CUREe-Kajima Research Project Final Project Report

Amplification of Strong Ground Motion Due to Local Site Conditions

Dr. Masato Motosaka Mr. Ariyoshi Yamada Mr. Y asuhiro Ohtsuka Mr. Masaki Kamata Mr. Yasukazu Tsuji

By

Report No. CK 92-01 February 1992

California Universities for Research in Earthquake Engineering ( CUREe)

Prof. AnneS. Kiremidjian Ms. Stephanie King Prof. Haresh C. Shah Dr. Masata Sugito

Kajima Corporation

(

• California Institute of Technology • Stanford University • University of California, Berkeley

( CUREe (California Universities for Research in Earthquake Engineering)

\

• University of California, Davis • University of California, Irvine • University of California, Los Angeles • University of California, San Diego • University of Southern California

Kajima Corporation

• Kajima Institute of Construction Technology • Information Processing Center • Structural Department, Architectural Design Division • Civil Engineering Design Division • Kobori Research Complex

( \

1.

CUREe-KAJIMA RESEARCH PROJECT

SIMPLE SITE-DEPENDENT GROUND MOTION PARAMETERS FOR THE SAN FRANCISCO BAY REGION

AnneS. Kiremidjian, Stephanie King and Haresh C. Shah

The John A. Blume Earthquake Engineering Center Department of Civil Engineering

Stanford University Stanford, California

and

Masata Sugito

Department of Civil Engineering Kyoto University

Kyoto, Japan

July 1, 1990- June 30, 1991

Chapter

List of Figures

List of Tables

Summary

1. Introduction

1.1 Overview

Table of Contents

1.2 Strong Ground Motion and Soil Parameter Data

1.3 Scope

2. Soil Parameter and Ground Motion Data Consolidation

2.1 Soil Boring Data

2.2 Soil Parameters Data

2.3 Strong Ground Motion Data Analysis

3. Ground Motion Amplification Factors

3.1 Methodology

3.2 Nonlinear Parameters and Applications

3.3 Sensitivity Analysis

4. Summary and Conclusions

5. Acknowledgements

6. References

ii

Page

111

Vl

Vlll

1

1

2

6

7

7

14

20

28

28

30

58

63

65

66

List of Figures

Figure Title Page

1.1 Locations of CDMG strong motion stations for recordings from the 3 October 17, 1989 Lorna Prieta earthquake (after Shakal et al, 1989).

1.2 Locations of USGS strong motion stations for recordings from the 4 October 17, 1989 Lorna Prieta earthquake (from Maley, et al., 1989).

1.3 Locations of bore-hole data from Gibbs et al. (1975, 1976, 1977). 5

2.1 Schematic representation of spatial data used in kriging. 8

2.2 Stanford University Parking Structure I and surrounding sites where 11 bore-hole data are available.

2.3 Blow count, N, with depth at the Stanford University Parking 13 structure I as obtained by kriging data from surrounding locations.

2.4 Peak ground acceleration values as functions of the rupture zone for 22 the Lorna Prieta earthquake of October 17, 1989 and the best fit of the Joyner and Boore (1988) attenuation law.

2.5 Peak ground velocity values as functions of the rupture zone for 22 the Lorna Prieta earthquake of October 17, 1989 and the best fit of the Joyner and Boore (1988) attenuation law.

3.1 Variation of peak ground acceleration amplification with peak ground 32 acceleration at rock surface for Data Set I.

3.2 Variation of peak ground velocity amplification with peak ground 32 velocity at rock surface for Data Set I.

' 3.3 Variation of peak ground acceleration amplification with peak ground 33 acceleration at rock surface for Data Set ll.

3.4 Variation of peak ground velocity amplification with peak ground 33 velocity at rock surface for Data Set II.

3.5 Variation of peak ground acceleration amplification with depth to 37 bedrock, dp. for Data Set I:

3.6 Variation of peak ground velocity amplification with depth to 37 bedrock, dp, for Data Set I.

3.7 variation of peak ground acceleration amplification with depth to 38 bedrock, dp. for Data Set II.

3.8 Variation of peak ground velocity amplification with depth to 38 bedrock, dp. for Data Set II.

iii

3.9 Variation of peak ground acceleration amplification with distance to 39 the rupture zone for Data Set IT.

3.10 Variation of peak ground velocity amplification with distance to the 39 rupture zone for Data Set IT.

3.11 Comparison of recorded and predicted amplification; (a) peak ground 46 acceleration amplification; (b) peak ground velocity amplification (Qpa sites).

3.12 Comparison of recorded and predicted amplification; (a) peak ground 47 acceleration amplification; (b) peak ground velocity amplification (Qha sites).

3.13 Comparison of recorded and predicted amplification; (a) peak ground 48 acceleration amplification; (b) peak ground velocity amplification (Qaf/Qhbm sites).

3.14 Comparison of amplification factors 13 using Data Set I and II and a 49 linear shear wave model; (a) peak acceleration amplification; (b) peak velocity amplification.

3.15 Comparison of amplifications to observed data for Qpa sites with 50 average depth to bedrock at 25m. (a) Peak ground acceleration amplifications (b) Peak ground velocity amplifications.

3.16 Comparison of amplifications to observed data for Qpa sites with 51 average depth to bedrock at 60m. (a) Peak ground acceleration amplifications (b) Peak ground velocity amplifications.

3.17 Comparison of amplifications to observed data for Qpa sites with 52 average depth to bedrock at 150m. (a) Peak ground acceleration amplifications (b) Peak ground velocity amplifications.

3.18 Comparison of amplifications to observed data for Qha sites with 53 average depth to bedrock at 50m. (a) Peak ground acceleration amplifications (b) Peak ground velocity amplifications.

3.19 Comparison of amplifications to observed data for Qpa sites with 54 average depth to bedrock at 125m. (a) Peak ground acceleration amplifications (b) Peak ground velocity amplifications.

3.20 Comparison of amplifications to observed data for Qha sites with 55 average depth to bedrock at 220m. (a) Peak ground acceleration amplifications (b) Peak ground velocity amplifications.

3.21 Comparison of amplifications to observed data for Qaf/Qhbm sites 56 with average depth to bedrock at 95m. (a) Peak ground acceleration amplifications (b) Peak ground velocity amplifications.

3.22 Comparison of amplifications to observed data for Qaf/Qhbm sites 57 with average depth to bedrock at 185m. (a) Peak ground acceleration amplifications (b) Peak ground velocity amplifications.

iv

3. 2 3 Sensitivity analysis of Data Set I ground motion amplification 59 factors to the parameters Stand dp.

3. 2 4 a Sensitivity analysis of Data Set IT ground motion amplification factors 60 to the parameters Stand dp.

3. 2 4 b Sensitivity analysis of Data Set IT ground motion amplification factors 61 to the parameters Sr and dp.

3. 2 4 c Sensitivity analysis of Data Set IT ground motion amplification factors 62 to the parameters Sr and dp.

v

List of Tables

Table Title Page

2.1 Blow count estimates for the five sites surrounding the Stanford 12 Parking Structure I as estimated by kriging data at each site by itself.

2.2 Rock surface and soil parameters for strong ground motion data from 17 the October 17, 1989 Lorna Prieta earthquake (Data Set I).

2.3a Rock surface and soil parameters for strong ground motion data from 18 the October 17, 1989 Lorna Prieta earthquake (Data Set II).

2.3b Rock surface and soil parameters for strong ground motion data from 19 the October 17, 1989 Lorna Prieta earthquake (Data Set II).

2.4 Soil and rock surface strong ground motion data for Data Set I. 23

2.5a Empirical, recorded and modified strong ground motions for Data 24 Set II.

2.5b Empirical, recorded and modified strong ground motions for Data 25 Set II.

2.6a Soil and rock surface strong ground motion data for Data Set II. 26

2.6b Soil and rock surface strong ground motion data for Data Set II. 27

3.1 Ground motion amplification data for Data Set II based on strong 29 ground motion records from the October 17, 1989 Lorna Prieta earthquake.

3.2 Regression constants for amplification of peak ground acceleration 31 defined by equation 3.4; (a) Data Set I= "Sugito" and (b) Data Set II= "modified".

3.3 Regression constants for amplification of peak ground velocity 31 defined by equation 3.4; (a) Data Set I= "Sugito" and (b) Data Set II = "modified".

3.4 Comparison of recorded and computed peak ground acceleration 40 amplification factors for soil type Qpa.

3.5 Comparison of recorded and computed peak ground acceleration 41 amplification factors for soil type Qha.

3.6 Comparison of recorded and computed peak ground acceleration 42 amplification factors for soil type Qaf/Qhbm.

3.7 Comparison of recorded and computed peak ground velocity 43 amplificationfactors for soil type Qpa ·

vi

3.8

3.9

Comparison of recorded and computed peak ground velocity amplification factors for soil type Qha.

Comparison of recorded and computed peak ground velocity amplification factors for soil type Qaf/Qhbm.

vii

44

45

SUMMARY

The amplification of earthquake ground motion is one of the most difficult parameters to quantify. The difficulties stem from (1) the lack of sufficient data on local soil parameters, (2) lack of sufficient strong motion data at different soil surface locations, (3) lack of strong ground motion data at various depths from the ground surface at the same location, (4) our inability to quantify the nonlinear characteristics of soils and (5) the use of approximate models to represent the nonlinear behavior of soils when subjected to dynamic forces.

In this report we present soil amplification factors for peak horizontal acceleration and velocity for earthquakes in the San Francisco Bay region. The method of Sugito (1986) was modified to reflect availability of data for this region. Primarily peak ground acceleration and velocity amplification parameters were evaluated. The ground motion amplification relationships are given in the following simple form: As = f3a Ar for peak ground acceleration

and Vs = f3v Vr for peak ground velocity. The amplification factors f3 are functions of the depth to bedrock dp, the ratio, Sr. of surface shear wave velocity and a standardized shear wave velocity, and the respective peak ground motion parameters

The project was accomplished through the following tasks:

*The strong ground motion data from the October 17, 1989 were used for estimating the amplification factors. In particular, initial analysis using data from 24 strong ground motion stations were used to determine the amplification from rock sites to soil sites. Subsequently, the analysis was repeated with strong ground motion for 52 sites. The grotind motions were modified for distance from the rupture zone and for azimuthal direction. The corrections for distance was based on the Joyner and Boore (1988) attenuation equation scaled to reflect the higher accelerations recorded during the Lorna Prieta earthquake.

* An extensive search was conducted to obtain soil parameter data at the locations of the strong ground motion stations. The parameters sought included variations of shear wave velocity or blow count with depth, depth to bedrock, and variations of density with depth. Initially data was obtained for 16 strong motion recording stations; Two months prior to the end of this project additional soil parameter data were made available by USGS for all the strong ground motion stations. Thus, the soil amplification analysis was repeated with more stations in the data

*A sensitivity analysis was performed to determine the variability of the amplification parameters f3 to changes in the S1 and dp parameters. The amplification factors appear to be more sensitive to variations in depth to bedrock than to variations in the Sr parameter.

Analysis of the data shows nonlinearity in soil amplification. This nonlinearity is reflected in the soil amplification parameters /3. Some of the nonlinearity may be attributed to resonance between the input motion and the soil deposits in the San Francisco Bay region. In order to determine the primary sources for the nonlinear behavior observed in this study a more detailed investigation is needed. Such an investigation, however, is beyond the scope of this study.

The parameters developed in this study can be used in development of micro zonation maps, for regional damage estimation, for rehabilitation decisions and resource allocation purposes.

viii

CHAPTER 1

INTRODUCTION

1.1 Overview

The amplification of earthquake ground motion is one of the most difficult parameters to

quantify. The difficulties stem from (1) the lack of sufficient data on local soil parameters, (2) lack

of sufficient strong motion data at different soil surface locations, (3) lack of strong ground motion

data at various depths from the ground surface at the same location, (4) our inability to quantify the

nonlinear characteristics of soils and (5) the use of approximate models to represent the nonlinear

behavior of soils when subjected to dynamic forces. Thus, current seismic design procedures use

four soil classes to describe the variation in ground motion. These include rock or stiff soils,

intermediate, soft to stiff clays and soft clays or soils with shear wave velocity less than 500 ft/sec.

Studies on the variations of ground motion due to local soil conditions indicate that peak

ground acceleration can be amplified by a factor of 1.5 to 3 depending on the properties of the local

soil (e.g., Borcherdt, 1990; ldriss, 1990; Seed et al, 1976; Trifunac, 1976). Furthermore,

examination of spectral accelerations at different surface soil conditions have shown that the

amplification is frequency dependent with a crossover period at approximately 0.2 sec (e.g., Seed

et al., 1976; Mohraz, 1976; Trifunac, 1976; Phillips and Aki, 1986).

Some of the factors that influence the amplification or deamplification of ground motion at a

specific site include the magnitude and the rupture process of the earthquake, the angle of incidence

and the frequency content of the incoming seismic waves, the topography of the source to site path

and that at the site, and the dynamic properties of the local soil deposits. In general, it is difficult to

include all of these characteristics for every site and all possible future earthquakes when

developing seismic codes. The same difficulties also exist when microzoning a region for planning

and disaster mitigation purposes.

Thus, it is necessary to develop a method for ground motion amplification that can be applied

widely. While theoretically more sound method may be desirable, amplification factors based on a

simple approach for generic soil conditions would enable their application to variety of sites.

1

In this report we present soil amplification factors for peak horizontal acceleration for

earthquakes in the San Francisco Bay region. The method of Sugito (1986) was adopted for this

purpose and modified to reflect conditions in the San Fransicso Bay area. The strong ground

motion data from the October 17, 1989 earthquake were used for estimating the amplification

factors.

1.2 Strong Ground Motion and Soil Parameter Data

There were 93 strong motion recordings obtained by the California Division of Mines and

Geology (CDMG) and 38 recordings were recovered by the United States Geological Survey

(USGS). Figure 1.1 shows the epicenter of the October 17, 1989 Lorna Prieta earthquake and the

CDMG strong ground motion station locations. Figure 1.2 shows the station locations for ,the

records obtained by USGS.

In order to develop soil amplification parameters for the local soils at strong ground motion

stations in the San Francisco Bay region, it is necessary to obtain data on various soil parameters at

the locations of strong ground motion stations. The soil parameters needed for analysis purposes

include

* depth to bedrock

* shear wave velocity variation with depth

* blow count variation with depth

*density variation with depth.

An extensive search was conducted to accumulate data on the various soil parameters

throughout the San Francisco Bay region. The search included CDMG and USGS information

files on strong ground motion stations and various publications discussing soil properties in the

San Francisco Bay area The CDMG files contained limited data which gave a general classification

of the soil type at their strong motion stations (Shakal et al, 1989). For 11 of the USGS stations

bore-hole data was available in Gibbs et al. (1975, 1976, 1977). Data were sought also from the

California Transit Authority (CAL TRANS), various geotechnical engineering firms and

researchers who may have obtained bore-hole data independently.

Although it was recognized that the data from the various sources may not be at the precise

locations of the strong ground motion stations, it was ·hoped that such data could be obtained in the

vicinity of these sites. Figure 1.3 shows the locations for which soil data were obtained in this

search.

2

•

37 s•

31.o'

111.5"'

•

• • • •

•

• • •

•

•

' "

N

l

' ' ' •, \

•' ' . '

•

\

11-1.•' ',

Figure 1.1 Locations of CDMG strong motion stations for recordings from the October 17,

1989 Lorna Prieta eanhquake (after Shakal et al, 1989).

3

PACIFIC OCEAN

• OIIOUND 11TH

• IUILDIHCI IITII

4 DAMS

·' & ' I tO

10 15 20 I I 1 ..

I I KM 20 30

. '~ • 17

•5 • 7

Figure 1.2 Locations of USGS strong motion stations for recordings from the October 17, 1989

Lorna Prieta earthquake ( from Maley, et al., 1989). ·

4

J7.5'

'"'·5'

~ .

"' t'pitf'fOitr ~

~,_.fLC.S

•

•

\

'\

JS

\ \

(l.I.C~ '

'·

Figure 1.3 Locations of bore-hole data from Gibbs et al. (1975, 1976, 1977).

5

During the last quarter of this project a report by Fumal (1991) was made available which

contains information on shear wave velocities for all the strong ground motion recordings from the

Lorna Prieta earthquake. The information in this report augmented data that had been made

available in the earlier reports from USGS (Gibbs et al., 1975; 1976; 1977). For seventy six of the

strong motion stations, however, the shear wave velocity profiles for the upper 30 meters reported

by Fumal were inferred based on the local geology in the vicinity of the site.

The information from the original search as well as from the Fumal ( 1991) report on soil

profiles was used to develop soil amplification parameters as discussed in Chapters 2 and 3.

1.3 Scope

Chapter 2 presents a method for consolidating bore-hole data and summarizes the data used for

soil amplification parameters.

In Chapter 3 the method for soil amplification computation is presented. Two types of data are

used for the purpose of computation of soil amplification factors. The first include the strong

motion stations at soil and rock sites for which bore-hole data are available. The second data set

includes the trrst data set plus strong motion station for which shear wave velocities with depth

were inferred by Fumal (1991 ). Synthesis of the strong motion data for uniform distances from the

rupture zone and azimuthal angle is discussed in this section. The soil amplification parameters

obtained from the two data sets are presented in Section 3.3. The sensitivity of these parameters in

discussed in Section 3.4

Chapter 4 presents a summary of the results and draws conclusions on the basis of the findings

in the report.

6

CHAPTER 2

SOIL PARAMETER AND GROUND MOTION DATA CONSOLIDATION

2.1 Soil Bore-hole Data

Soil parameter data were sought for the strong ground motion recordings identified in Chatper

1. In particular, data from Standard Penetration Tests (SPT) with information on blow count, N,

and density with depth are needed in order to apply the method of Sugito (1986). Such data are

available for a very limited number of sites. For the sites for which data are available, several logs

are usually obtained none of which are exactly at the site of the strong motion station.

In order to extrapolate the information from the locations of the logs to the strong ground

motion recording site a statistical averaging method called kriging is implemented (Journel and

Huijbregts, 1978). Although we were unsuccessful in gathering bore-hole data for the various

strong ground motion stations of the October 17, 1989 Lorna Prieta earthquake, the kriging method·

for spatial averaging of soil parameter data will prove useful when such data become available.

Thus in this section, we briefly summarize the underlying assumptions and the general

methodology for simple two-dimensional kriging.

Kriging is a linear regression technique used to estimate an unknown value Z(x0) at the location

x0, by a linear combination of n known values Z(xa) at locations X a, where a = I, ... , n.

Figure 2.1 shows the locations x a of the known parameter values and the location of the desired

parameters x0. Following Journel and Huijbregts (1978) and Journel (1989), the unbiased estimate

of Z(x0) is written as:

Z*(xo) = .E Aa Z(xa) with .E Aa = 1 (2.1)

and the error variance is given by:

di; = VAR{Z(xo)-Z*(xo)} = .E .E a~pC(xa-xp) (2.2)

with:

where a = I, ... , n

and

7

Figure 2.1 Schematic representation of spatial data used in kriging.

The covariance between two points Xa andx.13 is obtained by:

C(xa-xp) = COV{Z(xa), Z(xp)} = C(O)- g(xa-xp) (2.3)

where C(O) is a specified constant, and g(xa-xfiY is the variogram between the two points xa and

x.13 and is defined as:

(2.4)

Minimizing the error variance of the estimate under the constraint .E lla = 1, the ordinary

kriging system is used to solve for the desired ll values as follows:

L llpC(xa - xp) + m = C(xa- xo) for a= l, ... ,n (2.5)

and (2.6)

8

The corresponding error variance is:

ai; = E {[Z(xo)-Z*(xo)]} = C(O)- L AaC(xa-Xp)- m (2.7)

In matrix notation, the ordinary kriging system becomes:

KA+m=k and AA=1 (2.8)

where the (n x n) covariance matrix is:

K = (2.9)

Similarly, the data-to-unknown covariance column k ( n x 1) is:

The vector of weights is

A is a row vector of ones given by

A= [ 1, ... ,1]

and the vector of known data values is

R = [Z(aJ), Z(a2), ... , Z(an)J.

The estimate of the unknown value, Z*(xo), is now:

Z*(xo) =A R (2.10)

with an error variance of:

a;;= C(O)- K A (2.11)

A unique solution for this system will exist iff:

1.) K is a positive definite matrix

2.) There are no two totally redundant data locations, that is:

9

In order to illustrate the method, two-dimensional ordinary kriging is used to estimate the

standard penetration of resistance N of soils at the Stanford Parking Structure I. This linear

regression method involves a type of weighted averaging of the known soil data at five

surrounding locations on the Stanford University campus.

Soil blow count data were obtained for the Graduate School of Business, the Stanford

Hospital, the Stanford Medical Center, the Laboratory Surge Building, and the Research Animal

Facility. At each of these sites, there are between 5 and 13 boring logs giving blow count data at

specific depths. For each of these sites, the centroid of the bore holes is found. Figure 2.2 shows

the location of the Stanford Parking Structure I and the location of the bore-hole data surrounding

that site. The number in parenthesis indicates the number of bore holes available at that site. Two­

dimensional ordinary kriging is then performed at each site and depth level to find a best estimate

of the N value at that site and that depth.

The single best estimate of the blow count at each site and depth level is used in the ordinary

kriging process to find the best overall estimate for the blow count with depth at the parking

structure. At each depth level, the best estimate for the soil N value for each of the five

surrounding sites is used to find the krigged or weighted average soil N value with its associated

variance at the parking structure.

For each site for which blow count data were available, Table 2.1 shows the estimates of soil

blow count and associated error variance, and the parameters used in modeling the variogram in the

kriging process. These estimates were then used to determine the blow count variation with depth

at the Stanford Parking Structure I. A final plot of the best estimate of soil blow count versus depth

at the Stanford Parking Structure I is shown in Figure 2.3.

Blow count data at the various locations differ in the total depth to which the log was taken.

Some of the logs go to depths of 200 feet while others stop at 30 feet. In addition, at some depths

the variability in blow count is greater than for other depths. It is important to recognize that

kriging minimizes the error in the parameter estimates based on the distance from the location, but

does not take into account other sources of error. In the application to the Stanford Parking

Structure I, kriging was performed to depths corresponding to the shortest bore-hole data. It was

felt that extending the analysis to greater depths will introduce further error and nonhomogeneity in

the results.

10

"·

-~

l

scale: 1 : 10,000

distances: ~-(!) : .62km CD @-~: .41 km ~

''ID- (~) : . .32km ®

OIS1 ~

101 Lacllt l'aaanl Clllhlrwn'l -11&1

0 m

-wa::H Ro----<-

Graduate School of Business

Stanford Hospital (l&t)

Stanford Medical Center (I~

(7)

.@)-®: .12km @ Laboratoly Surge Building (!i)

®-®: .40km G) Research Animal Facility (7)

® Stanfom Parking Structure I

Figure 2.2 Stanford University Parking Structure I and surrounding sites where bore-hole data

are available.

11

Table 2.1 Blow count estimates for the five sites surrounding the Stanford Parking Structure

I as estimated by kriging data at each site by itself.

SITE DEPTH VARIOGRAM MODEL PARAMETERS KRIGED ESTIMATES (feet) c (silll A~ N (blow COUD!}_ variance

Graduare School of Business 2 29 93 I2 3I 4 45 110 8 4I 9 110 200 69 60 I4 2000 IOO 48 I996 I9 I8SO I60 45 I069 24 4SO IOO 83 515 29 6000 IOO I39 6643 34 8 I40 37 IO 39 226 I4I 59 235

· Stanford Medical Center 2 6IO I75 24 425 9 I40 200 42 86 I4 1700 200 I07 I04I I9 23500 I75 8S I6379 24 325 I75 40 227 29 22S I75 38 I 57 34 435 I75 37 303 39 4SO 260 39 2IO

Stanford Hospital 4 360 200 40 47 9 315 200 23 49 I4 8SO 200 42 110 19 soo I75 73 74 24 3200 I80 116 46S 29 2800 ISO 6I 485 34 3SO I85 60 49 39 4SO I30 117 90

I...aboratory Surge Building 2 IK) 200 I4 58 4 40 IOO 37 46 9 I200 IOO 54 I389 I4 70 IOO 44 8I I9 3S 60 45 47 24 26S ISO 79 335 29 4041 I75 I06 3403

Research Animal Facility 2 325 90 33 367 4 22S 200 46 I07 9 290 I60 48 180 I4 30000 IOO 99 32850 I9 51 SO 260 66 ~

24 5700 3SO I33 Im 29 425 200 72 245

Puking StiUcture I 2 68 36 I9 53 4 230 36 39 I92 9 9S 20 49 114 I4 I200 I5 82 IS75 19 300 40 58 207 24 1800 15 84 2362

29 1430 40 84 10I8 34 610 25 49 813 39 2030 5 49 3045

12

0

53

\ variance

-10

1575 --• • -- 207 :1: -20 1-D. w c

2362

0 20 40 60 80 100

BLOW COUNT (N)

Figure 2.3 Blow count, N, with depth at the Stanford University Parking structure I as obtained

by kriging data from surrounding locations

13

2. 2 Soil Parameters Data

The difficulties with obtaining soil parameter data were already discussed in the introduction of

this report. Because of these difficulties, the ground motion amplification parameters were

analyzed in two steps. The first analysis is based on data at 8 rock and 14 soil sites where strong

motion recordings were obtained from the October 17, 1989 Lorna Prieta earthquake. These data

were selected after the initial search for blow count or shear wave velocity was completed. Table

2.2 lists the soil and corresponding rock site stations used for this purpose.

The second analysis is based on 26 rock site and 26 soil site stations. The recently published

preliminary report by Fumal (1991) was used for the purpose of consolidating the second set of

data. The report provides average shear wave velocities for the top 30 meters of soil at each of the

strong ground motions stations. Table 2.3 lists the station locations and the parameters used in the

ground motion amplification analysis.

Since blow-count data were available for a very limited number of strong ground motion sites,

a simple soil classification was defined based on the generalized geologic map unit. The four

classes are

Rock:

Qpa:

Qha:

Qaf/Qhbm:

Include sandstone, shale, radiolarian chen and volcanic rock;

Pleistocene deposits (weakly to moderately consolidated, poorly sorted

deposits of irregular layers of clay, silt, sand and gravel, approximately 150

mdeep);

Holocene (younger) deposits- fine grain, medium-grain and coarse-grained

deposits; silts and clay rich in organic materials;

Younger bay muds - unconsolidated, water satruated, dark plastic clays.

The soil classifications for each strong ground motion record are listed in Tables 2.2 and 2.3.

In addition, two simple soil parameters are used in defming the soil amplification factor. The first

factor is the depth to bedrock, dp, in meters. This parameter is included to reflect the influence of

the low frequency characteristics of the ground motion at a specific site. The parameters dp listed in

Tables 2.2 and 2.3 have been obtained from contour maps of depth to bedrock in the San

Francisco Bay area (Khale, 1966; Bonilla, 1964; and Schlocker, 1968).

14

The second soil parameter used in this study is Sr. which represents the softness of the surface

layer at a specified site and includes relatively high-frequency characteristics of the ground

response. The parameter Sr is defined as follows:

(2.12)

where Vs is the shear wave velocity of the surface layer at a specified site, and vb is a standard

value of shear wave velocity for a very soft layer. Herein, the value of Vb is fixed at 88 m/sec to

correspond to the mean value of shear wave velocity of Bay mud (Borcherdt et al., 1979). The soil

parameters Sr listed in Table 2.2 have been obtained from Borcherdt et al. (1979). The Sr

parameters listed in Table 2.3 were developed using the shear wave velocities estimated by Fumal

(1991).

The soil parameter data in Table 2.2 is considered to be more reliable than the data in Table 2.3

primarily because the shear wave velocities for these data are based on actual measurements and are not inferred from geologic similarities. However, they constitute a rather small data set. Grou~~

motion amplification parameters will be estimated using both data sets since neither set is ideal. In

order to distinguish between the two data sets, we will refer to the data in Table 2.2 as Data Set I

and these in Table 2.3 as Data Set II. The results from the two analyses will be compared in

Chapter 3.

Review of the data listed in Tables 2.2 and 2.3a and b reveals that the shear wave velocities for

rock sites vary greatly. FotData Set I the shear wave velocities vary form 745 m/sec to 1,230

m/sec. For Data Set II most of the shear wave velocities range between 515 m/sec to 1230 m/sec.

The Corralitos station, however, in inferred to have a shear wave velocity of 395 m/sec. Such

wide variations in the rock shear wave velocities are expected to introduce errors in amplification of

ground motion. This wide variation in shear wave velocity also leads the authors to believe that the

amplification of ground motion should be a function of the rock shear wave velocity.

The ration of shear wave velocities S1 for the soft soil sites is also found to vary considerably.

For Qpa sites, S1 ranges from 0.12 to 0.24 indicating that these are relatively stiff sites. For Qha

locations, the value of S1 is from 0.24 to 0.44 reflecting a slight increase in the sofmess of these

sites relative to the Qpa sites. For the Qaf/Qhbm stations the parameter S1 ranges form 0.36 to

0.77. Table 2.3b also lists the thickness of the Bay mud deposits which vary considerably from

station to station. More importantly, however, the depth to bedrock in each case is also found to

vary greatly from station to station.

15

In order to reflect the variations in ground motion with shear wave velocities and depth to

bedrock, the amplification factor developed in Chapter 3 will be expressed as a function of the

parameters Sr and dp. The variations of rock shear wave velocities are reflected in the rock ground

accelerations and velocities. To account for these variation, the amplification factors will be

functions of the peak ground motions at rock as well.

