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IV
IlqA 1504 CIVIL ENGINEERING STUDIES .", STRUCTURAL RESEARCH SERIES NO. 504
ell •• .e~z Reference Room
University of Illinois BIOG NeEL
208 N. Romine street Urbana, Illinois 61801
UILU-ENG-83-2005
ISSN: 00694274
STOCHASTIC ANALYSIS OF LIQUEFACTION UNDER EARTHQUAKE LOADING
By J. E. A. PIRES
Y. K. WEN
and
A. H-S. ANG
Technical Report of Research
Supported by the
NATIONAL SCIENCE FOUNDATION
under
Grants CEE 80-02584 and 82-13729
UNIVERSITY OF ILLINOIS
at URBANA-CHAMPAIGN
URBANA/ILLINOIS APRIL 1983
50272 -101 i
So Recipient'. AccHalon No. REPORT DOCUMENTATION 1.1 .. ~:REPORT NO.
PAGE UILU-ENG-83-2005 I~ 4-Title and Subtitle
STOCHASTIC ANALYSIS OF SEISMIC SAFETY AGAINST LIQUEFACTION
5. Report Dete
APRIL 1983
~-----------------------.--------------------.--------------------~---- t---------------------------4 7. Author(s) 8. Performlne Oraanlzatlon Rept. No.
J. A. Pires, Y. K. Wen, A. H-S. Ang SRS No. 504 9. Performing Organization Name and Address
Department of Civil Engineering University of Illinois 208 N. Romine Street Urbana, IL 61801
12. Sponsoring Organization Name and Addr •••
National Science Foundation Washington, D.C.
15. Supplemflntary Notes
10. Project/T.sk/Work Unit No.
11. Contract(C) or Grant(G) No.
(C)
(G) NSF CEE 80-02584 NSF CEE 82-13729
13. Type of Report & Period Covered
r---------------------------~ 14.
i-----------------------------· -------. -------.----------.- ... - . - .. -- ------------------------1 i -16. Abstract (Limit: 200 words)
Liquefaction of sand deposits during earthquakes is studied as a problem of random vibration of nonlinear-hystere~ic systems. The random vibration results lead to the determination of the mean and standard deviation of the strong-motion duration of a random seismic loading with a given intensity that causes liquefaction. These statistics are used to define the seismic resistance curves against liquefaction in the deposit. It is assumed that liquefaction occurs when the excess pore pressure becomes equal to initial effective overburden pressure, i.e., when the shearing stiffness of the sand deteriorates to zero under repeated alternate shaking.
The random vibration results, together with the results from the uncertainty analysis of the soil properties, are also used to calculate the reliability of sand deposits against liquefaction under a random seismic loading with given intensity and duration. These conditional reliabilities, and the probabilities of all significant seismic loadings for the site over a specified time period, are then combined to obtain the lifetime reliability against liquefaction.
r-----------------------------------------------------------------------------------------------4 17. Document Analysis 8_ Descriptors
Earthquake engineering
Liquefaction
b. Identlfiers/Open·Ended Terms
c. COSATI Field/Group
18. Availability Statement
(See ANSI-Z39.18)
Hysteretic system
Random vibration
Me~z ~ererence Room Universlty o~ Illinois
B106 NCEL U 208 N. Romine Street rbana, Illinois 61801
19. Security CI.ss (This Report) 21. No. of Paees
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UNCLASSIFIED See Instructions on Reverse
22. Price
OPTIONAL FORM 272 (4-77) (Formerly NTIS-35) Department of Commerce
ii
ACKNOWLEDGMENTS
Tois report is based on the doctoral thesis of Dr. J. A. Pires,
submitted in partial fulfillment of the require~ents for the Ph.D. in
Civil Engineering at the University of Illinois.
The study is part of a broader progran 03 the safety and reliability
of structures to seismic haza'rd, supported by the National Scien~e
Foundation under grants CEE 80-02584 and GEE 82-13729. Supports for
this study are gratefully acknowledged.
CHAPTER
1
2
3
4
iii
TABLE OF CONTENTS
Page
INTRODUCTION ••••••••••••••••••••••••••••••••••••••••.••••• 1
1.1 Introductory Remarks ••••••••••••••••••••••••••••••••• 1 1.2 Review of Procedures for Stochastic Analysis of
1.3 1.4 1.5
Liquefaction ••••••••••••••••••••••••••••••••••••••••• 2 Purpose and Scope •••••••••••••••••••••••••••••••••••• 5 Organization ••••••••••••••••••••••••••••••••••••••••• 6 Notation ............................•.................. 7
THE RANDOM VIBRATION MODEL 10
2.1 2.2 2.3 2.4 2.5
Introduction •••••••••••••••••••••••••••••••••••••••• 10 The Smooth Hysteretic Model ••••••••••••••••••••••••• 11 The Equivalent Linearization •••••••••••••••••••••••• 14 Energy Dissipation Statistics ••••••••••••••••••••••• 17 DOF Reduction Technique ••••••••••••••••••••••••••••• 20
SOIL CHARACTERIZATION AND LIQUEFACTION MODEL 23
3.1
3.2
Dynamic Shearing Stress-Strain Relation for Soils 3.1.1 Introduction ••••••••••••••••••••••••••••••••• 3.1.2 Sands 3.1.3 Clays Pore Water Pressure and Stiffness Deterioration ••••• 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5
Introduction ••••••••••••••••••••••••••••••••• Uniform Cyclic Loading ••••••••••••••••••••••• Irregular and Random Dynamic Loading ••••••••• Stiffness Deterioration •••••• ' •••••••••••••••• Discussion •••••••••••••••••••••••••••••••••••
SEISMIC RELIABILITY ANALYSIS ............................... 4.1 4.2 4.3
4.4
Introduction Reliability Evaluation •••••••••••••••••• ~ ••••••••••• Uncertainties in Soil Properties •••••••••••••••••••• 4.3.1 Undrained Resistance To Liquefaction
Under Uniform Cyclic Stress Loading •••••••• 4.3.2 Additional Soil Properties ••••••••••••••••••• Earthquake Loading •••••••••••••••••••••••••••••••••• 4.4.1 Ground Mot ion Mode I •••••••••••••••••••••••••• 4.4.2 Uncertainties in Earthquake Load Parameters
23 23 24 26 27 27 28 33 36 41
45
45 46 49
49 53 55 55 57
5
6
iv
Page
ILLUSTRATIVE EXAMPLES .................................... 60
5.1 5.2
5.3
5.4
5.5
Introduction ........................................ Homogeneous Saturated Sand Deposit 5.2.1 Problem Description •••••••••••••••••••••••••• 5.2.2 Total Stress Analysis 5.2.3 Stiffness Deterioration •••••••••••••••••••••• Reclaimed Fill ••••••••••••••••.•••••• 5.3.1 5.3.2
................................. Introduction Reliability Evaluation
Case Studies •••••••••••••••••••• 5.4.1 Introduction .. " ............................. . 5.4.2 Hachinohe (Japan) •••••••••••••••••••••••••••• 5.4.3 Niigata (Japan) 5.4.4 Main Obseryations •••••••••••••••••••••••••••• Summary of Results ..................................
