. 3DV)......Cl

LLJ::::-...... YSI-c:x:: a-lLLJ0:::

lJ

90uOJVl........Eu

60.-lLLJ::::-LLJ

30::::-.......I-c:x::-lLLJ0:::

D

6115 rNuOJ

~til

........Eu Y3Duuc:x::LLJ 2lS f..I-:::>

i!-l0

~V)coc:x:: y

l]

0 ::; llJ or;; 20 2S... 1-

FREQUENCY, Hz

FIG. 5.27. Comparison of predicted response(intermediate) and computed response for 10 percentdamping, Ferndale City Hall, 1954--N46W.

251i35


b = Computed response (Trifunac et al., 1972)90

Eu

·(/)......t::::l

I.LJ>-~...JI.LJ0:::

0

90uellVI......Eu 50..·-!

UJ>UJ

3D>......l-ex:;-!UJ0:::

lJ

545 r-N I

U Iell

lVI......Eu Y30

·

~ fuuex:;

UJ 215I-

~:::>-!0V)

coex:;

0

11 c i£l is 2D 2S...FREQUENCY, Hz

FIG. 5.28. Comparison of mean plus one standarddeviation response (intermediate) and computed responsefor 10 percent damping, Ferndale City Hall, 1954--N46W.

252'is

r.a - Response at 30' level (this study)

Eu b - Computed response (Trifunac et al., 1972). 3D

(/)......Cl

I.LJ>...... isI-ex:-lI.J.J0:::

D

15uQ)(/)

........Eu SO-lI.J.J>I.J.J

2S>......I-ex:-lI.J.Jc:::

D

SIS rNu IQ)

l(/)

........Eu "'lID.

uuex:I.LJ 2IJ5I-:=l-l0(/)

coex:

lJ

0 5 l!J is 20 25

FREQUENCY, Hz

FIG. 5.29. Comparison of predicted response (hard)and computed response for 2 percent damping, Lake HughesStation 1, 1971--S69E.

Eu

3D

i5

IJ

15

253


b ~ Computed response (Trifunac et al., 1972)

l~~_""'-"----.l_--I..-_.--..l

o

~ 2S-l-e::(-Jl.I.I0:::

uC1JIII.....Eu

.-JL.I.J>

50 a

51SN

UC1JIII

E'U I.JIIJ.

UUe::(

l== 205::::>-JatI)coe::(

D

r

Jj 5 isFREQUENCY, Hz

20 2S

FIG. 5.30. Comparison of mean plus one standarddeviation response (hard) and computed response for2 percent damping, Lake Hughes Station 1, 1971--S69E.

254.t;I.-

a - Response at 30' level (this study)Eu b - Computed response (Trifunac et al., 1972)· III

if) aI-<

Cl

U.l::::-I-< 5l-e:(...JU.l0:::

0

l.fSu(1)Vl

........Eu 3D·-l

l..l.J::::-l..l.J is::::-I-<

l-e:(-Jl..l.J0:::

IJ

285 r-N

U a(1)Vl

........E Iu ISO r-·

Iuue:( ,l..l.J 95 rfI-::J-J0

~if)coe:(

lJ

lJ 5 iD 15 2D 25

FREQUENCY, Hz

FIG. 5.3l. Comparison of predicted response (hard)and computed response for 10 percent damping, Lake HughesStation 1, 1971--S69E.

255is



a

m

5

Eu

lJ

'isu<1.len

"Eu 3D....JW:::-w is:::-.......l-e::(

-'w0:::

IJ

285N

uOJen

"Eu i9D.

uue::(

LLJ 95I-::>....laV1coe::(

0

0 5 iO 15 20 25

FREQUENCY, Hz

FIG. 5.32. Comparison of mean plus one standarddeviation response (hard) and computed response for 10percent damping, Lake Hughes Station 1, 1971--569E.