16

STATION NAME

Yerba Buena Island

Piedmont*

Gilroy #1

Upper C. -Pulgas

So-Sterra Point*

San Fractsco-Rincoln Hill

Woodside

San Francisco-Telegraph Hill

Treasure Island

Oakland-2-story Bldg*

Oakland Outer Harbor*

Uilroy #2

Gilroy Gavilan co.*

Agnew

Foster Ctty

San Franci Int Airport

San Francisco 18story*

Sunnyvale South St

Hollister City Hall

Stanford Univ Parking

APEEL Array, Redwood*

San Fran 600 Montgo St

Table 2.2 Rock surface and soil parameters for strong ground motion data from the October 17,

1989 Lorna Prieta earthquake (Data Set I)

CODE N. w. GEOLOGIC MAP UNIT GENERA DISTANCE TO AVERAGE ST LAT. LONG. L RUPTURE SWV (m/s) VALUE

GEOLOG ZONE (top 30m) IC (km)

SYMBOL 58163 37.81 122.36 Sandstone rock 77 880 NA

5H338 37.823 122.233 Weathered-Sandstone rock 74 745 NA

47379 36.973 121.572 Sandstone rock 15 1230 NA

58378 37.49 122.31 Sandstone rock 44 435 NA

58539 37.674 122.388 Rock rock 65 910 NA

58151 37.79 122.39 Sandstone rock 76 745 NA

58127 37.429 122.258 Conglomerte rock 36 485 NA

58133 37.80 122.41 Sandstone rock 78 745 NA

58117 37.825 122.373 Fill Qhbm 79 88 1.00

58224 37.806 122.267 Alluvium Qps 73 267 0.33

58472 37.816 122.314 Bay Mud Qhbm 76 88 1.00

47380 36.982 121.556 Alluvium Qhac 16 163 0.54

47006 36.973 121.568 Terrace Deposit over Qpa 15 187 0.47 Sandstone

57066 37.:239 12l.Y52 Alluvtum· Qhaf 25 187 0.47

58375 37.55 122.23 Aluv., 210m; serpintine Qhbm 44 80 1.10

58223 37.662 122.398 Deep Alluvium Qhbm 60 88 1.00

58480 37.792 122.400 Fill over Bay Mud Qhaf 77 187 0.47

Qhaf 187 0.47

1575 36.851 121.402 Holocene fine-grained Qha 33 200 0.44 alluvium

Qa 187 0.47

1002 37.52 122.25 Artificial FilVHolocene Qhbm 44 88 1.00 Bay mud (9.5m)

1239 37.80 122.40 Qhafs (97) 187 0.47

17

DEPTH TO BEDROCK

(m)

NA

NA

NA

NA

NA

NA

NA

NA

86.7

85.3

97.5

140.2

25.9

231.6

201.2

164.6

58.8

181.7

55.8

36.6

91.4

43.6

Table 2.3a Rock surface parameters for strong motion data from the October 17, 1989 Lorna

Prieta earthquake (Data Set II)

STATION NAME CODE N.LAT. W.LONG. GEOLOGIC MAP UNIT GENERAL DISTANCE TO AVERAGE

GEOLOGIC RUP11JRE ZONE SWV (m's)

SYMBOL (km) (top 30m.)

ComliiOI 57007 37.239 121.952 Landslide derived from shale of Hia)l.land Wav rock I 395

Coyote Lake Dam abutment 57504 37.124 121.551 Serpentinite rock 21 515

Anderson Dam abutment 1652 37.166 121.628 Seroentinite rock 24 515

Gilroy Ill 47379 36.973 121.572 Sandstone of Franciscan a.sscmblll!e rock 15 1230

SuaiORa ·Aloha Ave. 58065 37.255 122.031 Santa Qara Fonnlllion rock 9 440

Santa cniz · ucsc 58135 37.001 122.060 Mewedimenwv rocks (schist) rock 16 670

Gilroy 116 • San Ysidro 57383 37.026 121.484 Upper Cm. and/or Paleo-Eoc:c:ne rocks rock 24 640

SAGO South 47189 36.753 121.396 Granodiorite rock 39 870

Stanford Linear Acceleralor 1601 37.419 122.20S Santa Qara Fonnalion/Miocene sandstone rock 34 320

Woodside Fire Station 58127 37.429 122.258 Butano ('?)sandstone rock 36 485

Monterey City Hall 47377 36.597 121.897 Granodiorite rock 44 870

APEEL lf7. Pul11as W1111:rTemple 58378 37.490 122.310 Butano ('?) sandstone rock 44 435

APEEL 1110 • SkyJine Blvd. 58373 37.465 122.343 Butano sandStone rock 44 405

·Hayward· CSUH stadium 58219 37.657 122.061 Leona Rhyolite roc:lt 53 525

So. San Francisco· Siena Pt. 58539 37.674 122.388 Sandstone at San BNno Mountain rock 65 910

San Francisco • Diamond HciRbts 58130 37.740 122.430 Sandstone and shale of Franciscan usemblll!e rock 73 745

Piedmont Jr. HiRb School 58338 37.823 122.233 Smdstone and shale of Franciscan usemblll!e rock 74 745

San Francisco • Rincon Hill 58181 37.790 122.390 Sandstone and shale of Franciscan a.sscmblage rock 76 745

Y c:rba Buena island 58163 37.810 122.360 Sandstone and shale of Franciscan assemblage rock 77 880

San Francisco • Pacific HciRhts 58131 37.790 122.430 Sandstone and shale of Franciscan uac:mblll!e rock 78 745

San Francisco • T ciCIIRI>h Hill 58133 37.800 122.410 Sandstone and shale of Franciscan usemblll!e rock 78 745

San Francisco· Presidio 58222 37.792 122.457 Se~"Pe~~tinite rock 79 515

Be:dl:cley • Lawrence Bezkclev Lab 58471 37.876 122.249 Upper Cretaceous shale rock 80 610

San Francisco • Ciff House 58132 37.780 122.510 Smdstone and shale of Franciscan usemblaRe rock 80 745

Golden Gate Bridge 1678 37.806 122.472 Surficial deoosiuJSc:rpc:ntinite rock 81 515

Point Bonita 58043 37.820 122.520 Sandstone and shale of Franciscan a.sscmblaRe rock 85 745

18

ST DEPTH TO

VALUE BEDROCK

(m)

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

NA NA

STATION NAME

Richmond Oty Hall ParlcinR lAt

Gilroy • Gavilan Colleac

Andc:non Dam downstream

CoYote Lake Dam downsu-eam

Gilroy 117 - M111tdli Ranch

Halls Valley

Fremont • Mission S111 J011e -

Fremont- Emerson Cowt

APm. #2E- John Muir School

Havwanl BART ParlcinR Lot

Capitola

Gilroy 112 • HWY 101 Mold

Gilroy 113 • Scw~~te T reaanent Plant

Gilroy 114 • San Ysidro School

Hollister Airport

AJti'C!f

Sunnyvl!e ·Colton Ave.

Hol.liJter CitY Hall

Hollister • South and Pine

Salinu

Olema

APEEJ... 112 • Redwood City

APEEL Ill ·Redwood Shorea

San Francisco lnL AirPort

Emeryville

Treasu!e Island

Table 2.3b Soil parameters for strong motion data from the October 17, 1989 Lorna Prieta

earthquake (Data Set ll)

CODE N.LAT. W.LONG. GEOLOGIC MAP UNIT GENERAL DISTANCE TO AVERAGE

GEOLOGIC RUP'I\JRE ZONE SWV (ow's)

SYMBOL (km) (top 30m.)

58505 37.935 122.342 Pli<>-Pleistocene alluvium Qpa 89 440

47006 36.973 12!.568 l...au: Pleistocene alluvimn Qpa IS 715

1652 37.166 121.628 Late Pleistocene alluvimn ()pa 24 365

57504 37.124 121.551 l...au: Pleistoc:a~e verv coane-IZnlined alluvium Qpa 21 505 . -57425 37.033 121.434 Late Pleistocene alluvium Qpa 28 455

57191 37.338 121.714 l...au: Pleistocene alluvimn Qpa 31 365

57064 37.530 121.919 l...au: Pleistocene fine--grained alluvium Qpa 39 285

1686 37.535 121.929 l...au: Pleistocene fmc-grained alluvium Qpa 40 285

1121 37.660 122.080 l...au: Pleistocene fme-grained alluvium Qpa 53 280

58498 37.670 122.086 l..a1e Pleistocene alluvimn Qpa 55 365

47125 36.974 121.952 Holocene alluvium Qha 14 285

47380 36.982 12!.556 Holocene coane-grained alluvium Qha 16 310

47381 36.987 121.536 Holocene fine-grained alluvimn Qha 18 355

57382 37.005 121.522 Holocene medium-IZnlined alluvium Qha 20 225

1656 36.888 121.413 Holocene fine-RJBined alluvimn Qh.a 34 200

57066 37.239 121.952 Holocene fine-IZnlined alluvimn Qh.a 25 240

1695 37.402 122.024 Holocene fine-»""i-ned alluvimn Qh.a 23 240

1575 36.851 121.402 Holocene fine-l!nlined alluviwn Oha 33 200

47524 36.848 121.397 Holocene fme-Rr.lined alluvium Qh.a 33 200

47179 36.671 121.642 Holocene alluvimn Qh.a 34 285

68003 38.043 122.797 Holocene coane-grained alluvium Qh.a 119 365

1002 37.520 122.250 Artificial fiiiJHolocene Bav mud (9.5 m) Qaf/Qhbm 44 130

58375 37.550 122.230 Artificial fill/Holocene Bay mud (9.5 m) Qaf/Qhbm 44 115

58223 37.622 122.398 Anificil! fiiiJHolocene Bav mud (6 m) Qaf/Qhbm 60 210

1662 37.844 122.295 Artificial fill/Holocene Bay mud (4 m) Qaf/Qhbm T1 245

58117 37.825 122.373 Artificial fiiiJHolocene Bay mud (13 m) OafiOhbm 79 130

19

ST DEPTH TO

VALUE BEDROCK

(m)

0.20 100.0

0.12 25.9

0.24 30.0

0.17 20.0

0.19 30.0

0.24 30.0

0.31 170.0

0.31 170.0

0.31 60.0

0.24 60.0

0.31 50.0

0.28 140.2

0.25 150.0

0.39 140.0

0.44 60.0

0.37 231.6

0.37 181.7

0.44 55.8

0.44 60.0

0.31 50.0

0.24 50.0

0.68 91.4

0.77 201.2

0.42 164.6

0.36 100.0

0.68 86.7

2. 3 Strong Ground Motion Data Analysis

The strong ground motion station locations for Data Sets I and II were identified in Tables 2.2

and 2.3. The geologic unit, the horizontal peak ground acceleration, Amax, and the peak ground

velocity, V max for both data sets are obtained from the processed records reported by the

California Division of Mines and Geology (Shakal et al., 1989) and the United States Geologocal

Survey (Maley et al., 1989). Tables 2.4 and 2.5a and b list all the station locations, the geologic

unit and the peak ground acceleration and velocity in the two horizontal directions. The two tables

correspond to Data Sets I and II, respectively.

Not all records were obtained at 0° and 90° directions from North. Thus, for the strong motion

records obtained at other directions, the time histories where converted to 0° and 90° by a simple

transformation of coordinates. The transformation was performed bor both data sets.

In order to remove the attenuation effects on the ground motion, the peak ground accelerations

at rock and soil surface sites should be compared at the same distance from the rupture zone. Thus,

Amax and V max for the rock sites listed in Tables 2.4 and 2.5 were scaled to similar distances as

the peak values for soil sites. The empirical values for A max and V max listed in Table 2.5a and b

were computed for each site using Joyner and Boore's (1988) attenuation functions given by:

log (Amax) = 0.43 + 0.23(M- 6)- log (r) -0.0027 (r) (2.13)

where

r= -J r~ + 64

ro = distance to the rupture zone (km)

M = magnitude of the earthquake ( = 7.1 for Lorna Prieta)

and

log (V max)= 2.09 + 0.49 (M- 6) -log (r) - 0.0026 (r) + 0.17 (2.14)

where

r= -J r~ + 16

ro = distance to the rupture zone (km)

M = magnitude of the earthquake ( = 7.1 for Lorna Prieta)

20

The empirical peak ground motion values, however, need to be scaled to show a better

agreement with the recorded values. For this purpose, the Joyner and Boore (1988) attenuation

equations are fitted to the data for the 26 rock sites by least squares analysis. The scaling factor for

the Amax values is found to be 1.189 with a coefficient of variation of 0.628. The scaling factor

for the V max values is found to be 0.489 with a coefficient of variation of 1.4. The scaling factor

for V max is governed by the data at the larger distances to the epicenter resulting in an unacceptable

error over the remaining V max values. Thus, the empirical V max values are left unmodified. The

scaled Amax values and the unmodified values of Vmax are listed in Table 2.5a and b, respectively.

Plots of the measured peak ground acceleration and velocity values along with the empirical

attenuation curves of Joyner and Boore are shown in Figures 2.4 and 2.5. Figure 2.4 also shows

the scaled attenuation function for peak horizontal acceleration. The data on these plots correspond

to the 26 rock sites of Table 2.5a and b.

The geographically closest rock site station is determined for each of the soft soil stations in

Data Sets I and II. Table 2.4 lists the soft soil and corresponding rock stations for Data Set I.

Tables 2.6a and b list the soil sites and the corresponding rock sites for Data Set II for peak ground

accelerations and velocities, respectively. The peak ground motion at each rock station is modified

to reflect the distance of the soft soil station from the rupture zone. This modification is done by a

simple ratio for both data sets as follows:

where

PGM(dJ)rec =

and

PGM(dJ)emp =

PGM(dJ)rec _ PGM(dJ)emp PGM( d2Jrec - PGM( d2Jemp

(2.15)

recorded peak ground motion (Amax or V max ) at distance

d 1 from the rupture zone,

empirical (i.e., modified Joyner and Boore (1988)

attenuation function) peak ground motion (Amax or V max )

at distance d1 from the rupture zone.

The distance d1 corresponds to the distance from the rock site recording station to the rupture

zone. The distance dz is the distance from the rupture zone to the soft soil site that is to be

compared to the rock site. The peak ground motions listed in Tables 2.6a and b were obtained

using the accelerations in Tables 2.5a and b and equation 2.15.

21

-s .1

II

• Joyner & Boore Modified Joyner & Boore Recorded data

Modification factor = 1.189 Standard deviation = .087

1 0

•

• •

dlat•nce to rupture zone (km)

• • • •

100

Figure 2.4 Peak ground acceleration data as functions of the rupture zone for the Lorna Prieta

earthquake of October 17, 1989; the original Joyner and Boore (1988) and the best fitted of the

attenuation laws.

1000

II Recorded data .......- Joyner & Boore

-u • 100 • ..... E u -> CJ Q.

10

1 0 100

dl•t•nce to rupture zone (km)

Figure 2.5 Peak ground velocity values as functions of the rupture zone for the Lorna Prieta

earthquake of October 17, 1989 and Joyner and Boore's (1988) attenuation law

22

Table 2.4 Soil and rock surface strong ground motion data for Data Set I

Soil Site Soil A max A max Vmax Vmax Rock S1te Rock A max A max 1 2 1 2 1 2

Name Type (90°) ( oo) (90°) ( oo) Name Type (90°) ( oo)

CDMG 1 . Treasure Island Qhbm -155.8 97.9 33.4 -15.6 Yerba Buena Island Sandstone -65.8 28.1

2. Oakland-2-story Bldg* Qps -166.6 130.5 19.4 8.9 Piedmont* Weathered- 97.5 49.7

3 . Oakland Outer Harbor* Qhbm -325.5 -253.6 50.7 -31.8 P1edmont* Serpentinite 97.5 49.7

4. Uilroy #2 Qhac 316.3 -344.2 -39.2 33.3 ( Uilroy #1 Sandstone 391.4 385.1

5. Gilroy Gavilan co.* Qpa 363.9 -312.6 -25.5 25.5 (Gilroy #I Sandstone 422.5 415.7

6. Agnew Qhal 157.6 163.1 -IH.2 30.~ (Upper C. -Pulgas Sandstone -165.6 -300.0

7. Foster City Qhbm 277.6 252.6 45.4 -31.8 Upper C. -Pulgas Sandstone -84.H -153.6

8. San Franci Int Airport Qhbm -352.8 -230.8 29.3 26.5 So-Sierra Pomt* Rock 77.1 -78.9

9. San Francisco 18story* Qhaf 137.4 163.7 -15.6 -16.9 San Fracisco-Rincoln Hill Sandstone 88.5 -78.6

usus 1. Sunnyvale South St Qhaf 208.0 211.8 34.1 -33.4 ( Upper C.-Pulgas Sandstone -162.3 -294.0

2. Hollister City Hall Qha 251.9 216.8 -38.6 -44.0 (Gilroy #1 Sandstone 179.4 176.5

3. Stantord Uruv Parkmg Qa -216.0 -255.0 -21.3 -33.2 Woodside Conglomerte 7~.7 79.5

4. APEEL Array, Redwood* Qhbm 23H.6 -244.2 49.6 36.4 Upper C -Pulgas Sandstone -84.8 -153.6

5. san Fran 600 Mom go St Qhats 119.4 -107.1 18.1 -9.6 San Francisco-Telegraph Hill Sandstone 90.5 -51.2

* : Peak ground motions have been obtained from time histories processed by the transformation of coordinates systems. ( ) :Peak ground motion have been modified according to Joyner & Boore's attenutation formulas.

23

Vmax Vmax 1 2

(90°) ( oo)

14.7 4.6

13.3 4.4

13.3 4.4

-29.5 27.9)

-32.6 30.8)

-27.1 35.0)

-13.6 17.6

7.0 -6.7

11.6 7.3

-26.5 34.3)

-12.2 11.5 )

-14.7 15.6

-13.6 17.6

9.6 6.5

Table 2.5a Empirical, recorded and modified peak ground acceleration for Data Set II

STATION NAME GENERAL EMPIRICAL RECORDED RECORDED EMPIRICAL GEOLOGIC I'GA PGA (90 dcg) PGA (360 dcg) I'GA (I!)

SYMBOL (g) (g) (g) mod. x 1.189 Conalitos rock .569 .478 .630 .616

Coyote Lake Dam abutmmt rock .187 .4(,0 .148 .222

Andc:Bon Dam abutmmt rock .163 .058 .087 .194

Gilrov II rock .255 .442 .435 .303 Saratoga- Aloha Ave. rock .371 .322 .S04 .442

Santa Cruz- UCSC rock .241 .409 .441 .287

Gilroy fl6 - San Ysidro rock .163 .. 170 .114 .194

SAGO South rock .095 .078 .O<i8 .112

Stanford Linear Accelerator rock .Ill .202 .288 .132

WOodside rue Station rock .104 .081 .081 .124

Monterey City Hall rock .082 .()(;2 .070 .097 APEEL 117- Pul~tas Water Temple rock .082 .086 .157 .097

AI'EELIIIO- Skyline Blvd. rock .082 .088 .103 .097 ' Hayward - CStnt stadium rock .064 .084 .074 .077

So. San Francisco - Sierra Pt. rock .049 .079 .080 .058 San Fnnciseo- Diamond HeiRhts rock .042 .113 .098 .049

Piedmcnt Jr. Higft_ School rock .041 .099 .051 .048 San Fnncisco- Rinccn Hill rock .039 .090 .080 .047

Y erba Duma bland rock .038 .O<i7 .029 .046 San Francisco - Pacific Heights rock .038 .061 .047 .045 San Francisco- TeleRraoh llill rock .038 .092 .052 .045

San Francisco - Presidio rock .037 .199 .100 .044

Berlceley- Lawrence Betlteley Lab rock .036 .117 .049 .043

San Francisco - ClifT House rock .036 .108 .075 .043

Goldm Gate Bridae rock .036 .243 .127 .042

Point Bonita rock .033 .076 .072 .039 Richmcnd City Hall Parlcina Lot Qpa .031 .121 .122 .037

Gilroy_- Gavilan College Qpa .255 .371 .319 .303

And=on Dam downstream Qpa .163 .232 .260 .194 Coyote Lake Dam downstream Qpa .187 .161 .153 .222

Gilroy 117 - Mantelli Ranch Qpa .138 .320 .210 .164 Halls Valley Qpa .123 .112 .131 .147

Fremont - Mission San J osc Qpa .095 .102 .120 .112

Fremont- Em men Court Qpa .092 .194 .145 .109 APEEL fi2E - John Muir School Qpa .064 .139 .170 .077

Havward BART Parking Lot Qpa .061 .122 .226 .073 C~tola Qha .270 .398 .472 .321

Gilrov 112 -IIWY 101 Motel Qha .241 .322 .351 .287 Gilroy 113 - SewaRe Treatment Plant Oha .216 .369 .542 .257

Gilroy 114 - San Y aidro School Qha .196 .214 .416 .233

Hollister Aitport Qha .Ill .281 .262 .132 Agnew Qha .156 .161 .166 .185

Sunnyvale- Colton Ave. Qha .170 .214 .219 .202

Hollister City Hall Qha .liS .257 .221 .137

Hollister - South and Pine Qha .liS .178 .369 .137

Salinas Qha .Ill .117 .086 .132

Olema Qha .019 .102 .161 .023

APEEL fl2 · Redwood City Qaf,Qhbm .082 . .243 .251 .097

AI'EEL Ill- Redwood Sham~ Qaf/Ohbm .082 .283 .257 .097

San Francisco lnt Ainlort Qaf,Qhbm .055 .332 .235 .065

Emeryville _Qaf,Qhbm .038 .259 .188 .046

Treasure Island Qaf/Ohbm .037 .159 .100 .044

24

Table 2.5b Empirical, recorded and modified peak ground velocity for Data Set II

STATION NAI\IE GENERAL EI\II'IRICAL RECORD!.;!) RECOIU.)EH El\fi'IRICAL GEOLOGIC rGV rGv (90 dcJ!) I'<~ V (:'60 dcJ!) rc:v (t'ml~>

SYMilOL (em/sec) (cmfsec) (cmfsec) unmodlncd Comlit05 rock 148.96 47.50 .55.20 148.96

Covote Lake Dam abutment rock 25.91 37.21 15.08 2S.91 Anderson Dam abutment rock 22.37 9.12 12.09 22.37

Gilroy~· mck 36.95 33.80 31.90 36.9.5 Sarato11a- Aloha Ave. rock 60.26 43.60 41.30 60.26 ---· Santa Cruz - UCSC mck 34.58 21.20 21.20 34 . .58

Vilroy 116- San Ysidro mck 22.37 13.90 13.10 22.37

SAGO South rock 12.70 10 . .54 9.08 12.70

Stanford Linear Accelerator mck 14.98 36.73 28.37 14.98

Woodside fire Station rock 13.99 14.70 1.5.60 13.99

Monterey Citv Hall rock 10.94 4.66 3.33 10.94

ArEEL n- l'ull!as Water Temple rock 10.94 13.60 17.60 10.94

AI'EEL II 10- Skyline Blvd. mck 10.94 21.80 13.30 10.94

Hayward - CSUllstadium rock 8.62 7.3.5 5.31 8.62 So. San Francisco- Sierra l't. rock 6.5.5 7.0.5 6.71 6 . .55

San Francisco - Diamond llciRhts rock 5 . .56 14.30 10.50 5 . .56 Piedmont Jr.llil!h School rock 5.4.5 13.27 4.3.5 S.4S

San Francisco- Rincon II ill rock 5.24 11.60 7.34 5.24 Y erba Ruena bland rock S.1S 14.70 4.61 s.1s

San Fnmcilsco - Pacific HciRht! rock 5.0.5 14.32 9.88 5.05 San Fnmci•co- Telej!raph llill mck 5.0.5 9.59 6 . .50 s.os

San Fnmcisco- Presidio rock 4.96 33.50 13.30 4.96 Berkeley- Lawrence Derlteley Lab rock 4.87 22.00 8.70 4.87

San Fnmcisco - Cliff llou•e rock 4.87 21.00 11.20 4.87 Golden Gate Bridge rock 4.78 3.5.49 18.02 4.78

roint Donita rock 4.44 14.48 8.77 4.44 Richmond Cit~ llall rarking Lot Qpa 4.1.5 1.5.19 1.5.13 4.1.5

Gilroy - Gavilan ColleRC Qpa 36.9.5 2SA9 2S.46 36.95 Anderson Dam downotrearn Qpa 22.37 21.'16 18.94 22.37

Corote We Dam downstream Qpa 25.91 19.20 16.54 25.91 Gilmy 117 - Mantelli Ranch Qpa 18.79 16.30 16.60 18.79

Halla Valley Qpa 16.70 13.70 12 . .50 16.70 Fremont- Mission San Jose Qpa 12.70 8.38 10.20 12.70 Fremont- Emenon Court Qpa .12.31 10.79 . 10.26 12.31

ArEEL "2E- John Muir School Qpa 8.62 13.00 13.60 8.62 llavward DART rarltinll Lot Qpa 8.21 9.08 13.12 8.21

Capitola Qha 39.63 30.70 36.10 39.63 Gilroy 112- IJWY 101 Motel Qha 34 . .58 39.20 33.30 34..58

Gilroy ~3- SewaRe Treatment Mant Oha 30 . .57 43.80 34 . .50 30..57 Gilroy "4 ·San Ysidro School Qha 27.32 38.20 39.10 27.32 .•

Hollister Aifl)Of1 Qha 14.98 34.40 43.25 14.98 A11new Qha 21.37 18.20 30.90 21.37

Sunnyvale- Colton Ave. Qha 23.4.5 34.08 33.43 23.45 llollister Cit~ Hall Qha 15..52 38.66 43.98 1.5.52

Hollister - South and rine Qha 1.5..52 30.90 62.80 1.5.52 Salinas Qha 14.98 1.5.37 10.24 14.98 Olema Oha 2 . .59 16.20 18 . .50 2 . .59

AI'Ec"L 112 - Redwood City QafiQhbm 10.94 49.67 36.40 10.94 AI'B:L Ill - Redwood ShOI'e.'l Qaf/Qhbm 10.94 45.40 31.80 10.94

San Francisco I nL Ailp<llt_ Qaf/Qhbm 7.30 29.30 26.50 7.30 rmeryville QafiQhbm 5.15 43.80 15 . .56 5.15

Treasure Island QafiQhbm 4.96 33.40 15.60 4.96

25

Table 2.6a Soil and rock surface strong ground acceleration data for Data Set II

STATION NAME GENERAL RECORDED RECORDED ROCK SITE FOR BEDROCK BEDROCK

GEOLOGIC PGA (90deg) PGA (360 deg) AMPLIFICATION PGA (90deg) PGA (360 deg)

SYMBOL (g) (g) (,g) (g)

Ric:bmond CitY Hall PulrinR Lot _Qpa .121 r .122 Piccimmt Jr. Hi~ School .075 I .038

Gilrov - Gavilan Colle110 Qpa .371 I .3\9 Gilrov 111 .442 .435

Andenon Dam downstream __Qpa .:!32 .260 GilJOy •1 .282 .:!78

Covote Lalce Dan downstream Qpa .161 .153 GilroY~~ .323 .318

Giltov #7 - Mantelli Ranch Qpa .320 .210 Gilrov #! .239 .235

Halls V allev Qpa .112 . 131 SaraiDn- Aloha Ave . .107 .167

Franmt- Mission San Jose Qpa .102 .120 S&nU>n ·Aloha Ave. .082 .128

Fremont • Emc:nm Coun Qpa .194 .145 S&nU>qa · Aloha Ave. .080 .l1A

APEEL 112E • John Muir School C-2!>a .139 .170 Piedmont J~. Hi~h School .157 .080

Hayward BART Par!cing Lot Qpa .122 .226 Piedmmt Jr. Hi_Kh School .!SO .076

Caoitola Qha .398 An Santa Cruz· l:CSC .459 .49S

Gilroy #2 - HWY I 0 I Motel Qha .322 .351 GilroY #l .418 411

Gilroy ll3 • Sewage Treatment Plant Qha .369 .542 Gilrov ill .315 .369

Gilrov #4 • San Ysidro School Qha .214 Al6 GilroY# 1 .339 .334

Hollister Airport Qha .281 .262 GilroY #I .192 .189

Al!llew Qha .161 .166 APEEL #7 · Pui2as Water Temole .165 .299

Sunn_)'Vale • Co1um Ave. _Qha .2\4 .219 SantORa • Aloha Ave. .148 .231

Hollister Citv Hall _Qha .257 .221 GilroY lll .199 .196

Hollister - South and Pine __Q!Ia .178 .369 Gilrov #I .199 .196

Salinas _Qha .!17 .086 SAGO South .092 .080

Olema Qha .102 i .161 Point Boruta .044 .042

APEEL #2 • Redwood Citv Qa!IQhbm .243 .251 APEEL #7 · Pu12as Wau:r Temole .086 .151

APEEL #I • Redwood Shores Qaf/Qhbm I .283 .257 APEEL #7 • Pu!.Ru Wue: Temple .086 I .!57

San Francisco Int. Ain>on I Qaf/Qhbm .. 332 .235 So. San Francisco - Siena Pt. .088 .090

Emervville Oaf/Qhbm .259 .188 Piedmau Jr. Hi~ School .094 .048

Treasure Island Qaf/Qhbm .!59 .100 Yctba Buena lsi and .065 .028

26

Table 2.(ib Soil and rock surface strong ground velocity data for Data Set II

STATION NAME GENERAL RECORDED RECORDED ROCK SITE FOR BEDROCK BEDROCK

GEOLOGIC PGV (90deg) PGV (360 deg) AMPLIFICATION PGV (90deg) PGV (360 deg)

SYMBOL (em/sec:) (em/sec:) (em/sec) (em/sec:)

!Uchmond City Hall Parl<in11l.ot Ooa 15.19 15.13 Piedmont Jr. Hilllt School 10.09 3.31

Gilroy • Gavilan Colleae Ooa 25.49 25.46 Gilroy rn 33.80 31.90

Andmon Dam downsaeam Ooa :!1.46 18.94 Gilroy II I ~0.46 19.31

Covote Lake Dam downstrellm Ooa 19.20 16.54 Gilrov Ill 23.70 2237

Gilroy 117 • Mantelli Ranch Ooa 16.30 16.60 Gilroy Ill 17.19 16.22

Halls V allev Ooa 13.70 1250 Sa.ratou. Aloha Ave. 12.09 11.45

Fremont· Mission San Jose Qpa 8.38 10.20 Sa.ratou • Aloha Ave. 9.19 S.70

Fremont • Emerson Court Ooa 10.79 10.26 Saratou. Aloha Ave. 8.91 S.44

APEEL #2E • Joltn Muir School Qpa 13.00 13.60 Piedmont Jr. Hird> School 10.98 6.88

HavwudBARTParl<inRI..ot Ooa 9.08 13.12 Piedmont Jr. Hird> School 19.98 6.55

Capitola Qha 30.70 36.10 Santa Cruz· UCSC 24.29 24.29

Gilrov 112. HWY 101 MOld Oha 39.20 33.30 Gilrov Ill 31.63 29.85

Gilroy 113 • Sewav;e Trulment Plant Oha 43.80 34.50 Gilroy Ill 27.96 26.39

Gilroy 114 ·San Ysidro School Oha 38.20 39.10 Gilroy #I 24.99 23.58

Hollister Airport Oha 34.40 43.25 Gilroy Ill 13.70 1293

AI!IIOW Oha !8.20 30.90 APEEL 117 • PulRU W01er Temple 26.57 34.38

SUMvvale • Colton Ave. Oha 34.08 33.43 Saratoll& ·Aloha Ave. 16.97 16.07

Hollister Cit v Hall Oha 38.66 43.98 Gilroy ill 14.20 13.40

Hollister· Soulh and Pine Oha 30.90 6280 Gilrov Ill 14.20 13.40

Salinas Oha l5.37 10.24 SAGOSoulh 1243 10.71

Olema Oha 16.20 18.50 Point Bonil& 3.44 5.1\

APEEL #2 • Redwood Ci1v Oai/Ohbm .t9.67 36.40 APEEL ~7 • Puigas Water Temple l3.60 17.60

APEEL ill • Redwood Shores Oai/Ohbm 45.40 3\.SO APEEL il7 • Puit~.u Water Temole 13.60 17.60

San Fnncisco lnL AirooR Oai/Ohbm 29.30 26.50 So. San Francisco • Siena Pt. 7.87 7.49

Emeryville Oai/Ohbm 43.80 15.56 Piedmont Jr. Hird> School 1253 ·UI

Treasure Island Oai/Ohbm 33.40 15.60 Y orb& Buena Island 14.16 4.44

27

CHAPTER 3

GROUND MOTION AMPLIFICATION FACTORS

3.1 Methodology

The model for ground motion amplification is based on the simple approach proposed

by Sugito (1986). In this method the soil surface ground motion is rated to rock surface

motion by the following relationship:

(3.1a)

(3.1 b)

where As and Ar are the peak ground accelerations at the soil and the rock surface,

respectively; Vs andVr are the peak ground velocities at the soil and rock surface, respectively; and f3a and f3v are the conversion factors from rock to soil surface motion for

peak ground accelerations and velocities, respectively.