60 61 61 61 63 64 64 66 67 67 68 69 70 70
SUMMARY AND CONCLUSIONS .................................. 72
6.1 6.2
Summary ..•••.•••...•••••.••••.•.•..••.•.•..••.••.•.• 72 Conclusions •••••••••••••••••••••••••••••• ~ •••••••••• 73
TABLES ............................................................. 76
FIGURES ............................................................ 90
APPENDIX
A EQUIVALENT LINEAR COEFFICIENTS 124
B ENERGY DISSIPATION STATISTICS 125
REFERENCES 128
v
LIST OF TABLES
Table Page
3.1 Function h(~) for a Sand with Dr = 0.54 •••.•••••••••••••• 77
3.2
3.3
3.4
Excess Pore Pressure with Cycles of Loading •••••••••••••• 77
Function h(i) for a Sand with D = 0.45 •••••••••••••••••• 78 r
Excess Pore Pressure with Cycles of Loading •••••••••••••• 78
4.1 Statistics of Parameters Defining Nt •.•.•.•••••.•••.•••• 79
4.2 c.o.v. of the Undrained Resistance to Liquefaction in Laboratory (a~oo = 4.8 psi) •••.••••••••••••••..••••.••• 79
4.3 c.o.v. of the Undrained Resistance to Liquefaction in the Field (a f = 4.8 s') 80 co p].. · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
4.4 c.o.v. of the In-Situ Relative Density ••••••••••••••••••• 80
4.5 c.o.v. of the Small Strain Shear Modulus G of Sands ••••• 81 m
5.1 Properties of Sand in Example I .•••••••••••.••••••••••••• 81
5.2 Lumped Mass Model for Example I •••••••••••••••••••••••••• 82
5.3 Statistics of TL for Several Load Intensities (seconds) .... ' .....................................•...... 83
5.4 J1.T And aT for a Mod ulaOt ed Load •••••••••••.••••••••••••• 84 L L
5.5a Response Statistics with 5 Elements •••••••••••••••••••••• 85
S.Sb Response Statistics with 7 Elements •••••••••••••••••••••• 85
S.Sc Response Statistics with 9 Elements •••••••••••••••••••••• 86
5.6 Lumped Mass Model for Example II (Mean Values) ••••••••••• 87
5.7a J1.T and a for Several wB ••••••••••••••••••••••••••••••• 87 L 't
5.7b Average ~ and G with Normal and Gamma L 't
PDF's for wB ••••••••••••••••••••.•••••••••••••••.••••••• 88
5.8 Lifetime Reliability for Layered Deposit ••••••••••••••••• 88
5.9 Historical Data on Liquefaction (Partial Data) ••••••••••• 89
vi
LIST OF FIGURES
Figure Page
2.1 Possible Hysteretic Shapes ••••••••••••••••••••••••••••••• 91
2.2 Lumped Mass Model •••••••••••••••••••••••••••••••••••••••• 92
2.3a O'E and &E for a SDF .................................... 93 T T
2.3b O'E and BE for a SDF .................................... 93 T T
2.4a SE for a 3 Degree-of-Freedom System ..................... 94 T
2.4b O'E for a 3 Degree-of-Freedom System ..................... 94 T
2.5 Statistics of ET for a Nonstationary Load ................ 95
3.1 Skeleton Curve of the Dynamic Shear Stress-Strain Relation for Soils •••••••••••••••••••••••••••••.••••••••• 96
3.2 Cyclic Shearing Stress-Strain Loops for Soils ••••••••.••• 97
3.3 Equivalent Viscous Damping Ratio D ••••••••••••••••••••••• 97
3.4a Model and Empirical Skeleton Curves for Sands •••••••••••• 98
3.4h Model and Empirical G/Gm with Y/Yr for Sands •••••••••••.• 98
3.5 Model and Empirical GIG with Y for Sands •••••••••••••••• 99 m
3.6 Model and Empirical GIG . with Y for Sands •.•••••••••••••• 99 m
3.7 Model and Empirical D with Y for Sands 100
3.8 Model and Empirical D with Y for Sands 100
3.9 Ramberg-Osgood's and Wen's Model D with Y for Sands ••••• 101
3.10
3.11
3.12
3.13a
3.13b
3.13c
Model and Empirical Skeleton Curves for Clays 101
Model and Empirical GIGm with Y for Clays ••••••••••••••• 102
Model and Empirical D with Y for Clays •.•••••••••••••••• 102
NQ. vs 't ••••••••••••••••••••••••••••••••••••••••••••••••• 103
Weight Function, h(i) ••••••••••••••••••••••••••••••••••• 103
Excess Pore Pressure for Uniform Loading •••••••••••.•••• 103
3.14a
3.14b
3.15
3.16
3.17
3.18
4.1
4.2
5.1
5.2
5.3
vii
Page
Nt vs t .................................................. 104
Excess Pore Pressure with Cycles of Loading (Table 3.2) •• 104
Sand Stiffness Deterioration ••••••••••••••••••••••.•••••• 105
Damage Parameters r E and rW with r ..................... u
Excess Pore Pressure with Cycles of Loading .............. Excess Pore Pressure with Cycles of Loading (Table 3.4) .. PSD Function for "Rock" Sites ............................ Kanai-Tajimi PSD Funct~on for Several wB
Homogeneous Sand Deposit
Cyclic Resistance Curves for Examp Ie I ..•..••...•........
Statistics of Time until Lique£ act ion ••••••••••••••••••••
105
106
106
107
107
108
109
109
5.4 Profile of Total Accelerations ••••••••••••••••••••••••••• 110
5.5 Modulating Function of the Base Excitation ••••••••••••••• 110
5.6a Expected Excess Pore Pressure Rise •••••••••••••••••••.••• 111
5.6b Stiffness Deterioration •••••••••••••••••••••••••••••••••• 111
5.7 Layered Deposit •••••••••••.•••••••••••••••.•••••••••••••• 112
5.8 Cyclic Resistance Curves for Example II •••••••••••••••••• 112
5.9 Lumped Mass Models for Example II •............•.......... 113
5.10a Reliability Against Liquefaction (IS-foot) ............... 114
S.10b Reliability Against Liquefaction (25-foot) ............... 115
5.11 Seismic Hazard For Eureka (California) ................... 116
5.12 Sensitivity of Reliability to ~D •••••••••••••••• e , ••••• 117 r
5.13 Sensitivity of Reliability to & ....................... 117 s u 5.14 Sensitivity of Reliability to the Uncertainties
in Soil Properties ••••••••••••••••••••••••••••••••••••••• 118
viii
Page
5.15a Historical Data of Liquefaction and No-Liquefaction (Partial Data) •.••••••••••••••••••••••••• 119
S.15b Historical Data of Liquefaction and No-Liquefaction (Partial Data) ••••••••••••••••••••••.•••• 119
5.16 Predicted Probabilities of Liquefaction for Some Historical Data •••••••••••••••••••••••••••••••.•.•••••••• 120
S.17a Sand Deposit for Case History 5 (Hachinohe) 121
s.17b Sand Deposit for Case History 6 (Hachinohe) 121
s.17e Sand Deposit for Case History 9 (Hachinohe) 122
s.lSa Sand Deposit for Case History 1 (Niigata) 122
S.lSb Sand Deposit for Case Histories 3, 10 and 11 (Niigata) ••• 123
S.lBe Sand Deposit for Case History 4 (Niigata) 123
1
CHAPTER 1
INTRODUCTION
1.1 Introductory Remarks
The Committee on Soil Dynamics of the Geotechnical Engineering
Division of the A.S.C.E. (1978) defined liquefaction of a saturated
cohesionless soil as the transfo·rmation of the soil from a solid state
to a liquid state as a result of the increase in porewater pressure and
attendant decrease in the effective stress. This phenomenon may be
caused by monotonic changes ~n shear stresses, cyclic vibratory
loadings, and shock waves such as those caused by earthquakes,
explosions or blast loads. Only the seismically induced liquefaction of
cohesionless sands is considered in this study.
Liquefaction produces a transient loss of shear resistance, but does
not always cause a long term reduction in shear strength. Because of
dilatancy, dense or medium sands harden or solidify during undrained
shear deformations; the loss of strength is temporary and the flow of
strains is also limited after some time. In loose sands, the flow of
strains and loss of strength will continue under undrained constant
total stress conditions. In either case, the surface deformations
caused by the flow of strains may have adverse effects on structures
supported on those soils, and the decrease in the effective stress may
result in loss of bearing capacity of the foundations.
Evidence of these adverse effects and loss of bearing capacity were
observed, for instance, during the Niigata earthquake of 1964, (Ohsaki,
2
1966), and the Montenegro earthquake of 1979 (Talaganov, Petrovski and
Mihailov, 1980). Liquefaction 1S an important factor 1n the aseismic
design and siting of embankment dams (Johnson, 1973; Seed, 1979b), as
well as in the stability consideration of natural and man-made slopes as
in the case of the Valdez landslides during the Alaska earthquake of
1964 (Seed, 1968 and 1979a). Seismically induced liquefaction is also
an important factor in the reliability assessment of lifeline systems,
such as pipelines and highways (Sazaki and Taniguchi, 1981), and is an
important factor in the aseismic design of bridges and waterfront
structures (Seed, 1979a). A survey of numerous case histories of
liquefaction failures was done by Gilbert (1976), and a reV1ew of the
history of earthquake-induced liquefaction 1n Japan was done by
Kuribayashi and Tatsuoka (1977).
