NBS BUILDING SCIENCE SERIES 138

Prediction of Pore Water Pressure Buildup

and Liquefaction of Sands During

Earthquakes by the Cyclic Strain Method

i si

. U53 . ,. NO.L^S PARTMENT OF COMMERCE • NATIONAL BUREAU OF STANDARDS

1932

c. J2_
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NBS BUILDING SCIENCE SERIES 138

SEP 1 5 1982

riot fiCiC

Prediction of Pore Water Pressure Buildup -p^'^^

and Liquefaction of Sands During

Earthquakes by the Cyclic Strain Method i^vz^

R. Dobry2

R.S. Ladd^

F.Y. Yokel'

R.M. Chung'

D. Powell-

'Cenler for Building Technology

National Engineering Laboratory

National Bureau ol Standards

Washington. DC 20234

-Department ot Civil Engineering

Rensselaer Polytechnic Institute Troy, NY 12181

-^Woodward-Clyde Consultants

Clifton. NJ 07012

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued July 1982
Library of Congress Catalog Card Number: 82-600561

National Bureau of Standards Building Science Series 138

Nat. Bur. Stand. (U.S.), Bldg. Sci. Ser. 138, 168 pages (July 19S2)

CODEN: BSSNBV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON; 1982

For sale by the Superintendent of Documents, Government Printing Office, Washington, DC 20402

Price (Add 25 percent for other than U.S. mailing.)
PREDICTION OF PORE WATER PRESSURE BUILDUP AND LIQUEFACTION OF SANDS

DURING EARTHQUAKES BY THE CYCLIC STRAIN METHOD

ABSTRACT

A cyclic strain approach for evaluating the buildup of excess pore water

pressures and the potential for liquefaction of level sandy sites during earthquakes Is proposed In this report. This strain approach Is based on the premise

that, for undralned loading of sand, there Is a predictable correlation between

cyclic shear strain and excess pore water pressure; also, that there Is a threshold shear strain below which there Is no sliding at the contacts between

sand particles and no pore water pressure buildup can occur. As the result, a sand deposit will not develop excess pore pressures If the Induced seismic shear

strain Is less than the threshold strain. Both theoretical evidence and experimental verification supporting the cyclic strain approach and the existence of

the threshold, are presented In the report. Based on all these findings, a specific design method is proposed for predicting if excess pore pressures

will develop at a specific site during a design earthquake.

Key words: cyclic strain; damping ratio; earthquake engineering; laboratory

testing; liquefaction; particulate mechanics; particulate model;

pore water pressure; sand; seismic loading; shear modulus; shear

strain; site stability.

COVER: A nnWioad mbankmznt u)kich moa totaZly de^tAoy2.d

diUUng an ^oAthquak^ in Japan by tiquz^^action

th<L undoAlyinQ looio. 6ata/iat2.d 6andi>.

iii

TABLE OF CONTENTS

Page

ABSTRACT iii

LIST OF TABLES vil

LIST OF FIGURES viii

NOTATION XV

1. INTRODUCTION 1

2. SCOPE 9

3. GENERAL APPROACH 11

3.1 State-of-the-Art (Cyclic Stress) Approach 11

3.1.1 General 11

3.1.2 The Simplified (Seed and Idriss) Procedure 13

3.1.3 Empirical Charts and Correlations 14

3.2 Proposed Cyclic Strain Approach 15

3.2.1 Problems with the Stress Approach 15

3.2.2 Why a Strain Approach? 18

3.2.3 Analysis of Available Cyclic Test Results 19

3.2.4 Proposed Cyclic Strain Method 22

4. A MODEL OF SPHERES FOR THE THRESHOLD STRAIN 49

4.1 General 49

4.2 Contact Between Elastic Spheres 50

4.3 Threshold Strain of Array of Quartz Spheres 52

4.4 Cyclic Stress-Strain Behavior at Very Small Strains 53

5. CYCLIC LABORATORY MEASUREMENTS 65

5.1 General 65

5.1.1 Sand Tested 66

5.1.2 Testing Technique 66

5.1.3 Test Program 69

5 .2 Shear Modulus and Damping Ratio 72

5.3 Modulus Degradation Under Cyclic Loading 75

5.4 Threshold Strain 76

5.5 Excess Pore Water Pressure 77

5.5.1 Comparison Between Au and Auj- 79

5.6 Pore Water Pressure and Modulus Degradation 80

6. EARTHQUAKE ACCELERATION AND THRESHOLD STRAIN 117

6.1 General 117

6.2 The Modulus at Small Strains, Gmax

6.2.1 Laboratory Results 119

6.2.2 In Situ Measurements 120

V
TABLE OF CONTENTS (continued)

Page

6.3 The Modulus Reduction Factor, G/Gmax 123

6.4 Parametric Study 124

7. SUMMARY AND FINDINGS 137

8. ACKNOWLEDGEMENTS 141

9. REFERENCES 143

APPENDIX A-1

vi
LIST OF TABLES

Page

Table 1.1 Some Modern Non-Japanese Earthquakes Which Have Induced

Liquefaction 3

Table 3.1 Some Factors Influencing the Cyclic Strength of Sands 17

Table 5.1 Index Properties for Monterey No. 0 Sand

(Mulilis et al., 1975) 67

Table 5.2 List of Cyclic Triaxial Tests 70

Table 5.3 Cyclic Triaxial Tests with " 2,000 psf 71

Table 5.4 Gj^^^ at 03 = 2000 psf for Monterey No. 0 Sand 73

vii
LIST OF FIGURES

^^|<v-i ' Page

Figure 1.1. Tilted Nllgata buildings after earthquake 6

Figure 1.2. Settlement of a dry sand in cyclic strain controlled

simple shear tests (Silver and Seed, 1971) 7

Figure 3.1. Cyclic shear stresses on a soil element during ground

shaking (Seed et al., 1975) 25

Figure 3.2. Typical form of the relationship between pulsating

shear stress and the number of cycles to cause failure - simple shear conditions (Peacock and Seed,

1967) 26

Figure 3.3. Cyclic stress method for evaluating liquefaction potential

(Seed et al., 1975) 27

Figure 3.4. Range of values of stress reduction ratio, rj, for

different soil profiles (Seed and Idriss, 1971) 28

Figure 3.5. Equivalent number of uniform stress cycles based on strongest components of ground motion (Seed et al.,

1975a) 29

Figure 3.6. Analysis of liquefaction potential at Niigata for

earthquake of June 16, 1964. (Seed et al., 1975) 30

Figure 3.7. Performance of saturated sands at earthquake sites

(Castro, 1975) 31

Figure 3.8. Correlation between stress ratio causing liquefaction

in the field and modified penetration resistance of sand, Nj

(Seed et al . , 1975) 32

Figure 3.9. Correlation between field liquefaction behavior of

sands for level ground conditions and modified penetration

resistance (supplemented by data from large scale

tests. Seed, 1979) 33

Figure 3.10. Cyclic stresses required to cause liquefaction and 20

percent strain in Sacramento river sand at different

densities - a^; = 1.0 kg per sq cm (Seed and Lee, 1965) ... 34

Figure 3.11. Cyclic stress ratio versus number of cycles for

different compaction procedures (after Mulilis et al.,

1975) 35

vlii
LIST OF FIGURES (continued)

Page

Figure 3.12 Effect of seismic history on cyclic strength of sand

(Seed, 1979) 36

Figure 3.13 Influence of overconsolldation on stress causing pore

water pressure ratio of 100 percent in simple shear tests

(Seed, 1979) 37

Figure 3.14 Influence of period of sustained pressure on stress

causing peak cyclic pore pressure ratio of 100 percent

(Seed, 1979) 38

Figure 3.15 Void ratio change for a sand as a function of cyclic shear

strain and number of cycles (Youd, 1972) 39

Figure 3.16 Grain size curves of sands used in testing 40

Figure 3.17 Effect of fabric on cyclic strength, stress-controlled

tests (Ladd, 1977) 41

Figure 3.18 Stress-strain curves for first compression and extension

excursions, stress-controlled cyclic triaxial tests.

Sand No. 2 (modified after Ladd, 1977) 42

Figure 3.19 Effect of fabric on cyclic strength after accounting

for sample stiffness, stress-controlled tests (modified

after Ladd, 1977) 43

Figure 3.20 Effect of fabric on cyclic strength, stress-controlled

tests (Park and Silver, 1975) 44

Figure 3.21 Stress-controlled cyclic triaxial tests of saturated

Crystal Silica sand (modified after Park and Silver, 45

1975)

Figure 3.22 Stress-strain curve for first cycle, cyclic triaxial

tests of saturated Crystal Silica sand (modified after

Park and Silver, 1975) 46

Figure 3.23 Strain-controlled cyclic triaxial tests of saturated

Crystal Silica sand (modified after Park and Silver,

1975) 47

Figure 3.24 Measured pore water pressure in saturated sands after

ten loading cycles, strain-controlled cyclic triaxial

tests (Dobry and Ladd, 1980) 48

Figure 4.1 Simple cubic array of equal spheres 56

ix
LIST OF FIGURES (continued)

Page

Figure 4.2 Elastic spheres under normal and tangential loads 57

Figure 4.3 Normal (a^) and tangential (tc) components of traction

on contact region between two spheres subjected to

normal force followed by a monotonlc tangential force

(Dereslewlcz, 1973) 58

Figure 4.4 Theoretical hysteresis loop due to oscillating tangential

force at constant normal force for two spheres In contact

(Dereslewlcz, 1973) 59

Figure 4.5 Experimental hysteresis loops obtained from cyclic tests

of bodies In contact (Johnson, 1955) 60

Figure 4.6 Tangential force - displacement relation for two elastic

spheres under constant normal force, N (Dobry and Grlvas,

1978) 61

Figure 4.7 Calculated threshold shear strain as a function of

Isotropic confining stress for a simple cubic array of

quartz spheres 62

Figure 4.8 Reduction of shear modulus as a function of shear

strain - comparison between calculated G/Gmax ^ simple cubic array and experimental range for sand 63

Figure 4.9 Damping ratio as a function of shear strain - comparison

between calculated X for a simple cubic array and

experimental range for sand 64

Figure 5.1 Grain size distribution of Monterey No. 0 sand 81

Figure 5.2 Typical correction factor for equipment compliance,

cyclic trlaxlal tests 82

Figure 5.3 Reduction of shear modulus as a function of cyclic shear

strain for Monterey No. 0 sand at = 2000 psf and

Dj. > 45 percent 83

Figure 5.4 Reduction of shear modulus as a function of cyclic shear

strains for Monterey No. 0 sand at =» 2000 psf and

Dp » 60 percent 84

X
LIST OF FIGURES (continued)

Page

Figure 5.5 Reduction of shear modulus as a function of cyclic shear

strain for Monterey No. 0 sand at = 2000 psf and

Dj. = 80 percent 85

Figure 5.6 Reduction of shear modulus as a function of cyclic

shear strain for Monterey No, 0 sand at » 2000 psf

and Dj. = 45, 60, and 80 percent 86

Figure 5.7 Damping ratio as a function of cyclic shear strain for

Monterey No. 0 sand at = 2000 psf and Dj. - 45, 60,

and 80 percent 87

Figure 5.8 Reduction of shear modulus as a function of cyclic shear

strain for Monterey No. 0 sand at Dj. = 60 percent and

= 533, 2000, 4000 psf 88

Figure 5.9 Shear modulus at very small shear strains (G^iax) a function of effective confining pressure (a^) for

Monterey No. 0 sand at Dj. = 60 percent 89

Figure 5.10 Normalized stiffness parameter as a function of cyclic

s^ear strain for Monterey No. 0 sand at » 60 percent,

03 * 2000 psf and various effective confining pressures .... 90

Figure 5.11 Degradation of shear modulus as a function of number

of cycles for Monterey No. 0 sand at Dj. = 45 percent,

- 2000 psf and various cyclic shear strains 91

Figure 5.12 Degradation of shear modulus as a function of number

of cycles for Monterey No. 0 sand at D^ = 60 percent

= 2000 psf and various cyclic shear strains 92

Figure 5.13 Degradation of shear modulus as a function of number of

cycles for Monterey No. 0 sand at Dj. = 80 percent, = 2000 psf and various cyclic shear strains 93

Figure 5.14 Degradation of shear modulus as a function of number

of cycles for Monterey No. 0 sand at = 2000 psf,

Y

= 3 X 10" percent and D^. = 45 and 60 percent 94

Figure 5.15 Degradation of shear modulus as a function of number

of cycles for Monterey No. 0 sand at 2000 psf,

Y

=» 1 X 10"^ percent, and D^ = 45, 60, and 80 percent 95

Figure 5.16 Degradation of shear modulus as a function of number

of cycles for Monterey No. 0 sand at » 2000 psf,

Y

= 3 X 10"^ percent, and Dj. = 45, 60, and 80 percent 96

xl
LIST OF FIGURES (continued)

Page

Figure 5.17 Degradation of shear modulus as a function of number

of cycles for Monterey No, 0 sand at = 60 percent,

Y

= 3 X 10"^ percent, and 03 = 533, 2000, and 4000 psf 97

Figure 5.18 Residual pore water pressure buildup after ten loading

cycles, as a function of cyclic shear strain for Monterey

No. 0 sand at = 2000 psf and = 45, 60, and 80

percent 98

Figure 5.19 Residual pore water pressure buildup after ten loading

cycles, as a function of cyclic shear strain for Monterey

No. 0 sand at = 533, 2000, and 4000 psf and = 45,

60, and 80 percent 99

Figure 5.20 Settlement In the first loading cycle as a function of

cyclic shear strain for dry Monterey No. 0 sand at various

relative densities and confining pressures, simple shear

tests (Pyke, 1973) 100

Figure 5.21 Pore water pressure buildup as a function of number of

cycles for Monterey No, 0 sand at = 2000 psf, Dj. = 45

, percent and various cyclic shear strains 101

Figure 5.22 Pore water pressure buildup as a function of number of

cycles for Monterey No. 0 sand at = 2000 psf, = 60

« percent and various cyclic shear strains 102

Figure 5.23 Pore water pressure buildup as a function of number of

cycles for Monterey No. 0 sand at = 2000 psf, = 80

percent and various cyclic shear strains 103

Figure 5.24 Pore water pressure buildup as a function of number of

cycles for Monterey No. 0 sand at = 2000 psf, y

=

3 X 10"^ percent, and Dj. = 45 and 60 percent 104

Figure 5,25 Pore water pressure buildup as a function of number of

)0 psf, Y

= percent 105

cycles for Monterey No. 0 sand at = 2000 psf, y

1 X 10"^ percent, and D_ = 45, 60, and 80 percen

Figure 5.26 Pore water pressure buildup as a function of number of

cycles for Monterey No. 0 sand at = 2000 psf, y

=

3 X 10"^ percent, and = 45, 60, and 80 percent 106

Figure 5.27 Pore water pressure buildup as a function of number of

cycles for Monterey No. 0 sand at Dj. = 60 percent,

Y

= 3 X 10~2 percent, and various effective confining

pressures 107

xii
LIST OF FIGURES (continued)

Page

Figure 5,28 Pore water pressure buildup as a function of cyclic shear

strain for Monterey No. 0 sand at Dj. » A5 percent, " 2000 psf and various numbers of cycles 108

Figure 5.29 Pore water pressure buildup as a function of cyclic shear

strain for Monterey No. 0 sand at Dj. = 60 percent,

2000 psf and various numbers of cycles 109

Figure 5.30 Pore water pressure buildup as a function of cyclic shear

strain for Monterey No. 0 sand at = 80 percent,

2000 psf and various numbers of cycles 110

Figure 5.31 Pore water pressure buildup after ten loading cycles, as

a function of cyclic shear strain for Monterey No. 0 sand

at = 2000 psf, and = 45, 60, and 80 percent Ill

Figure 5.32 Pore water pressure buildup after thirty loading cycles,

as a function of cyclic shear strain for Monterey No. 0

sand at = 2000 psf, and Dp = 45, 60, and 80 percent 112

Figure 5.33 Pore water pressure buildup in cyclic triaxial straincontrolled tests after ten loading cycles, as a function of

cyclic shear strain, for various NC saturated sands, - 60

percent and for various confining pressures 113

Figure 5.34 Comparison between Au and Auj. as a function of pore water

pressure ratio for Monterey No. 0 sand at " 45, 60 and

80 percent and = 533, 2000 and 4000 psf 114

Figure 5.35 Degradation of shear modulus as a function of pore water

pressure buildup for Monterey No. 0 sand 115

Figure 6.1 Simplified soil profile 126

Figure 6.2 Modulus reduction curves for sands (Iwasaki et al.,

1978) 127

Figure 6.3 Relation between normalized stiffness parameter, K2niax»

relative density (modified from Seed and Idriss, 1970) 128

Figure 6.4 Crosshole geophysical method (Hoar and Stokoe, 1977) 129

Figure 6.5 Normalized shear modulus parameter. A, measured for sands

in the field using geophysical techniques (Powell, 1979) ... 130

xlii
LIST OF FIGURES (continued)

Page

Figure 6.6 Coefficient of earth pressure at rest, Kq, as a function of

overconsolidatlon ratio, OCR (Hendron, 1963) 131

Figure 6.7 Influence of the coefficient of earth pressure at rest,

Kq , on the normalized shear modulus parameter, A 132

Figure 6.8 Reduction of shear modulus at different cyclic shear

strains, y» for sands (data from Iwasaki et al., 1978) 133

Figure 6.9 Liquefaction chart for threshold peak ground surface

acceleration, (ap)^. 134

Figure 6.10 Liquefaction chart for threshold peak ground surface

acceleration, (ap)^-, at z=20 feet 135

Figure 6.11 Lower and upper values of (ap)t to account for scatter

in (G/Giuax)Yt '^d ^=^0 ft 136

xiv
NOTATION

A Normalized shear modulus parameter

a Radius of contact area between two spheres

Hp Horizontal peak acceleration at the ground

surface

(ap)j. Threshold peak ground surface acceleration

c Propagation velocity of the relevant seismic

waves

Cj. Correction factor relating the cyclic shear

strength obtained in a trlaxial test to that

anticipated under typical field conditions

Relative density

e Void ratio

E Young's modulus

f Friction coefficient

6 Secant or effective shear modulus of soil

G^ax Shear modulus of soil at small strains

(Yj, " 10 percent)

g Acceleration of gravity

K Lateral earth pressure coefficient

Kq Coefficient of lateral earth pressure at

rest

1/2 K2 Normalized stiffness parameter G/(a^) '

M Magnitude of earthquake, Richter Scale

N Normal force

N Standard penetration resistance

N' , N^ Corrected or modified standard penetration

resistance for the effect of overburden

pressure

XV
NOTATIONS

n Number of cycles of a sinusoidal (cyclic)

shear stress or strain of uniform magnitude

applied to the the test specimen

R Radius of a sphere

R(=a(ic/2a5) Cyclic shear stress ratio in stress-controlled

cyclic triaxial test

rj, TjjCz) Stress reduction factor varying from a value

of one at z (depth) = 0 to values below 0,7

at z = 100 ft.

T Tangential force

u Initial pore water pressure

V Seismic ground particle velocity

Vg Shear wave velocity

AW Area enclosed by t - y hysteresis loop

z Depth of soil element below the ground surface

2^ Depth of groundwater table

y Shear strain, cyclic shear strain

Yc Seismic (Cyclic) shear strain

Yt Threshold cyclic shear strain

6 Tangential displacement between centers of

spheres

Ae Change of void ratio

Au, Au(t) Excess pore water pressure; excess peak pore

water pressure

Au^ Residual pore water pressure

Axial strain corresponding to the first

compression excursion (i.e., n = 1/4 cycle).

X Damping ratio

V Poisson's ratio

xvi
Mass density of soil

Normal contact stress

Initial vertical pressure

Initial vertical pressure

Initial effective vertical pressure

Initial total confining pressure; in

triaxial testing, 03 = 02 = confining

pressure in the chamber.

Initial effective confining pressure

Average effective stress = 1/3 (a]^ + +

aj)

Cyclic deviator stress

Peak cyclic deviator stress

Cyclic shear stress

Uniform magnitude of a sinusoidal (cyclic)

shear stress applied to the test specimen,

usually taken as a fraction of the peak

value , Tp

.

Peak seismic shear stress associated with

peak ground acceleration

Cyclic shear strength for a given relative

density, D^.