:iI1S

Eu

· 23lJ(/)......0

LJ..l>...... il5l-e::(-lLJ..l0:::

lJ

585u(1JU)

........Eu

39D·-l

LJ..l>LJ..l

ISS>......l-e::(-lLJ..l0:::

lJ

5880N

u(1JU)

........Eu 3520

·uue::(

LJ..l 1950I-:::>-l0(/)coe::(

[]

256



[] IIJ is 20

FREQUENCY, Hz

FIG. 5.33. Comparison of predicted response (hard)and computed response for 2 percent damping, Pacoima Dam,1971--S15W.

257



r230..Eu

·V1.....oUJ:>...... iiS~-ILL.!ex::

D

5B5uClJen

........Eu

391J·-I

LL.!:>

LL.!iSS:>......

I-<C-IWex::

lJ

5880 IN

UClJen

........Eu 3920

·uU<C

w i950I-:::>...J0V1co<::(

1]

D 5 ID IS 20 2S

FREQUENCY, Hz

FIG. 5.34. Comparison of mean plus one standarddeviation response (hard) and computed response for 2 percent damping, Pacoima Dam, 1971--S15W.

258i2!J

Eu

·Vl......ol.LJ:>......~-ll.LJ0:::

Ell

lJ



2S20isIlJl:...

ilIJ

255

·...Jl.LJ:>

l.LJ:> 55......~...Jl.LJ0:::

u(1J(/)

"Eu

IJ

2850N

u(1J(/)

"-Eu 1900

·uuc:(

l.LJ 950I-::>...J0Vlcoc:(

0

D

FREQUENCY, Hz

FIG. 5.35. Comparison of predicted response (hard)and computed response for 10 percent damping, Pacoima Dam,1971--S15W.

259i2fJ

a - Mean plus one standard deviation responseEu

b - Computed response (Trifunac et a1. ~ 1972)· fiD

Vl.....0 aLI.J:::-......

iilJI-<C-lLI.J0:::

lJ

255uQ)U)

"-Eu

illJ·-!

LI.J:>L.LJ

85:::-.....I-<C...JL.LJ0::;

0

2BSON

uQ)U)

"-Eu 1900

·uuc::(

w 950I-::l...J0Vlcoc::x::

0

l1 5 iD is 20 2S

FREQUENCY, Hz

FIG. 5.36. Comparison of mean plus one standarddeviation response (hard) and computed response for 10 percent damping, Pacoima Darn, 1971--S15W.

260i2!]

Eu

8)]

i.\t1



a

IJ

Stl

15uCLlVI

.........Eu

.-'UJ>UJ> 25......~-'UJ0::

0

210 ..N Iu

CLl

~til

.........Eu iBO. Iu

I h b

u.:::c:UJ 90I-::J

rAN~\-'0VJ -------co.:::c: M#0

0 t: iD is 20 25-J

FREQUENCY, Hz

FIG. 5.37. Comparison of predicted response (soft)and computed response for 2 percent damping, HollywoodStorage P.E. lot, 1952--vertical.

261

b - Computed response (Trifunac et alo~ 1972)

a - Mean plus one standard deviation responsei2fJ

Eu

· HD(/) -l-<Cl

UJ:>..... YDI-<l::...JI.LJ0:::

lJ

lSu(lJlJ)

.......Eu SD·....J

UJ>-UJ 2S:>.....I-<l::....J aUJ0:::

lJ

210 l"'

N !u(lJ

llJ)

.......E

iBOu

·uu<l::

UJ 90I-:::>....J0(/)co<l::

0

D iD is 20 25

FREQUENCY, Hz

FIG. 5.38. Comparison of mean plus one standarddeviation response (soft) and computed response for 2 percent damping, Hollywood Storage P.E. lot, 1952--vertical.

262"IS

ra - Response at 30 level (this study)

Eu

b - Computed (Trifunac et al., 1972)response. 31]Vl......Ci

lJ.J>...... isI-c:::r::-llJ.J a0:::

IJ

3D

ru(1)til......Eu

20

-llJ.J>lJ.J

iO>......I-c:::r::-llJ.J0:::

I]

i21] r aN I

U(1)til......E IU EO I

r.uuc:::r:: IIlJ.J YO bI-

Ilf::;,-I0

~IVlcoc:::r:: rf]

IJ r:; 10 is 20 ;J--' _:J

FREQUENCY, Hz

FIG. 5.39. Comparison of predicted response (soft)and computed response for 10 percent damping, HollywoodStorage P.E. lot, 1952--vertical.