According to Seed (1979a) there are basically two approaches for
evaluating the liquefaction potential of saturated sand deposits:
methods based on observations of the performance of sand deposits during
past earthquakes; and those based on the evaluation of the dynamic
response of the deposits, using laboratory data for determining the
conditions conducing to liquefaction.
1.2 Review of Procedures for Stochastic Analysis of Liquefaction
Attempts to quantify the probability of liquefaction of sand
deposits under earthquake loadings have been made during the past
decade. A summary of these procedures was presented by Christian (1980)
1n a state-of-the-art report on probabilistic methods in soil dynamics.
including liquefaction. Extensions of the methods for evaluating
3
liquefaction potential, based on the performance of sand deposits during
past earthquakes, to include the uncertainties in the soil properties or
in the earthquake loading were reported by Christian and Swiger (1975),
Yegian and Whitman (1978), and Davis and Berrill (1982). In these
methods, the historical data for the in-situ resistance of the soil are
plotted graphically against the intensity of the earthquake load. These
methods involve a definition of the boundary separating the data between
liquefaction and non-liquefaction. Yegian and Whitman (1978) obtained
this boundary by the method Qf least squares.
Christian and Swiger (1975) used a discriminant analysis to separate
the historical data. This is a technique of multivariate statistics,
and was chosen to minimize the overall probability of error. The
uncertainties caused by the scarcity and unreliability of the data used
in the analysis were not accounted for in the model. Youd and Perkins
(1978) prepared maps of ground failure potential, 1n terms of a
probabilistic measure based on seismic risk analysis, and maps of ground
failure susceptibility. According to Christian (1980) the two sets of
maps are used to assess the likelihood of liquefaction.
Davis and Berrill (1982) postulated that the porewater pressure
increase was proportional to the dissipated seismic energy density. The
pore pressure increase was related to the earthquake magnitude, distance
to the centre of energy release, initial effective overburden pressure,
and the SPT blowcount.
Haldar (1976) and Haldar and Tang (1979a) calculated the probability
of liquefaction by extending a method based on the determination of
stresses induced in the field by the seismic excitation, and laboratory
Me~z Reierence Room University of Illinois
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4
determination of the conditions causing liquefaction of the soils. The
deterministic simplified approach of Seed and Idriss (1971) was extended
to include the uncertainties ~n the earthquake loading and soil
properties. McGuire, Tatsuoka, et. a1 (1978) performed a probabilistic
assessment of liquefaction using a method based on the simplified
approach of Seed and Idriss (1971). Uncertainties in the soil strength
and in the c~rthq~ake ind~c~d she~r stresses ~ere included.
Faccioli (1973), Donovan (1971), and Donovan and Singh (1978),
applied a cumulative damage theory to evaluate the potential for
liquefaction using laboratory data and dynamic analysis of the deposito
In both cases only the uncertainty in the input ground motion was
considered. Donovan (1971) calculated the safety factor for deposits
us~ng the Miner's rule of fatigue, using an approximate probability
density function of the peak stresses (Rayleigh distribution); the
parameters were derived fram the peak ground acceleration, the duration
of the motion, and the dominant period of the deposit. Faccioli (1973)
used linear random vibration analysis to calculate the probabilities and
statistics of the seismic response of the deposits.
Fardis (1979) and Fardis and Veneziano (1981a, 1981b, 1982)
incorporated the uncertainties in the soil parameters and in the spatial
variation of the deposito The nonlinear-hysteretic and deteriorating
behavior of the saturated sand was included in the dynamic analysis, as
well as the effect of drainage in the porewater pressure buildup.
With few exceptions (e.g. Fardis and Veneziano, 1981), all the
previous studies are based on the assumption that the number of seismic
loading cycles causing liquefaction is known. A general procedure for
5
determining this number based on available geotechnical data remains to
be developed; this must include the determination of the equivalent
duration of an earthquake of given intensity.
1.3 Purpose and Scope
In this study a procedure is developed that represents the excess
pore pressure rise ~n saturated sand deposits under random seismic
excitations in terms of a continuous damage parameter. The damage
parameter is a function of the hysteretic shear-strain energy dissipated
and of the amplitude of the restoring shear stress, thus measuring both
the amplitude and number of loading cycles. This permits the study of
liquefaction of saturated sand deposits as a problem of random vibration
of nonlinear-hysteretic systems.
The random vibration results lead to the determination of the mean
and variance of the equivalent duration of an earthquake loading with a
specified intensity that causes liquefaction. It ~s assumed that
liquefaction occurs when the excess pore pressure ratio becomes equal to
one, i.e. when the shearing stiffness of the sand has deteriorated to
zero under repeated alternate shearing. The effect of stiffness
deterioration on the dynamic response of the soil is included 1n the
analysis. but the effect of drainage during the pore pressure buildup is
not taken into account.
The random vibration results together with the results from the
uncertainty analysis of the soil properties are used to calculate the
reliability of the deposits against liquefaction under random seismic
6
excitations with given durations and intensities. These conditional
reliability indices, and the probabilities of all significant seismic
loadings for the site over a specified time period, are then combined to
obtain the lifetime seismic reliability against liquefaction.
Current design procedures against liquefaction, regardless of their
degree of sophistication, are tipically deterministic. The factors of
safety are chosen such that, to the designer's opinion, an appropriate
degree of conservativism in face of the consequences of liquefaction is
assured. However, because the uncertainties in the design parameters
(e.g. the probabilities of all significant ground excitations during the
lifetime of the project) are not included in the design process, the
actual risk implicit ~n the design ~s not known. The proposed
probabilistic approach yields a measure of the likelihood of undesirable
performance such that the risk implicit ~n the design can be calculated,
and the relative risk between various design alternatives can be
compared~ Then, decisions concerning the design selection may be made.
1.4 Organization
In Chapter 2 a model for calculating the probabilities and
statistics of the resp~nse of nonlinear-hysteretic and degrading systems
under random seismic loadings is summarized. An additional development
of the model ~s presented for calculating the variance of the total
hysteretic shear strain energy dissipation.
In Chapter 3, a technique is developed for determining the excess
pore pressure generation in the sand, under undrained conditions, caused
7
by random seismic loadings. The technique allows the utilization of
data from conventional cyclic simple-shear tests or cyclic triaxial
tests in the study of deposits under irregular and random dynamic
loadings.
The methodology for the se1sm1C reliability evaluation against
liquefaction 1S formulated in Chapter 4. The rel~ability indices are
defined and the techniques for their calculations are described. The
uncertainties in the soil properties as well as in the earthquake
loading are included.
Examples of applications are illustrated for a homogeneous saturated
sand deposit and a layered deposit of sand and clay. The sensitivity of
the reliability of the deposit to the various sources of uncertainty is
examined.
The probabilities of liquefaction predicted with the proposed model
are compared with the observed field behavior during past earthquakes,
namely: three locations in the city of Hachinohe ,(Japan) during the
Tokachioki earthquake of May 16, 1968; and~ at several locations in the
city of Niigata (Japan) that showed evidence of liquefaction and
no-liquefaction during past earthquakes.
1.5 Notation
Throughout the text, the time derivative of any quantity will be
denoted with a dot over the symbol.
a
a m
= input base acceleration.
= peak ground acceleration.
A, a, ~, 0, r
A m
[ B ]
c
D
D r
e
E c
K
m
N
N~
q
r u
s o
8
= parameters controlling the hysteretic stress
strain relation.
= seismic intensity measure.
= covariance matrix of the random seismic loading.
= coefficient of viscous damping.
= equivalent viscous damping ratio.
= relative density.
= void ratio.
= hystereti~ energy dissipated per cycle.
= total hysteretic energy dissipated.
small strain shear modulus.
stiffness of each element of the system.
= mass of each element of the lumped mass system.
number of uniform cycles of loading.