Overconsolidation ratio

Cyclic shear stress ratio causing llquefacti

in the laboratory, for Dj. of 50 percent.

Modulus reduction factor of the soil at shea

strain y

xvii
FACING PAGE: TivU poAtial {^cuMiAd o^j thd louJeA. San ¥2A.nando

Vm, Mklch occ.uAA2.d duAlng tko, VohnxwAy 1971

doAtkquakt M(U tkd h.2A,iitt tiqazicKitioYi

tko. hydARutLccitly-- {^AlZed embayikm2.nt matoAlcLl.

EA,ghty thoix^md pdopZe. had to be tmpoH.aJuJiy

2va.du.at<Ld and coLitd havz lo6t thoJji tivu in

a total mbankmznt cotlap^d.

xviii
1 . INTRODUCTION

The liquefaction of saturated cohesionless soils during earthquakes is one

of the most important problems facing earthquake engineers. There has scarcely

been a major earthquake without at least some reported cases of liquefaction.

Sand boils, flotation to the ground surface of buried concrete tanks, cracking

of pavements, settlement and tilting of buildings and bridge supports, collapse

of waterfront structures, lateral spreading and cracking of slopes and embankments, and flow failures of natural slopes and earth dams have been some of

its manifestations.

Kuribayashi and Tasuoka, 1975 [42]^/ list 44 Japanese earthquakes between 1872

and 1968 for which liquefaction of sandy sites occurred. Of these, the best

^ Numbers in brackets refer to literature references in section 9.

1
known is the Niigata earthquake of 1964, where tilting and failure of multistory

buildings due to liquefaction of the foundation sand was widespread (see fig.

1.1) (Kishida, 1966 [36]; Koizumi, 1966 [38]; Ohsaki, 1966 [56]; Seed and

Idriss, 1967 [76]). A more recent Japanese earthquake which also caused extensive liquefaction occurred on June 12, 1978 in Miyagi-Ken-Oki (Kobayashi et al .

,

1978 [37]; Yamamura et al ., 1979 [89]).

Table 1.1 is a partial list of 14 other earthquakes outside Japan which have

occurred during this century and which induced liquefaction.

The high incidence of liquefaction during earthquakes, together with its

potential for damage, has made the phenomenon a prime subject of concern in

earthquake engineering. The seismic design of nuclear power plants and other

critical facilities routinely includes evaluation of the liquefaction potential

of saturated sandy or silty cohesionless soil layers. The design of new and

the inspection of old earth dams in seismic areas is carried out considering

the possibility of liquefaction of the dam and/or its foundation when sandy or silty cohesionless soils are involved. Due to its complexity, the mechanism

of the liquefaction phenomenon is not yet completely understood and a large

amount of liquefaction researcn is still being done, especially in the U.S.

and Japan. The recent upsurge in the construction of fixed offshore oil platforms throughout the world, where potential failure of the foundation due to

ocean wave induced liquefaction of the ocean bottom must be considered in the

design, has reinforced the interest in clarifying the liquefaction phenomenon.

Most research on liquefaction has taken place in the last 10 to 15 years. Some

significant publications, including recent summaries and discussions of the

state-of-the-art, are: Lee and Seed, 1967 [47]; Seed, 1968 [72]; Seed and

Idriss, 1971 [78]; Castro, 1975 [9]; Youd, 1975 [93]; Seed et al .

, 1975 [80];

Castro and Poulos, 1977 [10]; Seed, 1979 [74]; and Peck, 1979 [61]. Some of

these papers were presented at the ASCE Specialty Session on "Liquefaction

Problems in Geotechnical Engineering" (ASCE, 1976 [4]).

In the last few years, two aspects of the liquefaction problem have generated a great deal of discussion and motivated significant research. The first aspect

relates to the conditions necessary to produce unlimited flow of the liquefied

soil in the field under the action of gravity loads such as those occurring in

a slope or beneath a structure. There is now general consensus that, while

loose or very loose cohesionless soils can experience unlimited flow, dense

soils at usual confining pressures cannot, because of their dilative behavior

at large shear strains (Castro and Poulos, 1977 [10]; Seed, 1979 [74]).

The second aspect of the problem is related to the importance of relative

density on the rate at which excess pore water pressure builds up during an earthquake. Early work suggested that relative density is the key soil parameter controlling pore water pressure increases (Seed and Idriss, 1971 [78]).

Many engineering decisions have been based on the assumption that relative density is the key parameter, and pore water pressures measured on reconstituted

samples in cyclic laboratory tests have been taken to be representative of pore

2
Table 1 . 1 Some Modern Non-Japanese Earthquakes Which Have

Induced Liquefaction

Earthquake

San Francisco, California

Bihar-Nepal , India

El Centro, California

San Francisco, California

Coatzacoalcos , Mexico

Southern Chile

Alaska

Caracas , Venezuela

Borrego Mountain, California

San Fernando, California

Haicheng, China

Guatemala

Tangshan, China

San Juan, Argentina

Year

1906

193A

1940

1957

1959

1960

1964

1967

1968

1971

1974

1976

1976

1977

Reference

Lawson et al .

, 1908 [46]

Youd and Hoose, 1976 [97]

Geological Survey of

India, 1939 [25]

Ross, 1968 [67]

Ross, 1968 [67]

Diaz de Cossio, 1960 [17]

BSSA, 1963 [7]

Ross et al., 1969 [68]

Cluff et al., 1973 [13]

Youd and Castle, 1970

Dixon and Burke, 1973

Seed et al , 1975 [80]

Xie Junfei, 1979 [88]

Hoose, 1976 [32] cited

Youd, 1977 [94]

Xie Junfei, 1979 [88]

Bruschi, 1978 [6]

[96]

[18]

by

3
water pressures in the field during earthquakes. However, more recent research

has conclusively demonstrated that relative density is only one of several

factors involved (Seed, 1976 [73], 1979 [74]). Based on these findings, Peck,

(1979 [61]) has questioned the validity of laboratory cyclic tests as presently

performed for predicting liquefaction potential, and has instead suggested

reliance on empirical methods based on field exploration by standard penetration

tests

.

This report addresses the problem of pore water pressure buildup and

liquefaction during earthquakes at level sites. It is generally agreed that the cause of pore water pressure buildup in saturated sands or cohesionless silts is

the cyclic loading of the soil associated with the passage of seismic waves. Both loose and dense dry sands compact and settle when subject to cyclic shear

loading, as illustrated in figure 1.2 (Silver and Seed, 1971 [83]). If the soil

is saturated and the loading takes place in an undrained condition, the relative

incompressibility of the pore water makes the rapid compaction of the sand impos

sible. Instead, an excess pore water pressure develops whose value increases

with the duration of cyclic loading, and in many fine sands and silts these

pressures only start dissipating after the ground shaking has ended. Some manifestations of liquefaction in the field, such as the occurrence of sand boils,

and the differential settlement of structures due to uneven post-earthquake

compaction of the foundation soil , can be explained by the presence of excess

pore water pressures and associated water flow. Other manifestations of

seismically-induced liquefaction, which are associated with large or unlimited

shear straining of the soil, can be explained by the decrease in shear strength

associated with these excess pore water pressures. This shear strength decrease

while obviously a very important aspect of the liquefaction problem, is outside

the scope of this work. This report focuses on the pore water pressure buildup

common to all manifestations of liquefaction at level sites during earthquakes.

The approach to the liquefaction problem presented in this report is based on

the premise that pore water pressure buildup during cyclic shear loading of

sand is controlled mainly by the magnitude of the cyclic shear strain. This

premise leads to the conclusion that shear modulus, rather than relative

density, is the main parameter controlling pore water pressure buildup in the

field. An important practical consequence is that measurements of in situ

modulus at small strains, which can be obtained from geophysical measurements

of shear wave velocity, should be used for predicting pore pressures. This is

in contrast with the present use of in situ relative density which: a) is not

a clearly valid concept when applied to natural sand deposits because of their

stratification (Castro, 1975 [9]); and b) cannot be measured directly in the

field, but instead must be inferred from penetration tests. Therefore, the

proposed strain approach, based on seismic shear strains, in situ measurements

of shear modulus, and cyclic strain-controlled tests, is different from current

practice, which is based on seismic shear stresses, in situ penetration measurements for relative density determinations, and stress-controlled tests.

Chapter 3 of this report describes the main features of the present

state-of-the-art and discusses the need for the new cyclic strain approach.

Chapters 4, 5 and 6 present results of studies performed to develop the cyclic

4
strain method, including a theoretical analysis using a particulate model of

the sand, laboratory measurements, additional studies of the most Important

parameters used in the method, and a proposed engineering procedure to eliminate

the potential for liquefaction at level sites during earthquakes.

5
Figure 1.1 Tilted Niigata buildings after earthquake

6
2.0

1.5

^ 1.0 o

z

LU ^ ^

0.5 ^ Z

1 1 1 1 1 1 II 1 1 II II M -

1111

=

II

11

45 PERCENT

60 PPERCENT I

1 i 1 1 I 1 1 I 1

80 PERCENT ~

1 1 1 1 1 1 1

1

0.01 0.1 1.0 CYCLIC SHEAR STRAIN, PERCENT

10.0

Figure 1.2 Settlement of a dry sand in cyclic strain controlled simple

shear tests (Silver and Seed, 1971)

7
FACING PAGE: VoXallzd vim tkz iU>^ixM.2^ Indaczd by

tiqadiaction OjJ tho, hydAautlc ^-itZ am thz

San ToAnmdo Vam duAlng the. TzbHAxofiy 1971

toAthqaakt. The. n.malyiLng mbankme.nt

dou)yi^tn.2.<m the, IoaMjJiz had only a

4- It, iH-zebooAd oi highly {^AactuAed 6oZl

uikich MOA on the, veAgz {^aiZuAe and could

have. ioA^le-d the. e.aAthq[iakz had lai>te.d a lejxi mofie iecondi.

8
2. SCOPE

This report contains:

(1) A review of present cyclic stress methods for predicting liquefaction

potential of level sandy sites;

(2) A proposed new approach to predicting liquefaction potential based on the

correlation between cyclic strain and excess pore water pressure buildup;

(3) The documentation for the existence of a threshold cyclic shear strain (y^),

below which there is no excess pore water pressure buildup, and an explanation for the existence of Yt ^ particulate soil model;

9
(4) The results of 12 undrained strain-controlled cyclic triaxial tests on Monterey No. 0 sand which are aimed at developing the basic parameters

needed for the proposed strain approach to liquefaction, and a comparison

of these results with measurements performed by others; and

(5) A proposed design method for predicting the threshold peak ground surface

acceleration ,(ap)(- , below which a site does not build up pore pressures

and cannot liquefy.

Chapter 3 contains a review of the existing stress approach and a discussion

of problems associated with its application. It also discusses the rationale

for the proposed strain approach, as well as experimental evidence from prior

work documenting the existence of the cyclic threshold strain, y^, and of a consistent correlation between cyclic strain and excess pore water pressure.

Chapter 4 contains a particulate model where the sand is represented by a simple cubic array of quartz spheres, and which predicts values of Yt dose

to those observed experimentally.

Chapter 5 contains the results of 12 undrained strain controlled cyclic triaxial

tests on Monterey No. 0 sand, performed under the direction of the second author

of this report (Ladd) . The tests include measurements at very small cyclic

strains (y 10"^ percent) and precise measurements of the threshold strain.

Comparisons are also presented between the results of these tests and measurements performed by others.

Chapter 6 contains the derivation of a proposed design method based on the

threshold strain, and on the derived concept of a "threshold peak ground surface

acceleration," (ap)^, needed to start pore pressure buildup at a given site and

saturated sand layer.

FACIWG PAGE: Uqae^JacX^^on-^ndaced lU^6uJia> and mbanfmznt

Juno. 197S dOAthquRkz.

10
3. GENERAL APPROACH

3.1 STATE-OF-THE-ART (CYCLIC STRESS) APPROACH

3.1,1 General

The current state-of-the-art method to predict pore water pressure buildup and

liquefaction potential during earthquakes in level sites has been developed to a large extent by Seed and his coworkers (Seed and Idriss, 1971 [78]; Seed et

al., 1975 [80]; Seed, 1979 [74]). The following two main assumptions are made

in this method:

1) The pore pressure developed at any saturated cohesionless soil element such

as the one shown in figure 3.1(a) is caused by the cyclic shear stress, t. This shear stress acts in the horizontal and vertical planes and is caused

11
by the passage of vertically propagating seismic shear waves. Figure 3.1(b)

shows a typical variation of t with time, xCt), during ground shaking.

2) The loading of the soil by T(t) is undrained (the pore water pressure

dissipation and redistribution within the soil mass are disregarded within

the time frame of the event). Therefore, the pore water pressure in the

element in excess of the hydrostatic pressure, Au(t), increases with duration of shaking and is a maximum at the end of the shaking (t=30 seconds in

figure 3.1(b)). Thus, the minimum value of the effective overburden pressure occurs also at the end of the shaking, and is aQ-Au(30) where a© = initial effective overburden pressure. If Au(30) = Oq, there is no effective stress in the soil and, by definition, "initial liquefaction" of the

soil has occurred. If Au(30) < Oq "initial liquefaction" did not occur during the shaking.

Initial liquefaction has been extensively used as a criterion defining failure.

Other criteria, based on the strain developed during stress-controlled tests,

have also been used; however, the discussion herein is mainly restricted to

the initial liquefaction concept. The ideal way to obtain the value of Au(30)

would be: (i) to retrieve a perfectly undisturbed soil sample from the given

depth, (ii) to consolidate a specimen in the laboratory to the effective field

static pressures, Oq and KQao , and (iii) to subject the saturated specimen to

the seismic shear stress history, T(t), in undrained condition, and monitor the

development of the excess pore water pressure, Au(t). In practice this 3-step

method cannot be implemented, and is replaced instead by the following, more

manageable procedure:

a) T(t) is replaced by n cycles of a sinusoidal shear stress of uniform

amplitude, t^. This cyclic stress, is taken as a fraction of the peak

value, Tp of T(t). Usually, t^, = 0.65 Tp is used. Therefore, T(t) is

replaced by n cycles of x^j ai^d the value of n is selected so that Au at

the end of the n cycles is approximately equal to Au at the end of T(t).

b) A disturbed soil sample is retrieved from the depth of interest, and is

reconstituted in the laboratory to the same relative density, D^, it had in

the field. Field Dj. is usually estimated from the measured standard penetration resistance, N, using available correlations between N, Oq* and

such as that of Gibbs and Holtz, (1957 [26]).

c) The reconstituted sample is consolidated under stresses approximating the

free field effective pressures (usually this means isotropic consolidation

under Oq) .

Then, an undrained stress controlled test is performed where n cycles of

the uniform cyclic shear stress, x^,, are applied to the sample in an undrained condition, while monitoring the excess pore water pressure

buildup, Au. If Au = Oq at the end of the n cycles, the sample has

experienced initial liquefaction. This result is then used to predict the

occurrence of the initial liquefaction in the field.

12
Usually, several stress-controlled, cyclic laboratory tests such as described

in (c) are performed on identical reconstituted samples having equal D^. and

consolidated under the same Oq. The cyclic stress, T(., is varied between

tests, and the number of cycles, n, needed to produce initial liquefaction is

obtained from each test. The curve of versus n is used for the prediction

of liquefaction in the field. Figure 3.2 shows an example of such a curve obtained from cyclic simple shear tests. The value of from the curve for

a given n (also called the cyclic strength of the soil) is compared with the

average t^. developed by the earthquake, and the liquefaction potential in the

field is evaluated from this comparison. This comparison is illustrated in

figure 3.3.

The shear stress history, T(t), and the derived value shown in figure

3.1(b) are sometimes obtained from site response analyses. In those analyses,

assumed ground motions are input at rock or at some depth within the soil , and

a shear beam model of the soil profile is used for the computations of seismic

shear stresses, strains and accelerations at different depths within the soil

(i.e., Schnabel et al., 1972 [71]). In this case, the calculated T(t) is a function of the input motions and of the geometry and stress-strain properties

of the soil model.

3.1.2 The Simplified (Seed and Idriss) Procedure

A further simplification of the cyclic stress approach described in section

3.1.1 has been proposed by Seed and Idriss (1971 [78]). This simplified procedure is widely used in engineering practice. It has the advantage of using a limited number of parameters which are usually available, and not requiring

the use of a computer.

In this simplified procedure, the liquefaction potential of a soil element at a depth z is evaluated in three steps as follows:

Step 1 . Determination of and n. This is done by computing the stress

ratio, Tc/oq caused by earthquake by means of equation 3,1:

= 0.65

^rd

3.1

where: ap = horizontal peak acceleration at the ground surface

g = acceleration of gravity = 32.2 ft/sec^

Oq, <Jo - total and effective overburden pressures at depth z rj = rj(z) = stress reduction factor varying from a value of one at

z=0 to values below 0.7 at z=100 ft. (see fig. 3.4)

The earthquake is assumed to induce in the soil n cycles of uniform cyclic

stress, Tc» The value of n is related to the magnitude, M of the earthquake,

and is equal to about 10 cycles for M=7. Figure 3.5 presents the most recent

relationship between M and n proposed by Seed et al., (1975a [81]),

13
Step 2 . Determination of (T)Dr causing initial liquefaction (cyclic

strength of soil). The value of uniform cyclic stress causing initial

liquefaction in n cycles, (t)|)i- is assumed to be a function of n, and of

the relative density and grain size of the soil. (t)di- is obtained from

equation 3.2:

(•^)nr = • c • Dr 3*2

a; 2a^ 50 ^ 3U

where: (t)d /oq = the cyclic shear strength ratio for a given relative

^ density, D^.

a^jj, = the cyclic deviator stress

50 = signifies a relative density of 50 percent

Dj. = field relative density in percent

^ ^dc ^ = the shear stress ratio causing liquefaction in the laboratory

20^ in a stress-controlled cyclic triaxial test, for D^. = 50

percent

C]. = a correction factor relating the cyclic shear strength obtained

in a triaxial test to that anticipated under typical field

conditions.

(^dc/2ao)50 a^if^ obtained from appropriate charts once n, D^, and the

grain size of the sand are known, Dj. is obtained from the standard penetration

resistance, N, using the Gibbs and Holtz correlation.

Step 3 , Comparison between and (t)j)j-. The values of and (t)dj.

obtained from equations 3.1 and 3.2 are compared. If > (T)Dr- liquefaction at depth z is predicted by the method. If Tc < (T)Dr liquefaction

is predicted.

The simplified procedure has all the main features of the general stress approach

discussed in section 3,1,1. Note the importance given to the relative density

of the soil in this method.

3.1,3 Empirical Charts and Correlations

After the 1964 Niigata earthquake, it was observed that the occurrence and

degree of damage caused by liquefaction were well correlated with measurements

of the standard penetration resistance, N, performed before the earthquake

(Kishida, 1966 [36]; Ohsaki, 1966 [56]), Based on this observation, some empirical correlations were obtained which are summarized in figure 3,6,

Figure 3,6 is directly applicable to a site having subsoil conditions similar

to those in Niigata and experiencing a ground shaking similar to that which

occurred in Niigata in 1964, More general correlations and charts, applicable

to wider ranges of soil and shaking conditions, have been proposed by Whitman,

1971 [86]; Seed and Idriss, 1971, [78]; Castro, 1975 [9]; Christian and Swiger,

1975 [12]; Yegian and Whitman, 1978 [91]; and Seed, 1979 [74], In all cases,

these authors have calibrated their proposed correlations with documented case histories where liquefaction has (or has not) occurred. Tables containing the

values of N and of other basic parameters of up to 50 case histories have been

presented by Seed and Idriss, 1971 [77]; Seed et al ,

, 1975 [80]; and Yegian,

1976 [90],

14
Figures 3.7 through 3.9 present some of these empirical correlations. In all

of these figures, the stress ratio caused by the earthquake is obtained from

the peak ground surface acceleration using an expression such as equation 3.1.

Other parameters needed to use the charts are N, Oq, and the earthquake magnitude, M (for fig. 3.9). In these three figu res, the measured value of N must

be corrected for the effect of overburden pressure. In figure 3.7 the corrected

value, N', defined in the figure, is used. The corrected N]^ value used in

figures 3.8 and 3.9 is calculated using equation 3.3.