263'i5

a - Mean plus one standard deviation responseEu b - Computed response (Trifunac et al.~ 1972)

3D·(/)-0LLJ:> is>-I

l-e(-ll.LJ0::

[1

3DuC1.lVl.......Eu 20·-l

LLJ:>

LLJ lD:>I-<l-e( a-lLLJ0::

!J

i2D rN I

uC1.lVl.......E

SOu

·uue( aLLJ '10l-=>-l0V1coe(

0

0 5 l[l IS 20 25

FREQUENCY, Hz

FIG. 5.40. Comparison of mean plus one standarddeviation response (soft) and computed response for 10 percent damping, Hollywood Storage P.E. lot, 1952--vertical.

o

~ llS......le:(-JLl.J0:::

Eu,.•

C/)......oLl.J::>......le:(-JLl.J0:::

uQ)V1Eu·-J

Ll.J::>

"BD

215

o

3115

230

264

a - Response at 3a level (this study)

b - Computed response (Trifunac et a1., 1972)

NUQ)V1Eu·u

ue:(

Ll.JI~-JoVlCCle:(

i395

93D

'165

o

a

D iO is 20 2S

FREQUENCY, Hz

FIG. 5.41. Comparison of predicted response (soft)and computed response for 2 percent damping, E1 Centro,1940--vertical.

l.I3D

EiYS

Eu

·(/).....Cl

I.J..!>l-t 215I-<-II.J..!0:::

265


b - Computed response (Trifunac et al.~ 1972)

IJ .k:

345u(1)VI-...Eu 230·...J

I.J..!>I.J..!

i15>......

6=~...JI.J..!c:r::

LJ j ;:-=-m

1395 r('\Ju

,(1)Vl-...Eu 930

·uu<I.J..! 4EiSI-;::)...J0(/)co<

L1

0 5 iD is 2D 25

FREQUENCY, Hz

FIG. 5.42. Comparison of mean plus one standarddeviation response (soft) and computed response for 2percent damping, El Centro, 1940--vertical.

iaii

Eu

~ ieD(/)-Cl

u.J::>- ED~.....Iu.J0::

IJ

266

a - Response at 3a level (this study)


l.J

~ 35-!;;:.....Iu.J0::

uOJVl......Eu

......Iu.J::>

iDS

10

645N

U(1)Vl......Eu i.l30.

owowc:t:

u.J 215I-::::l.....IoVlcoc:t:

a

S 10 is 20 25

FREQUENCY, Hz

FIG. 5.43. Comparison of predicted response (soft)and computed response for 10 percent damping, El Centro,1940--vertical.

5D

ISO

Eu

~ i2JJV)......Cl

w:>.....!;;:-lI.LJ0:::

o

iDS

a

267


b = Computed response (Trifunac et al, 9 1972)

uQ)VI

...........Eu

,-lW:>

10

IJ

El.fSN

UQ)VI

...........Eu Y30

uuc::(

::: 215:::>...JoV)coc::(

[]

o 5 iD is 20 2S

FREQUENCY, Hz

FIG. 5.44. Comparison of mean plus one standarddeviation response (soft) and computed response for 10percent damping, El Centro, 1940--vertical.

268iDS

a - Response at 30 level (this study)Eu

b - Computed (Trifunac et a1. , 1972)~ 1]] response.~

U').....al.LJ::>

35.....~c:x:....l4J0:::

lJ

50u(lJtil,Eu "10.

....ll.LJ::>

l.LJ 20::>.....~c:x:....l4J0:::

[J

255

r<'J

~U(lJtil,E

~u ilD

II I~.uuc:x:4J 95

~r Vv~~::::l....l0U')

~ ---cec:x:[j

0 5 to i5 20 ;:1-.... ::.