= number of uniform cycles of loading capable of
causing liquefaction.
total restoring force ~n each element.
= the excess pore pressure.
damage parameter ratio.
= intensity scale of the double sided Kanai-Tajimi
PSD function for earthquakes.
= random duration of the strong motion part of the
earthquake loading.
= duration of the strong ground motion until
liquefaction occurs.
u
u s
z
, Uco
't
9
a:: relative displacement" of two consecutive lumped
masses or, the deformation of the hysteretic
spring.
z: deformation of the skeleton curve of the
hysteretic spring.
= hysteretic component of the displacement. The
hysteretic restoring force is Kz.
= reliabil1ty index against seismically induced
liqu,efaction.
= shear strain.
== parameters of the Kanai Tajimi power spectral
density function.
== effective confining stress.
= vertical effective stress.
= shear stress.
= normalized shear stress.
= standard deviation of variable X.
= coefficient of variation of variable X.
10
CHAPTER 2
THE RANDOM VIBRATION MODEL
2.1 Introduction
Soil deposits subjected to earthquake loadings are often likely to
undergo shear deformations of the order of 0.01 to 0.5 percent, which ~s
well in the nonlinear inelastic range. In addition, the behavior of the
material is hysteretic and the stiffness or strength are likely to
deteriorate with the number of oscillations. Thus, an accurate analysis
of the seismic response of such soil deposits requires a structural
model capable of including the nonlinear, inelastic, hysteretic and
deteriorating behavior.
A number of models capable of reproducing this behavior have been
proposed: namely, Hardin and Drnevich (1972a, 1972b); Finn, Lee and
Martin (1977); Martin (1975); Martin and Seed (1978, 1979); Pyke (1979);
Bazant and Krizek (1976); Richart (1975); Newmark and Rosenblueth
(1971); Katsikas and Wylie (1982); and Liou, Streeter and Richart
(1977). Faccioli and
horizontally layered
Ramirez (1976) analysed the seismic response of
soil deposits with a Ramberg-Osgood shearing
stress-strain behavior using a random vibration model, in which the
parameters of an equivalent linear system were obtained by a harmonic
linearization technique. Gazetas, Debchaudhury and Gasparini (1981),
obtained the statistics of the response of earth dams excited by strong
motions consisting of vertically propagating shear waves using a linear
random vibration model. In general, however, to obtain the statistics
11
and probabilities of the response with the above nonlinear models,
repeated time-history analysis, i.e., Monte Carlo simulation, must be
performed, which can be very costly. Recently, Wen (1976, 1980), Baber
and Wen (1979, 1981) proposed a .hysteretic restoring force model, and an
analytical method for the solution of the required statistics and
probabilities without statistical simulation. The following sections
will briefly describe this model including some of its most recent
developments and extensions.
2.2 The Smooth Hysteretic Model
The fundamental characteristics of the proposed hysteretic model may
be described for a single degree of freedom system. The equation of
motion is:
mu + eu + q(u,t) -ma (2.1)
and the restoring force is:
q(u,t) aKu + (1 - a) Az (2.2)
where (l-a)Kz 1S the hysteretic restoring force represented by:
. z (2.3)
in which
u = the relative displacement of the mass
z = the hysteretic component of the displacement.
m = the mass
12
a = the base acceleration
c = coefficient of viscous damping
K = initial stiffness, and
a,A,/3,o,r = parameters describing the shape of the hysteresis loop,
the elastic and inelastic deformation, and the maximum
strength for softening springs.
The total restoring force, q(u,t), has a hereditary property because of
the inclusion of the z term, which ~s the solution of the nonlinear
differential equation, i.e. Eq. 2.3. A large number of hysteresis
shapes can be described by varying the parameters A, .. /3, 0 and r, where
A, /3 and 0 must satisfy certain criteria to assure that the total energy
dissipated through a cycle ~s positive. Some of the possible
combinations are shown in Fig. 2.1. If different springs, each with
appropriate combinations of a, A, /3, 0 and r, are combined in series
and/or in parallel, additional hysteresis shapes such as shear-pinched
loops may be reproduced.
The skeleton curve, defined as the locus of the tips of the
hysteresis loops with different amplitudes, is given by:
z __ .....;;.d..::....~ __ + f A - (0 + S)sr 0
The incremental work done by hysteretic action is,
(1 - a.) Kzdu
(2.4)
(2.5)
and, the energy dissipated per cycle of amplitude z, Ec(z), is g~ven by
13
(Baber and and Wen ,1979),
E (z) = 2(1 - a) K[foZ A _ ?:;d?:; - ( ?:;d?:; J c (8 + o)~r 0 A - (0 - 8)sr
(2.6)
The ultimate hysteretic restoring force (l-~Kzu' defined as the
limiting value of (l-a}Kz as u approaches infinity is:
(1 - a) Kz u (2.7)
The total energy dissipated by hysteretic action, ET, is
t
ET = f (1 - a) K(zu)dt (2.8) o
Deterioration of the stiffness and/or strength of the material can
be included by specifying the system parameters to be functions of the
total hysteretic energy dissipated ET• The hysteretic restoring force
(l-a)kz is now represented by the following modified form of Eq. 2.3:
(2.9)
where v =V(ET ) and TJ =1](ET) account for the stiffness and strength
deterioration, respectively. The parameter A may also be defined as a
function of the total energy dissipated by hysteresis, ETe In this
form, monotonic reduction in A(ET) will represent degradations in both
the stiffness and strength.
14
2.3 The Equivalent Linearization
The response statistics, which reflect the random nature of the
loading, have been obtained by the method of equivalent linearization
(Wen, 1980; Baber and Wen, 1979). The special form of the nonlinear
hysteretic model presented in Sect. 2.2 permits the linearization of the
equations of motion ~n close form, without resorting to the
Krylov-Bogoliubov (KB) approximation.
Consider the mu1tidegree of freedom system in Fig. 2.2, which
represents a lumped mass model of a horizontally layered soil deposit
under vertically propagating shear waves. The equations of motion may
be writ ten as:
qi-1 qi m. qi+l ul
- (1 - c5';1) -- + - [1 + (1 - c5. ) -~-] - (I - c5. ) • mi - l mi ~l mi - l ~n mi +l
where:
and,
. z. ~
mi +l 2 x -- = - (- 2~ w U - wBuB) c5 1..1 m. B B B
~
C.u. + a.K. + (1 - a.)K.z. ~ ~ ~ ~ ~ ~ ~
r. A.u. -~ ~
r.-l (8.u.lz.1 ~
~ ~ ~ z. + c.u.lz.1 ~)v. ~ ~ ~ ~ ~
n. ~
in which:
i=l,n
i=l,n
i l,n
i=l,n
(2.10)
(2.11)
(2.12)
(2.13)
UB = the relative motion of the filter with respect
IS
to the ground (a Kanai-Tajimi filter is assumed)
WB'~B = parameters that characterize the filter transfer
function
temporal envelope that modulates the
stationary excitation if nonstationarity 1S
desired
5. 1 5. = Kronecker deltas. 1 ' 1n
The base excitation may be a white noise, tB (Caughey, 1960), or a
filtered white n01se (Amin and Ang, 1966, 1968; Shinozuka and Sato,
1967; Liu, 1970; Clough and Penzien, 1975). Possible forms for the
modulating function ¢l(t) for earthquakes were suggested by Amin and Ang
(1966, 1968) and Shinozuka and Sato (1967).
The rate of energy dissipated by hysteresis in the i-th spring is
given by:
ET
. = (1 - a.) K. (z. U . ) 1 ]. ]. 1
].
(2.14)
Only Eq. 2.13 needs to be linearized. The linearized form of
Eq. 2.13 was obtained by Baber and Wen (1979) as follows:
. z.
]. c .u. + K .z. e].]. eJ.]'