In order to develop the appropriate algebraic form of the conversion factors f3a and f3v,

the relatiol'l:ship between the soil and rock peak accelerations and velocities were

investigated extensively by Sugito (1986), Sugito et al. (1986) and Sugito and Kameda

(1990). Their studies were based on strong ground motions from earthquakes in Japan. In

addition, simulated records were utilized to investigate the nonlinear relationship of these

parameters. As a result of these studies, the amplification parameters were modeled as

nonlinear functions of rock surface motion, the thickness of various soil layers and the

blow-count with depth. For this purpose a new parameter was defined as follows:

s. = 0264 i"" exp {- 0.04 N(x)} exp {- 0.14x} dx- 0.885 (3.2)

where N(x) is the blow count at depth x and ds is the thickness of the soil layer. The blow

count is related to the voids ratio of the soil which has also been found to be highly

correlated to soil amplification and deamplification (Rogers et al, 1985).

28

Table 3.1 Ground motion amplification data for Data Set II based on strong ground

motion records from the October 17, 1989 Lorna Prieta eartquake.

STATION NAME GENERAL PGA (90 deg) PGA (360 deg) PGV (90 deg) PGV (360 deg)

GEOLOGIC MEASURED MEASURED MEASURED MEASURED

SYMBOL AMPLIFICATION AMPLIFICATION AMPLIFICATION AMPLIFlCA TION

Richmond City Hall Pmina Lot Ooa 1.605 3.176 1.505 4.574

Giliov • Gavilm Colleae Ooa 339 .733 .754 .798

Andenon Dam downsueam Ooa .822 .937 !.049 .981

Covote Lake Dam downsueam Ooa .499 .482 .810 .739

Gilroy 1*7 • !'.fantelli Ranch Ooa 1.339 .890 .948 1.023

Halls Valley Ooa 1.046 .782 1.134 1.092

Fremont - !'.fission San Jose Ooa t.i49 .935 .912 1.172

Fremont· Emenon Cowt Ooa 2438 1.165 1.211 1.216

APEEL H2E • John !'.fuir School Qpa .883 2117 .620 1.978

Hayward BART Pamnt~ Lot ()pa .817 2956 .455 2.003

Caoitola Qha .868 .953 1.264 1.486

Giliov #2 • HWY 101 Motel Oha .772 .854 1.239 1.115

GilroY #3 ·Sewage Trea!mcnt Plant Oha .984 1.469 1.566 1.307

Gilroy 114 ·San Ysidro School Oha .631 1.246 1.529 1.658

Hollister AirPort Oha 1.460 1.384 2.510 3.344

AIIIICW Oha .972 .SS6 .685 .899

Sunnyvale· Colton Avo. Oha 1.449 .948 2.009 2.080

Hollisu:r City Hall Oha 1.291 1.128 2.723 3.282

Hollister· Soutb and Plno Qha .893 1.883 2.177 4.687

Salinu Oha 1.!70 1.067 1.236 956

Olema Oha 2308 3.836 1.919 1617

APEEL ~2 • Redwood C!ty Oaf/Ohbm 2811 1.603 3.652 2.068

APEEL HI· Redwood Shores Oaf/Ohbm 3.!74 1.645 3.338 1.807

San Francisco inL AirPort Oaf/Ohbm 3.789 2620 3.725 3.539

Emervville Oaf/Ohbm 2762 3.928 3.496 3.789

Treaswe Island Oaf/Ohbm 2459 3.618 2359 3.513

29

As discussed in Chapter 2, data on blow count with depth was available only at a few

strong ground motion sites for the Lorna Prieta earthquake. Instead, average shear wave

velocities were obtained for Data Sets I and ll and a new parameter was defined to reflect the high frequency components of the ground response. The conversion factors f3a and f3v are defmed as follows:

(3.3a)

log f3v =lao- a1log V,]m- 1.5 (3.3b)

where ao, a1 and mare constants which characterize the nonlinear amplification effect of

surface layers depending on the input peak ground motion levels (Sugito et al., 1991).

These constants are expressed as functions of the local soil parameters S t and dp in the

following manner:

ao =co+ Cj Sr + Czlog dp m = do + d1 Sr + dz log dp a1 =eo+ e1 Sr + ezlog dp

(3.4)

The coefficients in equation 3.4 are developed using Data Sets I and II as discussed in

the next section.

3.2 Nonlinear Parameters for Ground Motion Amplification

The amplification ratio between soil and rock site peak ground motions were developed

initially for Data Set I and then for Data Set ll. Figure 3.1 shows a plot of the amplification

of peak ground acceleration as a function of rock site accelerations for Data Set I. The same

relationships for velocity amplification is shown in Figure 3.2. The results from the

analysis of Data Set ll are shown in Figures 3.3 and 3.4. Table 3.1lists the amplification

values displayed in Figures 3.3 and 3.4. The nonlinear character of ground motion

amplification due to local soils is evident in all of these figures.

The relationship between the amplification factor and the parameter dp was also

investigated Figures 3.5 and 3.6 show the relationship between the amplification factors as

functions of the depth to bedrock, dp, for all soil classes in Data Set I. The two figures

correspond to peak ground acceleration and velocity amplification, respectively. Figures

3.7 and 3.8 show similar plots for the amplification values obtained from Data Set ll. From

30

Table 3.2 Regression constants for amplification of peak ground acceleration defined

by equation 3.4; (a) Data Set I= "Sugito" and (b) Data Set II= "modified".

cl c2

-3.920 1.670

5.580 -7.800

dl d2

0.250 0.021

-0.286 0.729

el e2

SUGITO 1.080 -1.690 0.910

MODIFIED 3.680 1.350 -1.770

Table 3.3 Regression constants for amplification of peak ground velocity defined by

equation 3.4; (a)Data Set I= "Sugito" and (b) Data Set II= "modified".

cl c2

10 -2.620 0.100

-1.128 -0.797

dl d2

SUGITO 0.220 0.153 0.054

MODIFIED -0.439 1.552 0.603

eO el e2

SUGITO 3.350 10 0.650

MODIFIED 1.368 -0.368

31

-::l

·.>.-=~ ..... ~ -o-.. , -

~ ~ ... "To;

~ ...,.,J

~ I

< i

"'~

$ '$ ~ e $ 0

0 ".::l

=~ ~ ! .... ·~

0 •• 0 0 0

0&

c: -,.,._ 0 •.:l-

·.;::: ...__ 0

~0 0 0

~ '()-

t= Ul-0

.,.._ c.. §

"'~ I - I

,_ I

:: o'

peak ace. on rock surface, Ar ( cm/sec2)

Figure 3.1 Variation of peak ground acceleration amplification with peak ground

acceleration at rock surface for Data Set I.

-a

0..-1 co~

... ~l 2; I

=~ > 0 ·.::s

0

0 I r: c: -...J

o..-.9 CX), - ~~ ~ I.: Ll'l1

0

~ 0~ 0 0

0 ~0 $ 0 0

0 0~ ~0

0 ~0

0

c.. ""'1 ~ :J

7"l, I

2

peak vel. on rock surface, Vr (em/sec)

Figure 3.2 Variation of peclk ground velocity amplification with peak ground velocity at

rock surface for Data Set I.

32

c ..2 -• u ~ a. E •

c 0 -• ..2 ~

a. E • > CJ A.

10,-------------------------~----------------------~

• •• • •

.. .,. ... 'I ..,-.. . . 1;-------~----~~--~~~~~------~--~~~--~~~~

.01 . 1

rock PGA (cm/aec"2)

Figure 3.3 Variation of peak ground acceleration amplification with peak ground

acceleration at rock surface for Data Set II.

10

• • •• • fiJ .. .. • ..... •.., .... .. •• Ill - .. .. - ---~ rf • •

•

.1 10

rock PGV (cm/aec)

100

Figure 3.4 Variation of peak ground velocity amplification with peak ground velocity at

rock surface for Data Set II.

33

these figures no specific pattern could be recognized for the functional dependence of dp to /3. The amplification values for all three soil classes are shown in these figures. No

specific trends, however, were found when the amplification factor was plotted as a

function of dp separately for each of the three soil classes.

Investigation of the amplification of peak ground acceleration and velocity with distance

to the rupture zone shows an increasing trend in amplification value with increasing

distance. Figures 3.9 and 3.10 show the variation of amplification of peak ground

acceleration and velocity with distance to the rupture zone of the Lorna Prieta earthquake.

The increasing trend is more pronounced with peak ground acceleration amplification

factors than with peak ground velocity. In both figures, however, the scatter is very large

to make definitive conclusions. The increase in amplification is, in general, expected since

the amplitude of peak acceleration and velocity at rock surface decreases with increasing

distance. Thus the ratio of soil to rock peak motions is successively divided by a smaller

number as the distance from the rupture zone increases. It is also recalled that with

increased distance the higher frequency components of the ground motion attenuate more

quickly. Thus the amplification ratio may be taken between waves with different

frequencies. Review of the data also reveals that the softest sites are the furthest from the

rupture zone introducing a bias in the data.

The evaluation of the constants defined in equation 3.4 for the nonlinear amplification

of peak ground acceleration and velocity was performed slightly differently for the two data

sets. Data Set I is relatively small. Thus data from strong ground motions in Japan was

used to determine the constants in equation 3.4. These constants were then scaled to fit the

data given in Data Set I.

The ground motion amplification relationships based on Data Set II with 52 strong

ground motion stations were obtained by direct nonlinear regression analysis. The

commercial statistical package SAS (1983) was utilized for that purpose.

Table 3.2 lists the values of the constants for peak ground acceleration amplification

defined in equation 3.4 for both Data Set I and II. The constants for amplification of peak

ground velocity are listed in Table 3.3. The resulting constants are different because the

two data sets and the methods of analysis of each data differ substantially.

Figures 3.11 to 3.13 show a comparison of the amplification factors as functions of Sr

as computed using (a) Data Set I (labeled as "Sugito Beta"), (b) Data Set II (labeled

34

"modified Beta") and (c) measured data. The three figures correspond to the three soil

classes defined in this study. Part (a) of each figure is amplification of peak ground

acceleration and part (b) corresponds to peak ground velocity. The amplifications in the

figures are shown only for location for which the soil parameter S1 is known. It is difficult

to conclude from these figures which amplification coefficient is closer to the observed

amplifications. Tables 3.4 to 3.9 provide the numerical values for each station location and

the amplification factors as evaluated (a) from the strong motion recordings, (b) from the

amplification equation based on Data Set I and (c) from the amplification equation based on

Data Set II. Tables 3.4 to 3.6 provide the comparisons for amplification of peak ground

acceleration and Tables 3.7 to 3.9 give the comparisons for amplification of peak ground

velocity. Each table corresponds to a different generalized soil class. There is a

considerable difference between the predicted and the recorded amplifications for many of

the observations. However, for the majority of the strong motion stations the amplification

values appear to be better predicted by the f3 parameter computed from Data Set IT than

these predicted by the f3 parameter from Data Set I.

A comparison between the f3 parameters from Data Sets I and II and the amplification

obtained from a linear one dimension shear wave model (Scnabel et al., 1972) is shown in

Figure 3.14. The amplification from the shear wave model is close to the predicted f3

values from Data Set IT and considerably different from the f3 values from Data Set I. Only

two amplification values are available for this site from the recorded data, thus it is difficult

to prove which f3 predicts the behavior better.

In order to compare the amplification factors from Data Sets I and II to larger data

samples, average shear wave velocity ratios S 1 were computed for the soil groups Qpa, Qha

and Qaf/Qhbm. Then for each soil group plots of ground motion amplification were

developed at various depths.

Figures 3.15 to 3.17 show the amplification values f3a and f3v as functions of peak

ground acceleration and velocity for Qpa soils. The average shear wave velocity ratio for

Qpa sites was found to be 0.24 and the three figures correspond to depths to bedrock of

25m, 60m and 150m, respectively. The agreement between the amplification from the data

and the two equations is considerably better for the shallower deposits than for the deeper

deposits. The agreement between the predicted amplification and the data is better for peak

ground velocity than for peak ground acceleration for all three soil depths.

35

Figures 3.18 to 3.20 show the amplification of peak ground acceleration and velocity

for Qha sites. The predicted amplifications are for average shear wave velocity ratio, Sr. of

0.35 and depths to bedrock, dp, of 50m, 125m, and 220m. The same observations hold

for these figures as for the Qpa sites. The data for the Qha sites, however, appear to be

more scattered. The agreement between the predicted values and the data are particularly

poor for the average depths of 125m and 220m and peak ground acceleration amplification.

There is also a large difference in the predicted peak groun.d accelerations by the two

empirical formulas obtained from Data Sets I and II.

The amplification factors obtained for the San Francisco Bay mud deposits are shown

in Figures 3.21 and 3.22. In these figures the average S1 value is computed to be 0.58 and

the two figures correspond to depths to bedrock dp of 95m and 185m. For the deeper sites

the data are very few, thus it is dificult to draw conclusions. For the shallower sites, the

agreement between the predicted amplifications and the data appear to be quite good. The

amplification factor obtained form Data Set II appears to fit the data in Figure 3.21 better

than the relationship obtained form Data Set I.

36

c Bay 110d sites -N 500 0 Alluv1tlll sites ~ 10.1

"' 400 ~ ........ e 10 I - 300 b 8-10-v201 0

= 0 00 a c

0 0 t .,..

200 1;; zo I 0 .. , ~ \ Gl

20- .. 0, 0 c .....

II \, 0 u u "' 100 0

0 ~ t ra l 20 ..... !01

0 100 zoo 300 400 500 600 700

depth to bedrock (feet)

Figure 3.5 Variation of peak ground acceleration amplification with depth to bedrock, dp,

for Data Set I.

21t I

~ co 50 10 ..... 2G b - s'

~ 40 Oo 10

"' 0 0 4 tJ ........ 30 e Cl 0 - 211-IJO'

>. 20 \ 0 0 ~ c .,..

u 8 0 0 - \ .\ Gl > 20 20-30

c Bay IIUd sites ~ .., 10 o Alluviu• sites & c

0 100 200 300 400 500 600 700 depth to bedrock (feet)

Figure 3.6 Variation of peak ground velocity amplification with depth to bedrock, dp, for

Data Set I.

37

-!!! c

..2 -• ...

.! • 0 0 • , c ~ 0 ...

CJ .¥ • • D.

-0 • • -E 0 ->-= 0 0 "i > , c ~ 0 ...

CJ

.¥ • • D.

0.6

• • QB 0.5

• • Ola

• Oaf/Qhbm

0.4 • • II • -0.3 •

• • • • • • ..

• .. 0.2 •

• I I I II 0.1

•• • • • • • • • • • • • • • 0.0

0 100 200 300

Depth to Bedrock (m)

Figure 3. 7 Variation of peak ground acceleration amplification with depth to bedrock, dp,

for Data Set II.

70

• • QB 60 • Ola

50 • Oaf/Qhbm • .. • • • 40 • • • • • • • • • 30 •• • • • • •

• 20 • I I I • • • • I 10 • • I 0

0 100 200 300

Depth to Bedrock (m)

Figure 3.8 Variation of peak ground velocity amplification with depth to bedrock, dp, for

Data Set II.

38

c .2 -• 0 ~

c 0 -• 0 ;:

A. E •

Q.

E • > c:l Q.

10

• .. • • •• • II

' II • • II

• •• II jIll II •

'iP ....... rt. • • I

.1 1 0 100 1000

distance to rupture zone (km)

Figure 3.9 Variation of peak ground acceleration amplification with distance to the

rupture zone for Data Set II.

10

.1 10

• •

I I • •• ••• . -··. I I~ • • •

• I I .. II

I • • • •

• •

100 1000

distance to rupture zone (km)

Figure 3.10 Variation of peak ground velocity amplification with distance to the rupture

zone for Data Set II.

39

Table 3.4 Comparison of recorded and computed peak ground acceleration amplification

factors for soil type Qpa

:::::::::~:::::::::~:::::::::::sa.m;me.:tm~:::::::::::::::::::::::::::: STATION NAME StVALUE DPVALUE CHANNEL ROCKPGA RECORDED SUGITO

and NUMBER (m/sec) (m) AZIMUTH (cm/sec"2) AMP BETA

Gilroy - Gavilan College 0.123 25.9 90 433.602 .839 .500

47006 0 426.735 .733 .506

Coyote Lake Dam downstream 0.174 20.0 90 316.863 .499 .661

57504 0 311.958 .482 .667

Anderson Dam downstream 0.241 30.0 90 276.642 .822 .705

1652 0 272.718 .937 .711

Gilroy #7 - Manu:lli Ranch 0.193 30.0 90 234.459 1.339 .754

57425 0 230.535 .890 .761

Halls Valley 0.241 30.0 90 104.967 1.046 1.200

57191 0 163.827 .782 .952

Fremont· Mission San Jose 0.309 170.0 90 S0.442 1.249 1.482

57064 0 125.568 .935 1.072

Fremont· Emerson.Court 0.309 170.0 90 78.480 2.438 1.508

1686 0 121.644 1.165 1.098

APEEL #2E • Jolm Muir School 0.314 60.0 90 154.017 .883 .986

1121 0 78.480 2.117 1.475

Hayward BART Paricing Lot 0.241 60.0 90 147.150 .817 .972

58498 0 74.556 2.956 1.447

Richmond City Hall Paricing Lot 0.200 100.0 90 73.575 1.605 1.442

58505 J 0 37.278 3.176 2.113

40

MODIFIED

BETA

.716

.717

.514

.515

.801

.802

.852

.854

.885

.847

1.264

1.210

1.267

1.213

1.747

1.995

1.777

2.029

2.016

2.278

Table 3.5 Comparison of recorded and computed peak ground acceleration amplification

factors for soil type Qha _.:: : .. ::;:···::: : :

.=:=:=::. ... :. ... ::

STATION NAME StVALUE DPVALUE CHANNEL ROCKPGA RECORDED SUGITO

AND NUMBER (m/sec) (m) AZIMUTH (cm/secA2) AMP BETA

Capitola 0.309 50.0 90 450.279 .868 .472

47125 0 485.595 .953 .443

Gilroy #2 • HWY 101 M01el 0.284 140.2 90 410.058 .772 .376

47380 0 403.191 .854 .383

Gilroy #3 • Sewage TreaL PIIIIU 0.248 150.0 90 367.875 .984 .407

47381 0 361.989 1.469 .414

Gilroy 1#4 ·San Ysidro School 0.391 140.0 90 332.559 .631 .497

57382 0 327.654 !.246 .504

SuMyvale ·Colton Ave. 0.367 181.7 90 145.188 1.449 .981

1695 0 226.611 ~48 .669

Agnew 0.367 261.6 90 161.865 .972 .868

57066 0 293.319 .556 .482

Hollister City Hall 0.440 55.8 90 195.219 1.291 .918

1575 0 192.276 1.128 .927

Hollister • SOUib and Pine 0.440 56.0 90 195.219 .893 .918

47524 0 192.276 .1.883 _.227

Salinas 0.309 50.0 90 90.252 1.270 _1.356

47179 0 78.480 1-967 1.463

Hollister Airpon 0.440 60.0 90 188.352 1.460 .932

1656 0 185.409 1.384 .942

Olema 0.241 50.0 90 43.164 2.308 1.897

68003 0 41.202 3.836 1.940

41

MODIFIED

BETA

1.216

1.198

L300

1.303

1.117

1.119

1.658

1.664

1.279

1.21"

.501

.528

1.479

1.483

1.485

1.489

1.624

1.663

1.606

1.611

1.899

1.914

Table 3.6 Comparison of recorded and computed peak ground acceleration amplification