FREQUENCY, Hz

FIG. 5.45. Comparison of predicted response (inter-mediate) and computed response for 2 percent damping,Ferndale City Hall, 1952--vertical.

iDS269

a - Mean plus one standard deviation responseE

Computed response (Trifunac et al., 1972)u b -10·Vl

I-<Cl

W>- 35.....I-et:-I!.l.!ex:

aIJ

6DuClltil

.........Eu YO·-l

W::::-w 2D>1-1

~-lWex:

lJ

255N

uClltil

.........E

ilOu

· auuex:l.J.J 85I-:;:)-I0Vl =coc:(

lJ

!J 5 !!J is 20 25

FREQUENCY, Hz

FIG. 5.46. Comparison of mean plus one standarddeviation response (intermediate) and computed response for2 percent damping, Ferndale City Hall, 1952--vertical.

- 3lJ

Eu

21J·(/)

1-4a

iD

o

3Du(I)III.......o 2D

·....JWJ>WJ iD>.....le::(....JWJ0::

IJ

270

ra - Response at 3a level (this study)"


a

NU(I)III.......Eu

·UUe::(

WJI:::>....Jo(/)

coe::(

i2IJ

81J

YO

u

[l;~i!

a

'-------------------

IJ 5 [{] is 2IJ 25

FREQUENCY, Hz

FIG. 5.47. Comparison of predicted response (intermediate) and computed response for 10 percent damping,Ferndale City Hall, 1952--vertical.

2713D

20

Eu

·V'l.....oI.J.J:>...... iIJ!;:....lI.J.Jex:

a



lJ

3D

o

U<lJU'l

.........Eu 20·....l

LLJ:>

LLJIiJ:>.....

~ a...JLLJex:

IJ

i2D rNu<lJ IU'l

.........

lEu 8D

·uuc:(

I.J.J 'i0f-=>....l0V'lcoc:(

D t:... is 20 25

FREQUENCY, Hz


272

Eu

o



a

5D

15

.--!l.J.J>

~ 25t-<

I«-ll.J.JCt::

uOJtil......Eu

0

315 r('oJ au IOJtil

~ ,......E

210u

.uu«l.J.J iDSI-:::l

~J ' ~-l0Vl ----c.e«

0 ,1 I !

lJ 5 2D 25

FREQUENCY, Hz


Eu

135

so

YS

IJ

273


b - Computed response (Trifunac et al. ~ 1972)

b

15uOJl!l.......Eu

51]....J1.l.J:>1.l.J

25:>......!;;c-l1.l.J0::

IJ

315C'J

uOJl!l.......Eu 21lJ.

uuc:x::1.l.J iDSI-::;:)....J0Vlcoc:x::

D

a

I .~W-v~ _l I !

j] 5 is

FREQUENCY, Hz

2D 25


274

Eu

3D



·<.n>-<oLLJ

;:: is!;: a-lLJJ0::

o L:b=--===__....... --l. -'-- -..l. ----'

u(])VI

.........Eu 21]·-l

W>LLJ iO>>-<I-e:::c-lLJ.I0:::

IJ

iSOC'J

u(])VI

.........E

iOOu

a

rI

I

~I~---~o

30

50

·uue:::cLLJI:::>-lo<.nco0::(

o 5 is 20 25

FREQUENCY, Hz


l.fS

Eu

3D

is

IJ

3DuCl.IIII

........Eu

2D'"0

....JUJ::>UJ

j[J::>.....r-«-!LU0:::

lJ

iSDC'J

uCl.IIII

........Eu 100

uu«UJ SOr-:::>....J0(/)

00«0

275


b ~ Computed response (Trifunac et alo~ 1972)

o 5 is

FREQUENCY, Hz

2lJ 2S


276



o

iBD

Eu.. l2D.Vll-ICl

W> 50.....~ a-!wa::

0

iDS

w 230I-:=J..JoVlcce(

.uuc:(

U<lJ\I)

........Eu 10

0

..JW>LJ.J 35:>.....l-e(-JWa::

0

691]N

u<lJ\I)

........E Y50u

o in it:;.- 20...,-c:.