The equations for C . and K . are given in Appendix A. e1 e1
(2.15)
The linearized set of the governing differential equations of
motion, Eqs. 2.10, 2.11, and 2.15, may be represented by the following
system of first-order differential equations:
{Y} + [G]{y} (2.16)
in which:
__ ..:I
dUU
with,
16
[0 1 0 0 •.. 0]
T {y. } ~
[u.U.z.] ~ ~ 1
(2.17)
(2.18)
(2.19)
(2.20)
The matrix [G] ~s the matrix of the system coefficients, including the
C . and K '. e1 e1
The zero time lag covariance matrix, [8], of the system of equations
defined by Eq. 2.16 ~s the solution of the following differential
equation:
[8] + [G][S] + [S][G]T = [B] (2.21)
in which,
[Set)] = E[{y(t)}{y(t)}T] (2.22)
and,
[B] (2.23)
where [Wg~] 1S the constant excitation power spectral density matrix',
17
and So is the two sided power spectral intensity of the white noise.
The system of equations defined by Eq. 2.21 is a system of nonlinear
ordinary differential equations, because [G] depends on the response
statistics, and its solution requires numerical integration in the time
domain. The stationary solution for nondeteriorating systems, i.e.
[8] = 0, may be obtained iteratively using the algorithm reported by
Bartels and Stewart (1972).
2.4 Energy Dissipation Statistics
The expected rate of hysteretic energy dissipated by the i-th
spring, ~E (t) is given by: T.
1
ll· (t) = (1 - et.)K.E[u.z.] E.r. 1. 1 1 1
1
(2.24)
(in the following the index i will be suppressed for simplicity). The
value of E[uz] is an element of the zero time lag covariance matrix
[Set)], defined in Eq.2.22. The mean square value of ET(t) is given
by:
Assuming that z and u are jointly Gaussian, the expected value of
may be calculated by:
2 2 222 = (1 - a) K (1 + 2p. )0 a.
uz z u
The coefficient of variation of ET(t) is:
(2.25)
(2.26)
18
(2.27)
The mean total hysteretic energy dissipated, ~E (t), from time to to t T
is
t ~E (t) = (1 - a)Kf E[z(T)U(T)]dT
T t
(2.28)
o
whereas the corresponding mean square of the total energy dissipated is:
2 2 2 f t t . I E[~(T)] = (1 - a) K El!o!o [z(s)u(s)z(v)u(v)]dsdv (2.29)
Evaluation of the right-hand side of Eq. 2.29 requires the knowledge of
the two time joint probability distribution of u and z, and involves the
calculation of a six fold integral. However, if jointly Gaussian
behavior is assumed for the two time joint probability distribution of u
and z, calculations may be performed without difficulty for both the
stationary and nonstationary cases. With this latter assumption,
t t (1 - a)2K2j f {E[z(s)u(s)]E[z(v)u(v)] +
t t o 0
+ E[u(s)~(v)]E[z(s)z(v)] + E[u(s)z(v)]E[z(s)u(v)]}dsdv (2.30)
Hence, only the two time covariance matrix of the response, [S(s,v)], is
necessary. For the stationary case, the above expression is simplified
to,
19
E[~(t)] = (1 - a)2K2\2!: (t - T)E[u(T)U(O)]E[z(T)Z(O)]dT +
+ 2ft (t - T)E[U(T)Z(O)]E[Z(T)U(O)]dTj + ~~ (t) t T
o
The two-time covariance matrix is obtained from:
d ds[S(s,v)] =-[G(s)][S(s,v)] for s>v
with the initial conditions -[S(s,v)] = [S(v,v)]. s:v
Alternatively, the matrix [S(s,v)] may be determined from,
-1 [S(s,v)] = [¢(s)][¢(v)] x [S(v,v)] for s>v
(2.31)
(2.32)
(2.33)
where the matrix [~(t)] is the solution of the matrix differential
equation,
[ ¢ ( t) ] = -[ G (t) ] [¢ ( t ) ] (2.34)
with the initial conditions [~(to)] = [I], in which [I] is the identity
matrix. The derivation of Eqs. 2.32 and 2.33, as well as some
suggestions for the evaluation of Eq. 2.30 are summarized in Appendix B.
The coefficient of variation of ET is necessary for calculating the
variance of the strong motion duration that will cause liquefaction, and
for evaluating the seismic reliability against liquefaction.
The variance of ET as obtained with the procedure previously
presented was compared with the results of simulations. Results of this
investigation are summarized in Figs. 2.3a and 2.3b. The coefficient of
variation DE decreases rapidly with time for the first few seconds of T
Me~z Reierence Hoom University of Illinois
Bl06 NeEL 208 N. Romine Street
Urbana, Illinois 61801
20
excitation.
~E and 5E for each element of a three-degree of freedom system are T T
shown 1n Figs. 2.4a and 2.4b. The behavior is similar to that of the
single degree of freedom system. It is interesting to observe that the
coefficients of variation, BE ' for all elements of the system are T
almost equal. This is because the only source of uncertainty is in the
loading" which 1.S the same for all elements of the system, see Eqs. 2.35
through 2.36e. Similar behavior -of observed for
nonstationary loadings (see Fig. 2.5).
2.5 DOF Reduction Technique
The numerical integration of the matrix differential equation, Eq.
2.21, in toe nonstationary case, or - the iterative solution of the
remaining Liapunov matrix equation for the stationary case, is not an
easy task for a system with many degrees of freedom. The number of
unknowns in the zero time lag covariance matrix, [8], is equal to
(3n+2)(3n+3)/2 for a system with n degrees of freedom subjected to a
filtered Gaussian shot noise excitation.
The number of unknowns in the problem may be reduced if it is
recognized that the motion of the system may be described with a
combination of a few modes of vibration, usually the first or the first
and second modes. It has been shown that the response of embankments
under earthquake loadings 1S primarily in the first few modes of
vibration, Makdisi and Seed (1979). The same is true for horizontally
layered soil deposits that do not have strong inhomogeneities. An
21
iteration technique is used to take into account the changes in the
modal shape of vibration associated with the nonlinear and deteriorating
behavior of the material. For deteriorating systems, such as deposits
of saturated loose sands under random seismic excitations, a frequent
updating of the modal shape of vibration is necessary. The equations of
motion for the multidegree of freedom inelastic, nonlinear-hysteretic
system are written as:
[M]{v} + {Q({v},{v},t)} P (t)
where:
M .. lJ
v. l
i
I j=l
ll. 1.
Q. = q. - q. 1 l l l-
{p (t) }
m.o .. l lJ
[1 1
i=l,n
i=l,n
i=l,n
1]
j=l,n
with all the other quantities as previously defined.
(2.35)
(2.36a)
(2.36b)
(2.36c)
(2.36d)
(2.36e)
If only one mode is used, the displacement can be expressed as
{v} (2.37)
where {~l} is the mode shape and Xl lS the generalized displacement.
Then, the system of Eq. 2.35 together with 2.11 and 2.13 is reduced to
22
Eq. 2.11 and
= {W1}T[M]{1} •
(-2sB
WB
uB
-
{W1}T[M]{Wl}
(2.38)
and
c'. (v. - v. ) + K' .z. e~ ~ ~-l e~ ~
i=l,n (2.39)
The unknowns are reduced to u' . . 1 B' uB' xl' xl' Z i' 1 = , n • The total
number of unknowns ln the zero time lag covariance matrix is
(n+4)(n+5)/2. If two modes are used, the number of unknowns increases
to (n+6)(n+7)/2. For example, for a system with 10 degrees of freedom
(n 10), there would be 528 unknowns for the complete solution, 105 for
the one degree of freedom approximation, and 136 for the two degree of
freedom approximation. This technique is used also to calculate the
variance of the strain energy dissipated by hysteresis.
23
CHAPTER 3
SOIL CHARACTERIZATION AND LIQUEFACTION MODEL
3.1 Dynamic Shearing Stress-Strain Relation for Soils
3.1.1 Introduction
The factors that control the basic shearing stress-strain curve are
shown ln Fig. 3.1. At zero shearing strain the tangent to the curve
establishes the maximum value of the shear modulus, G • m The secant
shear modulus corresponding ·to any intermediate point on the curve, such
as point P in Fig. 3.1, is denoted by G. The maximum value of the
shearing stress obtained in a static test is T • Cyclic torsional tests m
of soils produce hysteresis loops similar to those shown ln Fig. 3.2.