N, = [1-1.25 log N

1 ^ ^ 2000^

where Oq is in psf.

It must be noted that the corrections used to calculate N* and in figures 3.7

through 3.9 are very similar except for a constant factor. For a wide range of

pressures, 500 psf < < A, 000 psf, = 0.5 N'.

The original use of N as a basis for the development of empirical liquefaction

correlations was based on two assumptions: (i) the paramount importance attributed to relative density in controlling the rate of development of excess pore

water pressures in the field, and (ii) the belief that N measures relative

density in the field. As discussed in section 3.1.1 of this report, both

assumptions (i) and (ii) have been challenged; however, this challenge does

not affect the proven success of N and of the empirical correlations as tools

to organize liquefaction case histories and to evaluate liquefaction potential.

Therefore, what is needed is an improved and more basic understanding of standard penetration test (SPT) measurements in cohesionless soils, and of the

relation between these measurements and the factors controlling liquefaction.

The results of recent research on the SPT along these lines by Kovacs, 1975

[40], and Kovacs et al., 1981 [41]; Schmertmann, 1977 [69] and Schmertmann and

Palacios , 1979 [70] represent a very promising start towards this objective.

3.2 PROPOSED CYCLIC STRAIN APPROACH

3.2.1 Problems with the Stress Approach

The current cyclic stress approach to liquefaction described above is based on

the premise that the pore water pressure buildup in a saturated sand, subjected

to a given cyclic shear stress history, is mainly a function of the relative

density D^- and the initial effective stresses acting on the sand. The influence

of the density on cyclic strength of reconstituted sand was first observed in

1965 (see fig. 3.10). Therefore, this parameter was incorporated by specifying

that the cyclic tests should be done on reconstituted samples compacted to the

estimated field density. The assumption that cyclic strength is mainly a function of relative density, is also used in the simplified procedure described

in section 3.1.2.

However, cyclic tests performed in the last few years have revealed that a number of other factors besides Dj- also influence significantly the results of

15
J stress-controlled tests. Some of these factors, which were recently discussed

in detail by Seed, 1979 [74], are listed in table 3.1.

Experimental results showing the significance of the last four factors of

table 3.1 on the cyclic strength of reconstituted sands are plotted in fig- ures 3.11 through 3.14. These figures show that the effect of these factors

can be even more significant than that caused by large variations in density.

Most of the evidence showing the influence of time under pressure, overconsolidation, prestraining and fabric on cyclic strength is from laboratory tests.

However, some limited evidence from the field suggest that the geological age

of the soil deposit influences liquefaction potential and should be considered

(Ohsaki, 1969 [57]; Youd et al., 1978 [98]; Finn, 1979 [24]). Seed, 1976 [73]

has pointed out that "...the liquefaction characteristics of in situ sand

deposits are determined by a number of complex factors, of which relative

density is only one, and careful evaluation of all these factors is required

in selecting soil characteristics for use in design."

The influence of all these factors on the cyclic strength of sands certainly

complicates the state-of-the-art and makes its practical use more difficult.

Efforts can be made to simulate as closely as possible the geological and seismic history of the soil when testing reconsolidated samples in the laboratory.

The specimens can be reconsolidated, prestrained, and aged under pressure prior

to cyclic loading. However, this complicates the tests and requires information that may not be available. Besides, there are limits to what can be done

on a reconstituted sample. Laboratory aging under pressure cannot possibly

simulate the hundreds or thousands of years of history of many soil deposits.

The fabric effect introduces an additional and serious problem, since there

is yet no reliable method to measure sand fabric in the field.

Testing undisturbed samples of cohesionless soils and performing the cyclic

tests on them rather than using reconstituted specimens would solve this

dilemma. Unfortunately, the factors included in table 3.1 appear to be very

sensitive to sampling and handling of sands prior to testing (Seed, 1979 [74]).

Peck, 1979 [61], has tentatively concluded that: "(1) unless the cyclic

loading tests used to evaluate liquefaction potential can be performed on absolutely undisturbed samples, which is manifestly impossible, the results

will probably indicate too great a likelihood of liquefaction; and (2) in many

instances the resistance to liquefaction in the field may be appreciably, even

spectacularly, greater than that determined on the basis of conventional cyclic

laboratory tests on reconstituted or even "undisturbed" samples if no allow^

ances are made for various possible beneficial effects such as time, repeated

small shearing forces, and stress history." Based on these conclusions. Peck

proposes at this time to rely more on empirical correlations based on field

standard penetration measurements, rather than using cyclic laboratory tests.

16
Table 3.1. Some Factors Influencing the Cyclic Strength of Sands

- Relative Density

- Method of Sample Preparation (Fabric Effect)

- Prior Seismic Straining (Prestraining or Preshaking Effect)

- Lateral Earth Pressure Coefficient (K ) and

o Overconsolidation Ratio (OCR)

- Increased Time Under Pressure (Aging Effect)

17
3.2.2 Why a Strain Approach?

The main premise of this report Is that a cyclic strain approach to the problem

of predicting pore water pressure buildup and liquefaction of saturated coheslonless soils would have significant advantages over the current cyclic stress

approach. Evidence substantiating this statement, which was available at the

outset of this research, Is discussed In this and the following sections.

Sliver and Seed, 1971 [83] showed experimentally that cyclic shear strain,

Yc = T^/G (G = secant shear modulus) rather than cyclic shear stress, t^, controls the denslf Icatlon of dry sands. Strain-controlled cyclic simple shear

tests were performed by Silver and Seed on Dry Crystal Silica No. 20 sand using

a range of relative densities, D^, of overburden pressures, Oq, and of cyclic

shear strains, Yc* found that the rate of settlement with number of

cycles depended on Dj- and Yc> Independent of Oq, and did not correlate

with Tq and G taken Independently. Some results of these tests are summarized

in figure 1.2. Based on the Seed and Silver results, Martin et al .

, 1975 [49]

successfully developed a cyclic strain, effective-stress model to predict pore

water pressure buildup in saturated sands during undralned stress-controlled

tests. All of these findings strongly suggest that Yc» rather than Tq,

controls both denslf Icatlon and liquefaction in sands.

Based on cyclic test results on dry sands, Drnevlch and Rlchart, 1970 [23]

Youd, 1972 [92] and Pyke, 1973 [64] concluded that there is a threshold cyclic

shear, y^j of ttie order of 10~^ percent, below which no denslf icatlon occurs

(see fig. 3.15). A value of y^ of about 10 ^ percent is also consistent with

the experimental results for dry sand shown in figure 1.2, and with straincontrolled tests results on saturated sands reported by Park and Silver, 1975

[59], and Dobry and Ladd , 1980 [20], and will be discussed in more detail in

section 3.2.3. A theoretical study of a simple granular model of a quartz sand,

originally proposed by Dobry and Swiger, 1979 [21], and presented in detail In

chapter 4, predicts a range of values for this threshold strain, 1 x 10"^ percent <. <. h X 10~^ percent for effective confining pressure between 500 psf

and 4,000 psf (24 and 192 kPa) . The existence of a threshold level at which

pore water pressure buildup starts is obviously very Important for liquefaction

prediction. The fact that this threshold has a more stable value when expressed

as a strain than when expressed in terms of stress is another argument in favor

of a strain approach to liquefaction.

The adoption of a cyclic strain approach should considerably simplify the

interpretation of cyclic laboratory tests on saturated sands. There is experimental evidence indicating that the factors presented in table 3.1 which

increase the cyclic strength of sands in stress-controlled tests, also increase

the shear modulus of sands (Seed and Idrlss, 1971 [78]; Drnevlch and Rlchart,

1970 [23]; Hardin and Drnevlch, 1972 [29]; Pyke, et al .

, 1974 [65]; Anderson

and Stokoe, 1977 [3J; Dobry and Ladd, 1980 [20]). This evidence suggests that,

if both T (cyclic shear strength) and G are similarly affected by the factors

listed in table 3.1, the ratio y = t/G should be less affected by these same factors. Therefore, the pore water pressure buildup in strain-controlled tests

should be less sensitive to those factors than in stress-controlled tests. A

more detailed discussion of this premise is presented in section 3.2.3.

18
The advantage gained by adopting a strain approach to liquefaction would be the

total or partial replacement of the parameters listed In table 3.1 by the shear

modulus, G. Unlike relative density or sand fabric, the shear modulus at small

strains, G^^x* directly measured In the field by means of the shear wave propagation velocity. Field measurements of Gpj^^ would automatically incorporate many of the characteristics of the soil deposits which are Important for

pore water pressure buildup and liquefaction during earthquakes. This approach

should, therefore, decrease the need for a detailed knowledge of the geological

and seismic history of the site which is presently required in the stress

approach.

There is still another argument in favor of the strain approach, which relates

to the advantages of running cyclic strain - instead of stress-controlled

tests of dense (dilative) sands. Castro (1975 [9]) has shown that, for cyclic

triaxial stress-controlled tests on these soils, there is a substantial redistribution of water content within the specimen, most of which probably occurs near the end of the test, when the cyclic strain becomes large. This redistribution affects the cyclic behavior of the dense sand specimen in such a way

that it ceases to represent the field situation; in particular, the "strains

measured in the laboratory in such a case are so conservatively large as to

make the test unusuable as a design tool," Castro and Poulos, 1980 [11].

Strain-controlled tests of dense sands, performed at smaller cyclic strains,

which are more representative of those in situ, should decrease the redistribution problem. Although more research is needed on the subject, it seems reasonable to expect that running strain-controlled tests of dense sands, at

those smaller representative strains will: (a) cause less water content redistribution before initial liquefaction occurs, and (b) provide more realistic

predictions of in situ pore pressures than those obtained from stress-controlled

tests (see also Peck, 1980 [62]).

3.2.3 Analysis of Available Cyclic Test Results

This section analyzes and discusses some available stress- and strain-controlled

cyclic triaxial test results on saturated sands from the viewpoint of the proposed cyclic strain approach. These results relate mainly to the fabric effect

listed in table 3.1, and were obtained from the files of one of the authors of

this report (Ladd) and from Park and Silver, 1975 [59].

The first data set was obtained from Ladd's files. It corresponds to

stress-controlled tests on a saturated sand compacted to Dj. = 83 percent by

different sample preparation methods. The sand used is the same as the soil

called "Sand No. 2" by Ladd, 1977 [43], and its grain size distribution is

shown in figure 3.16. The cyclic triaxial strength data for initial liquefaction are plotted in the usual way (i.e., cyclic stress ratio versus n) in figure 3.17 for the two sample preparation methods used: Moist Vibration and

Dry Tamping (see Ladd, 1977 [43] for a description of the two methods). Figure

3.17 shows again in a very dramatic way the effect of sand fabric, which was already Illustrated in figure 3.11. The moist vibration specimens have much

larger cyclic strengths than the dry tamping specimens. The explanation for

this is that the sand compacted by moist vibration was stiffer, and therefore

developed smaller cyclic strains than that compacted by dry tamping. This is

19
illustrated by the data in figure 3.18, which correspond to the same stresscontrolled tests of figure 3.17. Figure 3.18 shows the maximum amplitudes of

cyclic axial stress and strain, corresponding to the first compression and

extension excursions, for both moist vibration and dry tamping tests. Two

conclusions can be drawn from figure 3.18: (a) the specimens are stiff er in

compression than in extension for both compaction methods, and (b) moist

vibration specimens are stiff er than dry tamping specimens, with the difference

being much larger in extension. For example, the two dry tamping tests corre- sponding to cyclic stress ratios, R = 0.36 and 0.37, developed in their first

extension excursion axial strains of almost one percent and failed in only 8 to

12 cycles (see fig. 3.17); on the other hand, a moist vibration specimen

tested at a similar stress, R - 0.41, developed in its first extension excursion a lower strain ("0.4 percent) and failed in 24 cycles. In figure 3.19,

the same results of figure 3.17 have been replotted using the axial strain in

the first extension excursion, e^, as a parameter, instead of the cyclic stress

ratio of the test, R. The difference between dry tamping and moist vibration

data points is much less in figure 3.19 than in figure 3.17. Although there

is still considerable scatter in figure 3.19, it was possible to define a single curve representing all the data points. Therefore, an important reason

for the lower cyclic strength exhibited in figure 3.17 by the dry tamping

specimens is that they were less stiff, especially in extension, and were

thus subjected to larger cyclic strains starting from the very beginning of

cyclic loading.

Figures 3.20 through 3.23 present results from stress-controlled and

strain-controlled cyclic triaxial tests on saturated Crystal Silica No. 20

sand, performed by Park and Silver, 1975 [59]. The grain size distribution of

the sand used is shown in figure 3.16. All tests were conducted on specimens

compacted at Dj- = 60 percent using two preparation methods: Dry Vibration

and Wet Rodding. The effective confining pressure in all tests was = 2,000

psf (96 kPa).

The cyclic strength results from the stress-controlled tests are presented in

the usual way in figure 3.20. Again, the effect of fabric is apparent with the

Dry Vibration specimens being significantly weaker. For a given stress ratio,

the Wet Rodding specimens needed 15 to 20 times more cycles to fail than the

Dry Vibration specimens (e.g, at R = 0.30, n = 30 cycles and 2 cycles, respectively). Figure 3.21 gives additional information on pore water pressure

buildup during the same stress-controlled tests. Figure 3.21 again shows that

the rate at which pore water pressure built up was much slower for the Wet

Rodding specimens.

The reason for the differences in stress-controlled test results shown in

figures 3.20 and 3.21 is, again, that Wet Rodding specimens were stiffer, and

therefore developed smaller cyclic strains than the Dry Vibration specimens.

This difference is illustrated by the comparison of the stress-strain curves

in the first cycle plotted in figure 3.22. For this case, no information was available to plot separately the first compression and extension excursions,

as was done in figure 3.18 for sand No. 2. The difference between the curves

in figure 3.22 is similar to that between the curves in figure 3.18.

20
Therefore, for both Crystal Silica sand and sand No. 2, the effect of fabric on cyclic strength, as measured in stress-controlled tests, seeras to be largely

a stiffness effect. A stiffer fabric, which develops lower cyclic strains from

the beginning of cyclic loading, also develops less pore water pressures and,

thus, liquefies in a larger number of cycles.

The results of strain-controlled cyclic triaxial tests, performed by Park and

Silver on the same Crystal Silica sand, compacted to the same relative density,

using the same specimen preparation procedures and under the same confining

pressures as those used for the above-mentioned stress-controlled tests are shown in figure 3.23. This figure shows the rate of pore water pressure

buildup at different cyclic axial strains, e^, during the strain-controlled

tests, and here also, the effect of fabric is minor and has been reduced to a scatter measured by the width of the hatched areas in the figure. Figure 3.23

demonstrates quite clearly that the rate of pore water pressure buildup with

number of loading cycles is essentially the same for both Dry Vibration and

Wet Rodding specimens, provided that the same cyclic strains are used. A

comparison between figures 3.21 and 3.23 shows again that the fabric effect on pore pressure buildup is very pronounced for stress-controlled tests (figure

3.21), while it is practically nonexistent if strain-controlled tests are performed (figure 3.23).

The lowest value of cyclic axial strain, Sy, used by Park and Silver in their

strain-controlled tests was 3 x 10"^ percent. Using a Poisson's Ratio for the

saturated sand, v = 0.5, it yields a cyclic shear strain, Yc

Yc = 1«5 Sy 3.4

or, the lowest shear strain was Yq ~ ^ percent. For this value of

cyclic shear strain, the rate of pore water pressure buildup was very slow.

As shown in figure 3.23, for = 3 x 10"^ percent, A /a^ £ 0.20 even after

100 cycles. This, added to the shapes of the curves In figure 3.23, again

suggests the existence of a threshold strain, Yt r^ear 10" percent as discussed

in section 3.2.2.

All results presented in figures 3.20 through 3.23 were performed by Park and

Silver on fresh specimens, i.e., each cyclic test was conducted on a new sample.

They also performed strain-controlled staged tests on specimens compacted using

the Dry Vibration procedure. In each stage, 300 cycles of a given cyclic strain

were applied undralned, with the pore water pressure buildup being monitored.

After this cyclic loading, the drainage valves were opened and the sample was reconsolidated under the same confining pressure, = 2,000 psf (96 kPa).

The valves were then closed, and in this new undralned stage, 300 cycles of a larger cyclic strain were applied. The process was repeated at several cyclic

strains. A comparison between the results of the staged tests and those on fresh specimens indicates that the pore water pressure buildup versus number

of cycles, n, was essentially Identical if pore pressure in the previous stages

had been kept small, Au/a^ ^ 0.4. Therefore, the values of Au/a^ after n »

10 cycles for fresh specimens obtained from figure 3.23 have been plotted in

figure 3.24, together with those from staged specimens for which Au/a^ <^ 0.4. In

figure 3.24, shear strain Yc rather than axial strain, e^, has been plotted, with

21
equation 3.4 used to compute Yc* Similar results from staged strain-controlled

cyclic triaxial tests on sand No. 1, obtained from the files of one of the

authors of this report (Ladd) have also been superimposed on figure 3.24. The

grain size distribution of sand No. 1 is shown in figure 3.16, The specimens

of sand No. 1 were compacted to Dj. = 60 percent using the Moist Tamping technique, and isotropically consolidated to effective confining pressures of =

10 psi (69 kPa) and = 20 psi (138 kPa). Figure 3.24 includes results of

strain-controlled tests on sand No. 1 using cyclic shear strains, Yf. < 10"^

percent. These small strain measurements were done by the use of the technique

described in section 5.1.2 of this report. At these small strains, Au/ = 0

after n = 10 cycles, and figure 3.24 again suggests a threshold, y^ ~ 10"^

percent.

Figure 3.24, which was included in a recent publication by Dobry and Ladd, 1980

[20], is remarkably consistent. Although it includes results of cyclic tests

conducted on two different sands, on normally consolidated specimens prepared

at two different laboratories using different techniques, and for a range of

confining pressures between 1,400 and 2,800 psf (69 to 138 kPa), one single

curve fits all results reasonably well. The threshold strain, ^ 10 ^ percent

is one important feature of this curve. The clear and consistent picture of

pore water pressure buildup provided by figure 3.24 is simpler than data that can be obtained from stress-controlled tests on the same sands. Figure 3.24

gives a clear indication of the potential usefulness of the cyclic strain

method.

3.2.4 Proposed Cyclic Strain Method

Instead of using the seismic (cyclic) shear stress, (or the stress ratio

T(s/aQ)t it is suggested to use the seismic (cyclic) shear strain, Yc»

purpose of evaluating liquefaction potential. There seems to be three possible

ways of obtaining Yc ^ given depth z of a soil profile and for a given

seismic excitation:

a) From the equation Yc ~ ^c/G. This equation assumes that the seismic shear

stress T(, at depth z is known. The value of T(, can be computed from an expression such as equation 3.1 if the ground surface acceleration is known.

b) From site response studies where a model of the soil is subjected to an input earthquake motion, and the strains, stresses and motions within the

model are calculated (e.g., Schnabel et al .

, 1972 [71].

c) From the ground particle velocity, V, and using the expression y

~ V/c,

where c is the propagation velocity of the relevant seismic wave, and which

is often (although not always) taken as c = (G/p)l/2 (p = mass density of

the soil). This method has been used extensively to predict ground shear

distortions near pipelines during earthquakes (e.g., see Newmark, 1967 [55]).

A common feature of procedures (a) through (c) is that they all explicitly

include the stress-strain or stiffness properties of the soil in the calculation.

This is in contrast with the cyclic stress approach to liquefaction, where the

stiffness of the soil is not explicitly considered. Generally speaking, a stiffer soil having a larger value of G will experience a smaller cyclic strain

and will develop less pore pressures.

22
The present report will focus on procedure (a) to calculate Yc* °f

procedure (a) permits the formulation of a cyclic strain method for evaluating

liquefaction potential along the lines of the original Seed and Idriss (stress)

method, described in section 3,1.2. The steps of the proposed strain method

to evaluate the liquefaction potential of a sand layer at a depth z are the

following:

Step 1 . Determination of Yc ^' Yc calculated using equation 3.5

Yc = 0.65

^ % ^d 3.5

S ^^max (^/'^max^Y

c Equation 3.5 is similar to equation 3.1, however; equation 3.5 considers the

stiffness of the soil, G, while equation 3.1 does not.