factors for soil type Qaf/Qhbm

STATION NAME

AND NUMBER

APEEL #2 - Redwood City

APEEL #1 - Redwood Shores

S~ Francisco Int. Airpon

Emeryville

Treasure Island

St VALUE DP VALUE CHANNEL ROCK PGA RECORDED

(m/sec) (m) AZIMUTH (cm/sec"2) AMP

0.677 91.4 84.366 2.811

0 154.017 1.603

0.765 201.2 90 85.347 3.274

0 154.017

0.419 164.6 90 3.789

88.290

0.359 100.0 90 4 2.762

0 47.088 3.928

0.677 86.7 63.765

0 27 3.618

42

SUGITO MODIFIED

BETA BETA

1.165 2.308

1.774 3.727

1.117 3.091

1.502

1.478 1.936

1.385 2.460

2.866

2.043 2.699

3.305

Table 3.7 Comparison of recorded and computed peak grou~d velocity amplification

factors for soil type Qpa

~~~~frill -STATION NAME StVALUE DPVALUE CHANNEL ROCKPGV RECORDED SUGITO

and NUMBER (m/sec) (m) AZIMUTH (em/sec) AMP BETA

Gilroy - Gavilan College 0.123 25.9 90 33.800 .754 .715

47006 0 31.900 .798 .742

Coyote Lake Dam downstream 0.174 20.0 90 23.700 .S1Q_ .925

57504 0 22.370 .73~ .954

Anderson Dam downstream 0.241 30.0 90 20.460 1.049 1.049

1652 0 19.310 .981 1.083

Gilroy #7 - Mantelli Ranch 0.193 30.0 90 17.190 .948 1.112

57425 0 16.220 1.023 1.146

Halls Valley 0.241 30.0 90 12.090 1.134 1.377

57191 0 11.450 1.092 1.413

Fremont - Mission San Jose 0.309 170.0 90 9.190 .912 l.S60_

57064 0 8.700 1.172 1.926

Fremont - Emerson Coust 0.309 170.0 90 8.910 1.211 1.898_

1686 0 8.440 1.216 1.964

APEEL #2E • Jolm Muir School 0.314 60.0 90 20.980 .620 1.076

1121 0 6.880 1.987 2.025

Hayward BART Parlcing Lot 0.241 60.0 90 19.980 .455 1.053

58498 0 6.550 2.003 1.954

Richmood City Hall Paricing Lot 0.200 100.0 90 10.090 1.505 1.570

58505 0 3.310 4.574 2.791

43

MODIFIED

BETA

.611

.625

.737

.754

1.027

1.056

.984

1.010

1.319

1.353

1.869

1.921

1.898

1.949

1.336

2.477

1.161

2.094

1.574

2.851

Table 3.8 Comparison of recorded and computed peak ground velocity amplification

factors for soil type Qha

- .

STATION NAME St VALUE DPVALUE CHANNEL ROCKPGV RECORDED SUGITO

AND NUMBER (mlsec) (m) AZIMUTH (em/sec) AMP BETA

Capitola 0.309 50.0 90 24.290 1.264 .980

47125 0 :!4.290 1.486 .980

GWroy#2-HVVYI01Mo~ 0.284 140.2 90 31.630 1.239 .7f.'17

47380 0 29.850 LU_5_ .747

Gilroy lf.3 -Sewage Treat. Plaru 0.248 150.0 90 27.960 1.566 .766

47381 0 26.390 1.307 .808

Gilroy #4- San Ysidro School 0.391 140.0 90 24.990 J.529 .947

57382 0 :!3.580 1.658 .. J~

Sunnyvale - Colton Ave. 0.367 181.7 90 16.970 2.009 1.254

1695 0 16.f.'170 2.080 1.308

Agnew 0.367 261.6 90 26.570 .685 .830.

57066 0 34.380 ~2 .634

Hollister City Hall 0.440 55.8 90 14.200 ,2,.723 1.508

1575 0 13.400 3.282 1.559

HolliSier - South and Pine 0.440 56.0 90 14.200 _2.177 1.510

47524 0 13.400 4.687 1.562

Salinas 0.309 50.0 90 12.430 1.236 1.464

47179 0 t0.710 .956 1.586

Hollister Airpon 0.440 60.0 90 13.700 2.510 1.54!_

1656 0 12.930 3.344 1.594

Olema 0.241 50.0 90 3.440 1.919 1.697

68003 0 5.!10 . 3.617 2.143

44

MODIFIED

BETA

1.200

1.200

1.001

1.031

.993

1.023

1.375

1.412

1.488

1.524

1.134

1.036

2.232

2.306

2.232

2.306

1.732

1.879

2.271

2.345

1.778

2.304

Table 3.9 Comparison of recorded and computed peak ground velocity amplification

factors fa,r soil type Qaf/Qhbm

STATION NAME

AND NUMBER

APEEL #2 - Redwood City

APEEL #1 -Redwood Shores

San Francisco Int. Airport

Emeryville

Treasure Island

St VALUE DP VALUE CHANNEL ROCK PGV RECORDED

(m/sec) (m) AZIMUTH (em/sec) AMP

0.677 91.4 .600 3.

0 17.600 2.068

0.765 201.2 90 13.600 3.338

0 17.600 1.807

0.419 164.6 7

7.490 3.539

0.359 100.0 90 12.530•

0

0.677 86.7 2.359

0 4.440 3.513

45

SUGITO MODIFIED

BETA BETA

3.256

1.581 3.037

1.628 2.274

2.225 2.278

2.295 2.326

1.548 1.921

2.890

1.822

c _g -• ~ ~

Q. E • • 0 c:L.

c .!! • u -Q. E • > 0 c:L.

4

3 -

2

.

0 0.1

5

4

3

2

0 0.1

B •

G

•

0

8

• ••

• g a

0.2

(a)

II

0

G

a D

0.2

(b)

• measured data

• Sugito Beta 0 Modified Beta •

• 0 I 0 0

• • • • !a I '

0.3

St veluo

B measured data

• Sugito Beta 0 Modified Beta

' •• 0

• so g • mo Iii

1!1

0.3

St veluo

Figure 3.11 Comparison of recorded and predicted amplification; (a) peak ground

acceleration amplification; (b) peak ground velocity amplification (Qpa sites)

46

0.4

0.4

c .!! -• .£ = a. E c c CJ G.

c 0 :: • u :;:::

G. E c

4

3 -

2 -

1

0 0.2

5

4

3

2

0 0.2

•

• e

• B

•

•

0 • B. • 0 •

e 0 I • II I

• • 0.3

I!

• 0

0 0 I

I 8 • • 0.3

• measured data

• Sugito Beta 0 Modified Beta

II 0

I • 8 • • I I

I I

I

0.4

St velu•

(a)

measured data • Sugito Beta

Modified Beta

II

II 1!1

I B

B • g

I •

0.4

St velu•

(b)

Figure 3.12 Comparison of recorded and predicted amplification; (a) peak ground

acceleration amplification; (b) peak ground velocity amplification (Qha sites)

47

0.5

0.5

c 0 -• u E A. E 4(

> 0 G.

5

0 B measured data

• Sugito Beta 4 0 Modified Beta

B II El g Iii • e •

0 3 • • 0 0 1!1 0

2 • 0 • a

• • • 1

0.3 0.4 0.5 0.6 0.7

St Y81Ue

(b)

Figure 3.13 Comparison of recorded and predicted amplification; (a) peak ground

acceleration amplification; (b) peak ground velocity amplification (Qaf/Qhbm sites)

48

0.8

c .!! ~ • .. u u

<C

0 u • ... • .. Ill

~ .. 0 'i >

0 u • ... • 'i Ill

10~------------------------------------------------------~

Sugito Beta

Modified Beta

SHAKE program

• measured data

-----................

.1+-------------~~----~~--~------~--~----~--~~~ 10 100 1000

PGA on rock (crn/eec:•2)

(a)

" 10~------------------------------------------------------~

----·······-·~·~·~·~·~-~-~-~-~--=.:.:.:.::.::-.7:::.::.:::-::.-

SugrtoBeta

Modified Beta

SHAKE program

• measured data

10

PQV on rock (crnl•ec)

(b)

100

Figure 3.14 Comparison of amplification factors f3 using Data Set I and II and a one

linear shear wave model. (a) Peak acceleration amplification (b) Peak velocity amplification

49

c 0

~ u = a. E oct

c 0 :;: CIS ~ = Q. E oct

> CJ D..

PGA AMPLIFICATION vs. PGA ON ROCK Qpa SITES (St:.24, dp=25 m.)

10------------------------------------~-------------,

10

sugito beta -------- modified beta

• measured amplification

• • --------------------------------------

100

PGA on rock (cm/secA2) (a) .

PGV AMPLIFICATION vs. PGV ON ROCK Qpa SITES (St:.24, dp:25 m.)

1000

10----------------------------------------------------------,

sugito beta

-------- modified beta

• measured amplification

PGV on rock (em/sec) (b)

Figure 3.15 Comparison of amplifications to observed data for Qpa sites with average

depth to bedrock at 25m. (a) Peak ground acceleration amplifications (b) Peak ground

velocity amplifications

50

c .2 ~ u = a. E <(

c 0 ::; cu u ~ c. E <(

> CJ a..

PGA AMPLIFICATION vs. PGA ON ROCK Qpa SITES (St:.24, dp:60 m.)

10----------------------------------------------~

• _____________ .... ----------------------------

sugito beta -------- modified beta

• measured amplifiCation

.1~--------~----~~~~~~------------~----~~~~~~ 1 0 1 00 1000

PGA on rock (cm/sec"2)

(a)

PGV AMPLIFICATION vs. PGV ON ROCK Qpa SITES (St:.24, dp:60 m.)

10~--------------------------------------------,

.................. ..................... .......... ......

sugito beta

-------- modified beta • measured amplification

• •

.14----------~---~----~~~~~r---------~------~~~~~~~100 10

PGV on rock (em/sec)

(b)

Figure 3.16 Comparison of amplifications to observed data for Qpa sites with average

depth to bedrock at 60m. (a) Peak ground acceleration amplifications (b) Peak ground

velocity amplifications

51

c 0 ;: ~ (,)

:!:: a. E ct

c .E -('II (,)

:E 0.. E ~

PGA AMPLIFICATION vs. PGA ON ROCK Qpa SITES (St=.24, dp:150 m.)

10-----------------------------------------------------.

1 0

• • ------------------

sugito beta -------- modified beta

• rt;~easured amplification

100

PGA on rock (cm/sec"2)

(a)

PGV AMPLIFICATION vs. PGV ON ROCK Qpa SITES (St:.24, dp:150 m.)

1000

10~-----------------------------------------------------,

•

sugito beta -------- modified beta

• measured amplification

10 100

PGV on rock (em/sec)

(b)

Figure 3.17 Comparison of amplifications to observed data for Qpa sites with average

depth to bedrock at 150m. (a) Peak ground acceleration amplifications (b) Peak ground

velocity amplifications

52

c .2 -a:l (J = c. E <

c .2 -a:l .2 ::: Q. E <

> e, D.

PGA AMPLIFICATION vs. PGA ON ROCK Qha SITES (St:.35, dp:50 m.)

10------------------------------------------------------~

10

• --------- . ------------. ------------. -

••

sugito beta -------- modified beta

• measured amplification

100

PGA on rock (cm/sec"2)

(a)

PGV AMPLIFICATION vs. PGV ON ROCK Qha SITES (St:.35, dp:50 m.)

1000

10,---------------------------------------------------------·

--- ... ------- ..... __ . -----... . -- . ----...... __ . ... _

.............. _

•

• sugito beta

-----· ...... _ ...

-------- modified beta • measured amplification

1 0

PGV on rock (em/sec)

(b)

-- .......

100

Figure 3.18 Comparison of amplifications to observed data for Qha sites with average

depth to bedrock at 50m. (a) 'Peak ground acceleration amplifications (b) Peak ground

velocity amplifications

53

c. E ct

c .2 fti (,)

= a E ct

> CJ D.

PGA. AMPLIFICATION vs. PGA ON ROCK Qha SITES (St=.35, dp:125 m.)

10~--------------------------------------------~

------------------------

sugito beta -------- modified beta

• measured amplification

-......... . --------------. • •• •

PGA on rock (cm/secA2)

(a)

PGV AMPLIFICATION vs. PGV ON ROCK Qha SITES (St:.35, dp:125 m.)

10,----------------------------------------------------------, ------.. ---........ . ----------..... _ . ------- ....... ---.... ~ . ..... .._ ...... ..... __ _

sugito beta

-------- modified beta • measured amplification

10 100 PGV on rock (em/sec)

(b)

Figure 3.19 Comparison of amplifications to observed data for Qha sites with average

depth to bedrock at 125m. (a) Peak ground acceleration amplifications (b) Peak ground

velocity amplifications

54

c .2 -as g ::: a. E ct

> CJ D.

PGA AMPLIFICATION vs. PGA ON ROCK Qha SITES (St:.35, dp:220 m.)

10,-----------------------------------------------------,

1 0

•

sugito beta -------- modified beta

• measured amplification

100

PGA on rock (cm/sec"'2)

(a)

PGV AMPLIFICATION vs. PGV ON ROCK Qha SITES (St:.35, dp:220 m.)

1000

10-------------------------------------------------------,

----------------------.............. .. sugito beta

-------- modified beta • measured amplification

1 0 100 PGV on rock (em/sec)

(b)

Figure 3.20 Comparison of amplifications to observed data for Qha sites with average

depth to bedrock at 220m. (a) Peak ground acceleration amplifications (b) Peak ground

velocity amplifications

55

.,

c .2 -('IS u

= Q. E <

c 0 (; .2 = a. E <

> ~ D..

PGA AMPLIFICATION vs. PGA ON ROCK Qaf/Qhbm SITES (St:.58, dp:95 m.)

10,--------------------------------------------------,

10

... ._ . --------------.... . ----------..... __ _ ------------•

sugito beta ·-·-···- modified beta

• measured amplification

100

PGA on rock (cm/sec"2)

(a)

PGV AMPLIFICATION vs. PGV ON ROCK Qaf/Qhbm SITES (St:.58, dp:95 m.)

1000

10,----------------------------------------------------.

------------------~- -------.......... .. ......... _ .. __ ----................ ___ _

·-·

sugito beta

-··----- modified beta • measured amplification

1 0 100

PGV on rock (em/sec)

(b)

Figure 3.21 Comparison of amplifications to observed data for Qaf/Qhbm sites with

average depth to bedrock at 95m. (a) Peak ground acceleration amplifications (b) Peak

ground velocity amplifications

56

c 0 ;: CIS .2 = a. E 4(

c .2 -CIS (,)

= a. E 4(

> 0 D.

PGA AMPLIFICATION vs. PGA ON ROCK Qaf/Qhbm SITES (St:.58, dp:185 m.)

10~--------------------------~----------------------~

1

1 0

------- I --------------

sugito beta modified beta

• measured amplification

100

--------------•

PGA on rock {cm/sec"2)

(a)

---------

PGV AMPLIFICATION vs. PGV ON ROCK Qaf/Qhbm SITES (St:.58, dp:185 m.)

1000 .

10,--------------------------------------------------,

sugito 185

-------- mod 185 • amp 185

1 0 100

PGV on rock (em/sec)

(b)

Figure 3.22 Comparison of amplifications to observed data for Qaf/Qhbm sites with

average depth to bedrock at 185m. (a) Peak ground acceleration amplifications (b) Peak

ground velocity amplifications

57

3.3 Sensitivity Analysis

The sensitivity of the amplification factors to variations in the soil softness parameter Sr

and the depth to bedrock dp is investigated next. Figures 3.23 and 3.24 show the values of the amplification factors f3a and f3v for several combinations of local soil parameters. Figure

3.23 shows the results of the sensitivity analyses of the f3a and f3v values obtained using

Data Set I. Figure 3.24 show the sensitivity analysis for the f3a and f3v values obtained

using Data Set II.

Figures 3.23a and 3.24a illustrate the average trend of the amplification characteristics

with increasing depth to bedrock. The amplification factors from Data Set I appear, to be

relatively insensitive the variations with depth to bedrock. Greater variability is observed

with the amplification factors obtained from Data Set II.

The variability of the amplification factors with increased softness of the soil is shown

in Figures 3.23b and 3.24b. Data Set II amplification factors appear to be less variable

than the parameters obtained from Data Set I. When both Sr and dp are varied, the

amplification factors from Data Set II appear very unstable. Figures 3.23c and 3.24c show

the variability with changes in both Sr and dp.

From all the plots in Figures 3.23 and 3.24 it can be observed that the amplification

increases with the softer soil. However, the amplification for soft shallow layers is smaller

than that for deeper soft layer soils at small values of acceleration and velocities. This trend

is reversed at the higher acceleration and velocity values. These observations pertains

primarily to the amplification factors obtained from Data Set I. Although such reversal in

the relative amplification of ground motions appears to be suggested in the plots for Data

Set II, it is not explicit in Figures 3.24. The cross-over point in these figures appears to be

at very large acceleration values.

58

e CD. ... 0 u

Q c-------------------------

~ b c 0

'iii t ;> c 0 u

,. E

CD. .... 0 ti

--- 0~• 50.0 n, Slo 0.15 ---- 0~• I·OO.on, Sf• Q,ll

-·-·- o~. 16o.on. sr. p.11 0 L-~~~~~·~~-L-~-W~

10· tO' (cmJsecl) to' peak ace. on rock surface

0 ,-----------------------~

<0 • ..... 0

c 0

'iii .... u ;;. c 0 u

--- o~. so.o n. Sl• o.75 ---- 0~• IOO,on, Sf• 0.75

-·-·- DP• ISO.Ill. 51• 0.75

0 L-~~~~~-~~-~~~~ Ill' 1 o• (em/sec) 1 o•

peak vel. on rock surface

e CD. ... 0 ti .!S

0 ~~------------------------~ e

c::..

0 u ~

..

~ ~------------------------E ~ ~

~

c 0 0 g 0

·u; ·u;

~ c 8 OP• IOO.on. Sl• 0.50

OP• IOO.on. 51• 0.75 0~• IOO.on. 51• 1.00

0 L-___.._ .......... _._. ........... o.L.. l_..__~ .......... u..u.. 10' 10' (cm/secl) IO'

peak ace. on rock surface

0 c-----------------------~

--- OP• IOO.on. Sl• 0.50

DP• 100.1r1. 51• O.IS

OP• IOD.D1. 51• 1.00

0 ~,.___.___. ......... .L.L~·.I--1--J'-:--L-.':..L.I..J..I. ,J

10' to' (cmlsec) Ill'

peak vel. on rock surface

... ~ c 0 u

,. 8

CD. ... 0 u

--- OP• 30.0 M, 51• 0.50 D•• 120.U1. ST• 0,/S

- • -·- OP• 2oO.D1. S!o 1.00 o._ __ ,__._._._......_ ........ __ __.___,__._...ww.u.J

10' IO' (cm/sec2) 10' peak ace. on rock surface

0 r-----------------------~ ....

~ 0

c 0 ·u; t:; ;> c 0 u

OP• ]0.0 n, Sf• O.SO

UP• fl0.lr1. Sl• 0.1~ j' OP• 2oO.D1. Slo 1.00

:! --"' I I I I I 1.d_~u '-I ......I. I

tiJ' 1u· (cmlsec) 10'

peak vel. on rock surface

Figure 3.23 Sensitivity analysis of Data Set I ground motion amplification factors to the parameters 51 and dp.

59

> ~

CD -G)

c:D

MODIFIED BETA vs PGA ON ROCK

10~-----------------------------------------------------,

-·­

10

·-·­................ ·-............. .. ............. ..... _

100

dp = 1 00 m, St = .50 dp = 1 oo m, St = . 75 dp =100m, St = 1.00

·-·- ...... _ ...... ·-..... ·-·-·-­................

PGA on rock (cm/secA2)

MODIFIED BETA vs PGV ON ROCK

1000

10~-----------------------------------------------------,

--------. -·-·-· -·- .-::::-............. _ ·-·-·-·-·-.................... ·-·-·--·-·-·-·-·--·------------

10

--- .......... -----dp = 1 oo m., St = .25 dp = 1 00 m., St = .50 dp = 100m., St = .75

PGV on rock (em/sec)

100

Figure 3.24a Sensitivity analysis of Data Set II ground motion amplification factors to

the parameters S1 and dp.

60

c( CJ a.

CIS -Gl m ., Gl

:;:: ., 0

:E

> CJ a.

CIS -Gl m ., .! = ., 0 :e

MODIFIED BETA vs PGA ON ROCK

10,----------------------------------------------------, ····-·-·-·-· ··-·-·-

1 0

10

·-·-·--..... ---- ·-·-· ----.. -·-.......... .

--

----- .__ .... -- -·-·-·-·-----....... ·-·--..... -------- ·-·-.. --

dp = 30 m, St = .50

dp =120m, St = .75 dp =240m, St = 1.00

100

PGA on rock (cm/secA2)

MODIFIED BETA vs PGV ON ROCK

---- .............. ----__

............ _ -------........ ~ ·-·-·--·--·- .... _ ....

--.

-···-·····-···-·····-·-·-··· ·-·- --..... ---........._

1 0

dp = 30 m., St = .25

dp =120m., St =.so dp =240m., St = .75

PGV on rock (em/sec)

1000

100

Figure 3.24b Sensitivity analysis of Data Set II ground motion amplification factors to

the parameters Stand dp.

61

CIS -CD m "C

CD :;: "C 0

:iE

> <.:J 0.

CIS -CD m "C

CD :;: "C 0

:iE

MODIFIED BET A vs PGA ON ROCK

10-------------------------------------------------------,

1 0

-·-·-·-·-·-· -·-·----- ·--·-·-·--........ ----------- ·-·-·-·-·-·-·---.-----.. ·-·-·--... ------ .... _--·-- .............. . -....... -- ...... _.

dp = 50 m, St = . 75 dp =100m, St= .75 dp =150m, St = .75

100

PGA on rock (cm/sec"2)

· MODIFIED BET A vs PGV ON ROCK

1000

10~-----------------------------------------------------,

.................. ...... ............... ....., ...... ... ... ·-.... ', .............. .....,

dp = 50 m., St = .50 dp =100m., St =.50

dp =150m., St =.50

-........... ....., ·-........ ,, ·--. ...... ....

1 0

-......... ', .................... ....................... ....... ::: ..... ....

PGV on rock (em/sec)

.... ....

100

Figure 3.24c Sensitivity analysis of Data Set II ground motion amplification factors to

the parameters S1 and dp.

62

CHAPTER 4

SUMMARY AND CONCLUSIONS

Earthquake ground motion amplification factors are developed based on a simple

nonlinear relationship between peak acceleration and velocity at rock surface and peak acceleration and velocity at soil surface. The amplification factor {3 is considered to be a

function of the depth to bedrock which attempts to characterze the low frequency

characteristics of the ground motion at a site. In addition, a soil parameter St is introduced

to reflect the softness of the surface layer at the site.

Amplification factors for peak ground acceleration and velocity are developed for the

San Francisco Bay region. For that purpose the strong ground motion records from the

October 17, 1989 Lorna Prieta earthquake are utilized. The main difficulty in site

amplification studies is the lack of sufficient soil data at strong motion stations. Soil

parameter data was obtained late in the study. These data included information on inferred

shear wave velocities of the top 30 meters of soil at the strong ground motion recording

sites. The strong ground motion data was discriminated according to the soil classifications

provided in the Fumal (1991) study and based on geologic maps by Schlocker (1968) and

Gibbs et al.(1975, 1976, 1977).

Based on the analysis of the strong ground motion and soil parameter data the

following observations are made:

*

*

Strong ground motion amplification appears to be nonlinearly decreasing with

ip.creasing peak ground acceleration and velocity.

No specific trend could be observed between the ground motion amplification

factors and the depth to bedrock.

* Peak ground acceleration and velocity amplification factors increase with increasing

distance to the rupture zone.

* The amplification factors appear to be very sensitive to variations in the depth to

bedrock and less sensitive to the soil parameter Sr.

Two sets of strong ground motion data were discriminated. The first set, labeled as

Data Set I, includes information and shear wave velocity and depth to bedrock which are

63

based on direct investigations of the local recording sites. The second set of data, referred

to as Data Set II, includes many of the inferred shear wave velocities (Fumal, 1991). The

first set of data contains a total of 24 recording stations while the second set contains 52

recording stations. The amplification factors computed from the two stes of data differ

considerably. The amplification factors from Data Set I are less sensitive to variations in the

parameter Stand dp. However, the amplification parameters from Data Set II are found to

predict the recoreded amplifications more closely.

The amplification factors developed in this study are particularly useful for

microzonation, regional damage estimation, and rehabilitation decision purposes. It should

be recognized, however, that there is a considerable amount of uncertainty in any predicted

ground motions due to the following reasons:

*

*

*

*

*

*

The shear wave velocities for many of the stations are inferred rather than measured.

The thickness of each layer and the variation of shear wave velocity with depth are

not considered.

The depth to bedrock is estimated from contour maps and interpolated between

contours.

The strong ground motion data is from one earthquake event only and represents

only one magnitude. Thus the effect of the size of the earthquake event could not be

investigated.

The strong ground motion stations are scattered and for certain soil types only a few

recordings are available.

The form of the nonlinear equation of ground motion amplification may need to be

modified to reflect the variations of shear wave velocities and densities of soil layers

with depth.

In order to include the above characteristics of soil amplification, more data are needed

on the soil parameters as well as strong ground motion recordings.

64

ACKNOWLEDGEMENTS

The authors would like to thank Dr. Roger Borcherdt of USGS, Menlo Park for many

useful discussions and for providing data and guidance on the soil parameters. We also

thank Dr. Tom Fumal for providing us with shear wave velocities for the Lorna Prieta

strong ground motion stations. Early on in the study, Will Holmes helped with the

accumulation and consolidation of bore-hole data for several locations in the San Francisco

Bay area. We sincerely thank William Holmes of Rutherford and Chekene for providing

the bore-hole data for the Stanford Campus. Additional bore-hole data was also made

available by Dames and Moore, and Geomatrix Consultants, both of San Francisco. Our

thanks for their willingness to help.

65

REFERENCES

Bonilla, M.G. (1964). Bedrock-Surface Map of the San Francisco South Quadrangle, California, Miscellaneous Field Studies Map MF-334, U.S.G.S.

Borcherdt, R. D. , Gibbs, J. F. and Fumal, T. E. (1979). Progress on Ground Motion Predictions for the San Francisco Bay Region, California, Geologic Survey Circular 807, U.S.G.S.

Borcherdt, R. D. (1990). Influence of local Geology in the San Francisco Bay Region, California on ground motions generated by the Lorna Prieta earthquake of October 17, 1989, Proceedings of the International Symposium on Safety of Urban Life and Facilities, Tokyo, Japan, November 1-2.

Fumal, T. E. (1991). A compilation of the geology and measured and estimated shear-wave velocity profiles at strong-motion stations that recorded the Lorna Prieta, California, Earthquake, U.S. Geol. Surv. Open-File Report 91-311.

Gibbs, J. F., Fumal, T. E., and Borcherdt, R. D. (1975). In situ measurements of seismic velocities at twelve locations in the San Francisco Bay region, U.S. Geol. Surv. Open-File Rep. 75-564, 87 p.

Gibbs, J. F., Fumal, T. E., and Borcherdt, R. D. (1976). In situ measurements of seismic velocities in the San Francisco Bay region- Part II, U.S. Geol. Surv. Open-File Rep.76-731, 145 p.

Gibbs, J. F., Fumal, T. E., Borcherdt, R. D. and Roth, E. F. (1977). In situ measurements of seismic velocities in the San Francisco Bay region- Part III, U.S. Geol. Surv. Open-File Rep.77-850, 143 p.

ldriss, I. M. (1990). Response of soft soil sites during earthquakes, Proceedings from the Memorial Symposium to Honor Professor Harry Bolton Seed, Berkeley, California.

Journel, A. and Huijbregts, C. (1978). Mining Geostatistics, Academic Press, London, UK

Journel, A. G. (1989). Fundamentals of Geostatistics in Five Lessons, American Geophysical Union,

Joyner, W. B. and Boore, D. M. (1988). Measurement, characterization, and prediction of strong ground motion, Proceedings the ASCE Conf on Earthquake Engineering and Soil'Dynamics II­Recent Advances in Ground Motion Evaluation, Park City Utah, pp. 43-103.

Kahle, J. E. and Goldman, H. B. (1966). Contours on the Top of the Bedrock Underlying San Francisco Bay, Special Report 97, Plate 2, C.D.M.G.

Maley, R., Acosta, A., Ellis, F., Etheredge, E., Foote, L., Johnson, D., Porcella, R., Salsman, M. and Switzer, J. (1989). U.S. Geologic Survey Strong-Motion Records from the Northern California (Lorna Prieta) Earthquake of October 17, 1989, Geologic Survey Open-File Report 89-568, 85 p.

66

Mohraz, B. (1976). A study of earthquake response spectra for different geological conditions, Bull. Seism. Soc. Am. 66, No.3, pp. 915-935.

Phillips, W. S. and Aki, K. (1986). Site amplification of coda waves from local earthquakes in central California, Bull. Seism. Soc. Am., 76, pp. 827-648.

Rogers, A.M., Tinsley, J. C. and Borcherdt, R. D. (1985). Predicting relative ground response, in Evaluating Earthquake Hazards in the Los Angeles Region, ed. J. I. Ziony, U.S.G.S. Prof. Paper 1360, pp. 221-248.

S.A.S. (1983). SAS Institute Inc., Cary, Nonh Carolina, U.S.A.

Schlocker, J. (1961). Bedrock-Swface Map of the San Francisco North Quadrangle, California, Miscellaneous Field Studies Map MF-334, U.S.G.S.

Schlocker, J. (1968). The geology of the San Francisco Bay area and its significance in land-use planning, Assoc. Bay Area Gov'ts. Supp. Rept, 15-3, Regional Geology, 47.

Schnabel, R.B., Lysmer, J. and Seed, H.B. (1972). SHAKE - A Computer Program for Earthquake Response Analysis of Horizontally Layered Sites, EERC Report No. 72-12.

Seed, H. B. and Idriss, I. M. (1969). The influence of soil conditions on the ground motions during earthquakes, J. Soil Mech. Found. Engrg., 91,SM1, pp 93-137.

Seed, H. B., Ugas, C., and Lysmer, J. (1976). Site-dependent spectra for earthquake-resistance design, Bull. Seism. Soc. Am. 66, No. I, pp. 221-243.

Shakal, A., Huang, M., Reichle, M., Ventura, C., Cao, T., Sherburne, R., Savage, M., Dunagh, R., and Petersen, C. (1989). CSMIP strong-motion records from the Santa Cruz Mountains (Lorna Prieta), California earthquake of 17 October 1989, Calif. Strong Motion Instrumentation Prog. Rept. OSMS 89-06, 195 p.

Sugito, M. (1986). Earthquake motion prediction, microzonation, and buried pipe response for urban seismic damage assessment, 158 p.

Sugito, M., Goto, H., and Takayama,S. (1986). Conversion factor between earthquake motion on soil surface and rock surface with nonlinear soil amplification effect, Proc. 6th Japan Earthquake engineering Symposium, Tokyo, Japan.

Sugito and Kameda (1990). Nonlinear soil amplification model with verification by vertical strong motion array records, Proceedings of the Fourth U.S. National Conference on Earthquake Engineering, Palm Springs, CA.

Sugito, M., Kiremidjian, A. S. and Shah, H. C. (1991). A simple site-dependent ground motion estimation method including nonlinear amplification effect, Proceedings Fourth International Conference on Seismic Zonation, Stanford, CA, pp 221-228.

Trifunac, M. D. (1976). Preliminary analysis of the peaks of earthquake strong ground motion - · dependence of earthquake peaks on earthquake magnitude, epicentral distance and recording site conditions, Bull. Seism. Soc. Am., 66, pp. 189-219.

67

Kajima-CUREe Project

Soil Amplification Characteristics due to Local Site Effect

subjected to Stochastic Motion on Rock Surface

Final Report of Kajima Team

Masato Motosaka

Ariyoshi Yamada

Yasuhiro Ohtsuka

Masaki Kamata

Yasukazu Tsuji

August,1991

SOIL AMPLIFICATION CHARACTERISTICS DUE TO LOCAL SITE EFFECTS SUBJECTED TO STOCHASTIC MOTION ON ROCK SURFACE

PART-1

Sll\1PLIFIED NIETHODS FOR THE EVALUATION OF NONLINEAR SOIL AMPLIFICATIONS BASED ON RANDOM RESPONSE THEOREi\1

TABLE OF CONTENTS PAGE

SUMMARY

LIST 0 F FIGURES ii

LIST OF TABLES v

1. INTRODUCTION 1

2. REVIEW OF PREVIOUS PAPERS 2

2.1 NONLINEARITY OF THE SOIL AND STRONG OBSERVATION RECORDS 2

(1) EVIDENCE OF NONLINEARITY OF THE SOIL 2 (2) NONLINEAR SOIL AMPLIFICATION 7

2.2 METHODOLOGICAL REVIEW 10

3. METHODOLOGY FOR NONLINEAR RANDOM RESPONSE ANALYSIS 12

3.1 STRAIN-COlVIPATIBLE PIECEWISE LINEAR ANALYSIS 12 BASED ON MOMENT EQUATION

(1) OUTLINE OF METHOD 12 (2) BASIC EQUATIONS 12

3.2 STRAIN-COMPATIBLE PIECEWISE LINEAR ANALYSYS 18 BASED ON EVOLUTIONARY POWER SPECTRUM

(1) OUTLINE OF METHOD 18 (2) BASIC EQUATIONS 18

4. VERIFICATION OF ANALYSIS METHODS 25

5. APPLICATION ANALYSIS 32

5.1 OBJECTIVE SITES AND LOCAL SOIL DATA 32

5.2 STOCHASTIC 1\.WTION ON ROCK SURFACE FOR COMBINATIONS OF 35 MAGNITUDE AND EPICENTRAL DISTANCE

(1) STATISTIC ROCK MOTION 35 (2) SEISMOLOGICAL ROCK MOTION 36

5.3 RESULTS OF NONLINEAR SOIL AMPLIFICATIONS 38

(1) AMPLIFICATION CHARACTERISTICS DUE TO STATISTIC ROCK IviOTION 38 (2) AMPLIFICATION CHARACTERISTICS DUE TO SEISMOLOGICAL ROCK MOTION 51

6. CONCLUSIONS 70

·7. REFERENCES 71

SOIL AMPLIFICATION CHARACTERISTICS DUE TO LOCAL SI'l'E EFFECT SUBJECTED TO STOCHASTIC MOTION ON ROCK SURFACE

PART-1

SIMPLE METHODS FOR THE EVALUATION OF NONLINEAR SOIL AMPLIFICATIONS BASED ON RANDOM RESPONSE THEORE!v1

SUMl\IARY

OBJECTIVE Consideration of the nonlinear and hysteretic nature of the soil behavior under large am­

plitude cyclic loading is indispensable for realistic estimation of ground motion associated with local subsoil during a strong potential earthquake. Moreover, when estimating ground motion at a specific site, amplification factors are quite sensitive to input motions which are of an unpredictable nature. Therefore, no confidence can be achieved from the results of single de­terministic analysis using a recorded or artificially generated motion, especially in the higher frequency range. To avoid the expense and effort required multiple deterministic analyses, prob­abilistic (random response) methods have been developed. Such methods require a stochastic description of the earthquake excitation and directly provide probabilistic information on the ground motion, helping the designer to make rational decisions regarding the safety of the facility.

It is the purpose of this study to present a random response methodology to estimate the ground motion considering nonlinear amplification subjected to nonstationary stochastic motion on rock surface considering the source and the propagation path of a strong potential earthquake.

NONLINEAR RANDOM RESPONSE METHODOLOGY Two possibilities of simple nonlinear random response method based on piecewise linear

stochastic analysis, are described. One is the method based on moment equations, which requires the evaluation of a convolution integral. The covariance response which is necessary for the reliability estimation of the maximum response value can be obtained efficiently in computation by applying the recursive evaluation of a convolution integral. The other is the method based on an evolutionary power spectra. This leads not only the covariance response but also the evolutionary power spectra of the response considering nonlinear soil amplification. Numerical verification of these methods is performed for both linear and nonlinear cases through the simulation by the Monte Carlo Method.

RESULTS AND CONCLUSIONS As a practical application, the ground motions at a typical Holocene (alluYial) site in Japan

(Shiogama site) are investigated, where the evidence of the nonlinearity of the soil have been reported from the observation records due to the strong earthquakes with different combina­tions of magnitude and epicentral distance. The specified strain dependency of the subsoil is considered. Stochastic motions on rock surface of which the power spectral densities and the envelope time function are specified based on the previous studies by statistical regression approach and also seismological approach are used for strong earthquakes with the different intensity levels. Discussion is focused on the nonlinear soil amplifications for the stochastic input motions. It is found that the calculated nonlinear soil amplification characteristics are consistent with those of observation records.

Through the analyses it is concluded that the proposed nonlinear random response method­ology using the stochastic input motion tan be applicable to prediction of the ground motion of a soft soil deposit due to a strong potential earthquake. '

- 1 -

Fig. 2.1

Fig. 2.2

Fig. 2.3

Fig. 2.4

Fig. 2.5

Fig. 2.6

Fig. 2.7

Fig. 3.1

Fig. 3.2

Fig. 4.1

Fig. 4.2

Fig. 4.3

Fig. 4.4

Fig. 4.5

Fig. 4.6

Fig. 4.7

Fig. 4.8

Fig. 5.1

Fig. 5.2

LIST OF FIGURES

Boring Logs of Strong-Motion Sites (after Tokimatsu and Midorikawa,1987)