The skeleton curve shown by the dashed line in Fig. 3.2 describes the
variation of the secant shear modulus, G, of the loops, with the
respective shearing strain, y. The loop shape and width describe the
increase in damping with the shearing strain, y.
It is often convenient to approximate the strain softening behavior
of soils, as shown in Figs. 3.1 and 3.2, by analytical expressions.
Hardin and Drnevich (1972b), have adopted modified hyperbolic relations
as those shown in Fig. 3.1. Richart (1975), Streeter, Wylie and Richart
(1974), have used the equations proposed by Ramberg and Osgood (1943),
and other expressions have been suggested by Martin (1976). Pender
(1977) proposed a model based on the critical state theory of soil
behavior. In the present study the differential equations describing
the smooth hysteretic model presented in Chapter 2 are used to
characterize the behavior of the soil under random seismic loading. It
24
should be emphasized that with this hysteretic model the shortcomings
discussed by Pyke (1979) are all circumvented.
Using the results of different types of laboratory tests, Hardin and
Drnevich (1972a, 1972b), Seed and Idriss (1970), Tatsuoka, et ale
(1979), Anderson (1974), Silver and Park (1975), Sheriff, Ishibashi and
Gaddah (1977), Stokoe and Lodde (1978), and others, were able to
determine the ve~ieticn of the ratio __ ..1
Q.L1.U -_ .. .!_--,-_ .... --:---.. -C~ULVQ.LCU~ VLC~VUC
ratio, D, defined in Fig. 3.3, with the shearing strain, y, for a wide
number of sands and clays. In the next two sections the parameters of
the proposed hysteretic model necessary to represent these experimental
relationships are determined.
3.1.2 Sands
Hardin and Drnevich (1972b) characterized the skeleton curves of
sands and clays by the function,
G
where,
in which a reference
1 (3.1)
(3.2)
strain defined as T IG , and a and bare m m
empirical constants that represent the influence of various test
parameters. Values of a and b were reported by Hardin and Drnevich
(1972b) for saturated and dry sands, as well as for saturated clays.
Following Richart (1975), the relationship between TITm and y/Yr
for
sands subjected to 1,000 cycles of loading, obtained from Eqs. 3.1 and
3.2, is shown by the dashed line in Fig. 3.4a. In the same figure, the
25
skeleton curve for the hysteresis model, with r = 0.50, A = 1.0, and
s = ~, is shown by the solid line. The values of S and ~ control the
shape of the loop and the reference strain These curves are
replotted in Fig. 3.4b to compare the variation of the ratio GIG with m
Y/Yr between the model and the empirical data of Hardin and Drnevich
(1972).
The variation of GIG with the shearing strain m for the hysteretic
model is shown 1n Fig. '3.5 for r = 0.50, A = 1.00, 8 = ~,
Yr = 4x10-4 and Yr = 7.5x10-4 • The range of experimental data reported
by Seed and Idriss (1970) are also shown in Fig. 3.5 for comparison. In
Fig. 3.6, the same curves for the analytical model are compared with the
experimental results obtained by Tatsuoka et al. (1979) for a sand at a
confining pressure of 2,020 psf. On these bases, it may be concluded
that the variation of the shear stiffness with the shearing strain is
well represented by the model using r = 0.50, A = 1.0, and 8 = ~.
The variation of the equivalent viscous damping ratio, D, with the
shearing strain amplitude, y, as defined in Fig. 3.3, is shown in Fig.
3.7, for the model with r=0.50, A=1.00, -4 Y = 4x10 and r
-4 Y. = 7.5x10 • r The experimental data in Seed and Idriss (1970) is also
shown in Fig. 3.7 for comparison. The model appears to dissipate more
energy per cycle; this could be because of the shape of the loop, or
because of different reference strains, Y • r The comparison with the
results reported by Tatsuoka et a1. (1979) for drained tests of
saturated and dry sands, at confining pressures of 2,020 psf, is shown
in Fig. 3.8.
26
Richart (1975) used the Ramberg-Osgood equations to characterize the
dynamic shearing stress-strain relation for sand. The variation of D
with Y/Yr for the Ramberg-Osgood parameters chosen by Richart (1975) is
shown with the dashed line in Fig. 3.9. The corresponding variation of
D with Y/Yr for the hysteretic model using r = 0.50, A = 1.00, S = fi, is
shown with solid lines in Fig. 3.9. If a linear spring with stiffness
aGm, where a = 2.5 %,. is added in parallel with a hysteretic spring, the
values of D decrease noticeably for high values of y/y , as shown in r
Fig. 3.9. A similar effect may be obtained if a nonlinear spring with
A = a, and ~ = 0, 1S used instead of the linear spring.
It seems that the hysteretic model is capable of characterizing the
variation of D with Y for strains up to 0.1 %, and tends to overestimate
the value of D for y>O.l %. Other values of r, a,~, 5 and A, may
result in a better characterization of the dynamic shearing
stress-strain relation for a particular sand. A system identification
technique, e.g. as proposed by Sues, Mau, and Wen (1983), may be used to
choose the parameters of the theoretical model that may better represent
the properties of the soil, if sufficient experimental data including
shearing stress-strain hysteresis loops obtained in cyclic torsional
tests are available.
3.1.3 Clays
For saturated clays the relationship between T/T and Y/Y at 1,000 m r'
cycles, for the Hardin and Drnevich equations, is shown by the dashed
line in Fig. 3.10. ~ The same relationship is show~ in 3.11 by solid
lines for two springs with A = 1.0, o={3, and r = 0.25 and r = 0.20,
respectively. The decrease of the ratio G/Gm with the shearing strain,
27
)', is shown ~n Fig. 3.11 for A = 1.0, Y = 4x10-4, S ={3, and r = 0.25 r
and r = 0.20. Experimental results reported by Seed and Idriss (1970),
and the curve of Hardin and Drnevich for 1,000 cycles and
~ = 4xlO-4 are also shown in Fig. 3.11. 'r
In Fig. 3.12 the variation of D with )' ~s shown for r = 0.25,
A = 1.0, S={3, and -4 )' r = 4x10 and -3 Y = 3x10 • r The range of
experimental data given in Seed and Idriss (1970), and the experimental
results reported by Tsai, Lam and Martin (1980), for a kaolinite clay,
are also shown in Fig. 3.12 for comparison. On these bases it appears
that the hysteretic model has a tendency to overestimate the values of D
at high strains, but is capable of correctly dissipating energy by
hysteresis even at very low strains.
3.2 Pore Water Pressure and Stiffness Deterioration
3.2.1 Introduction
For the purpose of this study, a technique for predicting the
porewater pressure rise ~n the sand caused by ground shaking, compatible
with the random vibration analysis, ~s necessary. The fundamental
mechanisms of pore water pressure generation and the techniques for its
evaluation have been described elsewhere (Martin, Finn and Seed, 1975;
Seed, Martin and Lysmer, 1976; Ishihara, Tatsuoka and Yasuda, 1975). In
the following, a technique that represents the excess pore pressure rise
in uniform-stress cyclic shear tests in terms of a continuous damage
parameter is formulated; this permits the study of liquefaction of sand
deposits as a problem of random vibration of nonlinear-hysteretic
systems.
28
3.2.2 Uniform Cyclic Loading
Based on energy considerations, Nemat-Nasser and Shokooh (1979)
developed an approach for modeling the liquefaction of cohesionless
soils. The amount of energy r~quired to change the void ratio from e to
(e+de) is defined as:
dW - de - v f(l + r )g(e - e )
u m (3.3)
where
dW = work performed in rearranging the particles
jj = parameter that may depend on the effective confining
pressure, (J"c'o) but not on the void ratio, e.
e minimum void ratio for the sand. m
ru = ex~ess pore pressure ratio, defined as the excess pore pressure
normalized with respect to (J"' or to (J"' vo co·
It ~s required that:
The
in Eq.
f(l) 1 ~> 0 and dr -
u
dimensions of dW and v are the
3.4 are nondimensional. For the
eO' de = - __ c dr
1lw u
g (0) 0 ~ > 0 de -
(3.4)
same, and all other quantities
saturated undrained case,
(3.5)
where ~W ~s the bulk modulus of the water, and a~o ~s the initial
effective confining pressure. Then Eq. 3.3 becomes:
29
The onset of liquefaction occurs when r = 1. u
(3.6)
Neglecting the total
volumetric pore strain, the work performed in rearranging the particles,
~w, when the excess porewater pressure rises from zero to ru is given by
* e r dr' A _____ 0____ f U U uW = \)
gee -e ) 0 f(l+ru')
. 0 m
or by the differential equation
dr U
d/:).W
gee - e ) __ ~o_* ___ m=- x f(l + r )
. u \) e o
h * - , were e 1.S the initial void ratio, and v = vcr ITJW
• o co
(3.7)
(3.8)
The value of ~W is related to the shear strain energy that the sand
dissipates by hysteresis, neglecting the work done by volumetric
changes, which could be several orders of magnitude smaller. Using the
data reported by De Alba et al (1975) it was shown that the value of ~W
corresponding to the onset of liquefaction, i.e. r = 1 is a constant u '
for a given initial state of the sand.