The meaning of each symbol in equation 3.5 is given below:

peak horizontal acceleration at the ground surface

acceleration of gravity

total overburden pressure at depth z rjj(z) = stress reduction factor with depth plotted in

figure 3.4

shear modulus of the soil at very small cyclic strain,

Yj, - 10" percent

effective modulus reduction factor of the soil corresponding

to the cyclic strain, y^.*

The equivalent number of cycles n is obtained from the magnitude of the

earthquake, M.

Step 2 . Comparison between Yc threshold strain of the soil, Yt

•

If Yc ^ Yt» neither pore pressure buildup nor liquefaction will occur

and the evaluation ends here.

Step 3 . If Yc ^ Yt» values of Yc ^ should be used in

conjunction with experimental curves similar to that shown in figure

3.24, to estimate the value of the pore pressure buildup at the end

of the earthquake, Au/oq, where Oq = initial effective overburden

pressure at depth z.

Step 4 . The value of Au/oq estimated in step 3 is used to decide if the

site will experience initial liquefaction (Au/oq = 1.0) or not

(Au/a^ < 1.0).

For the case of Yc ^ Yt steps 3 and 4 above, (G/G^^x)Yc equation 3.5 is a function of both Yc current pore presure buildup, Lu/ Oq, Therefore,

the relation iG/G^^y^)y^ and Yc keeps changing during the earthquake. Obviously,

some additional research is needed to develop definite rules for computing Yc> as well as to refine other aspects of the proposed cyclic strain method, for the

case of Yc ^

g

^d

^max

(G/Ginax)Y^

23
The rest of this report presents results of studies and laboratory tests

conducted to develop the necessary information for the use of the proposed

cyclic strain method.

24
a. IDEALIZED FIELD LOADING CONDITIONS

oh

KqCJo

i It

t

Initial stresses Cyclic load sequence

b. SHEAR STRESS VARIATION DETERMINED BY RESPONSE ANALYSIS

00

CO

CO

U4

CO

CO

12 18

TIME, SECONDS

24 30

Figure 3.1 Cyclic shear stresses on a soil element during

ground shaking (Seed et al . , 1975)

25
0.3 MEDIUM MONTEREY SAND

INITIAL VOID RATIO, ej = 0.68 (Dr = 50%)

INITIAL EFFECTIVE CONFINING PRESSURE,

2.0 kg per sq cm

10,000

NUMBER OF CYCLES TO CAUSE FAILURE, n

Figure 3.2 Typical form of the relationship between pulsating shear

stress and the number of cycles to cause failure - simple

shear conditions (Peacock and Seed, 1967)

26
CYCLIC SHEAR STRESS,

Figure 3.3 Cyclic stress method for evaluating liquefaction potential

(Seed et al., 1975)

27
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 STRESS REDUCTION RATIO,

Figure 3. A Range of values of stress reduction ratio, rj, for different

soil profiles (Seed and Idriss, 1971)

28
5 6 7 8 9

EARTHQUAKE MAGNITUDE, M

Figure 3.5 Equivalent numbers of uniform stress cycles based on strongest components of ground motion (Seed et al.,

1975a)

29
\

i2

uJ 0.5

CO

CO

1.0

2.0

1

1 —^

LIGHT D

NO LIQU

AMAGEEFACTION

^

N

\

\\

\

N

\

HEAVY DAN

LIQUEF/

1A6E AND

ACTION

^^5::

B — B(

0

3UNDARY DETERMIN

]UNDARY DETERMIN

hISAKI

ED BY DAMAGE SUF

ED BY FIELD OBSER

IVEY (KISHIDA)

i/ATIONS (KOIZUMI)

10 20 30 40

STANDARD PENETRATION RESISTANCE, N-BLOWS/FOGT

Figure 3.6 Analysis of liquefaction potential at Niigata for earthquake

of June 16, 1964 (Seed et al., 1975)

30
Legend

* Cyclic mobility or liquefaction

o

N' =

No ground failure

5QN

a' + 10

o

o = Effective overburden pressure, psi

Figure 3.7 Performance of saturated sands at earthquake

sites (Castro, 1975)

31
0.5

0.4

0.3

Zl 0.2

7i 0.1

LOWER BOUND FOR SITES WHERE —^/

LIQUEFACTION OCCURRED

Ni = STANDARD PENETRATION RESISTANCE

CORRECTED TO AN OVERBURDEN PRESSURE

EQUAL TO 1 tsf (1 tsf = 96 kpa)

N

20 30

- BLOWS PER FOOT

40

• LIQUEFACTION; STRESS RATIO BASED ON ESTIMATED ACCELERATION

# LIQUEFACTION; STRESS RATI O BASED ON GOOD ACCELERATION DATA

O NO LIQUEFACTION; STRESS RATIO BASED ON ESTIMATED ACCELERATION

O NO LIQUEFACTION; STRESS RATIO BASED ON GOOD ACCELERATION DATA

Figure 3.8 Correlation between stress ratio causing liquefaction

in the field and modified penetration resistance of

sand, Ni, (Seed et al . , 1975)

32
<

LU

LU O

QC

Q- QC

LU OLI. QC° z:^ y 2

_J LU 0 I— >- o

^ z

s <

EE

CD 00

113

<

LU < I O 00

to 2

00

00

LU

^o*^ QC O

I- O

00

O>-o

MODIFIED PENETRATION RESISTANCE, N^ -blows/a

Figure 3.9 Correlation between field liquefaction behavior of sands

for level ground conditions and modified penetration

resistance (supplemented by data from large scale tests)

(Seed, 1979)

33
CO

LU

oc

I—

00

oco

CD »

<

CO

<

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

Cyclic stresses for initial liqueftction

•i = void ratio

•—^ ,. _

-o.^ 'i ^

ei = 0.61

0.71

ej = 0.78

a . _ A

"1 - u.o/

1 10 100 1000 10.000

NUMBER OF CYCLES, n

100,000 1,000.000

Figure 3.10 Cyclic stresses required to cause liquefaction and 20 percent

strain in Sacramento River sand at different densities -

0:5 = 1.0 kg per sq cm (Seed and Lee, 1965)

34
Curve No. Method of Compaction

1 High frequency vibrations on moist samples

2 Moist tamping

3 Moist rodding

4 Low frequency vibrations on dry samples

5 High frequency vibrations on dry samples

6 Pluviated-water

7 Pluviated-air

8 Dry rodding

Figure 3.11 Cyclic stress ratio versus number of cycles for different

compaction procedures (after Mililis et al, 1375)

35
0.35

0.30

0.25

0.20

0.15

0.10

0.05

Monterey No. 0 Sand

= 8 psi (55.2 kpa)

Dr =54% Pluviated

Sample No Pre-shaking

=55% Pluviated

Sample With Pre-shaking

1

10 20 50 100 200 500 1000

NUMBER OF CYCLES TO CAUSE PEAK CYCLIC

PORE PRESSURE RATIO OF 100%

Figure 3.12 Effect of seismic history on cyclic strength of sand (Seed, 1979)

36
0.4

1 —I

\ r—

I—I I I I

I

1

1 1—

I—I I I

I

;^ 0.3-

< 0.2 -

QC

00

CO

LU

fic 0.1 -

00

Q

1 I I I I I I I I I

I I I I I I I I

1 3 5 10 30 50 100

NUMBER OF CYCLES CAUSING PEAK CYCLIC

PORE PRESSURE RATIO OF 100%

Figure 3.13 Influence of overconsolidatlon on stress causing pore

water pressure ratio of 100 percent in simple shear

tests (Seed, 1979)

37
LU

CO

00

00

o

QC

LU

a.oc

LU

hLl_ <

<

00

CO

LU

QC

I—

00

<

CO

>-QCO

I—<

QCO

QQ<

00

oa.

LU

o

>-

00

UJ

QC

CD

LU

QC

I—

3.0

2.5

2.0

1.5

O

'A

-V

.0

0.5 —

01

TIME AFTER DEPOSITION - YEARS

1 10 100 1000

Laboratory test data-Monterey lllo. 0 Sand

Hydraulic sand fill from Upper San Fernando Dam

Hydraulic sand fill from Lower San Fernando Dam

South Texas sand . San Mateo sand A

10 10^ 10^ 10' 10' io6

TIME AFTER DEPOSITION - DAYS

Figure 3.14 Influence of period of sustained pressure on stress

f ^M P , causing peak cyclic pore pressure ratio of 100

percent (Seed, 1979)

38
SHEAR STRAIN, PERCENT

Figure 3.15 Void ratio change for a sand as a function of cyclic

shear strain and number of cycles (Youd, 1972)

39
10 1 0.1 0.01

SIZE (mm)

Figure 3.16 Grain size curves of sands used in testing

40
SAND i\lo. 2 LADD (1977)

0.10

1 10 100 1000 10,000

NUMBER OF CYCLES TO INITIAL LIQUEFACTION, n

1

Figure 3.17 Effect of fabric on cyclic strength, stress-controlled

tests (Ladd, 1977)

41
Figure 3.18 Stress-strain curves for first compression and extension

excursions, stress-controlled cyclic triaxial tests, Sand No. 2 (modified after Ladd, 1977)

42
T T T

10 100 1,000 10,000

NUMBER OF CYCLES TO INITIAL UQUEFACTION, n

I

Figure 3.19 Effect of fabric on cyclic strength after accounting for sample

stiffness » stress-controlled tests (modified after Ladd, 1977)

43
oo

b

V CO

CM

II o

CO

00

LU

I—

00

0.50

0.40

0.30

0.20

0.10

0

CRYSTAL SILICA No. 20 SAND Dr = 60% (PARK AND SILVER. 1975)

Au

FAILURE CRITERION, — = 1.0 (INITIAL LIQUEFACTION)

a 3c

1

WET RODDING

DRY VIBRATION

10 100 1,000 10,000

NUMBER OF CYCLES TO FAILURE, n

Figure 3.20 Effect of fabric on cyclic strength, stress-controlled

tests (Park and Silver, 1975)

44
DRY VIBRATION WET RODDING

Figure 3.21 Stress-controlled cyclic triaxial tests of saturated

Crystal Silica sand (modified after Park and Silver,

1975)

45
Figure 3.22 Stress-strain curve for first cycle, cyclic triaxial tests

of saturated Crystal Silica sand (modified after Park and

Silver, 1975)

46
n=100 CYCLES

10-2 3x10-2 10-1 3x10-1

AXIAL CYCLIC STRAIN, e^^ PERCENT

Figure 3.23 Strain-controlled cyclic triaxial tests of saturated

Crystal Silica sand (modified after Park and Silver,

1975)

47
FACIWG PAGE: Sand bolU ob62Avzd a^teA thz

ImpQAlal VoJUzy, CA, OcXobaA

1979 <LaAtkqaak(L . LiquU.{i^ed

^Im 6and& weAe poihed up

thAoagh thz ground AuAjJace

oij dfiy aoauz J>oll6 by exce^i^

pon.2. MCUt^ pn.2^6uA&,

1.2

1.0

b

i U

UIs

0.2 -

o Crystal silica sand 03 = 2000 psf, fresh samples, dry vibration A Crystal silica sand 03 = 2000 psf, fresh samples, wet roddint A Crystal silica sand 0'^ = 2000 psf, stated testing, dry vibration

Sand no.1 o'^ = 2800 psf, staged testing testing, moist tamping

vSand no. 1 03 = 1400 psf, staged testing, moist tamping

10

-3 10

-2 10

SHEAR STRAIN, 7, PERCENT

Figure 3.24 Measured pore water pressure in saturated sands after ten

loading cycles, strain-controlled cyclic triaxial tests

(Dobry and Ladd, 1980)

48
4. A MODEL OF SPHERES FOR THE THRESHOLD STRAIN

4.1 GENERAL

An Important aspect of the relationship between cyclic strain and excess pore

water pressure buildup is the existence of a threshold shear strain, Y|- , in

sands, below which no densif ication and, therefore, no excess pore water pressure buildup occurs. Section 3.2.2 summarized experimental evidence suggesting

that in sands this parameter seems to have a remarkably constant value, of the

order of 10"^ percent. The origin of and the parameters controlling it are investigated in this section by means of a theoretical model of spheres.

The model selected is that of a simple cubic array of identical quartz spheres.

Some of the results using this model have been presented elsewhere (Dobry and

Swiger, 1979 [21]). Even though the simple cubic array is a very simplified

49
model of a real sand, it is shown in the rest of this chapter that the study of

this model provides: (a) an explanation of the physical origin of Yt> ^ reasonably good prediction of the value ^ 10~ percent which has been

measured in actual sands; (c) insight into the influence of parameters such as confining pressure and grain size on y^; and (d) a reasonable prediction of

the measured cyclic stress-strain behavior of actual sands at strains below the

threshold, y < Yf

Figure 4.1 shows the model. The simple cubic array of elastic quartz spheres

is subjected first to an isotropic confining pressure, a, and then to a cyclic

shear stress, + t (only the stresses corresponding to the positive t are shown

in the figure). Associated with the cyclic stress there is a cyclic strain of

the array, + Y» It is assumed that y is smaller or at most equal to the

threshold value, Y ^ Yf will be seen that this is equivalent to assuming

that no sliding occurs at any of the contact points between the spheres. The

following elastic constants and friction coefficient for the quartz spheres were obtained from Larabe and Whitman (1969 [45]) and are used for the

calculations

Young's Modulus E = 11 x 10^ psi

Poisson's Ratio v = 0.31

Friction Coefficient f = 0.50

Section 4.2 presents a study of the shear force-displacement relation at the

contact points between the spheres, using the results of the Mindlin-Deresiewicz

theory. In section 4.3 this information is used to calculate the value of Yt*

In section 4.4 the stress-strain behavior calculated for the model at small

strains, y < Yt> compared with that of actual sands.

4.2 CONTACT BETWEEN ELASTIC SPHERES

Figure 4.2 shows the situation at any one of the four contact points around the

representative central sphere of figure 4.1. A normal force N and a tangential

force T must be transmitted through the contact. The relations between these

forces and the overall stresses o and t acting on the array shown in figure 4,1 are

a=JL 4.1

4r2

4r2

The normal force N produces an elastic shortening of the distance between the

centers of the neighboring spheres. This shortening, which translates into

normal and volumetric strains for the whole array, is of no interest for the

present calculations. On the other hand, the tangential force, T, produces a tangential displacement, 5, between the centers of the spheres. This tangential

displacement is the direct cause of the shear strain of the whole array:

50
4.3

where the two terms represent, respectively, the contributions to y of the

two spheres at left and right of, and above and below of, the representative

central sphere of figure 4.1,

Therefore, an understanding of the physical origin of 6 and a calculating of

its value are the key to understanding and calculating y and y^.

The following calculation of 5 is based on the work by Mindlln (1949 [51]),

Mindlin et al., (1951 [52]) and Mindlln and Dereslewlcz (1953 [53]), as summarized in a previous report by Dobry and Grivas (1978 [19]).

The contact point between the spheres in figure 4.2 is really a small circular

area of radius a < < R. The value of a is:

a = (1IZ^\ R)l/3

4 E

The normal force, N, is distributed over this circular area. The corresponding

normal contact stress, a^^, has the parabolic distribution shown in figure

4.3, where a^, is a maximum at the center of the contact area and zero at the

edge of the area.

The distribution of the tangential force, T, over the same contact area is of

special interest. If the tangential force has been raonotonically Increased

while keeping N constant, the elastic solution for the shear stresses, Tq,

within the contact area gives = «> at the edge of the contact area.

If the solid friction condition, _< fa^, Is imposed at all points within

the contact area, it is found that there is an annulus of Inside radius c and

outside radius a (see fig, 4.3), where t,, = fa^ and where slip occurs between the two surfaces and energy is lost by friction. As T Increases, c decreases, until, for T = f N, c = 0, the condition = fa^ prevails

over the entire contact area, and there is gross sliding of the two spheres

along their contact. If 5 = horizontal displacement between the centers of the

spheres, the force-displacement curve, T vs. 6, is of the yielding type, as shown by curve OP in figure 4.4.

If the tangential load T is cycled between two fixed values, T* and -T*, (T* <

f N) , while maintaining N constant, a hysteresis loop is formed, as shown in

figure 4.4. This hysteresis loop is similar to the experimental loops measured

in sands subjected to cyclic shear loading (e.g., see Seed and Idriss, 1970

[77]). The area enclosed by the loop measures the energy spent by friction in

the annulus of slip, and for the case considered here (T* < f N) , the loop is

stable (i.e., it repeats itself cycle after cycle).

Mindlin et al., (1951 [52]), Johnson (1955 [34] and 1961 [35]) and Goodman

and Brown (1962 [27]) verified experimentally the predictions of the Mindlln51
Deresiewicz theory, by pressing together glass and metallic bodies and then

applying cyclic tangential forces at the contact between them. All predictions

were verified, including the existence of the annulus of slip, as well as the

location of the tangential force-displacement curves for monotonic (curve OP

in fig. 4.4) and cyclic (hysteretic loops in figs. 4.4 and 4.5) loadings.

4.3 THRESHOLD STRAIN OF ARRAY OF QUARTZ SPHERES

The equation of the monotonic "backbone" curve (OP in fig. 4.4) predicted by

the Mindlin-Deresiewicz theory is:

« = 1 - (1 - _1 )2/3 4.5

6i fN

where

5 = 3(2-v) (l+v)f N 4.6

4Ea

Equation 4,5 has been plotted in figure 4.6. When the tangential displacement

6 = 6i, the tangential force T = fN and gross sliding of the contact occurs. Therefore, 5j is the threshold displacement at which there is a tendency for an overall change of the geometric arrangement of the spheres to occur. The

threshold strain, Yt» 1^ related to 6i by equation 4.3:

If the value of from equation 4.6 is substituted into equation 4,7, and the

resultant expression is combined with equations 4.1 and 4.4, the following

equation is obtained for Yt*

Y^ = 2,08 (2-v) (l+v)f ra^2/3 4.8

(l-v2) (E)

Finally, if the numerical values of the constants for quartz listed in

section 4.1 are used, the following simple expression is derived,

., Yt(^) = 1.75 X 10"^ (a)2/3.

Equation 4.9 gives the threshold strain, Yt > a function of the confining

pressure, a for a simple cubic array of quartz spheres. This equation is

plotted in figure 4.7. The result is extremely Interesting. It suggests that:

(a) for the range of confining pressures of most practical interest (500

psf < a < 4,000 psf), Yt 1^ 1^ range between about 1 x 10"^ percent and

4 x 10~2 percent, which is close to the experimental values reported for actual

sands; and (b) Yt 1^ independent of grain size (the radius of the spheres, R is

not present in equations 4.8 and 4,9. Finally, equation 4,8 offers a means to

study the Influence of the material constants E, v, and f for sands other

than quartz sands.

a in psf 4.9

52
4.4 CYCLIC STRESS-STRAIN BEHAVIOR AT VERY SMALL STRAINS

The simple cubic model of quartz spheres can also be used to predict the cyclic

stress-strain behavior of sands at very small strains, y < Yt • Measurements

during cyclic shear loading of sands have produced experimental hysteresis

loops such as those shown In figure 4,4, except that, for actual sands, stress

(t) Is plotted versus strain (y). Instead of T versus 6 used In the figure.

However, the curves In figures 4.4 and 4,6 can be readily converted Into t - y

plots valid for the simple cubic array and loading system of figure 4,1.

Specifically, equation 4.5, representing the backbone curve In figures 4.4 and

4.6, becomes:

Yt fcT^ 4.10

It Is of Interest to compute two normalized parameters for the simple cubic

array of quartz spheres at very small strains, y < Yt » ^"^^ to compare these

parameters with the values measured In actual sands. These parameters are:

(a) the modulus reduction curve, G/Gm^x versus y» and b) the damping ratio, X versus the shear strain y«

The expression of G/G^ax versus y can be obtained directly from equation 4.10

as follows:

3/2

T = fa [1 - (1 - ^) ] Yt

4.11

= T » fa n _ M _ Y

^^/2^ 4.12

sec G = G,,^^ = 1 = [1 - (1 - X_) ] Y Y Yt

G = (il ) = 1 l£ max ^-T- ' J — ^Y

Y=:0 ^ ^t

Y

3/2

G .2 ^-(^-V 4.13

^max 3 iHt

Equation 4.13 has been plotted as a dashed curve In figure 4.8 for a representative value Yt ~ 10" percent. The experimental range for sands presented

by Seed and Idrlss (1970 [77]) Is superimposed In the figure for comparison.