Variation of Spectra of Strong-Motion Records with the Amplitude Level (after Tokimatsu and Midorikawa,1987)

Strain Dependence of Shear Modulus Ratio (after Tokimatsu and Midorikawa,1987)

Peak Ground Acceleration vs. Shear Modulus Ratio (after Tokimatsu and Midorikawa,1987)

Peak Ground Velocity vs. Shear Modulus Ratio (after Tokimatsu and Midorikawa, 1987)

Values of Conversion Factor {3a and {3v for Typical Soil Conditions (after Sugio and Kameda,1990)

Variation of Ground Motion Amplification Ratio versus Rock Surface Ground Motion (after Sugito,Kiremidjian and Shah, 1991)

Conception of Piecewise Linear Stochastic Analysis

Flow Chart

Schematic of Structure (after Langley,1986)

Comparison of the results with those of the previous papers

Soil Profile and Dynamic model at Shiogama Site

Strain Dependency of Shear Moduli and Damping Factor of the soils at Shiogama Site

Power Spectral Density and Envelope Function of Stochastic Input Motion for the Verification of Analysis

Sample Input Waves for Monte Carlo Simulation

Examples of Hysteresis Loops

Verification of Analytical Method by Numerical Simulation

Evidence of Nonlinearity of the Soil at Shiogama Site (after Tokimatsu and Midorika.wa, 1987)

Observational Amplification Values at Shiogama Site

- 11 -

PAGE

3

4

5

6

6

8

8

22

23

25

26

27

27

28

29

30

31

33

34

Fig. 5.3

Fig. 5.4

Fig. 5.5

Fig. 5.6

Fig. 5.7

Fig. 5.8

Fig. 5.9

Fig. 5.10

Fig. 5.11

Fig. 5.12

Fig. 5.13

Fig. 5.14

Fig. 5.15

Spectral Intensities of Empirical Motion on Rock Surface by the Statistic Approach for Different Magnitudes (after Sugito and Kameda,198-5)

Spectral Intensities of Empirical Motion on Rock Surface by the Statistic Approach for Different Epicentral Distances (after Sugito and Kameda,1985)

RMS Inputs on Rock Surface for Different Magnitudes

RMS Inputs on Rock Surface for Different Epicentral Distances

Evolutionary Power Spectra on Rock Surface by the Statistical Apporoach

Comparison of RMS Responses on Ground Surface with those on Rock Surface for Different Magnitudes

Comparison of RMS Responses on Ground Surface with those on Rock Surface for Different Epicen~ral Distances

Evolutionary Power Spectra on Ground Surface considering Nonlinear Soil Amplification

Spectral Amplifications for Different Magnitudes and for Different Epicentral Distances

RMS Intensity Amplification for Different Magnitudes and Different Epicentral Distances

Comparison of Calculated Intensity Amplification with Observational Amplification Values at ShiogamaSite

Spectral Intensities on Rock Surface by Considering Source, Path and Deep Soil Amplification for Different Magnitudes

Spectral Intensities on Rock Surface by Considering Source, Path and Deep Soil Amplification for Different hypocentral Distances

40

41

42

43

44

45

46

47

48

49

50

53

54

Fig. 5.16 . Spectral Intensities on Rock Surface by Considering Source, Path and 55 Deep Soil Amplification for Different Azimuths

Fig. 5.17 RMS Inputs on Rock Surface for Different Magnitudes 56

Fig. 5.18 RMS Inputs on Rock Surface for Different Hypocentral Distances 57

Fig. 5.19 RMS Inputs on Rock Surface for Different Azimuths 58

Fig. 5.20 Evolutionary Power Spectra on Rock Surface by Considering Source, Path 59 and Deep Soil Amplification

Fig. 5.21 Comparison of RMS Responses on Ground Surface with those on Rock 60 Surface for Different Magnitudes

Fig. 5.22 Comparison of RMS Responses on Ground Surface with those on Rock 61 Surface for Different Epicentral Distances

Fig. 5.23 Comparison of RMS Responses on Ground Surface with those on Rock 62 Surface for Different Azimuths

- Ill-

Fig. 5.24

Fig. 5.25

Fig. 5.26

Fig. 5.27

Fig. 5.28

Fig. 5.29

Fig. 5.30

Fig. 5.31

Evolutionary Power Spectra on Ground Surface considering Nonlinear Soil Amplification

Spectral Amplifications for Different Magnitudes and for Different Hypocentral Distances

Spectral Amplifications for Different Azimuths

RMS Intensity Amplification for Different Magnitudes, Different Hypocentral Distances and Different Azimuths

Comparison of Calculated Intensity Amplification with Observational Amplification Values at Shiogama Site

Effect of Impedance rations on Intensity Amplification

Strain Dependency of Shear Moduli and Damping Factor

Effect of the different nonlinear characteristics on Intensity Amplification

-IV-

63

64

65

66

67

68

69

69

Table 2.1

Table 2.2

Table 2.3

Table 4.1

Table 5.1

Table .5.2

Table 5.3

Table 5.4

LIST OF TABLES

Site Characteristics for Analysis (after Tokimatsu and Midorikawa.,1987)

Strong-Motion Records Used (after Tokimatsu and Midorikawa,1987)

Rock Surface and Soil Surface Strong Motion Records Dataset

System Properties (after Langley, 1986)

List of Used Earthquake Records

List of Used Earthquake Records

List of Used Earthquake Records

Soil Parameters

- v-

PAGE

3

3

9

25

34

50

67

68

SOIL AMPLIFICATION CHARACTERISTICS DUE TO LOCAL SITE EFFECTS SUBJECTED TO STOCHASTIC MOTION ON ROCK SURFACE

PART-1

NONLINEAR RANDOl\1 RESPONSE METHODS FOR THE EVALUATION OF NONLINEAR SOIL AMPLIFICATIONS

1. INTRODUCTION

Consideration of the nonlinear and hysteretic nature of the soil behavior under large am­plitude cyclic loading is indispensable for realistic estimation of ground motion associated with local subsoil during strong potential earthquake. Moreover,when estimating ground motion at a specific site, amplification factors are quite sensitive to input motions which are of an unpredictable nature. Therefore, no confidence can be achieved from the results of a single de­terministic analysis using a recorded or artificially generated motion. To avoid the expense and effort required multiple deterministic analyses, probabilistic (random response) methods have been developed. Such methods require a. stochastic description of the earthquake excitation a.nd directly provide probabilistic information on the ground motion, helping the designer to make rational decisions regarding the safety of the facility.

It is the purpose of this paper to present a. random response methodology to estimate the ground motion considering nonlinear amplification subjected to nonsta.tiona.ry stochastic motion on rock surface for combinations of magnitude and epicentral distance.

In section 2, a literature review regarding the nonlinearity of the soil based on strong motion records is described. Previous papers showing the evidence of the nonlinearity of the subsoil and the nonlinear soil amplification from observation records are reviewed. A methodological review regarding random analyses is also described in this section.

In section 3, two possibilities of simple nonlinear random response methods based on piece­wise linear stochastic analysis are described. One is the method based on moment equations, which requires the evaluation of a convolution integral. The recursive evaluation of a con­volution integral is applied to lead computational efficiency. The other is the method based on evolutionary power spectra. Numerical verification of these methods is performed for both linear and nonlinear cases through the simulation by Monte Carlo Method in section 4.

In section 5, as apractical applications, the ground motions at a specific site considering the specified strain dependency of the subsoil are estimated for stochastic motion on rock surface used for strong potential earthquakes with different combinations of magnitude and epicentral distance.

Conclusions are stated in section 6.

- 1 -

2. REVIEW OF PREVIOUS PAPERS

2.1 NONLINEARITY OF THE SOIL AND STRONG OBSERVATION RECORDS

(1) EVIDENCE OF NONLINEARITY OF THE SOIL

Consideration of the nonlinear and hysteretic nature of the soil behavior under large am­plitude cyclic loading is reportedly indispensable for estimation of ground motion associated with local sediment sites, especially alluvial deposit, in earthquake engineering. A large body of literature exist on the laboratory and analytical studies of nonlinear behavior of the soil. There ha.ve also been some studies of the nonlinearity using the actual strong motion records, (e.g. Abdel-Ghaffer and Scott,1979, Tokima.tsu and Midorikawa.,1981, Tazo et al.,1987) where the strain dependent shear modulus G and damping factor h are discussed. Tokimatsu and Midorikawa (1987) showed the evidence of nonlinearity of the soil from observation records, where 4 typical Holocece sites in Japan are investigated using many strong earthquake records with different amplitude levels (ref. to Fig. 2.1,Table 2.1 and 2.2). They showed the variation of spectra due to the different intensity levels at Shiogama (where the impedance ratio of the surface layer and the bedrock is about 10) in Fig. 2.2. Predominant period becomes longer and the damping factor estimated from the shape of the spectra becomes larger with increase of the maximums ground acceleration. They also showed the strain dependency of the shear modulus calculated from the observation records based on their method as shown in Fig. 2.3, ·where the tendency of decrease of shear modulus agrees with that of the laboratory test. Decrease of shear modulus for the different intensity levels are shown for the peak acceleration of ground motion in Fig. 2.4 and for the peak velocity in Fig. 2.5. Their results show that when the peak acceleration is less than about 20 gal, the soil remains in the linear region and the shear modulus decreases by half when the peak acceleration of the records is from 150 gal to .500 gal and the peak velocity is from 10 kine to 20 kine. Sibata et al. (1988) discussed the effects of the nonlinearity of the soil using the observation records at Shiogama site.

- 2 -

SHIOG~MA HOSOSHIMA HIROO KUSHIRO J.M.A. DEPTH m S?T N-v•lue

0 ~10 50 20

40

60

. .

'1--T-..--.-riso r:::- o. so :-::· : ~ 1-r-+-+-....!......1 ~ :·· ... b ~· . . ... . . .•.. . ..

£ '-'-'-'---L-1

DroP SOiL

§cLAY

8s!LT []sAND

[DJSAND & 0 GRAVEL

Ill ROCK

Fig. 2.1 Boring Logs of Strong-Motion Sites (after Tokimatsu and Midorikawa,l987)

Table 2.1 Site Characteristics for Analysis (after Tokimatsu and Midorikawa,l987)

SITE SOIL DEPTH 1 p Vs 0 Vss (ml (5°) r t/m3J rm/sl rm/sl

SHIOGAMA SILT 15 0. 58 1. 50 103 n~ HOSOSH I t·1A SAND-CLAY 51 0.76 1. 80 268 HI ROO SAND 6 0.17 1.80 141 40d' KUSHIRO JMA SILT-SAND 14 0.26 1. 7 5 215 2400

* ESTIMATED VALUE

Table 2.2 Strong-Motion Records Used (after Tokimatsl1 and Midorikawa,l987)

SITE DATE r-1 D ( km)

SH IOGA~IA SEP. 25. 1972 5.5 50 FEB. 15. 1973 4.9 50 NOV. 19. 1973 6.4 50 FEB. 20. 1978 6.7 60 JUNE 12 1978 7.4 40

HOSOSHIMA APR. 01. 1968 7. 5 30 AUG. 06, 1968 6.6 40 APR. 21. 1969 6.5 10 JULY 26. 1970A 6.7 10 JULY 26. 1970B 6. 1 10 JUNE 22 1972 5.4 10

HI ROO OCT. 26. 1965 6.8 160 SEP. 19. 1967 6.2 110

' 1-IAY 16. 1968 7.9 0 SEP. 21, 1958 6.8 80 OCT. 08. 1968 6.2 60 JAN. 21 1970 6 7 sn

KUSH I RO J. ~1. A. NOV. 15. 1961 6.9 60 FEB. 21. 1962 6.2 80 APR. 23. 1962 7.0 60 JULY 18. 1962 5.9 60 JUNE 23. 1964 6.9 80 OCT. 26. 1965 6.8 160 NOV. 04. 1967 6. 5 20

D:Focal Deoth. X:Hyoocentral Distance - 3 -

X PEAK ACC. ( km l ( cm/sz l 100 23 160 17 130 56 110 12' 110 2i~ 100 242 130 45 60 98 60 122 60 49 80 21

290 189 220 97 180 308 100 163 100 185

sn liOS 120 111 110 170 120 5 13 100 87 190 109 200 272

60 357

6/12,1978

11/19,1973 I I

126cmts2

\

9/25,1972 '\ 56cmts

2

/0 ~h=l'ro

\V 23cm,s2

. I \

2/15, 1973 I? cmts2

.2 .5 I. 2. PERIOD sec

5.

Fig. 2.2 Variation of Spectra of Strong-Motion Records with the Amplitude Level (after Tokimatsu and Midorikawa,1987)

- 4 -

0 J. 0 .-----r---.---.-..------....A.. 1 1..!-:-:--~+-- 1A

<.9 + -- ··--... _1::,. -- .......... - ·m ........._ :"--- ',~ ~ ·-.. # " <.9 --~.......... ~ ··-... "

0

t 0::: (f)

30.5 ::::> 0 0 2 0::: <( w I (f)

eSHIOGAMA .&HOSOSHIMA OHIROO .AKUSHIRO J.M.A.

··-.. iii:--- ~ ·. ·•. " -F.......... ·-... "

·· .. "-+ ··.. " ··.. " ·. "

·.. "\.. ~ ··· ... _ " O··... "\.. ·. "-'\..

·.+.. "\.. • •·... " ·o o """ ··.. "-·. ' ·. "

+LA VILLITA DAM (after TOSHINAWA eta!.) ······ ... 0 '-... ··· ....•

---LABORATORY DATA FOR CLAY · .. ·· ... ',, ·· .. FOR SAND ··... ... ' '-

.............

10-5 10-4

EFFECTIVE SHEAR STRAIN Yeff

Fig. 2.3 Strain Dependence of Shear Modulus Ratio

(after Tokimatsu and Midorikawa,1987)

- 5 -

o I.Or--i--.-,-,--..--,-...--.-,.....,--l>----.--rl:l<r-r----. ~ ~ ',, r'.t. (!) • "'A.•\..9~ 0.~ 0 ... ',,....,./:,. \ \

~ \ \ \ a:: \ \ "A

~ ~\ ~

~.....JQ5~ ' \ ~ -· • SHIOGAfvi.A •, ',. ' o ------ •HOSOSHIMA 0~ 0',,0 ~ --- OHIROO 'e

-

a:: --- .AKUSHIRO J.MA. <t w I ~

1 I I 01 10 100 1000

PEAK GROUND ACCELERATION Amox cm;s-2

Fig. 2.4 Peak Ground Acceleration vs. Shear Modulus Ratio (after Tokimatsu and Midorikawa,1987)

o 1.0 .--,.-r-r-,---•,.----,,-___ •: ,, .... ,A-~.,.l:lr~-~r,-r,....,-~

~ • j~·~ ... "L

0

!(( a:: ~

30.5~ ~ 0 0 :2 a:: <t w I ~

•SHIOGAMA •HOSOSHIMA OHIROO .AKUSHIRO J.M.A.

... .IS.'. ... \

0

0

·. L

' .... 0 .... _

0 · ..•

-

o~~-LLL--~'~-L~--~~--~~~~~~~--0.1 I 10 100

PEAK GROUND VELOCITY Vmox cm1s

Fig. 2.5 Peak Ground Velocity vs. Shear Modulus Ratio (after Tokimatsu and Midorikawa,1987)

- 6 -

(2) NONLINEAR SOIL AMPLIFICATION

It is important to estimate the ground motion on the soil surface considering the nonlinear effect and the amplification factor for different intensity levels at the bedrock surface from the point of not only the site specific seismic design but also the micro-zonation purposes.

Sugito and Ka.meda (1990) showed the simple method for the estimation of ground motion on the soil surface by incorporating the amplification factor as "conversion factor" for different levels of acceleration at the bedrock surface based on the empirical (statistical) approach and which can be directly compared with the observed results of the strong earthquake, e.g. the Lorna Prieta earthquake. They defined the conversion factor f3a ((3.) as the ratio of peak acceleration (velocity) at the soil surface to that at the bedrock surface, and estimated f3a for typical soil conditions specified by geotechnical parameters Sn and dp (see also Sugito, 1986). Their results are shown in Fig. 2.6, where the flat part of value corresponds to the linear response region and the decrease from the flat level indicates the nonlinear characteristics of the soil. The amount of decrease depends on the soil parameters (Sn, dp) and the acceleration at the bedrock surface (A.) . The parameter dp denotes the depth to the bedrock where the shear velocity is 600-700m/sec, and the parameter Sn is calculated from the N-value profile obtained from the standard penetration test by the following formula

d, .

Sn = 0.2641 exp( -0.04 N(x)) · exp( -0.14x) dx - 0.885 (2.1.1)

where N(x) is the N-value at depth x (in meters) and d. is the depth (in meters) of the N-value profile.

Sugito et al. (1991) showed the corresponding observation from the Lorna Prieta earthquake as shown in Fig. 2.7. In Fig. 2.7, the ratios of the observed peak acceleration to that at the rock surface using the observation records at the CDMG and USGS sites (shown in Table 2.3) were plotted. Fig. 2.7 shows that the Lorna Prieta earthquake provide us the valuable data of which amplification factor is about one or more less and that amplification factor decreases with increase of the peak acceleration at the bedrock surface, which shows the nonlinear effect. In the study, they showed the similar simple estimation method for amplification factors using the soil parameter St and dp . The parameter St is given by v6/v. where v. is shear wave velocity of a surface layer and vb is standard value of shear wave velocity for a very soft layer, in the study, v6 is fixed at 88 m/s corresponding to the mean value of the shear wave velocity of Bay mud (Borcherdt et al., 1979).

- 7 -

"' ca 1.. 0 ...., u ., ..... c: 0 ·~

"' 1.. Cl> > c: 0 u

s.-~----~--------~ (very soft site) ·Sn=IJ.6, dp=lSOm

/ Sn=-0.2,dP.-=20m (hard site)

lU IUU (cm/sec 2 )

peak acceleration Ar

> ca 1.. 0 ..... u ., ..... c: 0 ·~ Vl 1.. OJ > c: 0 u

s~----------------~ 4

(hard site) Sn=-0.2, dp=20m

10 100

peak velocity Vr.(cm/sec)

Fig. 2.6 Values of Conversion Factors f3a and f3·v for Typical Soil Conditions. (after Sugito and Kameda,1990)

0 -0

o.- 0.~ ::::::~ ~J 0

-Q.; I.C'l-i 0 0 ... vi

0 < ...... ~ ~0 ~

~~ 0

< 0 0 0

.Q I 0 0

~

c::i 0

o~J 0 Oo .... c: 0.~ 0 Oo .Q ""- oo

c::i "--i 0

'0" u l..'"l~ t:: vJ

0.. ...... 1 ;:: C'\1...;

..,, "'~

~ v~ 8

00~ i 0

.:1 > 0 0oo 0

.Q 0 0

~ 0 000

~0

c: -- 0 O.-i

0 "'1 «:,"'

~ ~~ 0

t::

~j 0.. ,.., E ct:

~~

I b! 7

0 I

peak ace. on rock surface, Ar (cm/sec2) peak vel. on rock surface, Yr (em/sec)

Fig. 2. 7 Variation of Ground Motion Amplification Ratio versus Rock Surface Ground Motion (after Sugito, Kiremidjian and Shah,1991)

- 8 -

Table 2.3 Rock Surface and Soil Surface Strong Motion Records Dataset (after Sugito, Kiremidjian and Shah,1991)

Soil Site Soil St dp A max1 A max2 V max1 V max2 Rock Site Rock Ama;x:l Amax2 Vmax1 Vmax2

Name Type ( m ) (90') ( 0.) ( 90.) ( 0') Name Type (90') ( 0') (90.) ( 0.)

CD JiG

1. Treasure Island Qhbm 1. 00 86. 7 -155. 8 97. 9 33. 4 -15. 6 Yerba Buena lnd Sandstone -65. 8 28. 1 14. 1 4. 6

2. Oakland-2-story Bldg* Qps o. 33 85. 3· -166. 6 130. 5 19. 4 8. 9 Piedmont* leathered- 97. 5 49. 1 13. 3 4. 4

3. Oakland Outer Barber* Qhbm 1. 00 97. 5 -325. 5 -253. 6 so. 7 -31. 8 Piedmont* Serpentinite 97. 5 49. 7 13. 3 4. 4

4. Gilroy #2 Qhac o. 54 14 o. 2 316.3 -344.2 -39. 2 33. 3 (Gilroy #1 Sandstone 391. 4 385. 1 -29. 5 27. 9

5. Gilroy Gavilan co.* Qpa o. 47 25. 9 363. 9 -312. 6 -25. 5 25. 5 ( Gilroy #1 Sandstone 422. 5 415. 7 -32. G 30. 8 )

6. Agnew Qhaf o. 47 231. 6 157. 6 163. I -18. 2 30. 9 ( Upper C -Pulgas Sandstone -165. 6 -300. 0 -27. 1 35. 0 ) co 1. Foster City Qhbm 1. 10 20 }. 2 277. 6 252. 6 45. 4 -31. 8 Upper C -Pulgas Sandstone -84. 8 -153. 6 -13. 6 17. G

8. San Franci lot Airport Qhbm 1. 00 164. 6 -352. 8 -230. 8 29. 3 26. 5 So-Sierra Point* Rock 17. 1 -78. 9 7. 0 -6. 7

9. San Francisco 18story* Qhaf o. 47 58. 8 137. 4 163. 7 -15. 6 -16. 9 San Fracisco- Sandstone 88. 5 -78. 6 11. 6 1. 3

. Rincoln Hill

USGS

!.Sunnyvale South St Qhaf o. 47 181. 7 208. 0 211. 8 34. 1 -33. 4 ( Upper C -Pulgas Sandstone -162. 3 -294. 0 -26. 5 34. 3

2. Hollister City Ball Qha o. 44 55. 8 251. 9 216. 8 -38. 6 -44. 0 ( Gilroy #1 Sandstone 179. 4 176. 5 -12. 2 11. 5

3. Stanford Univ Parking Qa o. 47 36. 6 -216. 0 -255. 0 -21. 3 -33. 2 foods ide Cong1omerte 79. 7 79. 5 -14. 7 15. 6

4. APEEL Array, Redwood* Qhbm 1. 00 9 I. 4 238. 6 -244. 2 49. 6 36. 4 Upper C -Pulgas Sandstone -84. 8 -153. 6 -13. 6 17. 6

5. San Fran 600 Kontgo St Qhafs o. 47 43. 6 119. 4 -107. 1 18. 1 -9. 6 San Francisco- Sandstone 90. 5 -51. 2 9. 6 6. 5

Telegraph Bill

* Peak ground motions have been obtained from time histories processed by the transformation of coordinates systems.

Peak ground motion have been modified accoding to Joyner & Lloore' s attenuation formulas.

2.2 METHODOLOGICAL REVIEW

It has long been recognised that earthquake ground motions are most realistically mod­elled as a non-stationary non-white stochastic process. Madsen and Krenk (1982) have given a method of determining the response of a multidegree of freedom (:MDOF) system to a sud­denly applied stationary, non-white ground motion. Closed form expressions for the mean squared response are derived in terms of the stationary cross-correlation. Again, the excitation considered was a suddenly applied stationary, non-white ground motion. Gasparini and De­bchaudhury (1980) have considered the response of a MDOF system to modulated non-white ground motion. By representing the nonwhite excitation as filtered white noise the analysis was considerably simplified and analytical results were obtained for the case of a piecewise linear modulating function. Langley (1986) proposed an efficient numerical procedure for calculating the statistics of the response of a MDOF system subjected to non-stationary non-white ground motions by numerical integration of the moment equations.

Although classical random vibration theory and the above extensive methods applies only to linear elastic structures, several analysis techniques have also been developed for dealing with both linear and nonlinear structures. !wan and Mason (1980) have derived general expressions for the mean squared response of a nonlinear MDOF system subjected to nonstationary non­white excitation, using a minimum mean squared error linearization technique. Owing to the complexity of the resulting equations the method, which involves numerical integration of the moment equations, was applied to the relatively simple case of a Duffing oscillator subjected to modulated white noise. Using the same linearization technique, Di Paola et al. (1984) have developed an efficient computational procedure for calculating the mean squared response of a nonlinear MDOF system subjected to non-stationary non-white ground motion. The computational efficiency is achieved by expressing the original equations of motion in recurrence form, leading to a reformulation of the moment equations.

Few researchers have applied nonlinear random vibration techniques in computing the seis­mic response of continuous systems that are of interest in soil dynamics. Faccioli (1976) de­veloped an equivalent linearization random vibration formulation to study the one-dimensional amplification of seismic waves by deposits of soil obeying Ramberg-Osgood-Masing constitutive equation. Singh and Khatua (1978) attempted to assess the seismic safety of earth dams by using stochastic linearization and performing an iterative analysis in connection with a finite­element discretization of the dam. Vanmarcke (1977) suggested several possible applications of random vibration theory to solve soil dynamics problems, including determination of nonlinear soil response and assessment of liquefaction potential. Gazetas et al.(1982) develop a simple and rational piece-wise linear random vibration procedure to estimate statistics of the nonlinear hysteretic response of earth dams to strong nonstationary stochastic excitation characterized by a Kanai-Tajimi spectrum. The method utilizes published data relating soil modulus and damping to the level of induced shear strain.

The method developed by Gazetas et al. is very simple and useful, but the application of the method is limited to the input motion characterized by a Kanai-Tajimi spectrum and the specified envelope function for practical purposes. The method proposed in section 3.1 is able to cope with the arbitrary power spectral density and the arbitrary, however, frequency independent envelope function of the input motion based on the the moment equations and approximating the power spectrum by a ratio of two polynomials in w2 •

The described methods based on the moment equations can result the covariance responses but not result the evolutionary power spectra of the response. Some pioneering work have been performed to enable principally to get the evolutionary power spectra of the response. The linear stochastic response analysis based on the theory of the evolutionary power spectrum, originally developed in the work of Priestley (1965, 1967), has been applied to structural dynamics general problems by, for example, To (1982, 1984). This method is able to cope with the arbitrary power spectrum and the arbitrary frequency dependent envelope function. The time segmenting approach was applied to this theory by Holman and Hart (1972, 1974) only in the case oflinear stuructures.

- 10 -

The method proposed in section 3.2 is a significant extension of this segmenting approach in consideration of the nonlinearity of the system. The results of this method based on the evolutionary power spectrum is comletely consistent with those of the method based on the moment equations.

- 11 -

3. METHODOLOGY FOR NONLINEAR RANDOl\I RESPONSE ANALYSIS

3.1 STRAIN-COl\IPATIBLE PIECEWISE LINEAR ANALYSIS BASED ON l\IOl\IENT EQUATION

In this section, the random response methods based on the strain-compatible piecewise linear analysis are proposed. Conception of the piecewise linear stochastic analysis is described in Fig. 3.1. First, the duration of seismic excitation is devided into several time segments, and the input rock motion is assumed empirically in each time segment. Second, in each time segment, the linear random response analysis is performed and updating of the system matrix is done at the end of each segment based on the constitative law, i.e. the relation between the covariance of the strain and the effctive strain. The flow chart of the analytical method is shown in Fig. 3.2.

(1) OUTLINE OF METHOD

In this study, the random response analysis of a general .MCK-type system based on the moment equations is extended to the nonlinear analysis subjected to nonstationary stochastic input motion.

Although the evaluation method for the covariance response based on the moment equa­tions has been proposed, this method requires the evaluation of a convolution integral resulting from the frequency dependency of the power spectrum of stochastic input motion. This con­volution integral is global in time and requires large computational efforts to be calculated in the time domain for the extension to the nonlinear analysis, which is one of the difficulties to be applied to the practical problem. Therefore, application of the method has been limited to the input motion of which power spectrum is of simple form, e.g. Kanai-Tajimi spectrum. To break through the difficulty, the recursive evaluation of a convolution integral leading to great computational efficiency is applied for the nonlinear analysis in this study by approximating the power spectrum by a ratio of two polynomials in w2 , which is mathematically eqltivalent to the ARMA modeL

Nonlinearity of the system is simply taken into account in this study by applying a piecewise linear stchastic analysis, where the system matrix is assumed to be constant within a piecewised time section and linear stochastic analysis is performed.

(2) BASIC EQUATIONS

SYSTEM EQUATION

Starting from the first order differential equations obtained by Foss' method from the· equation of motion of MCK-type system, the equations to evaluate the covariance response of the system are formulated based on the piecewise linear stochastic analysis.

In the formulation, the duration of seismic excitation is devided into L time segments t,_ 1 S t S t, , n = 1, 2, ... , L. In each segment, the governing equations in terms of the state value {Z ... (t)} are described as follows.

(3.1.1)

where

{Z,.(t)} = { {:i:n(t)}} {x,(t)} (3.1.2)

[A ]- [-[M,J- 1[Cn] n - [ J] (3.1.3)

{V} = { ~{~} (3.1.4)

- 12 -

{Z,(t)}: state value vector in t,_ 1 ::; t::; t,,

{x,.(t)}: relative displacement vector in tn-l ::; t::; t,. ,

{x,(t)}: relative velocity vector in t,_ 1 ::; t::; t,. ,

[A,.] : system matrix in t,_1 ::; t ::; t,. ,

[ .. M,] : mass matrix in tn-l ::; t::; t, ,

[C,.] : damping matrix in tn-l ::; t ::; t,. ,

[K,.] : stij fness matrix in t,._l ::; t ::; tn ,

x9,.(t) : input acceleration in t,_ 1 ::; t::; t,. ,

{f} : load vector

It is noted that Eq .. (3 .. 1..1) is obtained from following equation of motion of MCK-type system.

(3.1.5)

General solution of Eq.(3.1..1) with the initial value {Z,.(t,._1)} at t = t,_ 1 is expressed as the following form ..

(3.1.6)

DESCRIPSION OF STOCHASTIC INPUT 1\!0TION

In this study the input motion x9,.(t) in Eq.(3.1..1) is assumed to be described as the following form considering the nonstationarity due to the envelope time function n(t) ..

x9 ,.(t) = n(t) · j,(t) (3.1.7)

where j,.(t) is stationary stochastic process in t,_ 1 ::; t::; t,. which has a zero mean Yalue, i.e. E[J,.(t)] = 0 .. It is noted that J,(t) can be changed in each time section.

MEAN VALUE AND COV ARIANCES OF RESPOl,lSE

Using the state value described in Eq(3 .. 1.6), the mean values and the covariances of the response can be considered. First, the mean values of the response are considered. The mean · value of Eq .. (3 .. 1.6) is given as follows.

E[ { Z,. ( t)}] = e [A.] (t-t.-d E[ { Z,. ( t,_l)}] +it" e [A.](t- .. ) {Y} E[ x9,. ( u)] du tn-1

(3.1.8)

Assuming that the. initial values of the system are zero at t = t0 and considering that j,.(t) has a zero mean value, the following equations result.

E(ig,.(u)] =n(t)-E(j,.(t)] = {0}

{Zl(to)} = { ~!~~!~n} = {o}

(3.1.9)

(3.1.10)

Considering Eqs.(3 .. 1..6) and (3.1.10), the state value {Z,(t,_1 )} at t = t,._ 1 is described as the following form.

n-2n-l-i t·

{Z,.(tn-l)} = L rr e[A.-r)(t._,-t._,_,) 1' e[A;](t;-"){V}xg;(u.)d1~ i=} T=l ti-l

+ ~~·-1 e[A.](t.-1-"){~r}xg·n-l(u)du tn-2

(3 .. 1.11)

- 13 -

With substitution of Eqs.(3.1.9) and (3.1.11) into Eq.(3.1.8), then the mean values of the response result :

n-2n-1-i t

E[{Z,.(t)}]= e!An)(t-tn-tl I: II eiAn-•J<tn-•-tn-•-tll' eiA.](t;-a)fV}E[xy;(u)]du i=l T=l t,-1

l

tn-1 + e (A.](t-t.-1) e (A.](t.-1-"l{V}E[xg·n-1(u)] du

'tn-2

+ ~~- 1 e!A.](t-"){V}E[x9,.(u)]du

= {0} (3.1.12)

Second, the covariances of the response are considered. Using Eq.(3.1.6), the covariance matrix of the state value [ m,.(t)] is described as the following equation.

[ m,.(t)] = E[ {Z,.(t)} {Z,.(t)}T]

= ~~- 1 ~~- 1 e (A.](t- .. ) {V}{V}T e (A.f (t-•)n( u)n( v)E[ fn(u)f,.( v) ]dudv

+ e [A.](t-t. -tl E( { Z,.( tn-1 )} {Z,. ( tn-1 )}T l e (A.]T (1-tn-tl

+ eiA.](t-t._,)~~- 1 E[{Z,.(t,._!)}x9,.(u)]{V}T e!A.f(t-u)du

+ ~~- 1 e!A.](t-"){V}E[x9,.(u){Z,.(t,._I)}T]du ·e(A.f(t-t._I) (3.1.13)

Denoting the cross corelat.ion function between the stationary stochastic proccess J;( u) and fi(v) as R;i(t) , the following equation results.

(3.1.14)

It is clear that R;i(t) becomes auto-correlation function when i = j in Eq(3.1.14).

Considering tha.t E[ {Z,.(t,._!)} {Z,.(t,._I)}T] of the second term in Eq.(3.1.13) is written as [ m,.(t,._ 1) ] and becomes the initial value of [ m,.(t) J in the time segment : t,._ 1 ::; t ::; t,. , Eq.(3.1.13) can be written as the following form.

[ m,.(t)] = [ mo,.(t)] + [ m.,.(t)] + [ Q,.(t)] + [ Q,.(t) f where

[ mo,.(t)] = J' [ e!A.](t-u){V}{V}T e(A.]T(t-•)n(u)n(v)R,.,.(u-v)dudv t"'-1 tn-1

( m.,.(t)] = e (An](t-t.-1} [ m,.(tn-1)] e (A.]T (t-t._l)

[ Q,.(t) ] = e (A.] (t-t._,) 1~- 1 E[ {Z,.(t,._I)} x9,.( u)] {V}T e (A.]T (t- .. ) d1t

(3.1.15)

(3.1.16)

(3.1.17)

(3.1.18)

By differentiating both sides of Eq.(3.1.15) with respect to time, the following moment equations are derived.

[ m,.(t)] = [ mo,.(t)] + [ m.,.(t)] + [ Q,.(t)] + [ Q,.(t) f =[A,.] [ mo,.(t)] + [ mo,.(t)] [A,.]T + [ Go,.(t)] + [ Go,.(t) f +[A,.] [ m.,.(t)] + [ m.,.(t)] [.4,.]T

+ [A,.j [ Q,.(t) J + [ Q,.(t)] [A,.f + [ Gqn(t)] +[A,.] [ Q,.(t) f + [ Q,.(t) f [An]T + [ Gqn(t) ]T =.[A,.][ m,.(t) ] + [ m,.(t) [A,.f + [ G,. (t) ] + [ G,.(t) f (3.1.19)

- 14 -

where

[ G,.(t)] = [ Go,.(t) 1 + [ G9,.(t) 1

[ Ga,.(t) 1 = 1~-1 e [An)(t-•){V} {V}T n(t)n(u)R,.,.(t- u) du

[ G9,.(t) 1 = e[An)(t-tn- 1 lE[ {Z,.(t,.-l)}x

9n(t)1 {V}T

(3.1.20)

(3.1.21)

(3.1.22)

Eq.(3.1.19) folds at all time steps of all time segments. Final solution of the moment equations can be obtained numerically at each time step by calculating [ Gn ( t) 1 specified as Eq.(3.1.20)"' Eq.(3.1.22). Applying Eq.(3.1.11) to Eq.(3.1.22),[ G,.(t) 1 is described as the following equation.

[ G,.(t)] = [ Ga,.(t)] + [ G9,.(t)]

= 1:_1

e [An)(t-•){V} {V}T n(t)n(u)R,.,.(t- u) du

(3.1.23)

To get the moment at each time step, at first [ G,.(t)] is calculated by estimating the convolution integrals appearing in Eq(3.1.23), and then Eq.(3.1.19) is solved. It is noted that in each time segment the system matrix and the correlation function must be the same, but that in the different segment the different system matrix and the different correration function can be permitted as seen from the description of Eq.(3.1.23). Therefore, the method can be applied to the nonlinear analysis based on the piecewise linear assumption in consideration of the nonstationarity of the input motion.

RECURSIVE EVALUATION OF CONVOLUTION INTEGRAL

Convolution integrals appearing in Eq.(3.1.23) are global in time and require large com­putational efforts to be estimated. This can be evaluated recursively in the time domain as applied in the field of the time domain soil-strunture interaction analysis. It is, in principle,

· only approximate. When the system function is expressed as the ratio of two polynomials in frequency, ·exact solution is obtaind. In this study the methodology of the recursive evaluation is applied.

In the case of this study, the convolution integrals appearing in Eq.(3.1.23) can be evaluated recursively when the power spectral density of the input motion corresponding to the Fourier Transformation of the correlation function R;i(t) is specified as the ratio of two polynomials in terms of s2 in Laplace domain, and/or when the Fourier Transformation of the time function e-[A•l'n(t) is specified as that form. In the following, the case is considered where the power spectral density of the input motion S( s) is specified as the ratio of two polynomia.ls in terms of s 2 in Laplace domain. By applying partial fraction expantion, the corresponding cross correlation function R;1(t) can be expressed explicitly in the paralell form. Then the recursive expression of the convolution integral can be obtained. ·

First, consider complex number s = iw + cr by extending w to complex number. \Vhen S(s) is the rational function of s2 , S(s2