Let the energy dissipated by hysteresis in one cycle of amplitude
T = rIa' be denoted as Ec(T). co The value of ~W after N cycles of
constant amplitude may be considered proportional to the number of
cycles of loading if the amplitude of shearing is large (Nemat-Nasser
and Shokooh, 1979); and thus,
30
~w = h(=t)NE (=() (3.9) c
where he:;:) ~s a function of the shearing stress amplitude. Equation 3.9
then becomes
* \) e r d'r = 0 f U u
gee - e ) f(l + r') o m 0 u
(3.10)
using the following simple forms-for f(l+r ) and gee - e ) u 0 m'
e )n
g (e - e ) = (e - m ,n > 1 o m 0
£(1 + r ) (1 + r ) 5 , 5 > 0 (3.11) U U
Eq. 3.10 yields,
* \) e = 0 1 [ 1 (1 r) 1-5 ]
gee - e ) x (s 1) - - U _ o m
For initial liquefaction, r = 1, u
* \) e _____ 0 ___ (1 _ 21- 5 )
(5 - 1) (e - e ) n o m
(3.12)
(3.13)
-where Nt is the number of cycles of constant stress amplitude, T,
capable of causing liquefaction of the sand for the given initial state.
The ratio rW = ~W(ru)/~W(ru = 1) is, according to Eqs. 3.10, 3.12
and 3.13, given by
where rN = N/N£.
31
1 + (1 + r )l-s u (3.14)
Seed, Martin and Lysmer (1976) have suggested that rN and ru might
be related by
. 26 nru rN = Sl.n <-2-) (3.15)
in which 8 1S an empirical parameter that varies from 0.50 to 0.9; a
value of e = 0.70 1S valid for a large range of data (Seed, Martin and
Lysmer, 1976). Equation 3.15 would be obtained if f(l + ru) in Eq. 3.10
is replaced by
f(l + r') u
1 6 ' , 29-1 1Tru nru
en sin (2)cos (-2-)
(3.16)
If the functions in Eqs. 3.11 and 3.16 are used for fC1+ru ) in Eq. 3.8,
the following respective incremental relations are obtained:
dr (1 + r )s (1 _ 21- s ) u u (3.17) -- = dr
W s - 1
and,
dr 1 u (3.18) -- = dr
W 28_18nru nru TIe sin (-2-) cos (2)
32
where,
1) (3.19)
Consider the T vs. N~ relationship shown in Fig. 3.13a. This cyclic
resistance curve ~s for a sand with a relative density D = 45 % at a r
confining pressure u~o = 8.0 psi, and was reported by De Alba et al
(1975). The knowledge of the cyclic shearing stress-strain relation for
this sand and the assumption that ~W(r = 1) = Nnh(T)E (7) is a constant u N c
for all pairs of T and N)1. on the cyclic resistance curve in Fig. 3.13a,
are enough to determine h(T) and the variation of r with r The value u W·
of Ec(r) is calculated for the initial state of the sand and the cyclic
shearing stress-strain relation is represented with the hysteretic model
proposed ~n Chapter 2 with r = 0.50" A = 1.0, and 6 =~. In Table 3.1
the values of N)1. and their corresponding stress ratios are shown, as
well as the values z/zu for the hysteretic model, where z corresponds to
T. The values of Ec(r) normalized with respect to Ec(T) for Nt = 3 are
shown ~n column 4 of Table 3.1, and the values of h(T) are shown in
column 5, where,
3EC
(:r3
)
N)1.Ec CENt) (3.20)
and T is the stress ratio corresponding to liquefaction in Nt cycles ..
The function h(T) is shown in Fig. 3.13b and, using Eq. 3.18, the
33
variation of the porewater pressure with the number of cycles is shown
in Fig. 3.13c.
3.2.3 Irregular and Random Dynamic Loading
Assuming that future values of ru depend only on its present value
and on future shear stress amplitudes, and using the theoretical
developments described earlier, the porewater pressure generation under
a nonuniform loading may be computed as follows:
(i) From the cyclic resistance curves for a sand, such as those
given in Fig. '3.13a, and with the shearing stress-strain
relation, calculate the function h(T) and the value of ~W for
ru = 1 as Nth(T)Ec(T).
(ii) For the i-th cycle of loading, calculate the value of ~W, and
the ratio rW
-and T. ~s the J
= ~W/~W(l), using E (T) and h(T), where c
6W. ~
i I E (T.)h(T.)
. 1 c J J J=
amplitude of the j-th cycle of loading.
(3.21)
(iii) The value of rW together with Eq. 3.18 are sufficient to
calculate the excess pore pressure ratio r u·
It is tacitly assumed that the sand does not deteriorate, therefore,
E (To) 18 always calculated for the initial state of the soil. The case c J
of a deteriorating material is considered subsequently.
The cyclic resistance curve of a sand with a relative density,
Dr = 45 %, at a confining pressure of u~o= 4000 psf, as reported by
Martin, et a1. (1975), is shown in Fig. 3.14a. Using the methodology
described by Martin, Finn and Seed (1975) the porewater pressure rise
for the nonuniform cyclic loading of Table 3.2 was calculated for this
34
sand; the results are shown in Table 3.2 and by the solid line in Fig.
3.14b. The porewater pressure, r u ' was also computed using Eq. 3.18
with 8=1.30; the corresponding results are shown by the dashed line in
Fig. 3.14h. The stress-strain curve for the sand was modeled with r
0.50, A = 1.0, 8 = f3. It should be mentioned that the stress-strain
curve of the sand in question is not accurately modeled with r = 0.50.
for this type of ex~mple~ and all the others in t:hiA r.h~nt.:),._ - -- - - - -- -.- - - - JI
the results shown by a dashed line in Fig. 3.14b are not sensitive to
differences ~n the stress-strain relation, or to the value of T , but m
are sensitive to the shape of the cyclic resistance curve of Fig. 3.14a.
The function h(T) accounts for these differences. The porewater
pressure ru seems to compare well with the results obtained by the more
fundamental approach.
As shown in Table 3.3 the function h(T) is quite uniform, whereas
the same function in Table 3.1 decreases rapidly as decreases. This
is, of course, the result of different slopes in the cyclic resistance
curves. The h(T) in Table 3.3 corresponds to the curve which has higher
slopes (that 1n Fig. 3.14a), and the h(T) in Table 3.1 corresponds to
the curve ~n Fig. 3.13a. Fardis (1979) and Fardis and Veneziano
(1981a,1981b), have shown that the average slope of the cyclic
resistance curves obtained in the laboratory is almost equal to the
slope in the curve of Fig. 3.13a. It appears that the greater the
amplitude of the loading cycles, the greater the percentage of the
shearing strain energy that ~s capable of contributing to the apparent
volume changes of the sand resulting from slip deformation. The
approach suggested by Martin, Finn and Seed (1975) implies that the
35
apparent reduction in the volume of the sand due to slip deformation for
ru = 1 , Evd ' is a constant for a given initial state of the sand. This
approach also shows that if the amplitude of the uniform cycles of
loading is below a threshold va1ue,o"the energy input into the sand, per
cycle of loading, will not be sufficient to cause the apparent reduction
in volume, EVd ' capable of producing the critical pore pressure ratio,
~.e. r = I.Oc ~
This threshold value for the sand of Table 3.3 with a
confining pressure of 4,000 psf would be approximately 130 psf.