FREQUENCY, Hz

FIG. 5.53. Comparison of predicted response (hard)and computed response for 2 percent damping, Lake HughesStation 1, 1971--vertical.

277

lJ

·....JI.l.J::::-

~ 3S-~....Jl.J.J0:::

b - Computed response (Trifunac et alo 9 1972)


1D

IBD

Eu.. i2lJ·V)-0

t.LJ::::-

50-~....JI.l.J0:::

alJ

iDSueuIII.......Eu

Ei9DN

UQ.lIII

.......5 YEO

·uu<t:

::: 230=:>....JoV)

~ oJJ 5 ID is 20 25

FREQUENCY, Hz

FIG. 5.54. Comparison of mean plus one standarddeviation response (hard) and computed response for 2percent damping, Lake Hughes Station 1, 1971--vertical.

ED278

D

~ is.......le:::....Jwc::

Eu

·V').......ol.J..J::>.......!;:....Jl.J..Jc::

u(IJen

.........Eu

·....Jl.J..J::>

21]

lJ

31]

a



315N

U(IJen

.........Eu 2lD

·uue:::W iDSI-::>....JoV')cce:::

D

!J [0 15 20 25

FREQUENCY. Hz

FIG. 5.55. Comparison of predicted response (hard)and computed response for 10 percent damping, Lake HughesStation 1. 1971--vertical.

279

ED

Eu

YD

2D

IJ



r

kYS

UQJ(J'l-.Eu

3D aQ

...JlJ.I::>l.l.J::> is.....I-e:t:...JLJ.Jc::

[}

315N

u(l.l(J'l-.Eu

21lJ.uuc:(

l.l.JI- iDS:::>-I0(/)

coe:t:

lJ

D t: in is 20 2S...

FREQUENCY, Hz

FIG. 5.56. Comparison of mean plus one standarddeviation response (hard) and computed response for 10percent damping, Lake Hughes Station 1, 1971--vertical.

280illlJ

a - Response at 30 level (this study)E

b - Computed response (Trifunac et al., 1972)ulYD

·(/).....0

I..lJ> 310..... aI-c:r:..JI..lJex: b

D

530uClJIJ)

.........E

Y20u

·..JI..lJ a>I..lJ 2lrJ>>-<I-c:r:..JI..lJex:

D

Y3S0 rNuClJIJ)

.........E 2900u

· buuex::I..lJ iY50I-~..J0(/)

coc:r: jJ

0 5 iO iC:; 20 251-

FREQUENCY, Hz

FIG. 5.57. Comparison of predicted response (hard)and computed response for 2 percent damping, Pacoima Dam,1971--verti ca1.

illlJ

Eu

lYD·V).....oU.J

;:: 31JJ~....JU.Jex:

lJ

630uQ)Vl.......E

Y20u

·-lU.J::::-U.J 210:::-.....l-c::r:-lU.Jex:

D

Y3SlJN

u(])Vl.......E 29DDu

·uuc::r:

281

a = Mean plus one standard deviation response


rI

~

~I J,j-1'I.

n 5 iD 'C:; 20 25f ...

FREQUENCY, Hz

FIG. 5.58. Comparison of mean plus one standarddeviation response (hard) and computed response for 2percent damping, Pacoima Dam, 1971--vertical.

300

Eu

· 2DDV)......Cl

w::>

iDD......f-<J:...JW0:::

lJ

iSSu(1)Vl.......Eu i3D

·...JW::>

w liS::>>-4

I-<J:...JW0:::

lJ

j920N

u(1)Vl.......E

i2BOu

·uu<J:

w El.jIJI-:::>-l0V)OJ<J:

U

r

l

282



a

o iO is 20 25

FREQUENCY, Hz

FIG. 5.59. Comparison of predicted response (hard)and computed response for 10 percent damping, Pacoima Dam,1971--vertical.