) = S(s) holds. Denoting the root of the numerator N(s) or the denominator D(s) as sk , then -sk becomes its root, where the real part of sk is negative. Complex conjugate of sk is also the root because the coefficients of the polinomials in s2 are the real numbers. S(s) can be thus expressed as the following form.

S(s) = H(s)·H(-s) = N(s)·N(-s) D(s)·D(-s)

- 15 -

(3.1.24)

where N(s) and D(s) have no common factor, and the order of N(s) is less than that of D(s).

Denoting the root of the denominator in Eq.(3.1.24) as sk ( Re( s~:) < 0, k = 1, ... , mp) , and denoting its order as n~:(k = l, ... ,mp), S(s) is described as the following partial expantion form.

mP "• !v! mp "• M S(s)="""" 1: I +"""" 1: I

~£I' ( s - sl:)1 ~£I' ( -s - sl:)1 (3.1.25)

where 1 d"•-l

1:M1 = ( I)' lim -d _1 (s- s~:)"• S(s) nk - . s-s.c snk

(3.1.26)

Considering that R;i(t) is the inverse Laplace Transfomation of S;j(s) and t 2': 0 ,and that s1 is located upper half of the complex plane, only the first term of Eq.(3.1.25) is enough to be considered. By the formula of the Laplace Transform, the following equation results

for t 2': 0 (3.1.27)

Expanding Eq.(3.1.27) with respect to u, the correlation function R;i(t-u) is expressed as the following form.

where

m~i n~i 1-1

R;i(t- u) = 2:2: 2: C~{mR;i,klm(t)ki,l:lm(u) 1:=11=1 m=O

•[ij ( 1)1-1-m c'i - kl•· l • -

l:lm - m! (I - 1 - m )! - 1j1 R;j,l:lm ( t) = tm e•.

R- ( ) l-1-m -s;i,. ij,l:lm u = u e •

"With substitution of Eq.(3.1.28) into Eq.(3.1.23), the following equation is obtained.

m;"' nk"' 1-1 t

[ G,.(t)] = 2:2:2: Ck't'!nn(t)Rnn,klm(t)e(An)t 1 e-(An)"n(u)Rnn,klm(u)du{V}{VV 1:=1 1=1 m=O tn-1

+ e [An){l-tn-1) [f"ff' e [An-r)(tn_,-tn-r-1)e [A;)t,

i=1 r=1

m~" ni?\ 1-1 t;

· 2:2:2: Ciimn(t)R;n,l:lm(t) l e-(A;)"n(u)R,,.,k/m(u)du 1:=1 1=1 m=O t,-1

] {V} {V}T

(3.1.28)

(3.1.29)

(3.1.30)

(3.1.31)

(3.1.32)

where the second term of the right side of Eq.(3.1.32) can be rewritten using the following initial value of [ Gn(t)] at t = tn-1 .

- 16 -

n-2m;"' nk" 1-1

[ G,(t,_l) I= L L L L [ G,;,um(t) I i=1 k=1 1=.1 m=O

n-2n-1-i m~" ni" 1-1 - [~ II e[Aft-rWft-r-tft_,_,)e(A;)t; ~~ ~ cin n(t )-R· (t ) - L L L L kim ft-1 •n,klm n-1

i=1 •=1 k=1 1=1 m=O

·it; e -[A;)"n(u)k,,klm(u)du · tt-l

"'-l·f'l. f'l.-l·ft m, "< 1-1

+ e [A.-t)t.-1 L L L c;,-;",.1""n(tn-1) R,.-1·n,klm(t,_1) k=1 1=1 m=O

(3.1.33)

Applying Eq.{3.1,33), Eq.{3.1.32) is expressed as the following recurrence form.

m;"' nk" 1-1 t

[ G,(t) I= L L L Cf,';,.n(t)R,,,klm(t)e[Aft]t 1 e-[A.]"n(u)Rnn,klm(u)du{V}{Y}T .1:=1 1=1 m=O t. -1

n-2m~"' ni"' l-1 -

+ e [A.](t-tft_,) ~ ~ ~ ~ n(t) . R;n,klm(t) [ G . . (t _ ) ] L L L L (t ) R ( ) n•,klm n 1 i=1 .1:=11=1 m=O n n-1 in,klm tn-1

(3.1.34)

Continuity condition of the state value a.t the end of each time segment leads to the following relation.

(3.1.35)

Using Eq.{3.1.34) and Eq.{3.1.35), [ G,(t)] at each time segment can be calculated recursively.

- 17 -

3.2 STRAIN-CO~JPATIBLE PIECEWISE LINEAR ANALYSYS BASED ON EVOLUTIONARY POWER SPECTRUM

(1) OUTLINE OF METHOD

In section 3.1, the evaluation method of the covariances based on the moment equations is. described, where the envelope time function is independent in frequency even though the time variation of the power spectrum can be taken into account by time segmenting. In the method, an evolutionary power spectrum of the response, SR(w, t) , representing the time variation of the power spectral density can not be directly obtained. In this section, a methodology of nonlinear random response of a general MCK system is formulated more generally than the method in 3.1 based on the piecewise linear analysis in order to get the evolutionary spectrum and the covariance response subjected to the nonstationary stochastic motion.

(2) BASIC EQUATIONS

The system equation of the general MCK-type system and its general solution in each time section ( t,._ 1 ::; t ::; t,. ) are written as follows.

where

{Z,.(t)} = e[A.)(t-t.-tl {Z,.(t,_I)} +it. e[A.)(t-"lfV}x9,.(u)du

t" -1

{Z (t)} = { {x,.(t)}} " {x,.(t)}

[A ] - [-[M,.]- 1[C,.J -[M,.J- 1[K,.]] " - [I] [ 0]

{V} = { ~{~} {Z,.(t)}: state value vector in t,._ 1 ::; t::; t,. ,

{x,.(t)}: relative displacement vector in t,._ 1 ::; t::; t,. ,

{x,.(t)}: relative velocity vector in t,._ 1 ::; t::; t,. ,

[A,.] : system matrix in tn-1 ::; t ::; t,. ,

[11!,.]: mass matrix in tn-1 ::; t::; t,. ,

[C,.J : damping matrix in tn-1 ::; t ::; t,. ,

[I<,.]: stij fness matrix in tn-1 ::; t::; t,. ,

x9,(t): input acceleration in t,._ 1 ::; t::; t,. ,

{f} : load vector

DESCRIPSION OF STOCHASTIC INPUT 'AfOTION

(3.2.1)

(3.2.2)

(3.2.3)

(3.2.4)

(3.2.5)

In this study the input motion x9,.(t) in Eq.(3.2.1) is assumed to be described as the following form considering the nonstationarity of the amplitude.

x9,.(t) = 1: n(w, t) · e iwt · dSw,.(w) (3.2.6)

where Sw,. is assumed to be an orthogonal proccess and n(w, t) is the envelope time function and x9,.(t) is a stochastic process in t,_ 1 ::; t::; t,. having a zero mean value, i.e. E[x9,.(t)] = 0. It is noted that x9,.(t) can be changed in each time section.

- 18 -

DERIVATION OF COVARIANCE AND EVOLUTIONARY POWER SPECTRU1l1

Assuming that the initial values of the system are zero at t = t0 , the following equation results.

{Z1(to)} = { ~;~~:~~~} = {0} (3.2.7)

Considering Eq.(3.2. 7), Eq.(3.2.2) is rewritten as the following form.

{Z,.(t)}= J' e[A.J(t-"){V}·x9,.(u)du

t"-1

(3.2.8)

With substitustion of Eq.(3.2.6) into Eq.(3.2.8) and changing the order of the variable of inte­gration, the following equation results.

{Z,(t)} = i: {e,.(w,t)}dSw,.(w)

+ [A.] (t-t.-tl 1"" {- ( t )} dS ( ) e -oo en-1 w, n-1 Wn-1 W

n-2 n..-1-i 00

+ e [A.] (t-t.-tl L II e [An-•] (t._,-t•-•-1) 1 {e;(w, t;)} dSw;(w) i=1 •=1 -oo

(3.2.9)

where {e,(w,t)}= it elA;J(t- .. ){V}·n(w,u)·e;"'1 du

1t-1

(3.2.10)

for t,._ 1 ::::; t ::::; t,. . Considering that dSw; is an orthogonal proccess, the following equation holds.

(3.2.11)

Using Eq.(3.2.9) and Eq.(3.2.1l),the covariance matri.x is described as the following form.

where

[ m,.(t)] = E[ {Z, (t, )} {Z,(t,.)}T]

= i: [SRn (w, t)] dw

[SRn(w,t)] = {e,(w,t)}{en(w,t)}T ·Sn,.(w)

+ e[A.](t-t.-tl {en-1(w, tn-1)} {en-1(w, tn-l)}T e [A.]T (t-t.-tl · Sn-l·n-1(w) n-2 n.-2 n-1-i

+ e[A.J(t-t.-tl L L( II e!A.-.](t.-.-~.-.-ll){e;(w,t;)}{ej(w,tj)}T n-1-j . ( II e[A.-•J<t·-•-t·-•-ll)T e[A.]T(t-t.-1). S;j(w) k=l

+ e [A.)(t-t._t) {en-!(w, tn-1)} {en(w, t)f · Sn-l·n(w)

+ {- ( t)} {- ( t )}T [An]T (t-t.-tl S ( ) en w, en-! w, n-1 e . n·n-1 w n-2 n-1-i

+ e[A.)(t-t.-tl L( II e[A.-.](t.-.-t.-•-tl){e;(w,t;)}{en(w,t)}T. S;n(w) i-1 •=1

- 19 -

(3.2.12)

n-2 n-1-i + L {e,.(w, t)} {e;(w, t;)V( rr e(An-•J<tn-•-ln-•-ll)T e [An]T (1-ln-ll . S,.;(w)

i=1 n-2 n-1-i

+ e [.4n] (1-ln-tl L {en-1 (w, tn-d} {e;(w, t;)}T ( rr e (An-•Wn-•-1n-•-d)T

i=1

n-2 n.-1-i

+ e [An](Hn-d L( IT e [An-•] (1n-•-1n-.-t)){e;(w, t;)} {en-1(w, tn-d}T

(3.2.13)

and [SR,. (w, tn-d] at t = t,.-1 is described as in the following form.

n-2 n.-2 n.-1-i

n-1-j . ( rr e[An-<J<tn-c-ln-<-tl)T. S;j(w) k=1

n-2 n-1-l

+ L {en-l(w, t,._i)}{e;(w, t;)}T ( II e (An-rWn-•-ln-•-ll)T · S,._ 1.;(w) i=l n-2 n-1-i

+ L( rr e[An-•Wn-•-ln-r-tl){e;(w,t;)}{e .. -l(w,t,._l)}T. si·n-1(w) (3.2.14) i=l T=l

Using Eq(3.2.14), Eq(3.2.13) is rewritten in the following form.

[SRn (w, t) j = e (An] (1-ln-tl [SRn (w, t,._I) j e [An]T (1-ln-1)

+ {e,.(w, t)} {en(w, t)}T · S,.,.(w) [An){l-ln_l) {- ( t )} {- ( t)}T S ( ) + e en-1 w, n-1 e,. w, . n-1·n w

+ {e,.(w, t)} {en-1(w, tn-1)}T · Sn·n-1(w) n-2 n-1-i

+ e [An](Hn-d L ( II e [An-•Wn-r-1n-•-1l)T {e;(w, t;)}{e,.(w, t)}T. S;.,.(w)

n-2 n-1-i

+ L {e,.(w, t)} {e;(w, t;)}T ( rr e [An-•Wn-r-ln-•-d)T e [An](Hn-tl . S,.;(w) (3.2.15) i=1

It is noted in Eq.(3.2.15) that the evolutionary power spectrum at each time section is calculated from the last value of [SR(w, t)] in the previous time section, cross power spectra of current and all past time sections, and power spectrum of current time section.

It is also noted tha.t the nonlinearity of the system and the nonstat.ionarity of the input motion can be taken into account in Eq.(3.2.1.5).

Special Case

As a special case , a nonlinear system subjected to a stochastic input motion specified as a stationary power spectral density and an envelope function n(t) is considered. The response value {Z,.(t)} in Eq.(3.2.9) is then discribed as the following form, which is calculated recursively.

{Z,.(t)} = L: {e,.(w, t)} dSw(w) (3.2.16)

- 20 -

where

{ e,.(w, t)} = e (A.]{H.-tl { e,. (w, t,._I)} + {e,. (w, t)}

{e,.(w, t)} = l'_, e (A;] (t-.. ) {V} · n(w, u). e iwt du

(3.2.17)

(3.2.18)

The evolutionary power spectrum and the covariances of the response are obtained by the following equations.

[SRn (w, t)] = {e,.(w, t)}{e,.(w, t)}T · S(w) [ m,.(t)J = E[ {Zn(t)} {Zn(t)}T]

= I: [SRn (w, t)] dw

EVALUATION OF CONVOLUTION INTEGRAL USING MODAL ANALYSIS

(3.2.19)

(3.2.20)

To evaluate the covaria.nces and the evolutionary power spectrum of the response, the convoluiton integrals appearing in Eq.(3.2.10) (Eq.(3.2.18)) must be calculated. Applying the modal analysis to the matrix [An] yields

(3.2.21)

where n).i and { .. 4>i} denote the eigenvalue of the matrix [A,.] and the corresponding eigen vector. Eq.(3.2.21) for all modes results in the following form.

where

[A,. ] [ <P,.] = [ <P,. ] [A,.]

[ <P,.] = [ {,.,Pi},···, {n4>2M} J

[A,.] = [\ ).i\]

(3.2.22)

(3.2.23)

(3.2.24)

From Eq(3.2.22), the matrix [A,.] is expressed as the following form when there is no double root.

[A,.]= [<P .. ] [A,.] [<P,.]- 1 (3.2.25)

Substituting Eq.(3.2.25) into Eq.(3.2.10), {e;(w, t)} is described as the following modal expres­sion form.

(3.2.26)

where

(3.2.27)

To calculate {e;(w, t)} from Eq.(3.2.26), the convolution integral in Eq.(3.2.27) is evaluated recursively for each mode corresponding to the eigen value ;>.i (j = 1 · · · 2M) . It is noted that in this method the shape of the power spectral density of the input motion : S;i (w) is arbitrary in frequency.

- 21 -

Input : Empirical Rock Motion I

Power Spectral Density j S1(f) '

(2)

System:

Updating System Matrix I ' I Gl' hl I ' I G2, h21 I I ~ '

I '

I

Linear Random Linear Random Analysis Analysis

t Covariance of strain

. O"y(t1)2

Constitutive Law

~ y eff(tl)= f2 a:y(tl)

G- Y eff' h- Y err relation

Fig. 3.1 Conception of Piecewise Linear Stochastic Analysis - 22 -

~ START I .--------------, Metil:xi-1

p Stochastic IniUt Motion: f(t) f(t) = n(t) · s(t)

1----~ ~ Initial S::li I Model Information on Method-2 Initial state ~?k----l Soi 1 Data

• ARMA model oonsidering source and path : s ( t)

s(t)~S(f) :Power S~trum I A(l/z) I 2

S (f)=So I B(l/z) I 2

A(l/z)=~aiz- 1 , B(l/z)=~biz-1

z=1exp(i2:d.6.t)

1

· Envelope furx::tion : n (t)

H Seismolosdcal aooroach · Power Spectrun : S (f) 1-

S(f)= I S(f)P(f)H(f) 1 2

P(f) :Transfer function of wave propagation path

H (f) : Transfer ftmction of the dee~r so i I

S(f)f-----..., S(f)= !SFo (f) : Fo : soorce ! spectrum : a;w-2

o'---::f'-c -----=f

· Envelq:>e function: n(f, t) T d=O. 5L(l/VR-COS e IVa)

n(f,t) Td K exp(,n-ft/Qc)/t

1

~Qca;f

0 ta t=n.6.t

Y Statistical aoorcach · Power Spectrum : S (f) =S (f, M, .6.) f­

. Envelq:>e function : n (f, t)

.-----;....:» matrix [A] Vs(z) , p (z) G-r,h-r

.-------~::: Each time step

\

Calculation of ampli­fication in frequency

\

Cal. of oovar iance : Ace. Vel. Strain

Effective strain t r~ff=cov C r)

New shear modu 1 us : G New damping factors: h

(G-r, h-r)

Change state matrix[A]

:<Nonlinear : Random I

: Response> : (Strain compatible ' piece-wise

linear analysis)

------------ ____________ !

power 1[\

t '-----+-,~

Output of covariance response and Evolu­tionary power s~ctrum f

Reliability estimation Intensity amplification ·Maximum response · Envelq:>e function Spectral amp! ification

\

jENDj

Fig. 3.2 Flow Chart

- 23 -

4. VERIFICATION OF ANALYSIS METHOD

To verify the described two methods and the developed computer codes, the following verification analyses are performed and it is confirmed that the described two methods lead to exactly the same results when the power spectral density has the form of a ratio of two polynomials in w2 as expected. Computaion time of the method based on the moment equations is a drastic reduction of order compared with that of the method based on evolutionary power spectra.

First, a linear random response analysis of the simple MCK-system, 3 story-frame as shown in Fig. 4.1, subjected to the nonstaionary non-white stochatic ground moiton investigated by Langley (1986) is performed. Kanai-Tajimi power spectral densidy is specified as the following form.

( 4.1)

and the parameters in the above spectrum are taken as v9 = 0.6 , w9 = 4• . As the envelope time functions (modulation function), the step function and the exponential function are used. The obtained results are compared with those by the reference as shown in Fig. 4.2. It is found that the described method leads to exactly the same results as those by La.ngley (1986).

Second, both linear and nonlinear random response analyses are performed using a typical soft soil on rock. Results are compared with those by the Monte Carlo 1-Iethod where Ramber~­Osgood type nonlinear spring are used. Fig. 4.3 shows the soil profile and analysis model (a lamped mass model with shear spring). Fig. 4.4 shows the strain dependency of the shear modulus and the damping factor ( h - 1 , G- 1 ) for silt and sand.

Fig. 4.5 shows the power spectral density in the form of a ratio of two I?olynomials in w2

(Yama.cla-Tayerniya(1970)) and the envelope time function (Amin-Ang(1968J) of the stochastic input motion. Update of the system due to the nonlinearity is done based on the following relation between the covariance of the strain and effective strain.

~·1?.1! =cov.(-y) ( 4.2)

Fig. 4.6 shows 5 waves among 100 sample waves which are used as input waves in the simulation by Monte Carlo Method. Fig. 4.7 shows the hysteretic loops in 1st layer and 5th layer clue to a sample wave.

Fig. 4.8 shows the covariance responses of the acceleration at two levels of the soil in both linear and nonlinear cases. Solid lines are the results of the random response described in section 3. Dotted lines are those by the Monte Carlo Method. It is found from these figures that the agreement is generally good though there are some discrepa.nsies in nonlinear case.

Through the described verification analyses, it is confirmed that the developed codes based on the describecl method lead to reasonable results with adequately high accuracy for the investigation of soil amplification factors.

- 24 -

u,

120kN/m2

Uz

240kN/m2

u3

360kN/m2

I - Ug TTT

Fig. 4.1 Schematic of Structure (after Langley,1986)

Table 4.1 System Properties (after Langley,1986)

Mode shape vector in Natural Modal thousandths of metres

frequency damping Participation (rad/s) ratio factor Mass I Mass 2 Mass 3

Mode I 4·58 0·05 -85·50 16·66 10·73 4·99

Mode 2 9·82 0·05 35·77 14·27 -8·52 -9·64

Mode 3 14·59 0·05 -19·41 4·65 -11·95 I 1·49

- 25 -

0·15 0 0 15 ....-..

('.l

~ ..__, ,...- ~ E C.f.)

~ 0·10 z 0 0 10 z 0 0

~ Cl. Vl .... C.f.) c:

~ .... > ~ ;::: < ~ _, .... > c: Oo05 005 ~

~ ~ ~

000~--~--~----~--~--~ 0 0 2·5 5·0 7·5 100 12-5

TIME !SECS.)

oooo~--~--~--~----~~

OoO 205 5o0 705 1000 1205 Time( sees)

(a) Step Modulating Function

015r-----.------,------.----..------,

N"" l:

~010 z 0 Cl. V) .... c: .... ~ 1-< _, .... c:

005

2°5 5·0 7·5 10·0 12·5

TIME !SECS.)

0015 ....-..

('.l

~ ..__, ~ Ul-U2 C.f.)

z 0 0 10 U2-U3 0 ~ C.f.)

ga U3-Ug ~ > 0005 ~

~ ~ ~ ~

oooo~--~--~--~----~~

000 205 500 705 1000 1205 Time( sees)

(b) Exponential Modulating Function

Fig. 4.2 Comparison of the results with those of the previous papers

- 26 -

S-wave Categories Density Velocity ----- (t/m3) (m/sec)

Silt 1.5 100

Sand __ .,. ___ ----· ----- --·----· ... -----

Mudstone 1.7 800 -

(a) Soil Profiles and Parameters '

Mass

Fig. 4.3 Soil Profile and Dynamic model at Shiogama Site

G/Gmax Damping Factor 1 . oooooo r--===;;;;;:;:;;;;~;::::-~--r----r--r-----~1 0.40

0.500000

\ ......... :.:.:. \ ....... :,:...-'' ...... ,, .. ; .,,

.......... :., ....... ...... , ...

0.20

... ' ,,

0 . 000000 ~---'=--=--=·=··_ ..... _ .... _ .. ~_~~-;.;;.L-;;:._:.:._:.:..!,_:.:._:.:._:. .. _ .... _....J..__!. ___ ..~....-__.J 0. 0 0.000001 0.000010 0.000100 0.001000 0.010000

Shear Strain

Fig. 4.4 Strain Dependency of Shear Moduli and Damping Factor of the Soils at Shiogama Site

- 27 -

,.-..., C"'l

1500.0 (.) <l) CZl

N-a (.)

"--"" ;::..-.. 1000.0 ~

'"'""" CZl ~ <l)

Q ~

ro ;..... 500.0 ~ (.) <l)

0.. r.Jj

;..... <l)

~ 0 0.0 ~ 0.0 2.5 5.0 7.5 10.0

Frequency (Hz)

(a) Power Spectral Density (Yamada and Takemiya, 1970)

1. 0

0.0~--~--~--~----~--~--~--~~~

0. 0 1 0 . 0 20. 0 30 . 0 40 . 0

· Time (sec)

(b) Envelope Function (Amin and Ang, 1968)

Fig. 4.5 Power Spectral Density and Envelope Function of Stochastic Input Motion for the Verification of Analysis Method

- 28 -

(cm/sec2)

100.0

-100.0

(cm/sec2)

100.0

-100.0

Max. Value (cm/sec2) 147. 405

Max. Value (cm/sec2) 132.420

Max. Value (cm/sec2) -149. 106

(cm/sec2)

1oo.o I ~~~~~~~~~~~~~~~~~~~~

-100.0

Max. Value (cm/sec2) 173.356

(cm/sec2)

1oo.o I ~~~~~~~~~~~~~~~~~--~~

-100.0 l ( cm/sec2

) Max. Value

wo.o LN~>~~~~~ h~~~~v~~~M~r4~~ -1oo.ol ·rv ~r~f~ ' 1~

.::;o .o • • I

Time( sec)

Fig. 4.6 Sample Input Waves for Monte Carlo Simulation

- 29 -

0.0 ~--······························ I I l I

································ 0.0

I I

-4 .OOOE-05 '----------~---------1~-<.000E-OS -4.000E-04 0.0 4.000E-04

Shear Strain

(a) Silt Layer

-B.OOOE-03 0.0 B.OOOE-03 3. OOOE-0<!

1

,-------------;-----------, 3. OOOE-OL

I

I i

. :

i I I

O.Or·······················-~,

I Jl/ ~ ,(Jf

i /!!. : I I :

! : ;

-3.000E-04 '--~--------~--------~ -3.000E-OL -S.OOOE-03 0.0 B.OOOE-03

Shear Strain

(b) Sand Layer

Fig. 4.7 Examples of Hysteresis Loops

- 30 -

,-..... N () Q) C/J ......_ E ()

'--"'

~ 0 ·-........ ro ~ Q)

.......... Q) () ()

~ U)

~ ~

,-..... N () Q) C/J ......_ E ()

'--"'

~ 0 ·-........ ro ~ Q) -Q) () ()

~ U)

~ ~

200.0

100.0

0.0 0.0

100.0

50.0

0.0 0.0

10.0 20.0

Analytical Method

Monte Carlo Simulation

30.0

Time (sec)

(a) Linear Case

Mass-I

Mass-5

10.0 20.0

Time (sec)

(b) Nonlinear Case

30.0

40.0

40.0

Fig. 4.8 Verification of Analytical Method by Numerical Simulation (Monte Carlo simulation is performed for 100 sample

waves based on non-linear analyses using R-0 type hysteretic loop.)

- 31 -

5. APPLICATION ANALYSIS

5.1 OBJECTIVE SITES AND LOCAL SOIL DATA

As a practical application, the ground motions at a typical Holocene (alluvial) site in Japan (Shiogama site) are investigated, where the evidence of the nonlinearity of the soil has been reported from the observation records due to the storong potential earthquakes. Fig . .5.1 shows the nonlinearity of the soil from the observation records at Shiogama site (Tokimatsu and Midorikawa, 1987). It is confirmed that the predominant period clearly becomes longer and the damping factor estimated from the shape of the spectra becomes larger with increase of the intensity level of the input motion.

In Fig . .5.2, the observational amplification values are plotted against the peak accelaration on the rock surface at Sendai Sumitomo Bldg. about 20 km away from Shioga.ma site. Used earthquake records are listed in Table .5.1. For the lack of records on rock surface near Shiogama site, the amplification values are calculated using the records obtained at basement floor of Sumitomo Bldg. on Neogene deposit with shear wave velocity of approximately 600 m/s. Fig. 5.2 shows the nonlinear effect of the soil, i.e. the decrease of amplificaton value with increase of the peak acceleration on rock surface. Though there may be the effect of the difference of epicentral distance, this fact must be due to nonlinear soil characterlistics.

Fig. 4.3 (in section 4) shows the soil profile and analysis model (a la.mpecl mass model with shear spring). The impedance ratio of the surface layer and the bedrock is about 10.

Fig. 4.4 (in section 4) shows the strain dependency of the shear modulus and the clamping factor ( h -1 , G- 1 ) for silt and sand at Shiogama site.

- 32 -

6/12,1978

11119,1973 (

I

\

9/25,1972 '\ 56cmJs

2

10 h= I cro ~ ,V 23cmJs2

I \

2/15,1973

. 2 .5 I. 2. PERIOD sec

Strong-Motion Records Used

SITE DATE M D X ( kml ( kml

SHIOGAMA SEP. 25, 1972 5.5 50 100 FEB. 15. 1973 4.9 50 160 NOV. 19, 1973 6.4 50 130 FEB. 20, 1978 6. 7 60 110 JUNE 12 1978 7.4 40 110

5 .

PEAK ACC. _icm/s2J

23 17 56

126 273

Fig. 5.1 Evidence of Nonlinearity of the Soil at Shiogama Site

(after Tokimatsu and Midorikawa, 1987)

- 33 -

4.00

3.00 •3 •S

~ 0 2.00 •4 ·-..... ro u ~ ·-.........

•6 ~ a ~ •1 •2

1. 00 0.90 0.80 0.70

0.60

3.0 10.0 100.0 800.0 Peak Acceleration on Rock Surface at Sendai

Sumitomo Bldg. on Neogene deposit (cm/sec2)

Fig. 5.2 Observational Amplification Values at Shiogama Site

Table 5.1 List of Used Earthquake Records

No Date Location Epicentra I Max. Ace. (cm/s 2 ) Average Magnitude Di st. (km) NS Camp. EW Camp. (cm/s 2 )

1 1978 2 20 Shiogama 98 106 129 118 M=6. 7 Sendai 119 105 100 103

2 1978 6 12 Shiogama 100 266 I

288 277 M=7.4 Sendai 105 250 245 248

3 1987 1 9 Shiogama 182 30 I

35 33 M=6.6 Sendai 193 13 10 12

4 1987 2 6 Shiogama 166 51 I

61 56 M=6. 7 Sendai 167 29 29 29

5 1987 4 7 Shiogama 135 68 86 77 M=6.6 Sendai 139 22 38 30 1987 4 23 Shiogama 148 41 41 41 6 M=6.5 Sendai 146 31 29 30

- 34 -

5.2 STOCHASTIC l\WTION ON ROCK SURFACE FOR COMBINATIONS OF MAGNITUDE AND EPICENTRAL DISTANCE

(1) 5T ATI5TIC ROCK A10TION

To predict eathquake ground motions is of great interest in earthquake engineering. Various lci.nds of prediction model for ground motion ha:ve been proposed in the last decade. One of the hopeful prediction model is that based on the statistical regression analysis of the observed earthquake records. Sugito and Kameda (1985) proposed such a prediction model for the motion on rock surface for given magnitude and epicentral distance. They, first, developed earthquake motion dataset on rock surface with the shear wave velocity v. = 700 m/ s and made a regression formula for evolutionary power spectrum on rock surface. Then, by using the dataset, the parameters in the formulas were obtained as a function of magnitude M and epicentra1 distance ~- The-formulas and the parameters are described as follows :

· where

jG(w, t) = jG(27r j, t) = O:m(f) · n(w, t)

= { 0

(f). t- t.(f) . ( - t- t.(f)) O:m tp(f) exp 1 tp(f)

log10 o:m(f) = Bo(f) + B1(f) · 111- B2(f) ·log1 o(~ +c) log10 tp(f) = Po(f) + P1(f) · M- P2(f) ·log10 (~ +c)

t.(f) = 5o(f) +51(!)·~

0 :::; t < t.

t. :::; t

Bo(f) = 0.1553 + 1. 750 ·log10 f - 0.336 (log10 J/ - 0.451 (log10 1)3

B 1 (f) = 0.506 - 0.0131 · log10 f B2(f) = 1.543 + 0.455 ·log10 f Po(f) = -1.312 + 1.054 ·log10 f + 0.227(log10 J? P1 (f) = 0.179 + 0.188 ·log10 f P2(f) = 0:344- 0.240 ·log10 f 5o(f) = 0.439- 0.878 ·log10 f 51(!)= (0.528- 0.242 ·log10 j- 0.889 (log10 1)2

) X 10-2

(5.2.1)

(5.2.2)

(5.2.3)

(5.2.4)

(5.2.5)

(5.2.6)

(5.2.7)

(5.2.8)

(5.2.9)

(5.2.10)

(5.2.11)

(5.2.12)

In above formulas, o:m(f) denotes the root of the power spectral density, and t.(f) and tp(f) are the starting time parameter and the duration parameter of G(w, t), respectively. The evolu­tionary power spectrum expressed by Eq.(5,2,1) is composed of the stationary power spectrum o:! (f) and the frequency dependent envelope time function n(w, t).

By using Eq.(5.2.1), the stochastic input motion on rock surface can be obtained easily for various combinations of magnitudes and epicentra1 distances.

- 35 -

(2) SEIS!o.!OLOGICAL ROC[( MOTION

Another prediction model has been proposed for stochastic motion on rock surface from the seismological point of view. Takemura and Kamata (1990) proposed the model estimating the ground motion by using the envelope time function and its amplitude determined on the basis of the theories of seismic-wave radiation and propagation. The amplitude of the wave is mainly determined from the duration time of the S-waves and the product of three parameters such as source spectrum, propagation path and local site effect. The source spectrum is assumed on the basis of the "w-square" model. The effect of the propagation path is evaluated from the Q.-structures of crust and the upper mantle. The local site effect is evaluated from the deep local subsoil. The basic formulas of the stochastic motion are described as follows:

Power spectal density

P(f) = ;d U(f?

U(f) = S(f) H(f) exp (- 11" ~;s) S(f) = {3. 411"

3 Mo . j4

X pVj 1 + (tr Td = 0.5L ( ~R - ~s cos e)

Envelope time function

{

0 1.0

n(f, t) = exp ( -7/-l) . (exp (-,.I c;;:T•))) -I t (ts + Td)

o::; t < ts is::; t < is+ Td

P(f) : power spectral density

U(f) : Fourie spectrum of the S- waves

ts : starting time of the S- waves

Qs : Q- val·ue of the S- waves

H(f): transfer function of the deep local subsoil

S(f) : source spectrum

lifo : seismic moment

X : hypocentral diatance

p: density

Td : duration time of the input motion

L: length of the fault

VR : velocity of the fault rupture

Vs : velocity of the S -waves

e: azimuth of the observaionpoint from the

strike direction of the fault

{3: factor representing the heterogeneity

of the fault rupture

Qc : Q- value of the coda- waves

fc : corner frequency

- 36 -

(5.2.13)

(5.2.14)

(5.2.15)

(5.2.16)

(5.2.17)

In this study, each parameter in above formulas is assumed to be constant or to be a function of the magnitude as follows.

ts = 2.0 sec

H(f) = 2.0

p = 3.0 gfcm3

VR = 3.3 km/ sec

Vs = 4.0 km/ sec

/3 = 1.0

Qs = Qc = 60 ·fLo log10 L = 0.5M - 1.88

{

1.5(M- 0.2) + 16.2 log10 Mo = 2.25M + 11.3

1.5(.111 + 0.2) + 16.2

1 log10 fc = -3(log1o Mo- 24.08)

M < 6.2 6.2 5 M < 6.9 6.9 5M

(5.2.18)

(5.2.19)

(5.2.20)

(5.2.21)

It should be noted that the described quality factor is specified based on the inYersion analysis (Kato et al., 1991)

- 37 -

5.3 APPLICATION ANALYSIS

(1) AMPLIFICATION CHARACTERISTICS DUE TO STATISTIC ROCK MOTION

Fig. 5.3 and Fig. 5.4 show spectral intensities of the motion on rock surface for different magnitudes and for different epicentral distances, respectively. As for the case of different magnitudes, the epicentral distance is 100 km. In turn, as for the case of different epicentral distances, the magnitude is 7.5.

The rms (root mean square) inputs on rock surface for different magnitudes and for different epicentral distances are shown in Fig. 5.5 and Fig. 5.6, respectively. In each figure, the nonstationarity of the input motion can be seen.

Fig. 5. 7 shows the evolutionary power spectra on rock surface for the case that maguni­tude is equal to 4.5 and 8.0. The assumed stochastic input motion on rock surface has the nonstationarity both in time and in frequency.

The proposed nonlinear random response methodolgy is applied to Shiogama site using the empirical rock motion shown in above figures.

The rms responses on ground surface are compared ·with those on rock surface in Fig. 5.8 and Fig. 5.9. As for the case that the magnitude is equal to 8.0 in Fig. 5.8, the rms response of acceleration on ground surface deary presents the nonlinearity, i.e. the maximum rrns response of acceleration on ground surface is less than that on rock surface in the main portion of excitation. As for the case that the epicentral distance is equal to 50km shown in Fig. 5.9, the same phenomenon can also be seen.

Fig. 5.10 shows the evolutionary power spectra on ground surface as for the case that magnitude is equal to 4.5 and 8.0. The predominant period becomes longer with increase of the intensity of the input motion.

The above results lead to the following amplification factors, i.e. spectral amplification factor f3s(f) and rrns intensity amplification factor {31• These amplification factors are defined as follows.

where

f3s(f) = J:: Ss(f, t)dt

J0T S(f) · n 2 (f, t) dt

f3I = Max[o-a.oud(t)] .1\1 ax[o-Rock ( t)]

Ss(f. t): evolutionary power spectrum of the motion on ground surface

S(f): power spectral density of the input motion on rock sur face

O"G•o•nd( t) : rms response on ground surf ace

O"Rock(t): rms response on rock surface

(5.3.1)

(5.3.2)

Fig. 5.11 shmvs that spectral amplifications for different magnitudes and for different epicentral distances are plotted against the frequency. It is found that the amplification factor {35 (1) decreases remarkably in high frequency range with increase of the intensity of the input motion in comparison with in low frequency range.

Fig. 5.12 shows that rms intensity amplifications defined by (5.3.2) for different magnitudes and for different epicentral distances are plotted together against the rms acceleration on rock surface. It is found that the amplfication factor smoothly decreases with increase of the rms accerelation on rock surface. In Fig. 5.13, this intensity amplification factor {31 is compared with the observational amplification values and the amplification factor f3a(f) proposed by Sugito (1986). The observational amplification values are denoted in section 5.1. In section 2.2, the

- 38 -

amplification factor proposed by Sugito is explained in detail, whish is specified by geotechnical parameters S,. and dp. Since the stochastic random response analysis cannot directly treat the peak acceleration, the peak factor about accceleration is assumed to be 3 both on rock surface a.nd on ground surface. As shown in Fig. 5.13, the calculated nonlinear soil amplification characteristics are consistent with those of observation records and those obtained by statistical method though there are some discrepancies, particulary in low intensity levels.

- 39 -

,-.... C""l () Q)

~ s () 10. 00

....... 0 0 ~

0.01~--~~~~~~--~~~~~~

0. 10 0. 50 1. 00

Frequency (Hz) 5.0010.00

Fig. 5.3 Spectral Intensities of Empirical Motion on Rock Surface by the Statistic Approach for Different Magnitudes (after Sugito and Kameda, 1985) [Epicentral Distance is equal to 100 km.]

- 40 -

,-...... ('f') g 100.00

CZl

~ 8 u

0 ·u; 10.00 ~ Q)

Q

0. 50 1. 00

Frequency (Hz)

5.0010.00

Fig. 5.4 Spectral Intensities of Empirical Motion on Rock Surface by the Statistic Approach for Different Epicentral Distances (after Sugito and Kameda, 1985) [Magnitude is equal to 7 .5]

- 41 -

M=8.0

20.0 M=6.0

·M=5.5 M=5.0

I ~~~~~~:E~~~M~=4:·~5~::::~~~==~~j 0.0~ 0.0 10.0 20.0 30.0 40.0 50.0

Time( sec)

Fig. 5.5 RMS Inputs on Rock Surface for Different Magnitudes [Epicentral Distance is equal to 100 km.]

- 42 -

,.......__ N u Q)

~ 100.0 u ...._.., ~ 0 ........ ...... ro ~ Q) -Q) u u ~ 50.0 (/.)

~

b.=50km

10.0 20.0 30.0 40.0 50.0

Time( sec)

Fig. 5.6 RMS Inputs on Rock Surface for Different Epicentral Distances

[Magnitude is equal to 7 .5]

- 43 -

o.oo

Power Spectral Density (crd-fsec3

) 16-67

Q.39

Q.26

0.13

Power Spectral Density (cm2/sec3

)

1211 .98

807-99

Q.OO Q.

so.oo

Frequency (Hz)

(a) M==4.5

Frequency(Hz) (b) M==8.0

Fig. 5.7 Evolutionary Power Spectra on Rock Surface

by the Statistical Approach (Hypocentral Distance is equal to 100 krn.1

- 44 -

o.oo

16-67

33-33

lOO.Or----,-----,-----,----,-----.-----.---~r----.----~----~

10.0

M=8.0

\ \ \

\ \

\ \

\ \

). ~

~ \

\ \

\ '\

'\

------ Rock Surface --- Ground Surface

'\ '\

' '\ ' ' ' ' ' ' ' ' ' ' ' ' .... ,

........ ........

............ ..... _ -- ....

20.0 30.0 40.0

Time (sec)

-------50.0

Fig. 5.8 Comparison of RMS Responses on Ground Surface with those on Rock Surface for Different Magnitudes [Epicentral Distance is equal to 100 km.]

- 45 -

,..--._ N (.) Q) C/:J --a 100.0 (.) '-"

~ 0 ·-.;.......>

ro ;....; Q)

,......; Q) (.) (.)

< 50.0 (.f)

~ ~

D.=50km /-,, I \ /

I \ / I \ / I \ //

! \~ I \ I \ I \ I I I I I I I I I I I I I I

10.0 20.0 30.0

Time( sec)

Rock Surface Ground Surface

40.0 50.0

Fig. 5.9 Comparison of RMS Responses on Ground Surface with those on Rock Surface for Different Epicentral Distances [Magnitude is equal to 7 .5]

- 46 -

Power Spectral Density o.oo (cm2/sec3)

o.oss 16.53

0.059

0.029 33.07

Frequency (Hz) 49.60

(a) M=4.5

Power Spectral Density o.oo (cm2/sec3)

162.75

16-53

108.50

54.25 33.07

Frequency (Hz) 49.60

(b) M=8.0

Fig. 5.10 Evolutionary Power Spectra on Ground Surface considering Nonlinear Soil Amplification [Hypocentral Distance is equal to 100 km.]

- 47 -

5.00

f3s(f) = I: Ss(f, t) dt I: S(f) · n2 (f, t) dt

0 . 1 0 '------l.-L-.J.......l.-L-'-l...l...l---'---...l..-1----'--.l....l...J..-'-f

0. 10

5.00

0. 50 1. 00

Frequency (Hz) 5.0010.00

(a) Different Magnitudes

D.=50km

\

f3s(f) = I: Ss(f, t) dt IoT S(f) · n2(f, t) dt

0 . 1 0 .______._.....__,_-'-L--L...I...J'-'-----'---'-L-L-.l-I-L-4J

0. 10 0. 50 1. 00

Frequency(Hz) 5.0010 00

(b) Different Epicentral Distances

Fig. 5.11 Spectral Amplifications for Different Magnitudes and for Different Epicentral Distances

- 48 -

I

1. 00

f3r = Maz[ua.,.ound(t)] M az[uRoek(t)]

0.50~----~--~----~--~------L-~

1. 0 10.0 100.0 1000.0

RMS Acceleration on Rock Surface ( cm/sec2)

Fig. 5.12 RMS Intensity Amplification for Different Magnitudes

and Different Epicentral Distances

- 49 -

4.00

3.00

2.00

1. 00 0.90 0.80 0.70

0.60

3 •5 f3a

/ (Sugito, 1986)

by EM-l(Statistical Approach)

L--L---J--------~----L---------L-~

3.0 10.0 100.0 800.0

Peak Acceleration on Rock Surface (cm/sec2)