The theoretical line generally coincides with the upper bound of the experimental range.

A similar calculation was performed for the hysteretlc damping ratio, X. The

usual definition of X Is:

X = i M 4.14

2ir TY

53
where AW = area enclosed by a t - y hysteresis loop. For the simple cubic array

X can be calculated using equation 4.14, or alternatively directly from the

theoretical T-6 hysteretic loop shown in figure 4,4. If this last procedure

is used, the damping ratio is:

X = J_ (AW) contact 4.15

2t\ T*5*

where ( contact is the area of the loop in figure 4.4 (energy dissipated at one contact during one loading cycle) , and T* and 6* are the maximum values of

the tangential load and displacement, respectively, during the cycle.

Equations 4.14 or 4.15 give identical results and equation 4.15 is used here

for convenience.

Goodman and Brown (1962 [27]) calculated the value of (AW) contact:

(AW)contact . jS (2-v)( 1+v) (fN) ^ n . (i . T^)5/3

5 Ea ^ fN

- 5 T* [1 + n T*)2/3]l 4.16

6 fN fN

By combining equations 4.5, 4.3 and 4.7, equation 4.17 is obtained:

11 = 1 - (1 - 51)3/2 = 1 - (1 - r_)3/2 4.17

fN 6i Yt

where y - maximum shear strain during the loading cycle.

By substituting equation 4,17 into equation 4.16, the following expression is

obtained for (AW) contact:

(AW)contact = jS (2-v)(l+v)(fN)2 | j . (j. y/y^)^^^

- 5(2-1 ) [1 - (1 - Y/yt)'^^} "^-1^

6 Yt

Equation 4.18 provides the value of (AW) contact needed to calculate X in

equation 4.15. The product T*5* for equation 4.15 is obtained as follows:

From equation 4,17:

T* = fN[l - (1 - I_)3/2] 4.19

Yt

From equations 4,6 and 4,17:

^* = y = 3(2-v)(l+v)fN y 4.20

y^ 4Ea y^

54
Finally, if equations 4.18, 4.19 and 4.20 are corabined with equation 4.15,

the following expression is obtained for X:

12 (1 - Y/Yt)^^^ 5 (2 - Y/Yt)

X = { 1 - TT^ — } ^'21

5Tr (Y/Yt)[l-(1-Y/Yt)^ ] 6 y/Yc

Equation 4.21 has been plotted as a dashed curve in figure 4.9 for a value Yt " 1.5 X 10~2 percent. The experimental range for sands given by Seed and Idriss

(1970 [77]) is again included for comparison. The theoretical equation for the

simple cubic array coincides approximately with the lower bound of the experimental results for most of the range of strains, Y<Yt=l«5x 10~^ percent.

The comparisons presented in figures 4.8 and 4.9 further verify the crude simple

cubic model used to compute Yf These figures show that the model predicts in

a general manner the main features of the cyclic stress-strain behavior of sands

at very small strains, y < Yf

Another interesting feature of equations 4.13 and 4.21 is that both G/Gj^g^ and y are unique functions of the normalized strain parameter, Y/Yf This is similar

to the hyperbolic stress-strain model for cyclic loading of soils proposed by

Hardin and Drnevich (1972) [29], where G/Gmax a function of Y/Yr» Yr " reference strain. In the simple cubic array, the threshold, Yt» plays the role

of a reference strain, and in actual sands, perhaps Yt ^'^^ Yr also related.

In that respect, it is interesting to note that measurements in sands and other

soils show that, if the confining pressure, a, is increased, both Yr

(G/G^3x)y ^ given y also increase (see also Richart, 1980 [66]). This is

similar to the prediction of the cubic array model: equations 4.8 and 4.9 and

figure 4.7 illustrate the increase in Yt with a, while equation 4.13 predicts

an increase in G/G^^j^ as Yt (and therefore as a) increases. For example, for

Y =10"^ percent and a = 500 psf, equations 4.9 and 4.13 predict (G/G )^ ^ lQ-2 percent = 0.71 for a cubic array of quartz spheres. If a = 4,000 psf, the same trend and very similar values of (G/G^q^x^y = 10~^ percent have been measured

in several sands by Iwasaki et al., (1978 [33];.

55
Figure 4.1 Simple cubic array of equal spheres

56
Figure 4.2 Elastic spheres under normal and tangential loads

57
Tcfno slip)

27ra3

Normal (a^) and tangential (t^,) components of traction

on contact region between two spheres subjected to a normal force followed by a monotonic tangential force

(Deresiewicz, 1973)

58
Figure 4.4 Theoretical hysteresis loop due to oscillating tangential

force at constant normal force for two spheres in contact

(Deresiewicz , 1973)

59

Figure 4.6 Tangential force-displacement relation for two elastic spheres

under constant normal force, N (Dobry and Grivas, 1978)

61
THRESHOLD SHEAR STRAIN, 7t. PERCENT

Figure 4.7 Calculated threshold shear strain as a function of isotropic

confining stress for a simple cubic array of quartz spheres

(Dobry and Swiger, 1979)

62
SHEAR STRAIN, 7. PERCENT

Figure 4.8 Reduction of shear modulus as a function of shear strain - comparison between calculated G/G^g^ ^ simple cubic

array and experimental range for sand

63
FACIWG PAGE; G^iound cAaciu which dovdiopdd

duAlng tkd San T^nando ¥zbn.aan.y

9, 1971, toAthqaako. In compactzd

{^iXZ dbovQ, tooi)<i 6atuAat^d 6and

at tkd paKtlalty comptoXtd J2,n6zn

VAjJjiatlon Vtant.

30

lO"'* 10-^ 10-^ 10'^

1 SHEAR STRAIN, 7 , PERCENT

Figure 4.9 Damping ratio as a function of shear strain - comparison

between calculated X for a simple cubic array and experiI mental range for sand

64
5. CYCLIC LABORATORY MEASUREMENTS

5.1 GENERAL

This chapter presents the results of a program of undrained cyclic triaxial

tests performed on specimens of saturated Monterey No. 0 sand. The tests were performed during the summer of 1979 , as part of the development of the cyclic

strain approach to evaluate liquefaction potential, described in this report.

For the reasons discussed in chapter 3, all cyclic tests were of the straincontrolled type. A key parameter needed for the cyclic strain approach is

the shear modulus of the soil, G. Therefore, a major objective of the tests was to obtain both G at small strains (G^g^^) » the variation of G and G/G-^^

with cyclic shear strain amplitude, + y» with number of loading cycles,

n. Another key parameter needed in the approach is the threshold strain, y^-

,

which was also measured during the tests. The measurements of G at small

65
strains and of Yt involved the use of an improved experimental technique

recently developed by one of the authors (Ladd), Finally, the development of

excess pore water pressure, Au, with number of cycles, n, of strain-controlled

loading, was also measured during the tests and is reported in this chapter.

In addition, the influence of relative density (Dj.) and of initial confining

pressure (a^) on G, G/Gmax* Yt studied and is discussed in this

chapter.

5.1.1 Sand Tested

The particle size distribution curve and the selected index properties of the

Monterey No. 0 sand, obtained by Mulilis et al., (1975 [54]) are shown in fig- ure 5.1 and table 5.1, respectively. The sand is a commercially available

washed uniform medium-to-fine beach sand (SP), composed of quartz and feldspar

particles. The maximum and minimum dry unit weight determinations were performed in accordance with the ASTM Test for Relative Density of Cohesionless

Soils (D 2049-69) and Kolbuszewski ' s (1948 [39]) method, respectively. The

specimens tested had initial relative densities, D^, of approximately 45, 60,

and 80 percent, and were prepared using the moist tamping compaction method

(Ladd 1978 [44]). It should be noted that the same type of sand was used at

the University of California at Berkeley to perform a number of studies on sand liquefaction and densif ication during earthquakes (e.g., see DeAlba, et

al., 1975 [14] and Pyke et al., 1974 [65]).

5.1.2 Testing Technique

The techniques used for specimen preparation and testing include unique features

such as the undercompaction of the lower layers of the specimen to achieve a more uniform density, and the capability to measure modulus and pore water

pressure response at very small strains (y - 10~^ percent). Details of the

undercompaction moist tamping technique are given by Ladd (1978 [44]).

The improved technique which has allowed extending the testing capability of

cyclic triaxial equipment from y - 10"^ percent to y - 10"^ percent includes:

(a) a frictionless loading system with precise axial alignment (air bushing

and specially machined and ground components),

(b) precise coupling between porous stones and top and bottom plattens

(individually lapped and indexed) and test specimen and porous stones

(refined compaction techniques),

(c) a correction for equipment compliance (see figure 5.2), and

(d) very sensitive recording systems (load to 0.01 pound and deformation to

1 x 10-6 inch).

Additional details on the specimen preparation and testing techniques are given in the following paragraphs, while the method of performing the calculations is included in the appendix. An electrohydraulic closed-loop loading

66
Table 5.1 Index Properties for Monterey No. 0 Sand

Mulilis et al . (1975)

Unified Soil Classification System Group symbol SP

Mean Specific Gravity 2.65

Particle Size Distribution Data

D5o» mm 0.36

C^,^^) 0.9

Dry Unit Weight Data

Maximum, pcf 105.7

Minimum, pcf 89.3

(1) = (D-^n) /(D^n x Din)

67
system and specially designed and manufactured triaxial cells, were used in

all tests.

The strain-controlled cyclic triaxial tests were performed in general accordance

with the procedures outlined by Silver (1975 [82]) and Park and Silver (1975

[59]).

The key points followed in performing the tests are:

1) Each specimen was reconstituted using the moist tamping method as outlined

by Ladd (1978 [44]), using a compaction mold attached to the base of the

triaxial cell. This method ensures a "perfect" contact between the speci- men and the loading platens. In addition, in these tests, some of which

involved low relative densities, it was found that, to obtain a "perfect"

contact: a) the bottom layer had to be placed and compacted in two parts,

and b) the top layer had to be partially compacted, then scarified, the top

stone inserted and twisted to get it seated properly and then compacted to

the prescribed density, with all equipment in place, by striking the top

of the loading piston.

2) Each specimen was saturated by backpressuring (backpressuring is done by

gradually increasing the backpressure and cell pressure simultaneously) at an effective stress of 5 psi (34.5 kPa). The test specimen was considered

to be saturated if the pore pressure response (B-parameter) was equal to or greater than 95 percent.

To assist in the saturation of the specimen, carbon dioxide (CO2) and deaired

water were percolated through the specimen prior to backpressuring. A backpressure of 70 psi (483 kPa) was applied in all tests.

3) Each specimen was isotropically consolidated in increments to the final

effective confining pressure of the test, a^, on the day prior to performing the cyclic test.

4) During backpressuring and consolidation, the triaxial cell was completely

filled with water (which had been deaired at the start of the test) and

axial deformations and volume changes of the specimens were recorded. In

addition, a small axial load was applied to the piston screwed into the

top cap, sufficient to maintain the specimen in an isotropic state of

stress

.

5) Prior to cyclic loading, the triaxial cell was transferred from the

consolidation area to the cyclic loading apparatus. During this stage,

the applied values of cell pressure, axial load, and backpressure were maintained constant.

6) The specimen was cyclically loaded without drainage using the

electrohydraulic closed-loop loading system. The system applied a sinusoidally-varying cyclic load or deformation at a frequency of 1 Hz.

Just prior to cyclic loading, an air pocket was formed at the top of the

cell and the 0-ring seal (which was attached to the bushing assembly) was

68
removed, thereby switching over to an air bushing housed in the bushing

assembly. During cyclic testing, changes in axial load, axial deformation,

and pore water pressure were recorded on a 7-in. oscillograph recorder.

These values were typically recorded within a resolution of two percent of

the recorded maximum value. In addition, an x-y recorder was used to

obtain hysteresis loops of selected loading cycles.

5.1.3 Test Program

A total of 12 undrained strain-controlled cyclic triaxial tests were performed

on saturated specimens of Monterey No. 0 sand. The list of tests is presented

in table 5.2.

With the exception of test 12, which was a staged test, all other tests used

fresh specimens. These 11 tests were all very similar, with the confining

pressure, a^, relative density, Dj., and cyclic shear strain, y, of the test

being varied between tests. Most of the tests were conducted with = 2,000

psf, except for tests 10 and 11, where = 533 psf and 4,000 psf, respectively. Three relative densities, 45, 60, and 80 percent and three values of

Y, 3xl0~2, lxl0~l and 3x10"^ percent, were used. Table 5.3 shows in a matrix

form the values of Dj. and y corresponding to each test for = 2,000 psf.

The typical undrained cyclic testing sequence for each test was as follows:

a) Measurements at very small strains . Measurements of G and pore water

pressure response (Au) were made at very small cyclic strains, 10"^ percent

< y < 10~2 percent. These measurements were done by applying cyclic loads.

Several levels (stages) of cyclic loads were typically applied with five

loading cycles being applied in each stage. In addition to measuring the

pore water pressure during cyclic loading, Au, the residual pore water

pressure, Auj., was also measured after cyclic loading was stopped. All these

measurements were nondestructive, as verified by the fact that Auj- = 0, and

also by the repeatability of the values of G at the given y, irrespective

of the previous history of small strain cyclic loading.

b) Measurement of yf . This was done by applying 10 cycles of a value of cyclic

strain, y, slightly larger than y , usually in the range 1 x 10"^ percent

< y < 2 x 10"^ percent. Both Au during cyclic testing and Au^. after the

10 cycles were recorded. Invariably the measured values of Au and AUj. were very small. The shear modulus, G, was also measured during these 10 cycles;

it usually varied little between the first and last cycle. After measuring

Auj., the drainage valves were opened and reconsolidation of the system

was allowed for.

c) Measurements at very small strains . Same as in step (a) above. This was done to verify that the G values at very small strains had not been significantly affected by step (b), thus, confirming the assumption that step

(b) could be considered nondestructive for practical purposes.

69
Table 5.2 List of Cyclic Triaxial Tests

Test No. Confining Pressure

.

psf

Relative Density,

percent

Cyclic Shear Strain, y

percent

1 2,000(1) 45 ?

A. 10-2

2 2,000 45 1 X 10-1

3 2,000 45 1 X 10-1

4 2,000 45 •J3 X 10-1

5 2,000 60 -JJ X 1 0—2

6 2,000 60 1X A. 1 n-1

7 2,000 60 3 X 10-1

8 2,000 80 1 X 10-1

9 2,000 80 3 X 10-1

10 533 60 3 X 10-2

11 4,000 60 3 X 10-2

12

1

533-944-

2,000-4,000

(Staged Test)

45

(1) 1 psf = 47.8 pascal

70
Table 5.3

Cyclic Triaxial Tests with aX = 2,000 psf^^^

Relative

Density

D^, percent

Cyclic Shear Strain, y. Percent

3 X 10 21 X 10 ^ 3 X 10 ^

45 Test 1 Test 2

Test 3 Test 4

60 Test 5 Test 6 Test 7

80 Test 8 Test 9

(1) 1 psf = 47.8 pascal.

71
d) Testing at y ^ Y t: > (destructive testing) . Strain-controlled cyclic testing

was performed at the cyclic strain of the test, y* This Is the cyclic

strain reported In tables 5.2 and 5.3. In most cases the test was carried

to 100 cycles or to Initial liquefaction (Au = o^), whichever occurred

first. However, tests 1, 10, and 11 were carried to 1,000 cycles. During

each test the shear modulus, G, the damping ratio, X, and the maximum pore

water pressure, Au, during the cycle were measured as a function of the

number of cycles, n. In addition, the test was stopped at selected numbers

of cycles to allow for measurement of the residual pore water pressure, Au,.,

and then restarted without reconsolldatlon.

Of special concern during the planning of the testing program was the assumed

nondestructive character of step (b). To further verify this assumption, tests

2 and 3 were conducted. These two tests are Identical In all respects, except

that step (b) was skipped In test 2, and step (a) was followed Immediately by

the destructive testing (step d). The results of the two tests 2 and 3 were essentially identical, thereby verifying the nondestructive character of

step (b)

.

Test 12 was a staged test, with stages at = 533 psf, 994 psf, 2,000 psf,

and 4,000 psf, respectively. Cyclic loading was performed undralned at each

stage, and excess pore water pressures were dissipated by reconsolldatlon

between stages. Except for the first stage at 533 psf, the results of this

test were obviously affected by the reconsolldatlon process and associated

curing period and are not included in the detailed presentation of results

Included in this chapter. Further research is definitely needed on the feasibility of staged cyclic tests for determining and Auj-.

5.2 SHEAR MODULUS DAMPING RATIO

Figures 5.3 through 5.10 present the experimental results for the shear modulus,

G, and the damping ratio, X. In all cases, G was measured as the secant modulus between the compression and extension peaks within the same cycle.

Figures 5.3, 5.4, and 5.5 summarize the values of G at = 2,000 psf, and

for Dj. = 45, 60, and 80 percent, respectively. In these figures, G Is plotted

versus shear strain, y, at n = 1 cycle and n = 30 cycles. The data points for

strains below or about 10"^ percent were determined during the nondestructive

very small strain measurements (step (a) in section 5.1.3), while the data

points at larger strains were obtained during the destructive measurements in

step (d).

Estimated values of G^ax we^^e obtained using the Hardin and Drnevich (1972

[29]) equation for sands isotropically consolidated under a presure a^:

Gmax = 1230 lll9Z3_Z_£)!(ap,^^^ where G^^^, in psi 5.1

1 + e

These G^a.x estimates are included in table 5.4 and have been superimposed on figures 5.3 to 5.5. The comparisons in these figures indicate excellent agreement between the values of G measured at very small strains (y - 10"-^ percent)

72
Table 5.4 G at a\ = 2,000 psf^^^ for Monterey No. 0 Sand

percent

e Hardin-Drnevlch G^g^y,^^^

45 0.72 1,940 ksf(l)

60 0.68 2,070

80 0.63 2,230

(1) 1 psf =47.8 pascal

1 ksf = 47.8 kpa

W = 1230 (^.)l/2^ ^^^^

i + e and in psi from Hardin and Drnevlch (1972)

73
during the cyclic triaxlal tests on Monterey No. 0 sand, and G^iax calculated

using equation 5.1.

The comparison between the curves for n = 1 cycle and n = 30 cycles in figures

5.3 through 5.5 confirm that G at y ^ 10~2 percent is independent of number of

cycles. For larger strains, the discrepancy between the two curves increases,

indicating that modulus degradation occurs during cyclic loading, with the

degradation increasing as y increases.

Figure 5.6 compares the three experimental curves of figures 5.3 through 5.5,

without the data points and only for n = 1.

Figure 5.7 shows the results of measurements of the damping ratio, X, during

the destructive testing in step (d), for = 2,000 psf and for all three

relative densities tested. Since X could not be measured during the first

cycle, the results presented in the figure are for n = 2 and n = 30 cycles.

For Y - 3 x 10~2 percent, X = 7 percent, with negligible influence of n. For

Y

=^ 3 X 10"^ percent, X = 20 to 30 percent, with a tendency to decrease with

the number of cycles. The trend of increase of X with y t^e numerical

values plotted in figure 5.7 are in general agreement with the results reported

for sands by other authors (e.g., see Seed and Idriss, 1970 [77]).

Figure 5.8 presents the influence of on the measured values of G versus y»

for Dj- = 60 percent and n = 1 cycle. The corresponding values of G^^x calculated using equation 5.1 have also been included in the figure, and again there

is good agreement between G at very small strains measured during the tests and

Hardin and Drnevich's expression.

The comparisons in figures 5.3 through 5.8 between measured modulus reduction

curves and G^^-^ values estimated with eq. 5.1, are very encouraging. They suggest that cyclic triaxial tests can be used to measure G^q^, if the improved

testing techniques described herein are used. The band of experimental results

for G/Gfn3x versus y compiled by Seed and Idriss for sands, and included in

figure 4.8, indicates that, at y = 10"-^ percent, G/G^^^ = 0.95 to 0.98.