Eu

3lJD

2lJD

iDD

Da

283



uQ)II)

........Eu

i3lJ....JI.I.J::>WJ

55:>.-l-e::(-!I.I.J0:::

IS21JN

UQ)II)

........Eu i281J.

uuc:::r:I.I.J 5Y[JI-:::>--'0Vlecc:::r:

D

[

11 r:;.... is

FREQUENCY, Hz

20 25

FIG. 5.60. Comparison of mean plus one standarddeviation response (hard) and computed response for 10percent damping, Pacoima Dam, 1971--vertical.

CHAPTER 6

SUMMARY AND CONCLUSIONS

6.1 SUMMARY

A statistical study is used to estimate a time-dependent power

spectral density of recorded earthquake accelerograms. The study as

sumes that the strong motion segments of accelerograms form a locally

stationary random process whose members exhibit a time-invariant fre

quency structure. In Chapter 3, in a pilot study, the validity of

this assumption is examined. The pilot study shows that by selecting

the strong motion segment of the record, the power spectral density

estimates of the subsegments of that record exhibit shapes and frequency

structures similar to that of the strong motion segment of the record

itself. Using the strong motion segment of the records a time-dependent

power spectral density is estimated which consists of three parts:

a normalized power spectral density which describes the frequency struc

ture of the ensemble and remains the same for the subsegments of records

considered; a time-dependent scale factor which describes the variation

of local mean square value; and finally the mean square value itself.

Normalized power spectral densities and scale factors for horizontal and

vertical components of accelerograms recorded on soft, intermediate and

hard geology are presented in Chapter 5. Correlation of RMS values with

a variable reflecting the four most commonly used design parameters,

peak ground acceleration, earthquake magnitude, epicentral distance, and

284

285

the duration of strong motion are obtained and presented in Chapter 4.

Such correlations will make it possible to estimate a power spectral

density for a specific site and earthquake parameters. Correlations

were obtained for several classifications of records; however, no sig

nificant changes on the correlation coefficients were observed due to

site geology or component classification. The estimated power spectral

densities are used to predict the response of a single gegree of freedom

system at several sites and the results are compared with spectral rela

tive displacement, relative velocity, and absolute acceleration computed

directly from the records.

The results of the study support the viability of using the random

vibration theory in earthquake resistant design of structures. There

are two features that make this study attractive for seismic analysis

and design. First, the results can be used to predict the response of

a system for a given probability that it may exceed a certain level, and

second, the prediction incorporates site geology as well as earthquake

magnitude, peak ground acceleration, epicentral distance and the duration

of strong motion. Finally, the findings can be used in the study of

artificially generated earthquake motion.

6.2 RECOMMENDATIONS FOR FURTHER STUDY

The following is a list of possible topics for future studies:

1. Inclusion of site geology in predicting the RMS values. Although

correlations between RMS and peak acceleration, earthquake magnitude,

epicentral distance and duration of strong motion are presented for

different geological classifications in this study, the possibility

286

of obtaining a single correlation which would include the site

geology should be investigated.

2. Expressions for power spectral densities and scale factors. Ana

lytical expressions should be developed for power spectral den

sities and scale factors presented in Figs. 5.1-5.12. Such expres

sions together with the regression equations in Chapter 4 for es

timating RMS values would be useful in seismic analysis and design

of structures and equipments.

3. Generation of acceleration-time history from the power spectral

densities. A procedure which is widely used in seismic analysis of

complex structures is to subject the structure to an acceleration

time history to compute its response. Scaled acceleration-time

histories from various seismic events as well as acceleration-time

histories generated from design response spectra have been used for

this purpose. Acceleration-time histories generated from the power

spectral densities would be valuable to design engineers and should

be considered.

4. Use of the entire acceleration-time history to estimate power spectral

density. In this study we used the strong motion part of the rec

ords to obtain the power spectral densities. An extension of this

study would be to consider the entire record length and estimate

the power spectral density in which both the magnitude and frequency

structure are time-dependent. One possible approach could be to

estimate power spectral densities for different portions of the

records.

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