Fig. 5.13 Comparison of Calculated Intensity Amplification with Observational Amplification Values at Shiogama Site [Peak factor is assumed to be 3.]

Table 5.2 List of Used Earthquake Records

No Date Location Epicentral Max. Ace. (cm/s 2) Average Magnitude Di st. (km) NS Comp. EW Comp. (cm/s 2 )

1 1978 2 20 Shiogama 98 106 129 118 M=6. 7 Sendai 119 105 100 103

2 1978 6 12 Shiogama 100 266 288 277 M=7.4 Sendai 105 250 245 248

3 1987 1 9 Shiogama 182 30 35 33 M=6.6 Sendai 193 13 10 12

4 1987 2 6 Shiogama 166 51 61 56 M=6. 7 Sendai 167 29 29 29

5 1987 4 7 Shiogama 135 68 86 77 M=6.6 Sendai 139 22 38 30

6 1987' 4 23 Shiogama 148 41 41 41 M=6.5 Sendai 146 31 29 30

- 50 -

(2) AMPLIFICATION CHA.RACTERISTICS DUE TO SEISllfOLOGICAL ROCK MOTION

Figs. 5.14, 5.15 and 5.16 show spectral intensities of the motion on rock surface for different magnitudes, for different hypocentral distances and for different azimuths, respectively. In this model, the effect of the azimuths of the observation point from the strike direction of the fault is considered. In Figs. 5.14 and 5.15, the azimuth is assumed to be 90°. The other values of the parameter are assumed as the same manner as (1).

The rms inputs on rock surface for different magnitudes, for different hypocentral distances and different azimuths are shown in Figs. 5.17, 5.18 and 5.19, respectively. In each figure, the nonstationarity of the input motion, i.e. the main shock composed of the S-waves and the following decreasing portion composed of the coda-waves, can be seen.

Fig. 5.20 shows the evolutionary power spectra on rock surface for the case that magni­tude is equal to 6.5 and 8.0. The assumed stochastic input motion on rock surface has the nonstationarity both in time and in frequency.

The proposed nonlinear random response methodolgy is applied to Shioga.ma site using the empirical rock motion shown in above figures.

The rms responses on ground surface are compared with those on rock surface in Fig .. 5.21, 5.22 and 5.23. As for the case that the hypocentral distance is equal to 50 km. in Fig. 5.22, the rms response of acceleration on ground surface deary presents the nonlinearity, i.e. the maximum rms response of acceleration on ground surface is less than that on rock surface in the main portion of excitation.

Fig. 5.24 shows the evolutionary power spectra on ground surface in the case that magu­nitude is equal to 6.5 and 8.0. The predominant period becomes longer with increase of the intensity of the input motion.

The above results lead to the same amplification factors {35 (1) and {35 (1) as defined by Eqs.(5.3.1) and (5.3.2).

Figs. 5.25 and 5.26 show that spectral amplifications for different magnitudes, for different hypocentral distances and for different azimuths are plotted against the frequency. It is found in all cases that the amplification factor !3s(J) decreases remarkably in high frequency range with increase of the intensity of the input motion in comparison with in low frequency range.

Fig. 5.27 shows that rms intensity amplifications for different magnitudes, for different epicentral distances and for different azimuths are plotted together against the rms acceleration on rock surface. It is found that the amplfication factor smoothly decreases with increase of the rms accerelation on rock surface. In Fig. 5.28, this intensity amplification factor {31 is compared with the observational amplification values, the amplification factor !3a(f) proposed by Sugito (1986) and also with the results of the last section (1), in which the statistical stochastic input motion is assumed. As shown in Fig. 5.28, the characteristics of the calculated nonlinear soil amplification are consistent with those of observation records and those obtained by statistical method though there are some discrepancies, particulary in low intensity levels. In particular, the two results calculated using the statistical rock motion and the seismological rock motion coincides with each other. This fact means that the intensity amplification factor defined by Eq.(5.3.2) is independent of the shape of the input spectrum on rock surface. The shape of spectra of the empirical rock motion used in this study is wide-band, then the characteristic of the sysytem, i.e. transfer function of the system mainly governs the charcteristics of the amplifications. If the shape of the input spectrum on rock surfa~e is narrow-band, then the amplification factor will be highly dependent on the input motion.

Additional two analyses are performed for the purpose of the investigation of sensitiYity on the nonlinear soil amplification characteristics.

- 51 -

First, to investigate the sensitivity of the amplification factor to the impedance ratio of rock and surface layer, the shear wave velocity of the bed rock is changed in the manner as shown in Table 5.4. The resulting intensity amplification factors are shown in Fig. 5.29. It is found that the impedance ratio of rock and surface layer affects directly the amplification factor, especially in lower intensity level, therefore the definition of 'rock' should be clarified when the amplification factor is discussed.

Second, to investigate the sensitivity of the amplification factor due to the strain dependency of the soil, that of the bay-mud is utilized for the nonlinearity of the soil at Shiogama site. Fig. 5.30 shows the strain dependency of shear moduli and damping factor of bay-mud (after H ryciw et al.,1991). The resulting intensity amplification factors are shown in Fig. 5.31. It is clearly found that the strain dependency of the soil affects the decrease of the amplification factor more importantly in higher intensity level.

- 52 -

~ M u (!.) 00

N--a u

'-"' ~ ....... -~ 00 ~ (!.)

Q ......... ~ ;....; ....... u (!.)

0.. (/.)

;....; (!.)

~ 0

A1 ~ 0

....... 0 0 ~

10.000

1.000

0. 100

0.010

0.001~--~~~~~~--~--~~_L~

0. 10 0. 50 1. 00

Frequency (Hz) 5.0010.00

Fig. 5.14 Spectral Intensities on Rock Surface by Considering Source, Path and Deep Soil Amplification for Different Magnitudes [Hypocentral Distance is equal to 100 km. Source spectrum is assumed to be Omega-square model.

Path information is based on the inversion analysis.

Deep soil amplification is assumed to be 2.] - 53 -

,..--._ cu 1 0 0 . 0 0 ...-----.---.--.-,--,-.,.--,--,-r----,----r---r--r--r--r-r...,.-,

Q) Cl".l

~ 5 z~~~s_ok_m __________________ ~

:>-. -~ 10.00 x~lOOktn ~ -~~-----------------~

Q x~lSOkm

1. 00 x~zookrn

0.10~--~~~~~~--~~~~~~

0. 10 0. 50 1 . 00

Frequency(Hz) 5.0010.00

Fig. 5.15 Spectral Intensities on Rock Surface by Considering Source, Path and Deep Soil Amplification for Different hypocentral Distances [Magnitude is equal to 7.5. Source spectrum is assumed to be Omega-square model. Path information is based on the inversion analysis. Deep soil amplification is assumed to be 2.]

- 54 -

,.-._ ~

~ 100000.---.--,-,~~~--~--~~~~ r::r.J

~ a u

~ ....... "c;) 10 0 00

a:s Q ~

~ ~ ....... u (])

0-4 (/) 1 0 00 [) ~ 0 ~ 4-1 0 ....... 0 ~ 0 010

0 0 10 0 0 50 1. 00

Frequency (Hz) 5.0010000

Fig. 5.16 Spectral Intensities on Rock Surface by Considering Source, Path and Deep Soil Amplification for Different Azimuths [Hypocentral Distance is equal to 100 km. Magnitude is equal to 7.5. Source spectrum is assumed to be Omega-square model. Path information is based on the inversion analysis. Deep soil amplification is assumed to be 2.]

- 55 -

50.0 ~ N u (].) CI'J

8 u ~ 40.0 ~ 0 M=8.0 ·-~ ~ ~ (].) M=7.5 -(].) u

20.0 M=7.0 u <t:: U')

~ 0.0

0.0 10.0 20.0 30.0 40.0 50.0

Time( sec)

Fig. 5.17 RMS Inputs on Rock Surface for Different Magnitudes [Hypocentral Distance is equal to 100 km.]

- 56 -

iOO.O ,-..... <'I u

X=50km C) C/) --a u

'--"

t:: 0 ·~ ~

ro 50.0 ~ C)

.....-4 C) u u ~ X=lOOkm Cl)

~ X=150km

/X=200km ~ /

y 0.0

0.0 10.0 20.0 30.0 40.0 50.0

Time( sec)

Fig. 5.18 RMS Inputs on Rock Surface for Different Hypocentral Distances

[Magnitude is equal to 7 .5]

- 57 -

,..-..,. C'l u (]) r.n --a u ,__., ~ 0

• ....-4 ~

~ ~ (]) ........ (]) u u ~ tl)

~ ~

60.0

e = oo

40.0 e = 45°

e = 90° /8~135'

e = 180° 20.0

0.0 0.0 10.0 20.0 30.0 40.0 50.0

Time( sec)

Fig. 5.19 RMS Inputs on Rock Surface for Different Azimuths

[Magnitude is equal to 7.5. Hypocentral Distance is equal to 100 km.]

- 58 -

Power Spectral Density (cm2/sec3

)

1Q.\.l8

3.Lt9

Density (cm2/sec3

)

106·90

71 .27

35.63

Q.OO o.

Frequency(Hz) (a) M==6.5

Frequency(Hz)

I I I

10.00

o.oo

16.67

33.33

so.oo

o.OO

16.67

10.00

(b) M==8.0

Fig. 5.20 Evolutionary Power Spectra on Rock Surface · by Considering Source, Path and Deep Soil Amplification

(Hypocentral Distance is equal to 100 Jan.}

- 59 -

SO.Or---~----~---r----,----,---.----.----.----.---~

------- Rock Surface -- Ground Surface

20.0

10.0 20.0 30.0 40.0 50.0

Time (sec)

Fig. 5.21 Comparison of RMS Responses on Ground Surface with those on Rock Surface for Different Magnitudes [Hypocentral Distance is equal to 100 km.]

- 60 -

100.0 ,.-..... N

C) <!) Cl':l -- ----- Rock Suface a C)

-- Ground Surface '--"'

~ 0 ·-~ 50.0 ~ X=lOOkm <!) -<!) C) C)

< t:/.)

~ ~

10.0 20.0 30.0 40.0 50.0

Time( sec)

Fig. 5.22 Comparison of RMS Responses on Ground Surface with those

on Rock Surface for Different Epicentral Distances [Magnitude is equal to 7 .5]

- 61 -

------- Rock Surface Fault Plain

--- Ground Surface

Observation Point

10.0 20.0 30.0 40.0 50.0

Time (sec)

Fig. 5.23 Comparison of RMS Responses on Ground Surface with those on Rock Surface for Different Azimuths [Magnitude is equal to 7.5. Hypocentral Distance is equal to 100 km.]

- 62 -

(i.OO

power spectral Density (cm?-fsec3

)

16·S3

i_.08

o./2.

o.oo Freq_uencJ (fl:z.)

(a) }.1:::::6.5

o.oo

power Spectral Density (crn2fsec3

)

i.6·S3

33·62 33·07

11 . 21 4.9·60

o.OO

(b) 1\11?8.0

Fig. 5.2.4 tsolutionarY power Spectra on G~o~nd. surface considering Nonlinear Soil j\rnphfteatlon U:lypocenttal Distance is equal to 100 ktn·1

- 63 -

,....._ "+--. ..._

<l"l '0...

$:1 0 ........ .......

5.00

C\:$ u 1.00 ·-~ -­.....-( 8 0.50

f3s(f) = f; Ss(f, t) dt J: S(f) · n2(f, t) dt <

0 . 1 0 '----'-.1....-J..--1.....1--'--L...L..I....--..l..--.....1--1-J....l.....L.l....l+J

0. 10

Frequency (Hz)

(a) Different Magnitudes

I I I I I II

X=50km

\

f3s(f) = f; Ss(f, t) dt foT S(f) · n2(f, t)dt

0 . 1 0 .____.._.i..__..C__t__!......:....!...J....J.._-_.!..___J__...J........J......!.....J...~

0. 10

Frequency (Hz)

(b) Different Epicentral Distances

Fig. 5.25 Spectral Amplifications for Different Magnitudes and for Different Hypocentral Distances

- 64 -

I I I II

T f3s(f) = fo Ss(f, t) dt

foT S(f) · n2(f, t) dt

0 . 1 0 L------1-l-1.-L.JI...!..L.l...!...--_--l.._--L._...L.....L...J....J....J...LJ

0. 10 0.50 1. 00

Frequency(Hz) 5.0010.00

Fig. 5.26 Spectral Amplification for Different Azimuths [Magnitude is equal to 7.5. H ypocentral Distance is equal to 100 km.]

- 65 -

1 . 00

0.50L-----~--~----~~~------~~

1 . 0 10.0 100.0 1000.0

RMS Acceleration on Rock Surface ( cm/sec2)

Fig. 5.27 RMS Intensity Amplification for Different Magnitudes, Different Hypocentral Distances and Different Azimuths

- 66 -

4.00

3.00 3 •5 {3 a

2.00 (Sugito, 1986)

~~ 1. 00 0.90 0.80 0.70

by EM-l(Statistical Approach)\

0.60

3.0

\ \

by EM-2(Seismological Approach)

10.0 100.0 800.0

Peak Acceleration on Rock Surface ( cm/sec2)

Fig. 5.28 Comparison of Calculated Intensity Amplification with

Observational Amplification Values at Shiogama Site

[Peak factor is assumed to be 3.]

Table 5.3 List of Used Earthquake Records

No Date Location Epicentral Max. Ace. (cm/s 2 ) Average Magnitude Dist. (km) NS Comp. EW Comp. (cm/s 2)

1 1978 2 20 Shiogama 98 106 129 118 M=6.7 Sendai 119 105 100 103

2 1978 6 12 Shiogama 100 266 288 277 M=7.4 Sendai 105 250 245 248

3 1987 1 9 Shiogama 182 30 35 33 M=6.6 Sendai 193 13 10 12

4 1987 2 6 Shiogama 166 51 61 56 M=6. 7 Sendai 167 29 29 29

5 1987 4 7 Shiogama 135 68 86 77 M=6.6 Sendai 139 22 38 30

6 1987 4 23 Shiogama 148 41 41 41 M=6.5 Sendai 146 31 29 30

- 67 -

Table 5.4 Soil Parameters

Rock Surface Layer Case

Vs (m/sec) p (t/m3) Vs (m/sec) p (t/m 3

)

1 500.0 1.7 100. 0 1.5

2 800.0 1.7 100.0 1.5

3 1200.0 1.7 100. 0 1.5

4.00

3.00

~ 0 2.00 ....... ...... eel 0 c.c .......

.....-1

0.. s <t::

1. 00 0.90 0.80 0.70

0.60

3.0 10.0 100.0 800.0

Peak Acceleration on Rock Surface (cm/sec2)

Fig. 5.29 Effect of Impedance ratios on Intensity Amplification [Peak factor is assumed to be 3.]

- 68 -

GIG Damping Factor max

! . 000000 r--!!!!'!53;~;:::::::::===-r--r---....--~---.------, 0.30

,·

0.500000

Shiogama-Silt , ., Shiogama-Sand ·',,, _,;;::.;;/' ~

\ .. ,~ I~';'

~ ~'"'/

-----:::::::::::::::::::;:;;;;;;:::=-:.:O'->- 0-0 0. 000000 L---===:L----L---1...---..l.....----'----'----'

0.000001 0.000010 0.000100 0.001000 0.010000

Shear Strain

Fig. 5.30 Strain Dependency of Shear Moduli and Damping Factor

4.00

3.00

~ 0 2.00 ·-....... ro u ·-~

Bay-mud model

/ ·-...--.4

~

8 -<C

1. 00 0.90 0.80 0.70

0.60 I I

3.0 10.0 100.0 800.0 Peak Acceleration on Rock Surface ( cm/sec2)

Fig. 5.31 Effect of the different nonlinear characteristics on Intensity Amplification [V s of Rock and Surface layer are assumed to be

1200m/s and lOOm/s. Peak factor is assumed to be 3.]

- 69 -

6. CONCLUSIONS

Concluding remarks in this study are summarized as follows.

The proposed nonlinear random response methodology using the empirical ground motion on rock surface can be applicable to prediction of the ground motion considering nonlinear amplification of local subsoil.

Through the application to a typical Holocene site in Japan (Shiogama site), it is fotrnd that the calculated nonlinear soil amplification characteristics are consistent with those of observation records and those obtained by statistical method.

In soft soil deposit with high contrast, the amplification curve for the intensity on rock surface does not depend on the shape of the spectral intensity of rock motion, but depends on the impedance ratio of rock and surface layer and on the nonlinear characteristics of the surface layer.

The impedance ra.tio of rock and surface layer affects directly the amplification factor, especially in lower intensity level. Therefore, the definition of 'rock' should be clarified when the amplification factor is discussed.

The strain dependency of the soil affects the decrease of the amplification factor more importantly in higher intensity level.

- 70 -

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- 71 -

Sugito,M. and Kameda,H., {1985). Prediction of nonstationary earthquake motions on rock surface, Proc. of JSCE, SE/EE, Vol.2, pp.l49-159. .

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Tokimatsu,K. and Midorikawa,S. (1987). Strain-dependent dynamic properties of surface soils estimated from strong motion accelerograms, Journal of Structural and Construction Engineer­ing (Transactions of AIJ), Vol.388, 131-137.

Vanmarcke,E.H. (1977). Random vibration approach in soil dynamics, The Use of Probab. in Earthq. Engrg., ASCE, pp.143-176

Yamada,Y. and Takemiya,H. (1970). Statistical estimation of the maximum response of struc­tures subjected to earthquake motion, Proc. of JSCE, No.182, pp.llS-132.

- 72 -

SOIL AMPLIFICATION CHARACTERISTICS DUE TO LOCAL SITE EFFECTS SUBJECTED TO STOCHASTIC MOTION ON ROCK SURFACE

PART-2

GROUND MOTION CHARACTERISTICS CONSIDERING TOPOGRAPHICAL AND SUBSURFACE IRREGULARITIES OF THE SOIL

SUBJECTED TO STOCHASTIC MOTION

TABLE OF CONTENTS

SUMMARY

PAGE

LIST OF FIGURES 11

LIST OF TABLES iii

1. INTRODUCTION 1

2. NONSTATIONARY RANDOM RESPONSE ANALYSIS STARTING FROM THE ARBITRARY 2 TRANSFER FUNCTIONS

(1) OUTLINE OF l\1ETHOD 2 (2) BASIC EQUATIONS 2

3. APPLICATION ANALYSIS 6

3.1 AMPLIFICATION OF GROUND MOTION IN SEDIMENTARY BASIN 6

(1) ANALYSIS MODEL AND SOIL DATA 6 (2) STOCHASTIC INPUT MOTION 6 (3) RESULTS OF SOIL AMPLIFICATIONS 9

3.2 Ai\'IPLIFICATION OF GROUND MOTION IN FOLDED SUBSURFACE IRREGULARITY 17

(1) ANALYSIS MODEL AND SOIL DATA 17 (2) STOCHASTIC INPUT MOTION 17 (3) RESULTS OF SOIL AMPLIFICATIONS 20

4. CONCLUSIONS 24

5. REFERENCES 25

SOIL AMPLIFICATION CHARACTERISTICS DUE TO LOCAL SITE EFFECT SUBJECTED TO STOCHASTIC MOTION ON ROCK SURFACE

PART-2

GROUND MOTION CHARACTERISTICS CONSIDERING TOPOGRAPHICAL AND SUBSURFACE IRREGULARITIES OF THE SOIL

SUBJECTED TO STOCHASTIC J'viOTION

SUl\.fMARY

OBJECTIVE Ground motions on a sedimentary deposit are reportedly characterized as lager amplitude

and longer duration than those on rock surface. These amplifications are strongly dependent on the local site conditions. Many studies have suggested from the theoretical approach and observations that the topographical and subsurface irregularities play an important role in estimating ground motion. Moreover, amplification factors are quite sensitive to input motions which are of a deterministically unpredictable nature.

It is the purpose to show amplification characteristics of ground motion considering specified irregularities of the soil subjected to nonstationary stochastic input motion based on random response theorem.

METHODOLOGY Nonstationary random response analysis starting from the arbitrary transfer functions (gen­

eralized Green's function) obtained from 2D/3D wave propagation analysis is described. The covariance response and the nonstationary power spectrum can be obtained. These results give information on the elongation of the duration and the amplification of amplitude for the stochastic input motion.

RESULTS AND CONCLUSIONS Two application analyses are performed and the qualitative characteristics of the response

due to the specified stochastic input motion are investigated. First, the amplification character­istics of ground motion in an idealized sedimentary basin are investigated for stochastic input motions. It is found in the application that the RMS response (root of covariance response) at the sedimentary basin is affected by the generated surface wave, which is characterized as the later phase. The obtained evolutionary power spectrum shows that the later phase is mainly composed of the frequency contents around the airy phase of the surface wave. Second, the am­plification characteristics of folded subsurface irregularities are similarly addressed. It is found in the application that the RMS response on the syncline axis is amplified due to the focusing effect compared to the horizontally layered model (one dimensional model at the syncline axis), while the response on the anticline axis is not so different from that of the one dimensional model at the anticline axis.

- 1 -

Fig. 2.1

Fig. 3.1

Fig. 3.2

Fig. 3.3

Fig. 3.4

Fig. 3.5

Fig. 3.6

Fig. 3.7

Fig. 3.8

Fig. 3.9

Fig. 3.10

Fig. 3.11

Fig. 3.12

Fig. 3.13

Fig. 3.14

Fig. 3.15

LIST OF FIGURES

Flow Chart

Sedimentary Basin Model

Dispersion Curve of Love Wave (Fundamental Mode)

Transfer Function at the Center Point in the Sedimentary Basin Subjected to the Incident SH Wave

Spatial Variation of RMS Velocity Responses in the Sedimentary Basin for M=6.5

Comparison of RMS Velocity Responses at the Specific points in the Sedimentary Basin for M=6.5

Evolutionary Velocity Power Spectrum at the Center Point in the Sedimentary Basin for M=6.5

Spatial Variation of RMS Velocity Responses in the Sedimentary Basin for M=7.5 ·

Comparison of RMS Velocity Responses at the Specific points in the Sedimentary Basin for M=7.5

Evolutionary Velocity Power Spectrum at the Center Point in the Sedimentary Basin for M=7.5

Comparison of RMS Velocity Responses of the Center Point in the Sedimentary Basin for M=6.5 and M=7.5

Folded Subsurface Irregularity Model

Transfer Function at the Specific Points on the Ground Surface Subjected to the Incident SH and SV Waves

Comparison of RMS Acceleration Responses at the Specific Points on the Ground Surface Subjected to the Incident SH Wave for M=7.5

Comparison of RMS Acceleration Responses of Horizontal Component at the Specific Points on the Ground Surface Subjected to the Incident SV Wave for :rv1=7.5

Comparison of R1v1S Acceleration Responses of Vertical Component at the Specific Points on the Ground Surface Subjected to the Incident SV Wave for M=7.5

- 11 -

PAGE

5

7

8

8

10

11

12

13

14

15

16

18

19

21

22

23

LIST OF TABLES

Table 3.1 Soil Parameters

- lll -

PAGE

7

SOIL AMPLIFICATION CHARACTERISTICS DUE TO LOCAL SITE EFFECTS SUBJECTED TO STOCHASTIC MOTION ON ROCK SURFACE

PART-2

GROUND MOTION CHARACTERISTICS CONSIDERING TOPOGRAPHICAL AND SUBSURFACE IRREGULARITIES OF THE SOIL

SUBJECTED TO STOCHASTIC MOTION

1. INTRODUCTION

Ground motions on a sedimentary deposit are reportedly characterized as lager amplitude and longer duration than those on· rock surface. These amplifications are strongly dependent on the local site conditions. Many studies have suggested from the theoretical approach and observations that the topographical and subsurface irregularities play an important role in es­timating ground motion. Moreover, amplification factors are quite sensitive to input motions which are of a deterministically unpredictable nature. Therefore, no confidence can be achieved from the results of a single deterministic analysis using a recorded or artificia1ly generated mo­tion. To avoid the expense and effort required multiple deterministic analyses,. probabilistic (random response) methods have been developed. Such methods require a stochastic descrip­tion of the earthquake excitation and directly provide probabilistic information on the ground motion, helping the designer to make rational decisions regarding the safety of the facility.

It is the purpose to show amplification characteristics of the ground motion considering specified irregularities of the soil subjected to nonstationary stocha-;tic input motion based on random response theorem.

In section 2, nonstationary random response analysis starting from the arbitrary transfer functions obtained from 2D/3D wave propagation analysis is described.

In section 3, two application analyses are performed. First, the amplification characteristics . of ground motion in idealized sedimentary basin are investigated for stochastic input motion. Second, the amplification characteristics of folded subsurface irregularities are similarly ad­dressed. In both applications, amplifications in intensity and duration due to the differnt locations are discussed.

Conclusions are stated in section 4.

- 1 -

2. NONSTATIONARY RANDOM RESPONSE ANALYSIS STARTING FROM THE ARBITRARY TRANSFER FUNCTIONS

(1) OUTLINE OF METHOD

In this study, the random resp·onse analysis starting from the arbitrary transfer functions (generalized Green's function) obtained from 2D /3D is described.

Although the evaluation method for the covariance response of a general.i\!CI<-type system has been proposed, it sometimes requires great computational effort to get the covariance re­sponse when the system has a large degree of freedom. Then, the random response method using the FFT procedure (after Perotti, 1991) is applied for the linear system of which transfer func­tion has been already obtained, and it remarkably reduces the computational time. The method based on the theory of the evolutionary power spectrum, originally developed by Priestly (1965, 1967), can also obtain the evolutionary power spectrum of the responses and is able to adopt the frequency dependent envelope time function. The flow chart of the analytical method is shown in Fig. 2.1.

(2) BASIC EQUATIONS

PRESCRIBED TRANSFER FUNCTION

In this method, the prescribed transfer function vector {H(w)} defined as the relative displacement to the input acceleration x9 (t) is considered. Cosidering the impulse response { h(t)} corresponding to the inverse Fourier Transfomation of { H(w)} and its derivative {i~(t)} , i.e. the impulse response of the relative velocity, the state vector of the response { Z(t)} is described as the following convolution integral form.

(2.1)

where

N-1

{ h(t)} = { L H(iwk) eiw;t} (2.2) k=O

N-1

{i~(t)}= {L iwk ·H(iwk)eiw;t} (2.3) k=O

N is the number of the discretization of the frequency range ofinterest.

DESCRIPSION OF STOCHASTIC INPUT MOTION

In this study the input motion x9n(t) in Eq.(2.1) is ~sumed to be described as the following form considering the nonstationarity of the amplitude.

xg(t) = n(t). J.(t)

( ) 1 ;= iwt ( ) = n t- e dSw w 2;r -oo

1 ;= ( ) iwt ( ) =- n t · e dSw w 2;r -oo

(2.4)

where Sw is assumed to be an orthogonal process and n(t) is the envelope time function and J.(t) is a stationary stochastic process which has a zero mean value, i.e. E[fn(t)] = o. It is noted tha.t J.(t) can be changed by time segmenting.

- 2 -

i\fEAN VA.LUE AND COV ARIANCES OF RESPONSE

Using Eqs.(2.2) and (2.3), the state value of the reponse is described as following form.

(2.5)

The mean value and the covariance of the response are described as the following equations.

E[{Z(t)}]= ('~ {{iwk·H(iwk)}}e'"'•<t-.. l ·E[x (u)]du Jo k=O {H(wk)} 9 (2.6)

= {0}

Denoting [m(t)] = E[{Z(t)}{Z(t)VJ and considering E[x9(u)x9 (v)] = n(u)n(v)R(u-v), Eq(2.6) is described as the following form.

N-1 N-1

[ m( t)) = L L [ Akl] · mkr (2.8) k=O 1=0

where

(2.9)

( 2.10)

It requires much efforts to calculate directly Eqs.(2.8), (2.9) and (2.10), for the number N is usually la.rge to obtain the covariance of the response with adequate accuracy.

Instead of the direct calculation of the covariance of the response, the FFT can be applied based on the following procedures. First, considering Eq.(2.4), Eq.(2.1) is rewritten as the following form.

{ Z(t)} = l { g~~ = :n} ;_: n(u.). e'"'". dSw(w) du

= ;_:{e.(w,t)}dSw(w) (2.11)

where

{ ( t)} ( ( ) { { h(t- u)}} '"'" d e5 w, = Jo n u {h(t-u)} e u ( 2.12)

- 3 -

Using Eq(2.12) and considering that Sw is an orthogonal process, the covariance response of the state value is described as the following form.

[ m(t)] = E [ { Z(t) }{ Z(t) }T]

= 1: {e.(w,t)}{e;(w,t)V ·S(w)dw (2.13)

where * denotes the complex conjugate. From Eq.(2.13), the evolutionary power spectrum is described a.s the following equation.

[SR(w,t)] = {e.(w,t)}{e;(w,t)}T · S(w) (2.14)

{ e.(w, t)} in Eq.(2.14) can be evaluated by the FFT. Considering that n(u) = 0 (t :::; 0) , and h(t- u) = 0 (t:::; u) , the integral section in Eq.(2.12) is extented to from -oo to oo in time, and the following equation results.

{ ( )} j oo ( ){{i~(t-u)}} '""'d e.w,t = _00

nu {h(t-u)} e u

Eq.(2.15) is rewritten as the following form.

{ ( )} -joe"'( ) {{iaH(ia)}} ;01td e. w,t - -oo" a,w · {H(ia)} · e a

where

1 ;= . . N(a,w)=- n(u)·e'w"e-'""-du 2;r -oo

{H(ia)} = 2_ joo { h(t- u) }e -iOt(t-u) du 2;r -co

Eqs. from (2.16) to (2.18) can be expressed as the the following discretized form.

N-1

{ H( iak)} = ~ "' { h( m.0.t) }e -iOt<(mAt) N~

m=O

(2.15)

(2.16)

( 2.17)

(2.18)

(2.19)

( 2. 20)

( 2. 21)

It is noted that N(ak>w) is the Fourier Transformation ofthe product of n(t) and e-iwt and {e.(w,t)} is calculated by the inverse Fourier Transformation of the product of N(ak>w) and the prescribed transfer function vector. i::l.t is the discrete time increament. Once { e.(w, t)} is obtained, the evolutionary power spectrum is calculated from Eq.(2.14). The covariance responce is then calculated from Eq.(2.13) using the obtained evolutionary power spectrum. }iioreover, the method using the FFT procedure can admit the frequency dependency of the envelope time function, i.e. n(w, t).

- 4 -

S(f)!-------.. S(f)= !3Fo (f) Fo : soorce

spectrum ccw-2

o'------'f'-c ---f

· EnvelQJe function: n(f, t) T d=O. 5L(l/VR-cffi e IVa)

n(f,t) Td ~ exp (- n-ft/QJ /t

Qcccf

t=n.6. t

Statistical a roach ·Power Spectrllll1 : S(f)=S(f, M . .6.) · EnvelQJe function : n(f, t)

2 D or 3D IE---I Information on Wave Pro(:agation Mcx!el · Soi 1 IJ:ita

Calculation of Trans­fer function : Ace. Vel.

Vp(z) , p (z) Vs(z), h(z)

· Topography · Subsurface

Irregularity

i-----------1

Cal. of covar iaoce : Ace. Ve 1. Strain

\

Output of covariance response and Evolu­tionary power spectrum

\I

Nonstationary random response starting fran arbitrary transfer fuoctions

power

Reliability estimation Intensity amplification · Maximl]!l response · Enve!QJe function Spedral amp! ification

Fig. 2.1 Flow Chart

- 5 -

3. APPLICATION ANALYSIS

As practical applications, the ground motion in sedimentary basin and in folded subsurface irregularity are investigated, of which transfer functions have been already obtained from two­dimensional deterministic wave-propagation analysis. The empirical rock motion explained in PART-1 is asssumed as input motion. The system considerd here is linear, and the nonstation­arity of the envelope time function is considered.

3.1 AMPLIFICATION OF GROUND MOTION IN SEDIMENTARY BASIN

(1) ANALYSIS MODEL AND SOIL DATA

As a first application, the ground motion in sedimentary basin is investigated, where the transfer funtion has been already obtained by using the discrete wavenumber method (Idoto­saka, 1990).

Fig. 3.1 shows the sedimentary basin model, which is composed of the sedimentary deposit and bed rock around the deposit. Soil parameters of this model is shown in Table 3.1. The deterministic wave propagation analysis have been previously perfomecl subjected to the plain SH incident wave with the angle 30°. Figs. 3.2 and 3.3 show the dispersion curve of Love-waves and the transfer function at the center point insedimentary basin. In this two dimensional anti-plain analysis, there can be seen not only the body wave of the shear waves but also the generated Love wave propagating in the sedimentary deposit with the later phase. The transfer function shown in Fig. 3.3 contains the characteristics of the propagation of the Love-wave as well as the body wave.

(2) STOCHASTIC INPUT MOTION

The input stochastic motion is assumed empirically using the seismological rock motion explained in PART-1. In this study, the hypocentral distance and the azimuth is assumed to be 100 km and 90°, respectively. The intensity levels of the input motion varies with increase of the magnitude.

- 6 -

-:]

Incident SH Wave

(Angle = 30 degree)

~ WL ~ _2D--=-(l_Ok=m'.!_!_) ) ___ ~_.... I· WR ~

~dimentary Depo~itJH (0.4km)_) ___ / ~

Rock

Fig. 3.1 Sedimentary Bas in Model

Table 3.1 Soil Parameters

WL= WR= 1.6km

S-wave Velocity Density Damping Factor (m/sec) ( tjm3 ) (%)

Sedimentary Deposit 500 2.0 1.0 - Rock 1000 2.3 0.5

c ........ u 0 -~

~ Phase Velocity

Group Velocity

0~--~---L--~--~ 0 0. 5 1. 0

Frequency (Hz)

Fig. 3.2 Dispersion Curve of Love Wave

(Fundamental Mode)

a)

"'0 ;::l ....... ...... ........ 0.. 8

--<

1 2

1 0

8

6

4

2

0 0

/2D-Model

lD-Model \

0. 5

Frequency (Hz)

/ '

Fig. 3.3 Transfer Function at the Center Point

in the Sedimentary Basin

Subjected to the Incident SH Wave

- 8 -

(3) RESULTS OF SOIL AMPLIFICATIONS

The linear random response analysis is performed and the results are shown m Figs. 3.4-3.10.

Fig. 3.4 shows the spatial variation of the rms velocity responses in the sedimentary basin subjected to the input motion with magnitude of 6.5. In the figure, the solid line indicates the results for the sedimentary basin model shown in Fig. 3.1, and the dotted line indicates the results for one-dimensional model at each point. It is found that the body wave propagates towards right in the basin. The generated Love wave can also be seen propagating more slowly than the body waves. The generation of the Love wave is consistent with the results obtained from the deterministic analysis. The results obtained from one-dimensional analysis indicates only the propagation of the body wave. ·

Fig. 3.5 shows the comparison of the rms responses at the three specific points in the sedi­mentary basi. It is also found that the Love wave propagates towards right and the amplitude decreases during the propagation.

The evolutionary power spectrum at the center point in the sedimentary basin is shown in Fig. 3.6. The difference between the exicitation of the frequency clue to the body wave and that due to the Love wave can be seen remarkably. The later phase is mainly composed of the frequency contents around the airy phase of the Love wave as sho"'n in Fig. 3.2.

Fig. 3. 7 shows the spatial variation of the rms velocity responses in the sedimentary ba.sin subjected to the input motion with magnitude of 7.5. In this figure, the body wave and the generated Love wave cannot be distinguished, but the duration of the ground motion becomes longer clue to the secondary surface wave than that resulted from the one-dimensional analysis. The same fact can be seen in Fig. 3.8.

The evolutionary power spectrum at the center point in the sedimentary basin, however, repersents the nonstationarity of the frequency contents due to the body wave and the generated Love wave as shown in Fig. 3.9.

In Fig. 3.10, noted is the difference of the shape of the rms velocity in time series as well as the difference of the level of that.

It is concluded that the rms response in the sedimentary basin is affected by the generated surface wave characterized as the later phase and that the evolutionary power spectrum shows the later phase which is mainly composed of the frequency contents around the airy phase of

- the surface wave.

- 9 -

>-' C>

Max. Value (em/sec)

~- --- - ::--.:-:-:-

~-~--~---~~~---~--~---~---~M-~--~======~

~----------

~~-~---~---~--~--~--=--========= ~~-~--~---~--~---~--~--~---~========= ~~ _J_~:::::::--::::=:::::::-:::::::--=--='-=;:;;::;;::;;;:;;;::;;:;;::;~======-~~~