Therefore, G^^^y. was estimated using equation 5.2:

G = ^^^Y = 10"^% 5.2

max 0.95 to 0.98

The values of G^n^x ^r = ^0 percent and = 533 psf, 2,000 psf and 4,000

psf, were calculated from the values of (G)y - iq-3 percent in the experimental

curves in figure 5.8 using equation 5.2. These values of G^ax were plotted versus as data points in figure 5.9. Figure 5.9 also includes two other plots

of Gpjg^x versus for comparison. The dashed line was obtained from Hardin

and Drenevich, equation 5.1. The solid lines were obtained from the Round Robin

resonant column test program on Monterey No. 0 sand (Drnevich, 1979 [22]). In

the Round Robin test program, G^^^ was measured by nine laboratories on speci- mens of dry Monterey No. 0 sand all using an identical sand placement procedure

and testing technique. The two solid lines in figure 5.9 correspond to the

range of values of G^j^^ obtained by the nine laboratories (Drnevich, 1979 [22]).

The comparison between G^gx obtained herein and the results of of the Round

74
Robin test program confirm that, with the improved testing techniques used here,

cyclic triaxial test measurements at y = 10"^ percent are feasible and can produce reliable values of G^jg,^.

An additional check of the cyclic triaxial measurements of G is presented in

figure 5.10. The data points in figure 5.10 are the same as presented in figure

5.8, for Dj. = 60 percent and n = 1 cycle, except that in figure 5.10, K2 is

plotted versus y K2 is a normalized parameter, obtained from Seed and Idriss'

(1970 [77]) equation for G:

G = 1,000 K2 (a^)^/^ where G, in psf 5.3

Therefore, K2 = G/{a'^)^^^ if G is expressed in ksf and 03 in psf. In the

Round Robin resonant column testing program, tests were performed at a\ = 1,040

and 6,250 psf, respectively. The average values of G^j^x ^^^^ ^he nine laboratories at these pressures, in conjunction with equation 5.3, gave values of

K2max of 50.4 and 47, respectively. These values of K2inax were plotted in

figure 5.10, together with the corresponding curves of K2 versus y predicted

using:

K2

(K2)y - K2max )y 5.4

^2max

where the curve of (K2/^2max)y ~

('^/^max^ versus y selected for the

calculations is the average curve for sands suggested by Seed and Idriss (1971

[77]) (average of the experimental band in figure 4.8).

The agreement in figure 5.10 between the data points and the curves obtained

combining the Round Robin's results with those of Seed and Idriss curve is

excellent at both small and large strains. The only exceptions are the data

points for = 4,000 psf and y > 10"^ percent, which plot somewhat higher

than the curves and the rest of the data points, with increasing discrepancy at

larger strains. This discrepancy would tend to confirm the tendency of G/Gj^^^^

to be somewhat higher in soils at larger confining pressures, as discussed by

Iwasaki et al., (1978 [33]), and Richart (1980 [66]). A similar effect was already discussed for the simple cubic array model in section 4.4. In any case,

the comparison in figure 5.10 further validates the experimental values of G

obtained in this research with the cyclic triaxial technique.

5.3 MODULUS DEGRADATION UNDER CYCLIC LOADING

In the destructive part of the strain-controlled tests (y > Yt)» there was both

pore water pressure buildup and degradation of the modulus with number of

cycles, n. This modulus degradation effect is presented in figures 5.11 through

5.17, as experimental curves of G/Gj versus n. In all cases, Gj is the secant

modulus measured in the first cycle conducted at the cyclic strain of the test,

y. The influence of y on the curves of G/G]^ versus n is presented in figures

5.11, 5.12, and 5.13, for Dj. = 45, 60 and 80 percent, respectively. These

figures show that G/Gj is significantly affected by both n and y, with G/G^

decreasing rapidly as y increases above 10"^ percent.

75
Figures 5.14, 5.15, and 5.16 illustrate the effect of Dj. on the curves of G/Gi versus n for y = 3 x 10"^, 1 x 10"-^, and 3 x 10~^ percent, respectively. These

figures show that G/Gj is significantly affected by relative density, with

modulus degradation being more pronounced at the lower relative densities.

Figure 5.17 shows the influence of confining pressure, for Dj. = 60 percent and

Y

= 3 X 10~2 percent. Other things being equal, this figure suggests that

modulus degradation is more significant at lower values of

5.4 THRESHOLD STRAIN

Figures 5.18 and 5.19 show the results of the threshold strain measurements in

steps (a) and (b) of the tests (see section 5.1.3), while figure 5.20 presents

relevant results on Yt f^'^ '^^V Monterey No. 0 sand obtained by Pyke (1973 [64]).

Figure 5.18 includes the values of the residual pore water pressure, Au^ after

n = 10 cycles, for = 2,000 psf and for the three relative densities tested.

Note that the values of Auj./a^| in the figure are very low and smaller than

0.1 (a value of Au/a^ - 1.0 would indicate initial liquefaction). Therefore,

figure 5.18 permits determining the value of the threshold strain with a high

degree of precision. Figure 5.18 demonstrates that:

a) For the sand tested and for = 2,000 psf, the threshold strain is Yt

~ 1.1 X 10~^ percent. This value of y^ independent of relative density in

the range 45 percent < Dj. < 80 percent.

b) For values of strain slightly larger than y^ ^ 10 percent <. y < 3 x

10~^ percent, the residual pore water pressure, Au^, increases rapidly with

strain, and the value of Auj. is again independent of relative density

for the range studied.

Figure 5.19 shows the influence of confining pressure, on Yt ^^'^ ^^r

strains up to y = 3 x 10~^ percent. The curve for = 2,000 psf from figure

5.18 has been superimposed for comparison. The data points in figure 5.19

corresponding to = 533 psf were obtained from test 10 (open triangles,

Dj- = 60 percent) and from the first step of test 12 (black triangles, = 45

percent). The data points for = 4,000 psf were obtained from test 11,

Figure 5.19 suggests that the value ofYt-l'l'^^l'^^ percent is valid for

the range of pressures, 533 psf < < 4,000 psf, and that the same curve of

Auj./a^ versus y Is valid for = 533 psf and = 2,000 psf, with this curve being independent of relative density. Although the evidence presented

is not conclusive, figure 5,19 seems to suggest that Auj./a^ at small strains

above the threshold is somewhat smaller for = 4,000 psf than for 533 psf

_< 2,000 psf.

Figure 5,20 presents evidence on Yt from cyclic, strain-controlled simple shear

tests on dry Monterey No. 0 sand, conducted by Pyke (1973 [64]). The tests were performed on specimens placed at relative densities, D^., between 40 percent

and 80 percent, and normally consolidated to vertical pressures, a^, between

800 psf and 3600 psf. The plot shows the settlement in the first loading

cycle, versus strain, y« It can be seen that the settlement depends strongly

on Y and on D^., but it does not depend on Oy. This is consistent with the

76
conclusion from similar tests on other sands discussed in section 3.2.2. In

figure 5.20, the settlement in the first cycle becomes zero at y = 0.01 = 10"^

percent, independent of and a^, thus, again suggesting - 10"^ percent.

Therefore, based on the evidence presented in figures 5.18 through 5.20, it can be concluded that - 10 ^ percent for normally ( Isotropically and anisotropically) consolidated Monterey No. 0 sand, with this value being valid over

a wide range of relative densities and confining pressures of practical interest, for both dry and saturated sand and for triaxlal and simple shear cyclic

loading conditions. This independence of 10~^ percent from variations in

the confining pressure is unexpected, as the simple cubic array model predicts

an increase in Yt increases (see section 4.3 and figure 4.7).

5.5 EXCESS PORE WATER PRESSURE

Figures 5.21 through 5.33 present the experimental results for the excess pore

water pressure, Au, measured during the strain-controlled cyclic triaxial tests.

Note that all these plots depict Au, the maximum value measured for the corre- sponding cycle during cyclic loading, rather than Auj-, the residual value

measured after stopping the cyclic loading. Figure 5.34 attempts to relate Au

and AU]..

Figures 5.21, 5.22 and 5.23 summarize the experimental results for = 2,000

psf, as plots of Au/a^ versus n for Dj- = 45, 60, and 80 percent, respectively.

These figures show that Au increases significantly as both y and n increase.

Figure 5.24, 5.25 and 5.26 illustrate the effect of Dj- on the curves of Au/a^ versus n, for y = 3 x 10"^^ i x 10~1 and 3 x 10"^ percent, respectively. As it

could be expected, the pore water pressure increases as D|- decreases. However,

the effect is less marked than it could be expected from plots of densif ication

of Monterey dry sand under cyclic loading, such as shown in figures 1.2 and

5.20. At a small number of cycles, n _< 10, Au is not generally affected or is

only moderately affected by D^..

The reason why pore water pressure buildup in saturated sand is less affected

by relative density than by densif ication of the same dry sand is not difficult

to understand. If (Au)]^ is the pore pressure increment for saturated sand

corresponding to one cycle of cyclic strain, and (Ae^ol)!* is the volumetric

strain decrement corresponding to the same dry sand having the same relative

density and subjected to the same cyclic strain, then (Au) and (Ae^ol)

related approximately as follows:

(Au)i = (Aevol>l

where E^. = drained tangent modulus of one-dimensional unloading curve of the

sand (Martin et al., 1975 [49]). Although equation 5.5 was originally developed for simple shear tests, it will be assumed here for the sake of this discussion, that the same expression, or a similar one, also applies to triaxial

tests. For the case of cyclic triaxial tests, (Au)i should strictly be interpreted as the residual value, Auj-, rather than the Au values included in the

plots. However, the difference between Au and Au^ does not seem to be affected

by Dj., and therefore, this should not affect the present discussion. The

77
Important point about the theoretical equation 5.5 is that if Dj. increases,

(Ae^Q2^)j^ decreases but increases (a dense sand is stiffer than the same sand in a looser state). Therefore, (Au)i is bound to be less affected by Dj.

than (Aeyol)l» which is exactly what the experimental results show.

Figure 5.27 shows the influence of confining pressure, for Dj- = 60 percent and

Y 3 X 10~2 percent. Other things being equal, normalized pore water pressure

buildup at this low y Is faster at lower values of a^. The effect is not very

significant at low numbers of cycles, but it becomes quite dramatic at n = 1,000

cycles. An interesting corollary is that if figure 5.27 were denorraalized (i.e.,

Au were plotted versus n) , the difference would almost disappear, with the curves plotting very close to each other.

Figure 5.28, 5.29, and 5.30 summarize the results for = 2,000, as plots of

Au/a^ versus y for different numbers of cycles, and for Dj- = 45, 60, and 80

percent, respectively. The format of these figures is very useful for the purposes of the cyclic strain approach to liquefaction, and is the same used for

other sands in figures 3.23 and 3.24.

Figures 5.28 through 5.30 show again that the pore water pressure buildup for

Y < Yt - 10"^ percent is insignificant, and that this conclusion is independent

of number of cycles and is valid for the three relative densities shown. Au is

not exactly equal to zero for y < 10"^ percent in the plots, due to the difference between Au and Auj- (compare fig. 5.28 with fig. 5.18). Figures 5.28

through 5.30 show that Au increases significantly as both y and n increase.

Figures 5.31 and 5.32 show the effect of Dj- on Au/a^ for = 2,000 psf and

for n » 10 and 30 cycles, respectively. As discussed before, there is a slight

but not dramatic influence of Dj. on Au. For some practical purposes, a representative band of results could well be taken from either figure and used irrespective of Dj.. This is a very important practical conclusion. It is usually

very difficult to estimate relative densities in the field and, therefore, the

ability to predict pore pressure development without knowing the relative

density can be very valuable.

It Is of interest to compare the data on excess pore water pressures in Monterey

Nb» 0 sand, presented here, with experimental data for other sands. Figure 3.24

compiled results for various sands and placement techniques, obtained for Dj. = 60 percent and n = 10 cycles during strain-controlled cyclic triaxial tests.

All the data in figure 3.24 were for a range of = 1,400 to 2,800 psf.

Figure 3.24 is reproduced in figure 5.33, where the data points for Monterey

No. 0 sand have also been added. The data for Monterey No. 0 sand, obtained

In this study and included in figure 5.33, are for Dj. = 60 percent and = 2,000 psf. The agreement in figure 5.33 between the old curve and the new data

points for Monterey No. 0 sand is outstanding. This reinforces the conclusion

that the curve in figure 5.33 is valid for most clean, normally consolidated,

saturated sands subjected to strain-controlled cyclic triaxial testing for

- 1,400 to 2,800 psf, Dj. ^ 60 percent, and n = 10 cycles. It must be reemphaslzed that the data points in figure 5.33 correspond to three different sands,

placed using three different methods, and that the tests were conducted

independently at two different laboratories.

78
5.5.1 Comparison Between Au and Auj.

As discussed before, two different types of excess pore water pressure were measured during the cyclic triaxial tests reported here. They were: a) the

peak cyclic pore 'water pressure during cyclic loading, Au, and b) the residual

pore water pressure, measured after cyclic loading had stopped, Auj- (see also

section 5.1.3). In those cases where both Au and Auj- were available, invariably Au >^ AUj.. In particular, for nondestructive testing at strains below

the threshold, y < 10~ percent, Au^ = 0 while Au > 0.

It seems reasonable to assume that the difference between Au and Au^ corresponds

to an "elastic" pore water pressure response, associated with the increase in

volumetric stress generated by the cyclic loading. Therefore, as a first

approximation, the following expression is assumed valid:

Au = Aur + (Au)eiastic 5.6

If the soil skeleton is assumed to be both elastic and isotropic, (Au)eiastic

should be proportional to the cyclic deviator stress, Aa^ (Lambe and Whitman,

1969 [45]) or:

(^'^^elastic = 3-^<^l 5.7

Combining equations 5.6 and 5.7, the desired relation between Au and Auj. is

obtained.

Au = Au^ + 1 Ac 5.8

^ 3 1

In particular, at strains lower than 10" percent, where Au^ = 0 and the

behavior of the soil could be expected to be close to being elastic and

isotropic, Au = — Aoi is predicted.

3

^ The expression (Au-Auj.)/ ( 1/3 Aaj) was computed for tests 1 through 11, for

all cyclic strains and numbers of cycles for which both Au and Au^ were available. It was found that, at small strains, above and below the threshold, y i

3 X 10"^ percent, and for moderate pore pressure buildup, 0.01 _< Au/ 0.20,

the expression has a fairly constant value, which is ( Au-Aui.)/(l/3 Aaj.) = 0.42

+ 0.07, as shown in figure 5.34. For values of Au/a^ outside this range the

values are more erratic. Therefore, for small strain^ testing (y < 3 x 10"

percent), above and below the threshold, and for Au/a3 ^ 0.20, the plots of

Au/a^ presented in this section could be approximately converted into plots of

AUj./a5 by means of equation 5.9: —Au-. Au 0.14 Aoi

T = — L 5.

I I f ^3 ^3 ^3

where + Aa^ is the cyclic deviator stress. The fact that ( Au-AUj.)/(l/3 Aai) is

not equal to 1.0, as predicted by equation 5.8, but instead is equal to 0.42,

is probably due to the membrane compliance effect (Martin et al., 1978 [50]).

79
5.6 PORE WATER PRESSURE AND MODULUS DEGRADATION

Sections 5.3 and 5.5 discussed the modulus degradation and the development of excess pore water pressure during cyclic loading, respectively. Modulus degradation was studied using the normalized parameter G/Gj, which is 1.0 at the

beginning of the destructive cyclic loading and subsequently decreases to values

between 0 and 1. Excess pore water pressure buildup was studied by means of the

normalized parameter, Au/a^, which is zero at the beginning and subsequently

increases to values between 0 to 1 . Both modulus degradation and pore water

pressure increase are affected significantly by y and n, and to a lesser degree

by Dj. and a^, and the effect of all these factors is very similar for both

G/Gi and Ln/a^y i.e., the factors which decrease G/Gj increase Au/a^ and

vice versa. Furthermore, at strains below the threshold, y < 10~2 percent,

G/Gj = 1 and Au/a^ - 0, i.e., they both stay constant, independently of n,

Dj. and a^.

It seems reasonable from the above discussion to assume that G/G^ and Au/a^

are directly related. To test this hypothesis, the two parameters were plotted

together as shown in figure 5.35. Figure 5.35 is reasonably consistent, con- sidering the diversity of test conditions. A single curve could be fitted to

the data points as shown in the figure. This relation between Au/a^ and G/Gi

for Monterey No. 0 sand under cyclic triaxial conditions is of considerable

theoretical and practical interest. It suggests that for pore water pressure

buildup and liquefaction analyses, the modulus G, can also be calculated using

a relation such as that shown in the figure if the pore water pressure, Au, is

known at any time during cyclic loading.

The largest amount of scatter in figure 5.35 occurs near the middle of the plot,

for (1 - (Au/a^)) - 0.4 to 0.7. There, the lowest data points, having some- what lower modulus degradation for a given pore pressure buildup, correspond

to tests 1, 5, 10, and 11, all run with a low cyclic strain, y = 3 x 10"^ percent. Conversely, the highest point corresponds to test 4, run with a high

cyclic strain, y - 3 x 10~^ percent. If needed, this influence of y could be

used to refine the correlation of figure 5.35 and decrease its present scatter.

80
GRAVEL SAND

SILT OR CLAY COARSE 1 FINE COARSE 1 MEDIUM | FINE

Diameter UNIFIED SOIL CLASSIFICATION SYSTEM

GRAIN SIZE IN MILLIMETERS

Figure 5.1 Grain size distribution of Monterey No. 0 sand

81
0.6 -

2,000 10,000 100,000

Euc IN psi (UNCORRECTED)

Figure 5.2 Typical correction factor for equipment compliance,

cyclic trlaxlal tests

82
Figure 5.3 Reduction of shear modulus as a function of cyclic shear strain

for Monterey No. 0 sand at = 2000 psf and Dj. = 45 percent

83
Figure 5.4 Reduction of shear modulus as a function of cyclic shear strain

for Monterey No. 0 sand at = 2000 psf and Dr = 60 percent

84
10'* 3 5 10' 3 5 10'^ 3 5 10' 3 5 1

SHEAR STRAIN, 7, PERCENT

Figure 5.5 Reduction of shear modulus as a function of cyclic shear strain

for Monterey No. 0 sand at = 2000 psf and Dp » 80 percent

85
Figure 5.6 Reduction of shear modulus as a function of cyclic shear strain

80 percent'' ' ' '''' °r » 45, 60, and

86
Figure 5,7 Damping ratio as a function of cyclic shear strain for Monterey

No. 0 sand at =« 2000 psf and = 45, 60, and 80 percent

87
Figure 5.8 Reduction of shear modulus as a function of cyclic shear strain

for Monterey No. 0 sand at Dj. = 60 percent and aj = 533, 2000,

and 4000 psf

88
10000

EFFECTIVE CONFINING PRESSURE, a' , psf

Figure 5.9 Shear modulus of very small shear strains {G^^y^) as a function

of effective confining pressure (aj) for Monterey No. 0 sand

and Dj. = 60 percent

89
60

SHEAR STRAIN, 7 , PERCENT

RR ROUND ROBIN RESONANT COLUMN TESTING PROGRAM

(Prnevfch, 197^ [22]

Normalized stiffness parameter as a function of cyclic shear strain

for Monterey No. 0 sand at = 60 percent, = 2000 psf and various effective confining pressures

90
T 1

1 1

1

1 1 —

Monterey no. 0 sand

NUMBER OF CYCLES, n

Figure 5.11 Degradation of shear modulus as a function of number of cycl^

for Monterey No. 0 sand at D,. =- 45 percent, - 2000 psf and various cyclic shear strains

91
Figure 5.12 Degradation of shear modulus as a function of number of cycles for Monterey No. 0 sand at = 60 percent, a\ = 2000 psf and various cyclic shear strains

92
500 1000

NUMBER OF CYCLES, n

Figure 5.13 Degradation of shear modulus as a function of number of

cycles for Monterey No. 0 sand at D^. = 80 percent, = 2000 psf and various cyclic shear strains

93
Figure 5.14 Degradation of shear modulus as a function of number of cycles for Monterey No. 0 sand at aX = 2000 psf, y = 3 x lO"^

percent and Dj. = 45 and 60 percent

94
Figure 5.15 Degradation of shear modulus as a function of number of cycl

for Monterey No. 0 sand at = 2000 psf, y = 1 x

10"-^ percent and Dj. = 45, 60, and 80 percent

95
NUMBER OF CYCLfS, n

Figure 5.16 Degradation of shear modulus as a function of number of cycles

for Monterey No. 0 sand at = 2000 psf, y =3 x 10"^ percent

and Dp = 45, 60, and 80 percent

96
1.0

0.8 -

5= 0.6 C9

0.4

0

Monterey no. 0 sand

Or =60%

y =3x10-2%

o'^ Symbol

5W

2000 o

4000

1 psf = 47.8 pascal

1 10 30 50 100

NUMBER OF CYCLES, n

300 500 1000

Figure 5.17 Degradation of shear modulus as a function of number of cycles

for Monterey No. 0 sand at D = 60 percent, y = 3 x 10"^ percent, and = 533, 2000, and 4000 psf

97
V CO

s

CO

s

0.06

0.05

0.01

g 0

10

Monterey no. 0 sand

= 2000 psf

0.04 n = 10 cycles

Dr Symbol

0.03 45 o

60

0.02 80

1 psf = 47.8 pascal

I bo ^—

3 5 10-^ 3

SHEAR STRAIN,7, PERCENT

10-^

Figure 5.18 Residual pore water pressure buildup after ten loading cycles, as

a function of cyclic shear strain for Monterey No. 0 sand at = 2000 psf and D,- = 45, 60, and 80 percent

98
Figure 5.19 Residual pore water pressure buildup after ten loading cycles

as a function of cyclic shear strain for Monterey No. 0 sand

at = 533, 2000, and 4000 psf and Dj- = 45, 60, and 80

percent

99
0.20

SHEAR STRAIN, 7 , PERCENT

Figure 5.20 Settlement in the first loading cycle as a function of cyclic

shear strain for dry Monterey No. 0 sand at various relative

densities and confining pressures, simple shear tests (Pyke,

100
Monterey no. 0 sand

NUMBER OF CYCLES, n

Figure 5.21 Pore water pressure buildup as a function of number of cycles

for Monterey No, 0 sand at = 2000 psf , Dj- = 45 percent

and various cyclic shear strains

101
1 1 1 1 1

1 r

Monterey no. 0 sand

1 3 5 10 30 50 100 300 500 1000

NUMBER OF CYCLES, n

Figure 5.22 Pore water pressure buildup as a function of number of cycles

for Monterey No. 0 sand at = 2000 psf, Dj- = 60 percent

and various cyclic shear strains

102
Figure 5.23 Pore water pressure buildup as a function of number of cycles

for Monterey No. 0 sand at = 2000 psf, Dj. = 80 percent

and various cyclic shear strains

103
Figure 5.24 Pore water pressure buildup as a function of number of cycles

, for Monterey No. 0 sand at = 2000 psf, y = 3 x 10"^

percent, and Dj. = 45 and 60 percent

104
Figure 5.25 Pore water pressure buildup as a function of number of cycles

for Monterey No. 0 sand at = 2000 psf, y = 1 x 10~^

percent, and Dj. = 45, 60, and 80 percent

105
Pore water pressure buildup as a function of number of cycles

for Monterey No. 0 sand at aX = 2000 psf, y = 3 x 10"^

percent, and Dj. = 45, 60, and 80 percent

106
^ '

1

' ^ 1 r

Monterey no. 0 sand

NUMBER OF CYCLES, n

Figure 5.27 Pore water pressure buildup as a function of number of ^ycles

for Monterey No. 0 sand at = 60 percent, y = 3 x 10 percent and various effective confining pressures

107
Figure 5.28 Pore water pressure buildup as a function of cyclic shear strain

for Monterey No. 0 sand at Dj. = 45 percent, = 2000 psf and

various numbers of cycles

108
1 psf = 47.8 pascal

I I I

I _j I I I 10-3 3 5 10-2 3 5 10-' 3 5 1

SHEAR STRAIN,7, PERCENT

Figure 5.29 Pore water pressure buildup as a function of cyclic shear strain

for Monterey No. 0 sand at Dj- = 60 percent, = 2000 psf and

various numbers of cycles

109
SHEAR STRAIN, 7, PERCENT

Figure 5.30 Pore water pressure buildup as a function of cyclic shear strain

for Monterey No. 0 sand at Dj- = 80 percent, = 2000 psf and

various numbers of cycles

110
Figure 5.31 Pore water pressure buildup after ten loading cycles, as a function of cyclic shear strain for Monterey No. 0 sand at

0^ = 2000 psf and Dp = 45, 60, and 80 percent

111
Figure 5.32 Pore water pressure buildup after thirty loading cycles, as a function of cyclic shear strain for Monterey No. 0 s^nd at

= 2000 psf and Dj. = 45, 60, and 80 percent

112
to

1.2

1.0

2- 0.8 I—<

UJE 0.6

UJ

OS

Q.

1 1

1

g 0.4

Q.

0.2

O Crystal silica sand = 2000 psf, fresh samples, dry vibration A Crystal silica sand = 2000 psf, fresh samples, wet rodding

Crystal silica sand o'^ = 2000 psf, staged testing, dry vibration

Sand no. 1 as = 2800 psf, staged testing, moist tamping

V Sand no. 1 a 3

= 1400 psf , staged testing, moist tamping

Monterey no. 0 sand o\ = 2000 psf, fresh samples, moist tamping

10-3

n = 10 cycles Dr = 60%

SHEAR STRAIN, 7, PERCENT

Figure 5.33 Pore water pressure buildup in cyclic triaxial strain-controlled

tests after ten loading cycles, as a function of cyclic shear

strain, for various NC saturated sands at Dj. = 60 percent and

for various confining pressures

113
0

0.10

Au /

0.20

Figure 5.34 Comparison between Au and Au^. as a function of pore w, pressure ratio for Monterey No. 0 sand at Dj. = 45, 60

and 80 percent and = 533, 2000, and 4000 psf

114
Figure 5.35 degradation of shear modulus as a function of pore water pressure buildup for Monterey No. 0 sand at = 2000 psf

Pressure

115
FACIWG PAGE: Lanchlldo. on thd ^koKz Lake, We^ced

induced bt/ tkd tiqaiilaction Zoo6^

-i>cutaAatQ,d 6 and. diiAA,ng thii 1957 San

¥A.ancl6co , CA, dOJvthqaako.

.

116
6. EARTHQUAKE ACCELERATION AND THRESHOLD STRAIN

6.1 GENERAL

Laboratory and analytical results have been presented, thus far, to support the use of a cyclic strain approach for predicting liquefaction potential. The

basic equation of the proposed method is equation 3.5 as restated below:

= 0.65 i 3.5

« w (-^ \ max ' c where the symbol a-^ is used instead of Oq (used in section 3.2) to denote total

vertical pressure. The rest of the symbols in equation 3.5 are the same as defined for equation 3.5 originally.

117
Equation 3.5 can be used in principle to compute the equivalent seismic cyclic

shear strain, y,,, acting on a layer of sand located below the groundwater table.

An element of this sand layer is sketched in figure 6,1. The soil is subjected

to a peak ground surface horizontal acceleration, ap, which induces the seismic

strain at depth z.

When using equation 3,5, ap is assumed known, and Oy can be obtained from the

unit weights of the layers between the ground surface and depth z, if the

depth to groundwater level, z^, is also known. The value of r^j can be obtained

from a plot such as figure 3.4. The other two factors in equation 3.5 are G^^^

and (G/GjQax)^^ "^^^ shear modulus at small strains, Gniax> ^® measured in

the field by means of geophysical techniques. The measuring in situ of Gjjj^x is one of the key aspects of the proposed cyclic strain method, and one of its main

advantages. The main source of uncertainty in equation 3.5 is (G/Gxaax^yc which

will be discussed in detail in the following paragraphs. (G/Ginax)Yc a function or curve giving G/G^^j^ once y^, is known. Typical measured curves of

(G/G^gx) versus y^, for sands are given in figure 6.2. Therefore, the determination of y^, using equation 3.5 will, in general, involve iterating.

Two different cases may arise when using equation 3.5 to compute y^,:

(a) The computed value is smaller than or about equal to the threshold strain,

i.e., y^ < y^ ^ 10"^ percent = 10"^, This will occur for a "stiff" sand

(^max '^^g^) and/or a small acceleration, ap. In this case, the use of

equation 3.5 is straightforward. At these small strains (y < 10"^

percent), (G/G^jjaxHc unity (see fig. 6.2) and G^^^ a very reliable predictor of the secant shear modulus, G, at the strain y^..

In addition, there is no pore water pressure buildup in the sand layer

during shaking, and neither G nor (G/Gj^ax^Yc change during cyclic loading.

This is illustrated by the test results for y^ < 10~ in figures 5,3, 5,4,

and 5.5,

(b) The computed value is significantly larger than the threshold, i.e.,

y^ > y^ =i 10~^ percent. This condition will occur for a "flexible" sand

(GjQ3x low) and/or a large acceleration, ap. In this case, the use of

equation 3.5 involves additional uncertainties due to the increased uncertainty in the value of (G/Gnj3x)Yc» ^® reasons for this is the

reduction of G/G^^x values significantly less than unity at large

strains (see fig. 6.2), with the corresponding increase in the uncertainty

of the calculated (Gj^^x^Yc* other words G^^x a less reliable predictor of the secant modulus G, at large values of y^,, than at small values

of y^. A second source of uncertainty for (G/Ginax)Yc that for yc > Yt

»

there is a pore water pressure buildup, and due to that the values of G

and (G/G^3x) ^ reduced with duration of cyclic loading (see results in

figs. 5.3, 5.4, and 5.5). Both G and (G/G^ax) Yc a function of the

number of cycles, and hence, of the duration of shaking, thus, further

complicating the use of equation 3.5 and adding to the uncertainty of the

calculated y^,. These problems, arising from the use of expressions such

as equations 3.5 for strains above the threshold, have also been recently

discussed by Seed (1980 [75]).

118
The rest of this chapter focuses on case (a), and specifically on the conditions

under which the seismic strain is equal to the threshold value, Yc ~ Case

(b) requires further research, and is not further discussed herein.

The available evidence for the existence and value of the threshold strain, y^-

,

is discussed elsewhere in this report and includes experimental results reported

by several authors, analytical results using a model of spheres in chapter 4,

and a very precise measurements of in Monterey No. 0 sand presented in

figures 5.18 and 5,19. All these results are remarkably consistent, and suggest

that Yt ~ percent is a realistic estimate of the threshold for normally

consolidated sands over a wide range of confining pressures and relative densities. For this reason, a value of y^- = 10"^ percent = 10 will be used for

the calculations in the rest of this chapter.

If the value y (, ~

y

^ ~ 10"^ is placed in equation 3.5, the peak ground surface

acceleration which Induces the threshold strain in the sand layer can be

computed. We call this acceleration the "threshold peak ground surface

acceleration" and label it "(ap)^":

(a_)t _4 ^max ^^/^max^^t 6 , -H-I = 1.538 X 10 ^

If (ap)(. is measured in g's, equation 6.1 can be rewritten:

, . , ^max (G/Graax)Yt

(ap)t = 1.538 X 10--^ (g's) 6.2

Throughout the rest of this chapter, equation 6.2 is used to compute (Sp)^.

Section 6.2 reviews available values of G^^^ ^cir sands measured in the laboratory and in situ, and the modulus reduction curve, (G/Graax^y versus y» is discussed in section 6.3. In section 6.4, equation 6.2 is used as the basis for

a parametric study of the value of (ap)t; for different sand stiffnesses and

depths, as well as for different water table elevations in the field.

6.2 THE MODULUS AT SMALL STRAINS, Gn,^^

6.2.1 Laboratory Results

Hardin and Drnevich (1972 [29]) performed an extensive study of G^ax

laboratory, using the resonant column technique, and they proposed the expression for G^ax shown in equation 5.1, which was used in chapter 5 to evaluate

the triaxial measurements of G at small strains in Monterey No. 0 sand. Seed

and Idrlss (1970 [77]) modified equation 5.1, and suggested the use of the

expression

:

Gmax = I'OOO ^2ma^ ^max,^;

^'^

119
where a^= (a J + 02 + cjO is the average effective normal stress, and

^2max ^ function of the relative density of the sand, Dj-. Equation 6.3 is

certainly valid for isotropically consolidated sands, in which 0^=0!=

= a^. There is also evidence suggesting its applicability to the case of

anisotropically consolidated sands, and to sands subjected to a static compressive deviator stress in cyclic triaxial tests (Hardin and Black, 1968 [28];

Tatsuoka, et al., 1979 [85]).

Figure 6.3 shows the function K2niax versus Dj., proposed by Seed and Idriss.

For a loose sand with a relative density, Dj. ^ 30 percent, ^2max " ^'^^ a very dense sand with Dj- = 90 percent, K2niax " Therefore, figure 6.3

predicts that, for a given state of stresses, G^^^^ will approximately double

for dense sand as compared with loose sand.

6.2.2 In Situ Measurements

Several geophysical (seismic) techniques have been used to measure G^ax

soils in situ. In all these techniques, the shear wave velocity, Vg, at small

strains is measured in the field and G^^^ i^ obtained from the expression:

W = P 6.4

where p = mass density of soil layer = total unit weight/acceleration due to

gravity. The geophysical techniques used for this purpose include the crosshole method, the downhole method, the refraction method and the Rayleigh wave method (Anderson and Espafia, 1978 [2]; Woods, 1978 [87]). Of these, the

most reliable one is the crosshole technique sketched in figure 6.4. In this

method, a vertically polarized shear wave impulse propagates horizontally, and

the travel time of the impulse between drillholes is measured to compute Vg

.

Powell (1979 [63]) performed a literature review of available in situ measurements of Gmax i'^ sands which had been obtained using these geophysical methods.

As these measurements were made at depths varying between 10 and 130 ft at

sites having different groundwater elevations, the values of effective vertical

overburden pressure, and of average effective stress, 0^ varied widely.

It would have been useful to normalize these measured G^j^x values by means of

equation 6.3, thus, obtaining K2ijiax*

1,000 (a;)^/2

V - max (L c '^2max -

In field conditions, ^v^ ^ ^ ^^o^» nevertheless, was generally

not measured at those sites. Therefore, Powell normalized G^^^ by (a^^)^ , as the value of could be eas A was defined instead of K2inax'

max ^^v' as the value of could be easily estimated in all cases and a coefficient

nmax ^'

816.5

120
In equation 6.5, both G^g^^ and are in psf and the units of A are the same

as these of Kr>^^ax'> (Ib^'^/ft). The numerical coefficient 816.5 In equation 6.5 was selected so that, for = 0.5, = (^) [ l+(2)(0. 5) ]

,

A = K2niax ^^'^ equations 6.4 and 6.5 become one and the same. A value of Kq = 0.5 is a reasonable estimate for normally consolidated, freshly deposited,

noncompacted sand deposits.

Figure 6.5 shows the data compiled by Powell, as a plot of A versus depth z, for 10 sandy sites consisting of clean sand and silty sand deposits without

gravel or clay. The values of A range from 35 to 240 with most of them between

35 and 150.

It is interesting to compare the values of A from the field in figure 6.5 with

the values of K2niax f^^oTM the laboratory in figure 6.3. The lower bound of

A ^ 35, coincides well with the lower bound of ^2max ~ -^^» corresponding to

loose sands. This observation is reasonable, as it could be expected that the

lower values of Gm^x (and of A) in the field should correspond to loose,

normally consolidated sands having a low value of Kq 0.5). On the other

hand, the upper bound of the A values in figure 6.5, which is at least 150 and

may be as high as 240, is much above the highest value, K2niax ~ figure

6.3. Therefore, while the laboratory results might suggest that, for a given

state of stress, sands may have values of G^gx differing by a factor of only

about two, the field results suggest that this ratio may be as high as four or seven.

From the viewpoint of the proposed strain approach to liquefaction, this wide

variation of the A value from field results is of great importance. A value

of A = 35 would define a "flexible" sand, while a value of A = 150 or 200 would

define a "very stiff" sand. The practical implications for liquefaction of a sand being "flexible" or "stiff" will be demonstrated in section 6.4. For the

purpose of this study, a range of values of A between 35 and 150 is used.

The possible reasons for this discrepancy between the highest measured values

of K2ijiax ^ will now be examined.

One possibility for the discrepancy is that the actual range of K2niax

different sands is larger than the ratio of 2 suggested by figure 6.3. In fact,

equation 6.3 and figure 6.3 are somewhat simplified versions of Hardin-Drnevich

equation 5.1. G^^^^ is really a function of the void ratio e, rather than a function of relative density, D^-. Therefore, different sands having different

grain size distributions and silt contents, such as those summarized in figure 6.5, may have quite different values of e, and thus may, as a group, cover

a wider range of G^ax than that suggested by figure 6.3.

Another possible explanation is that Kq > 0.5 in the field due to

overconsolidation or other factors, in which case A ^ ^2max' From equations

6.4 and 6.5, the relation between K2niax ^ value of Kq is

^2max = (_^^_)^^'^ 6.6

A 1 + 2Kq

121
For an overconsolidated sand, Kq is a function of the overconsolidation ratio,

OCR. As shown by the typical data in figure 6.6, for OCR = 1 , Kq = 0.4, which

is close to Kq = 0.5 assumed here. For OCR 7, Kq 1, while for much large

values of OCR, Kq can even approach 1.6 or 1.8. Equation 6.6 is plotted in

figure 6.7 for the range of Kq between 0.4 and 1.6. It can be seen that for a sand with OCR = 7 and Kq =^ 1 , K2niax ~ 0.8A. Therefore, if some of the sands

having A =^ 150 in figure 6.3 were consolidated with Kq = 1, yi2max ~ (0.8)(150)

= 120, and the factor between maximum and minimum K2iiiax ^'^ field would be

3.4 instead of 4 obtained before.

The variation discussed above for the void ratio, e, and for Kq of sands in the

field may serve as a partial explanation of the difference in ranges between

figures 6.3 and 6.5. However, they do not explain all the differences since

other factors also seem to play an important role. These other factors, which

have been shown to increase G^g^y^ of sands in the laboratory and yet were aot

considered, neither in the original Hardin-Drnevich equation (eq. 5.1) nor in

the modified Seed-Idriss version (eq. 6.3), include: (i) seismic prestraining,

and (ii) time under pressure.

The seismic prestraining effect was originally discussed by Drnevich and

Richart (1970 [23]), when performing resonant column tests on dry sand. They

found that a large number of cycles of high amplitude shear straining could

cause a large increase in the value of G^^j^ if the amplitude is above the

threshold strain, y > - 10~^ percent. The increase in G^^^ was significant

for a few thousands of cycles (an Increase of about 30 percent) while for one million cycles G^j^^x increased by a factor of two or three. This large

increase in G^ax could not be explained by changes in void ratio, and was attributed by Drnevich and Richart to wear and stiffening of the contacts

between the sand grains. Another possible explanation of the effect of prestraining in sand has been suggested by Youd (1977a [95]). In his hypothesis,

cyclic straining produces changes in the packing of the sand by means of the

collapse of the more unstable grain arrangements. These collapses have a negligible or small influence on the overall relative density or void ratio

of the sand, but they do produce a more stable and stiffer structure or fabric

of the soil. A large number of high amplitude oscillations of the soil may

occur in situ because of man-made operations, e.g., compaction of sand with

vibrating equipment, vibrations due to nearby operating machinery, traffic,

etc., or in geologically old natural soil deposits located in active seismic

areas

.

The effect of time under pressure on Gj^^^ soils has been studied

systematically by Afifi and Richart (1973 [1]) and Anderson and Stokoe (1977

[3]). The test results discussed by these authors show that G^^x increases

with time of secondary consolidation in all soils. The increase for G^^^ in

sands in the laboratory is typically of the order of one percent per log cycle

of time after 1,000 minutes. An extrapolation of this rate of increase would

suggest a significant increase in G^j^^ geologically old sand deposits.

122
6.3 THE MODULUS REDUCTION FACTOR, G/Gm.

The modulus reduction curve (G/Gniax)Y versus cyclic shear strain, y, Is critical

for the application of the proposed strain approach. Of special interest is the

value of (G/Gj^g )y,. at the threshold strain, Yt " 1^"^ percent. Hardin and

Drnevich (1972 [29]) and Seed and Idriss (1970 [77]) discussed the curve or (G/Gjnax^Y ^^^^us y for sands. After reviewing the experimental evidence available at the time, Seed and Idriss proposed the curve shown in figure 6.2, with

an experimental band to take into account the scatter of the results. This band was previously shown in figure 4.8. Both the curve in figure 6.2 and the band

in figure 4.8 are independent of the relative density of the sand and of confining pressure. At the threshold strain, Yt - 10"^ percent, (G/G^g^^)y^ « 0.75

with the band giving a dispersion range between 0.65 and 0.85.