~~~====~~~==~-I ----===============~~~-------------------------

'-·--··-·----'----·-··-··--J .. __________ L ______ ~_.l _____ l U ~:. I U I ~~ 20 25

Time (sec)

30 35 __j

<10

0.74

0.71

I. 17

I .26

I .28

I .28

I .27

I .25

I .19

I .69

I. 44

0.78

0.77

Fig. 3.4 Spatial Variation of RMS Velocity Responses in the Sedin1entary Basin for M=6.5

3.0

Point A Point B Point C

....--. """""0 8 07 u

1 (\) (/)

........... 2.0 8

u ..._, :>-. ~ •M u 0 .......

/ PointB ~ U')

..... ~ 1 . 0 ..... ~

10.0 20.0 30.0 40.0 50.0

Time (sec)

Fig. 3.5 Comparison of RMS Velocity Responses at the Specific points

in the Sedimentary Basin for M=6.5

...... tv

Power Spectral Density

(cm2/sec)

3.65

2.114:

1 . 22

o.oo 0· o. 0

Frequency (Hz)

"" 0 7 A-----o.oo

16.15

Time (sec)

32·29

118·44

Fig. 3.6 Evolutionary Velocity Power Spectrum at the Center Point in the SedimentarY Basin for M=6.5

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

...... w

Max. Value (em/sec)

3. 19

3.05

4. 41 -___ ;~_-=-==-=-=-=--===-~~===~~~~~~~~~~~ ------ ------------- - -- -- -

5.32

~-~ .............. ......,. ---- -------

~----------~H-~~---=-======~~~ ~---- ---------- - ------ - 5. 47

6.21 --------~--------------_----------------~~-~--~-~--~~~~~~~~~~~ --- - -- - - - -- - - - - - - - - - --- -

5.98

"'::""""------~-

~---~

~==~-----_--_-~_---~<--~~-=---~---=----~--~---~---~-~--~----~-- 5.69

5.72 ~-- --- -~-- ---

-- ------ --------- -----~-----------------~

~------ --- ----------- ------- ----- -- -- 6.93 Jv --~

5.39

3.37

3.32

Time (sec)

Fig. 3.7 Spatial Variation of RMS Velocity Responses in the Sedimentary Basin for M=7.5

..... "'"

,-.... 0 <!)

~ 8 0 "-"

0 ....... 0 0 ........

~ U')

::E 0:::

6.0

4.0

2.0

/ ,-Point A

,, /' , ' //

I I j/ : , ______ ... _ _. ______ .... , I \ I I I I I I I I

------~ \ \

Point A

"""=0

,, Input '~'"'

\ \ ' ... ' " ' ...

\ ' /PointE

' ... ' ....... ' ... ', ' ...

' ..., ',

' .................

Point B Point C

0

O.O~~J_--~ ____ L_ __ _L __ _J ____ ~-==r==~L===3===~ 0.0 10.0 20.0 30.0 40.0 50.0

Time (sec)

Fig. 3.8 Comparison of RMS Velocity Responses at the Specific points

in the Sedimentary Basin for M=7 .5

....... 01

Power Spectral Density

(cm2/sec)

14:5·22

96·81

4:8·4:1

o. 0 Frequency (Hz)

o.oo

16·15

Time (sec)

32·29

Fig. 3.9 Evolutionary Velocity Power Spectrum at the Center Point in the Sedimentary Basin for M-7 .5

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

,-.., 6.0 0 ""'"~--0 __ ~7

Q) CJ)

8 0 .._..,

M=7.5 ;:>-. ..... ...... 4.0 0 0 -~

('/.) ...... ~ (J)

~ 2.0

10.0 20.0 30.0 40.0 50.0

Time (sec)

Fig. 3.10 Cotnparison of RMS Velocity Responses of the Center Point

in the Sedimentary Basin for M=6.5 and M=7.5

3.2 AMPLIFICATION OF GROUND MOTION IN FOLDED SUBSURFACE IRREGULARITY

(1) ANALYSIS MODEL AND SOIL DATA

In this section, the ground motion in folded subsurface irregularity is investigated, where the transfer funtion has been already obtained by using the discrete wavenumber method (Sugawara et al., 1991).

Fig. 3.11 shows the folded subsurface irregularity model, which is composed of the surface layer and underlying bed rock having an irregular boundary. Soil parameters of this model is also shown in Fig. 3.11. The deterministic wave propagation analysis have been previously perfomed subjected to the plain vertical incident S-wave. In Fig. 3.12, the transfer functions at the specific points on ground suface are shown for incident SH wave and for the incident SV wave. As for the case of the incident SV wave, the vertical component of the ground motion is genara.ted due to the subsurface irregularity though the input motion ads horizontally. In each figure, there are transfer functions at the three points on ground surface, i.e. at the center point, at the syncline axis and at the anticline axis, respectively. As for the horizontal component of the transfer function, the amplitude of the syncline axis is larger than that of the others in higher frequency range of interest. As for the vertical component of the transfer function for the incident SV wave, the amplitude of the center point is slightly larger in all frequency range of interest.

In this application, the effect of the change of the transfer function, i.e. the effect of the wave type of the input motion, on the rms responses is investigated by the described random response method.

(2) STOCHASTIC INPUT MOTION

The input stochastic motion is assumed empirically using the seismological rock motion explained in PART-1. In this study, the magnitud,e the hypocentra.l distance and the azimuth are assumed to be 7.5, 100 km and 90°, respectively.

- 17 -

L=3200m X

V8 = · 500 m/s j • I p = 1.8 tjm3

V 1600 I ------~----Z I 250m 11= I mo p = m s D = 1 00 m I ~~(

~--------------------- -------~---------------vs = 1000 m/s p = 2.0 t/m3 V = 2500 m/s H = 800 m Il h = 2 %

r . I

..... 1 CXl

Incident Wave (SH,SV)

Fig. 3.11 Folded Subsutface Irregularity Model

....... co

Anticline Axis Syncline Axis

----~~----------~--·----------Center Point

6.0~--~--~----~--~----. Syncline Axis

Anticline Axis

0.0~--~--~----~--~----~ 0. 0 Frequency (Hz)

5. 0

(a) Incident SH Wave

6.0~--~--~----~--~--~

~~ Anticline Axis Syncline.Axis

!\' ~

j. "I I I\ ~ I

, I I \ , /

' I .A \.J"·',. '' ;·/1./\f \. \ 'v-\r' \.._;\ l.;::_j ~· \1

. I .J' \ ' I 1\ ~ """"'·""-. w ... ' ', i"

Center Point ' ' '\ 1

'.'' ,J, O.OL---~--~----~--~--~

0 · ° Frequency (Hz) 5

· O

(b) Incident SV Wave (Horizontal Component)

3.0~--~--~~--~----~--~

/ I , Syncline Axis

Anticline Axis

Frequency (Hz) 5. 0

(c) Incident SV Wave (Vertical Component)

Fig. 3.12 Transfer Function at the Specific Points on the Ground Surface

Subjected to the Incident SHand SV Waves

(3) RESULTS OF SOIL AMPLIFICATIONS

The linear random response analysis is performed and the results are shown Figs. 3.13-3.1.5.

Fig. 3.13 shows the spatial variation of the rms acceleration responses on the ground surface subjected to the incident SH wave. In the figure, the solid line indicates the results of the folded subsurface irregularity model shown in Fig. 3.11, and the dotted line indicates the results of one-dimensional model based on the vertical composition of the layer a.t each point. It is found that the syncline axis is amplified due to the focusing effect compared to the one-dimensional model a.t the syncline axis, and contrasively tha.t the rms response at the anticline axis is not so different from tha.t of one-dimensional model at the anticline axis. This fact reflects the characteristics of the transfer function as denoted in section (1) and coincides with the results obtained from the deterministic wave propagation analysis.

Fig. 3.14 shows the spatial variation of the rms acceleration responses of horizontal compo­nent on the ground surface subjected to the incident SV wave. The same phenomena denoted above can be seen in this figure.

The generation on the vertical motion due to the subsurface irregularity is found in F.ig. 3.15. It is also found that the rms responses of the vertical motion is amplified significantly at the center point, and those at the syncline axis and at the anticline axis are not amlifiecl, which are consistent with the characteristics of the transfer functions.

- 20 -

N ......

ell .......

~ ~ ....... ....... 0 I=: ;;:.....

Cl)

ell ....... »<:

--< Q)

c ....... ....... 0 ....... .......

~

....... I=: ....... 0

0-.! 1-4

~ 0 Q)

u

Max. Value ,-~----------- -------~ (cm/sec2)

j - ----· 17.48

r==----------------------------------------- -- ---- --------------- 27 81 .

40

Time (sec)

Fig. 3.13 Comparison of RMS Acceleration Responses at the Specific Points

on the Ground Surface Subjected to the Incident SH Wave for M=7 .5

t-v t-v

Cl.l ....... ><

<( <1) s:: ....... ....... (..) s:: ;>..,

Cl)

Cl.l ....... i><:

<( <1) s:: ....... ....... (..) ...... ...... s::

<(

...... c: ....... 0 ~

1-< 2 s:: <1)

u

Max. Value • --- u =- o= - ------ (cm/sec2)

L 18.93

e.----------- 19.12

~--------------- --- 20.30

~--

Time (sec)

Fig. 3.14 Comparison of RMS Acceleration Responses of Horizontal Cotnponent

at the Specific Points on the Ground Surface Subjected to the Incident SV Wave for M=7.5

N <:..)

(/) .......

~ ~ !=: .......

..--< 0 !=: >-..

(/)

(/) ....... ~

~ ~ !=: .......

..--< 0 ....... ~

!=: ~

~

!=: ....... 0 ~ H

~ !=: ~

u

Max. Value (cm/sec2)

4.79

~--------------------------------============== 3.61

...r- 6.71

~ 3.86

~ 4.91

I I__ __________ L I _____ L I I I 0 5 I 0 15 20 25 30 35 40

Time (sec)

Fig. 3.15 Cotnparison of RMS Acceleration Responses of Vertical Component

at the Specific points on the Ground Surface Subjected to the Incident SV Wave for M=7.5

4. CONCLUSIONS

Conclusions obtained in this study are summarized as follows.

(1) Amplification characteristics of sedimentary basin due to stochastic input motion :

RMS response at the sedimentary basin is affected by the generated surface wave charac­terized as the later phase.

Evolutionary power spectrum shows that the later phase is mainly composed of the fre­quency contents around the airy phase of the surface wave.

(2) Amplification characteristics of folded subsurface irregularities due to stochastic input motion:

RMS response on the syncline a.xis is amplified due to the focusing effect compared to the horizontally layered model (one dimensional model at the syncline a.'l:iS).

RMS response on the anticline axis is not so different from thet of the one dimensional model at the anticline a.xis.

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5. REFERENCES

Motosaka,M. (1990). Analytical investigation of the diffracted surface wave generated in a sedimentary basin, The 18th Symposium on Ground Vibrations, Tokyo, Japan, pp.61-70.

Perotti,F. (1990). Structural response to non-stationary multiple-support random excitation, Earthquake Engineering and Structural Dynamics, Vol.19, pp.513-527.

Priestly,M.B. (1965). Evolutionary spectra and non-stationary processes, J. Roy. Statist.. Soc., B27, pp.204.

Priestly,M.B. (1967). Power spectral analysis of non-stationary random processes, Jm.1~nal of Sound and Vibration, Vol.6, pp.86-97.

Sugawara,O., lviotosaka,M. and Kamata,M. (1991). Analysis of wave propagation characteris­tics in irregular site with folded subsurface structure, Summaries of technical papers of annual meeting, Architectural Institute of Japan, pp.435-436.

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