More recent results have confirmed these values reported by Seed and Idriss.

Figure 6.2 includes a comparison of (G/Gj^ax^ curves obtained by different

investigations, which was compiled and originally published by Iwasaki et al .

,

(1978 [33]). At Yt = 10"^ percent, (G/G^^j^)Yt in figure 6.2 ranges from 0.75

to 0.90. Figure 6.8, which was also published by Iwasaki et al . , (1978 [33]),

includes results for 13 sands having different grain size distributions. The

factor B in the figure is a constant characteristic of each sand. Of special

interest in figure 6.8 is the value of (G/G^^^)y^ for Yt ~ 10~^ percent which

is notably constant and equal to 0.75 for the 13 sands used. It should be

noted that these 13 sands were tested by Iwasaki et al. in a dry state by a combination of the resonant column and torsional shear techniques, and for a confining pressure of 2,000 psf (=1 kg/cm^). Iwasaki et al . also performed

tests at other confining pressures in the range from 550 to 4,000 psf and

found similar results to those presented in figure 6.2. They noticed a tendency for (G/Gijiax)Y increase with confining pressure; however, all

(G/Gniax)Yt values were in the range from 0.70 to 0.85. This influence of

confining pressure on (G/Gniax)Yt is consistent with the discussion by Richart

(1980 [66]) and with the results for Monterey No. 0 sand summarized in figure

5.10.

The results discussed above strongly suggest that (.G/G^y,)y^- - 0.75 for Yt ~ 10~2 percent, with an experimental scatter between about 0.65 and 0.85. These

numbers seem to be independent of relative density and to be generally representative for the range of confining pressures of practical interest. Tatsuoka

et al. (1979 [85]) showed that these conclusions for (G/G^^x^Yt valid for

both isotropically (K = 1) and anisotropically consolidated in the range 0.33<K

<1) sand specimens. Very recently, Canales (1980 [8]) presented results showing that, although G^j^x is strongly affected by prestraining , the curve

(G/Gxaax^y value (G/Gin2ix)Yt threshold are about the same before

and after prestraining. With respect to the influence of time under pressure

on G/Gjnax» Anderson and Stokoe (1977 [3]) have suggested that G/G^ax

increase somewhat due to this effect. Specifically, at the threshold,

(G/G^3x)Yt increase from about 0.75 without the time effect, to 0.80 or 0.90 after long time under pressure.

For the purposes of this study, a representative value of (G/G^g^^)y^ = 0.75 is

adopted for sands at Yt ~ 10" percent, with lower and upper bounds of 0.65 and

0.85, respectively.

123
6.4 PARAMETRIC STUDY

This section presents the results of a parametric study of the threshold peak

surface acceleration, (ap)t» based on equation 6.2 and on the results discussed

in sections 6.2 and 6.3. For simplicity, in equation 6.2, G^q^^ is replaced by

A as defined by equation 6.5, and a total unit weight = 115 Ib/ft-^ is assumed

for the soil both above and below the groundwater table. For the field condition sketched in figure 6.1, = 115 z (Ib/ft^), and = 115 z^ + (115-62.4)

(z-z^). Finally, and for (.G/Gj^^ax^yt ~ 0*75, equation 6.2 becomes

(a„)^. . (62.4 z„ + 52.6z)^/2 _P_i = 8.2 X 10"^ 5^ (a^)t in g's. 6.7

A z r^ P ^

Equation 6.7 was used for the parametric study. In the calculations, the

following values of r^j, obtained from figure 3.4 were used. ^ (feet) ^jd

10 0.98

20 0.96

30 0.92

The calculations were performed for values of z below the water table, z > z^.

Equation 6.7 has been plotted in figure 6.9 as a function of z^ and for depths

z = 10, 20, and 30 feet, which covers the range of depths where liquefaction

most frequently occurs.

Equation 6.7 is also plotted as (ap)^ versus z^ in figure 6.10 for the depth,

z = 20 ft and for A = 35, 100, and 150.

Figures 6.9 and 6.10 clearly show the influence of the parameters z, z^, and A

in determining the value of (ap)^-. In figure 6.9, for the same stiffness. A,

there is a large decrease in (ap)(- between z = 10 ft and z = 20 ft while the

decrease is much smaller from 20 to 30 ft. As expected, (ap)t- increases when

the depth to groundwater, z^, increases.

The effect of the stiffness parameter. A, on (ap)(- is very dramatic. As (ap)^

is directly proportional to A, the value of (ap)t: should more than quadruple

when going from a "flexible" (A = 35) sand to a "stiff" (A = 150) sand, other

conditions being equal. This is illustrated by figure 6.10. In a sand layer

having a measured A = 35 and located at 20 ft, and for shallow groundwater,

z^ = 0, the threshold is (ap)^ 0.05g. If the sand is very stiff with A = 150,

then (ap)t - 0.21g. This difference is very significant since a peak surface

ground acceleration of 0.20g is quite strong and can even be higher than the

design acceleration in many low seisraicity areas. As shown by figure 6.10,

for z = 20 ft, (ap)t can be substantially larger than 0.20g if A > 150 and/or

the groundwater is located at some depth.

124
For an example of application using figures 6.9 and 6.10, let us assume that we have two adjacent soil profiles, both with the groundwater level at z„ = 10 ft

and potentially liquefiable sand layers at z = 20 ft. The design surface acceleration is also the same, = 0.15g which corresponds to a magnitude 8 (long

duration) earthquake. In site 1, the sand layer has a value of A = 35, measured using the crosshole technique. Therefore, (ap)^ - 0.06g and this layer

will most probably liquefy. In site 2, the sand layer has a measured A = 150.

Therefore, (ap)(- = 0.26g. The sand layer in site 2 will not even start

developing an excess pore water pressure, let alone liquefy during the design

earthquake

.

Figures 6.9 and 6.10 also suggest that sand deposits will not liquefy for peak

ground accelerations less than about 0.05g, even for the worst soil conditions,

shallow water table, and for large earthquake magnitudes causing the longest

durations of shaking. Seed et al., (1975 [81]) compiled a list of thirty-eight

liquefaction case histories. According to that list, the smallest value of ap

to cause liquefaction is 0.08g which occurred during the 1933 Tohnankai earthquake in Japan, which had a magnitude 8.3 and a long duration of shaking. Based

on a review of about 100 liquefaction failures in Japan during the last century,

Kuribayashi and Tatsuoka (1975 [42]) concluded that the minimum intensity in

the Japanese Intensity Scale, JMA for which liquefaction has occurred is five,

which corresponds to a range of peak acceleration between 0.08g and 0.25g.

Finally, liquefaction is usually associated with earthquakes having Modified

Mercalli Intensities MMI of VI or larger. The MMI of VI corresponds approximately to a ground acceleration of 0.05g. Therefore, the available evidence

indicates that the results of the parametric study present herein are generally

consistent with reported cases of liquefaction during earthquakes.

In order to evaluate the uncertainty in (ap)(- introduced by the scatter of

values of r^ and (G/Gp,ay)Yt- » the chart of figure 6.10 was recalculated to

obtain lower bound and upper bound curves, as follows.

Lower Bound Curve 0.98 0.65

Average Curve (figure 6.10) 0.96 0.75

Upper Bound Curve 0.94 0.85

The corresponding values of (ap)t> calculated using modified versions of

equation 6.7, are plotted in figure 6.11 for z = 20 ft, and for A = 35 and A =

150. Although the numerical values of (ap)(. change somewhat when the variations in rj and (G/Gmax^^'t considered, the main conclusions reached above

on the influence of stiffness on (ap)^ do not change. For z^ = 0, if A = 35,

(ap)t varies between 0.04 and 0.06g, while if A = 150, (ap)(- varies between

0.18g and 0.24g. For the example of sites 1 and 2 discussed above, with z„ =

10 ft and z = 20 ft, the ranges of (ap)^ are 0.05 to 0.07g and 0.22 to 0.30g

for sites 1 and 2, respectively. Therefore, for a design acceleration of 0.15g

and a long duration earthquake, site 1 may liquefy and site 2 will not, as concluded previously using average values of r<j and (.G/Gxaax^yf

125
Figure 6.1 Simplified soli profile

126
Figure 6.2 Modulus reduction curves for sands (Iwasaki et al., 1978)

127
RELATIVE DENSITY, Dr, PERCENT

Figure 6.3 Relation between normalized stiffness parameter, K2niax» ^^'^ relative density (modified from Seed and Idriss, 1970)

128
a.-PLAN VIEW

(Not to Scale)

b.-CROSS-SECTIONAL VIEW

Figure 6.4 Crosshole geophysical method (Hoar and Stokoe, 1977)

129
SHEAR MODULUS PARAMETER, A

173 208 242

Gmax

816.5

-(Ib^^/ft)

Figure 6.5 Normalized shear modulus parameter, A, measured for sands in

the field using geophysical techniques (Powell, 1979)

130
2.0

OVERCONSOLIDATION RATIO (O.C.R.j

Figure 6.6 Coefficient of earth pressure at rest, Kq , as a function of

overconsolidation ratio, OCR (Hendron, 1963)

131
Figure 6.7 Influence of the coefficient of earth pressure at rest, Kq

the normalized shear modulus parameter, A

132
1.0 7 = 10'^ inch/inch

p

= 1.0 kg/cm^

^ 0.5 CO

7 = 10 —

0.0

7 - 5x10'

4 4 a—-u^ • < VO A

O-vo7 -

10-3 <3-I_'53-v—-

7 = 3x10"^

0.5

O

0 o

_l_

1.0B

Legend

Toyoura sand • Model-test sand O Ban-nosu sand A A Ban-nosu sand B

Ban-nosu sand C V Ban-nosu sand 15 O Ban-nosu sand 18 < Ban-nosu sand 21 < Iruma sand Ohgi-Sima sand ^ Kinjo sand 1

Kinjo sand 2

\2 Monterey sand no. 0

Figure 6.8 Reduction of shear modulus at different cyclic shear strains,

Y, for sands (data from Iwasaki et al., 1978; B is a coeffi-*

cient characteristic of each sand tested)

133
Figure 6.9 Liquefaction chart for threshold peak ground surface acceleration,

(ap)t

134
Figure 6.10 Liquefaction chart for threshold peak ground surface acceleration

(ap)t» at z=20 feet

135
FACIWG PAGE: Sand bolU In nlcz ^laJid

mcUL Old KAXakaml RlveA.

The, ^and boUU weA.e caoAed

by ttquz^actlon duAying the.

Mlyag^-kzn-oki, Japan,

Jane 12, 197S, zafithqaakt.

5 10 15 20 25

DEPTH OF GROUNDWATER TABLE, Z^, FEET

Figure 6.11 Lower and upper values of (ap)t to account for scatter In

(G/Gmax)Yt and r^j at z=20 ft

136
I

7. SUMMARY AND FINDINGS

We have reached the following conclusions from the work presented in this

report, on methods for predicting pore pressure buildup and liquefaction potential of saturated sands at level sites during earthquake:

1) Data from cyclic stress-controlled tests on sands accumulated in the last

few years demonstrate that a number of factors besides relative density

influence the value of the cyclic strength. These factors include fabric,

overconsolidation , prior seismic straining and time under pressure. The

findings raise serious doubts about the present practice of using stresscontrolled cyclic tests on disturbed samples reconstituted to the estimated

field density.

137
Cyclic test results have demonstrated that there is a predictable

correlation between cyclic shear strain and the pore water pressure buildup

of saturated sands. An example of this correlation is presented in figure

5.33. Also, more consistent results are obtained if strain-controlled

rather than stress-controlled tests are conducted. In particular, fabric,

which has a large influence on cyclic strength, does not influence significantly the pore water pressure developed during strain-controlled tests.

It is suggested that the influence of relative density and of the other

factors listed in point (1), on the cyclic strength, is due to a large

extent to differences in stiffness between specimens in both compression

and extension, which, in turn, induce very different shear strains during

stress-controlled tests. This would explain why strain-controlled tests

give more consistent results that stress-controlled tests.

Results of strain-controlled tests on normally consolidated dry and

saturated sands by several investigations, using a number of testing techniques, have consistently suggested the existence of a threshold cyclic

shear strain, - 10" percent. For strains below this threshold, there

is neither densif ication nor prestraining of dry sands and there is no pore water pressure buildup in saturated sands. An analytical model of the

sand constituted by a simple cubic array of quartz spheres predicts similar

values of Vf (y^- = 1 x 10"^ to 4 x 10~^ percent for the range of confining

pressures of practical interest). A series of undrained cyclic straincontrolled triaxial tests on saturated Monterey No. 0 sand reported herein

measured a value y^- = 1»1 x 10~ percent. The experimental data in figures

5.18 and 5.19 indicate that this value of Yt sand tested is independent of relative density and of confining pressure for the range between

about 500 psf and 2,000 psf. This proof of the existence of Yt > well as its constant value are powerful arguments in favor of a strain approach to

liquefaction.

Based on the conclusions above, a cyclic strain approach to liquefaction

is proposed. The basic equation of the suggested method (eq. 3.5) requires

estimating both the seismic strain induced in the sand layer and the effective shear modulus of the layer during the earthquake. The proposed method

is based on measuring the shear modulus in situ at small strains, G^j^x*

using geophysical techniques, and on performing cyclic strain-controlled

tests in the laboratory to determine: (1) the modulus reduction values,

G/Gniax> (ii) the value of Yt » (iii) the pore water pressure buildup

Au, versus cyclic strain y ai^d number of cycles n.

A series of 12 undrained strain-controlled cyclic triaxial tests on saturated Monterey No. 0 sand specimens was performed. The tests included

nondestructive, high precision measurements at very small strains (y =^ 10"^

percent) using an improved technique recently developed by the second author

(Ladd) , which allowed the measurement of G^iax Yt the sand specimens.

In addition to G^jax Yt » the values of G/G^^j^ and Au needed for the

cyclic strain approach were also measured. The results of this test program

are presented in chapter 5 where the influence of relative density and

confining pressure on G^gx Yt » cyclic strain and number of

straining cycles on G/G^^g^ presented and discussed in detail.

138
The results are also compared with similar data obtained by other researchers

on Monterey No. 0 sand and other sands, with good agreement,

6) Two outcomes ace possible when applying the proposed strain approach to a specific site under a given design peak surface acceleration, ap. In the

first outcome, which will occur for a low value of a^ and/or a stiff sand

layer (G large), ^ Yt ~ ^^"^ percent, the application of the method

Is straightforward with very small uncertainty, and the method predicts that

the risk of liquefaction is negligible. In the second outcome, which will occur for a high value of a^ and/or a flexible sand layer (Gj^ax small), the

seismic strain, > = 10~^ percent, and there is risk of liquefaction.

In this case, the uncertainty in the application of the method will increase

as y^. Increases above y^-, due to the uncertainty in the value of G/G^^j^.

7) Finally, a simplified cyclic strain approach to liquefaction is proposed,

aimed at determining, for a given site and depth of the sand, the value of

the surface peak acceleration inducing the threshold strain in the layer.

This is called the threshold peak ground surface acceleration (a^)^. If the

design acceleration, ap < (ap)t» the danger of liquefaction can Se discarded.

If ap > (ap)|-, further studies are needed. Simplified charts were developed

and are presented in chapter 6 to compute (ap)^. for a given site. The use

of these charts require knowing G^j^x* the depth of the layer, the overburden

pressure and the depth to groundwater table. These charts are consistent

with the historic experience of seismic liquefaction, and are recoraended

for preliminary site-specific evaluations. These charts, shown in figures

6.9 through 6.11, Indicate that (ap)(- may be as low as 0.04g for a site with

low Gniax ^'^^ shallow water table, and as high as 0.20g or 0.30g for a stiff

site having a high measured G^g^^ ^ deep water table.

139

8 . ACKNOWLEDGMENT

The authors gratefully acknowledge the contributions by Mr. Paul Dutko and his

associates at Woodward- Clyde Consultants, Clifton, NJ, in carrying out the

laboratory testing program presented in this report. The contribution of

Mr. Gregory Thomas at Rensselaer Polytechnic Institute, who plotted the laboratory results shown in chapter 5, is also gratefully acknowledged. The authors

also want to thank Dr. Rune Dyvik and Captain William Pierce at Rensselaer

Institute, who reviewed the manuscript and made useful suggestions. A special

note of thanks to Mrs. Brenda Thompson and Ms. Ulesia Gray of the Word Processing Center of CBT for the typing and performing the many revisions of this

manuscript.

141

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,

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148
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78. Seed, H. B. and Idrlss, I. M. , (1971) "Simplified Procedure for Evaluating

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150
APPENDIX

Calculations of Cyclic Triaxial Tests Results

From the measured peak axial loads and axial deformations within a given cycle,

cyclic deviator stresses and axial strains are computed using the specimen

dimensions after consolidation. No corrections were made for the affect of

the rubber membrane.

The shear strain amplitude is calculated from the axial strain amplitude

using the following equation:

AL

+Y = +e (1+v) = PP x (1+v) = + 1.5e

2H,^c

whe re

:

+Y = shear strain amplitude (in/ in) +e = axial strain amplitude (in/in)

^Lpp = peak-to-peak axial deformation measured within a given

loading cycle using the oscillograph recorder

Hf. = height of specimen after consolidation

V = Poisson's ratio; a value of 0.5 was used in all tests.

The shear modulus is calculated using the following equation;

G =

E p„„ X H

= ^PP c

2 (1+v) 3Ac X Lpp

where

:

G = shear modulus

E = Young ' s modulus

Ppp = peak-to-peak axial load measured within a given loading cycle using

the oscillograph recorder

Aj, = area of specimen after consolidation

Calculated values of shear strain amplitude and shear modulus were also

corrected for sample setup compliance using the following equations:

±Yc = ±Y X CF

Gc = G/CF

where

:

+Y(, = shear strain amplitude corrected for equipment compliance

G^ = shear modulus corrected for equipment compliance

CF = correction factor for equipment compliance obtained from a curve such as that presented in figure 5.2.

A-1
The data presented in figure 5.2 represent the results of a series of special

tests in which each test was individually corrected for equipment compressibility. This correction consisted of correcting each recorded value of ALpp

by subtracting away the peak-to-peak axial deformation of a steel cylinder

grouted into the cell in the same manner as the text specimens and at a con- fining pressure and peak-to-peak axial load similar to that which was recorded

when ALpp was determined.

During the cell calibration and from test to test, the same stones, platens,

etc., are used. In addition, these items were indexed in such a manner that

they are in the same position from test-to-test.

A-2
NBS.n4A (REV. 2-80) U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publication Date BIBLIOGRAPHIC DATA

REPORT NO. SHEET (See instructions) NBS BSS 138 July 1982

4. TITLE AND SUBTITLE

Prediction of Pore Water Pressure Buildup and Liquefaction of Sands

During Earthquakes by the Cyclic Strain Method

5. AUTHOR(S)

R. Dobrv, R.S. Ladd, F.Y. Yokel. R.M. Chung, and D. Powell

6. PERFORMING ORGANIZATION (If joint or other thon NBS, see /n struct/on

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State. ZIP)

Saine as 6.

10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 82-600561

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most si gnificant information. If document includes a significant

bi bliography or literature survey, mention it here)

A cyclic strain approach for evaluating the buildup of excess pore water pressures

and the potential for liquefaction of level sandy sites during earthquakes is proposed in this report. This strain approach is based on the premise that, for

undrained loading of sand, there is a predictable correlation between cyclic shear

strain and excess pore water pressure; also, that there is a threshold shear strain

below which there is no sliding at the contacts between sand particles and no pore

water pressure buildup can occur. As the result, a sand deposit will not develop

excess pore pressures if the induced seismic shear strain is less than the threshold'

strain. Both theoretical evidence and experimental verification supporting the

cyclic strain approach and the existence of the threshold, are presented in the

report. Based on all these findings, a specific design method is proposed for

predicting if excess pore pressures will develop at a specific site during a design

earthquake.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names- and separate key words by sem/colonsj

cyclic strain; damping ratio; earthquake engineering; laboratory testing; liquefaction; particulate mechanics; particulate model; pore water pressure; sand;

seismic loading; shear modulus; shear strain; site stability.

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