The extension of the me~hodo1ogy described above to random loadings
may be described as follows:
(i) From the cyclic resistance curves for sand such as those ~n
Figs. 3.13a and 3.14a and from the shearing stress-strain
relation of the soil under cyclic loading, calculate the
function h(T) and the value of ~W for r = 1 as u '
~W(r = 1) = Nnh(T)E (T) U N c (3.22)
(ii) At time t, the value of ~W(t) is calculated as,
t . ~W(t) f X(s)E(s)ds (3.23)
t 0
where t designates the starting time of the excitation, 0
ET(t) is the expected rate of hysteretic energy dissipated by
the sand due to shearing, and X(t) is a function analogous to
h(T) that may be calculated by;
X(t)
T
f max h(T')E (T')p-(T' ,0-,0.,t)dT' c T T T o (3.24)
E (T')p-(T' 0- o~ t)dT' c T' T' T'
Me~z Reference Room University of IllinOis
Bl06 NCEL 208 N. Romine Street
Urbana, IllinOis 61801
36
where PTC • ) is the probability density function of the
peaks of the normalized hysteretic restoring shear stress
T/~;O at time t CKobori and Minai, 1967; and Lin, 1976).
(iii) The value of ru(t) is then computed using a modified version
of Eq. 3.18. The derivatives are:
drw 1 X(t)E(t) dt = b.W(r = 1) (3.25)
u
and,
dru 1 1 --=-x dt 1T0'. TIr 2 1 TIr u 0'.- u
cos (-2-) sin (2)
. X(t)E(t)
x b.W(r = 1) u
(3.26)
Steps (ii) and (iii) continue until liquefaction occurs or until the
excitation stops. The above procedure does not include the
deterioration of the material properties caused by the porewater
pressure increase, and is appropriate for a total stress analysis. In
the case of a stationary excitation, implementation is simple because
X(t) and [T(t) are constants for any given loading. The porewater
pressure r1se is then entirely described by Eq. 3.26 after the values of
XCt) and ET(t) have been obtained from the random vibration analysis.
The solution of Eq. 3.26 for this case is known.
3.2.4 Stiffness Deterioration
The porewater pressure development in saturated sands under
vibratory loading leads to a decrease in the value of the shear modulus
at low strains, G as defined earlier. m' When the pore pressure
37
approaches the initial effective confining stress, the stress path is in
the failure envelope for a large portion of the cycle, and the
stress-strain behavior becomes very complex. In this study, only the
tilting of the hysteresis loops caused by the stiffness deterioration
will be considered, because the general stress-strain relation of the
sand at high pore pressure ratios is not well defined. Other
researchers, e.g. Katsikas and Wylie (1982), Finn, Lee and Martin (1977)
and Fardis (1979)", have also considered the deterioration of the maximum
shear strength, Tm. In this study, only the deterioration of stiffness
is considered; there is evidence that good results can be obtained on
this basis (Martin and Seed, 1979). The value of G at time t after the m
start of the excitation is related to G and to the square root of the mo
effective confining stress, as follows:
(3 .27)
Therefore, the shear modulus at low strains, Gm(t), is given by;
G (t) = 11 - r (t) G m u mo
(3.28)
Let ~ 1n Eq. 2.9 be defined as:
n 1 (3.29)
11 - r u
Then, the energy dissipated by a cycle of normalized shearing stress
-amplitude T, when the porewater pressure has risen to r u '
given by (Baber and Wen, 1979),
38
E (T,n) = nE (T) C C
(3.30)
The gradual deterioration of the hysteresis loops according to Eqs. 3.27
through 3.30, may be seen in Fig. 3·.15.
If the sand 1S submitted to N cycles of loading of constant
shearing stress amplitude, the total energy dissipated by hysteresis
would be,
(3.31)
In order to apply the same procedure described in Sect. 3.2.1 for
problems with deterioration, Eqs. 3.17 or 3.18 must be modified. The
equation relating the excess pore pressure ratio to the number of cycles
of loading should be kept the same. This equation and its incremental
form are
r = 1 arcsin(rNI/Z6) u iT 1:
(3.32)
and
~ 1 'ITr 'ITr ~o- U u
drN = eITsin (-Z-)cos(--2-)dru (3.33)
The equation for ~, obtained from substituting Eq. 3.32 into Eq. 3.29,
1S
1 (3.34)
/ 2 . 1/29
1 - TI arcsl.n(rN )
Now, define the following quantity,
39
flEer ) u
(3.35)
after N < NQ, cycles of loading with uniform shearing stress amplitude,
where rN is related to ru through Eq. 3.32. This equation defines the
value of ~W(r ) when the deterioration of Gm is taken into account; its u
value is always larger than ~W(r ) u for the same porewater pressure
ratio. Dividing both sides·of Eq. 3.35 by ~W(r = 1), u
flEer ) u
t:.W(r = 1) u
Define r E as
'ITr 26 1 'ITr u - u
fru 9'ITCOS C-
2-) sin (-2-)
------~~--------~~ dr II - r U o u
flEer ) u
flW(r = 1) u
(3.36)
(3.37)
The differential equation relating r E to ru may be derived from Eqs.
3.36 and 3.37; thus,
dr 11 - r ~ = _____________ u ________ __ (3.38) drE nru 29-1 nru
9'ITcos (-2-) sin (-2-)
The solution of Eq. 3.38 is shown by the solid line in Fig. 3.16 and
may be compared with the solution of Eq. 3.18 which is shown in dashed
line. For the early stages, the curves are almost coincident because
40
very little stiffness deterioration has yet occurred,
and ~E(r ) are very similar. u
1.e. ~l{r ) u
In the derivation of Eq. 3.38 it was assumed that the number of
cycles of loading, N, is a continuous parameter. Multiplying the
right-hand side of Eq. 3.38 by drE/dN, the equation governing the
porewater pressure increase under nonuniform loading may be writen as,
dr u
dN
where,
/1 - ru 1 dE (T (N) ) ---=-=-1-) h (T (N) ) C dN
1Tr 29 I 1Tr 6W(r u - u u e1Tcos(--2--)sin (--2-)·
T(N) = the shearing stress amplitude of the N-th cycle.
(3.39)
The porewater pressure buildup determined through Eq. 3.39 for the
problem previously described in Sect. 3.2.2 and summarized in Tables 3.2
and 3.3, is shown in Fig. 3.17 1n dashed line; the results may be
compared with the solid line of Fig. 3.14b. The same problem was solved
again, this time with the cycles of loading applied in a reverse order.
The results are shown 1n Table 3.4 for the fundamental approach of
Martin, Finn and Seed (1975), and plotted in Fig. 3.17 with a solid
line. The solution of Eq. 3.39 for this loading is shown in Fig. 3.18
in dashed line. The results still compare reasonably well with the
theoretical ones; the differences are mainly the result of the
particular choice of the function f(l + ru) as given by Eq. 3.16.
The procedure described in Sect. 3.2.2 for calculating the porewater
pressure buildup under random seismic loading, using a total stress
analysis, may be extended to include the effect of stiffnes·s
41
deterioration of sand. The first step remains the same as before, and
two additional steps are necessary as follows:
(ii) At time t, the value of ~(t) is calculated by
~E(t)
t
f X(s)E(s)ds t
o
(3.40)
However, ET(t) 1S not· the same as in Eq. 3.23 because the
reduction in the stiffness and the resulting changes 1n the
rate of hysteretic energy dissipation are taken into account,
as indicated in Ch. 2, with Eq. 2.9, where TJ is defined by
Eq. 3.29. The function X(t) is defined 1n the same manner as
before, i.e. by Eq. 3.24.
(iii) The value of ru(t) is now computed using a modified version
of Eqs. 3.39 and 3.38 as follows:
dr u
"dt= II - r
u rrr 2e 1 Trr
rrecos( 2u )sin - ( 2u)
X(t)E(t) ~W(r = 1)
u
Eq. 3.41 1S obtained with Eq. 3.38 and
dr 1 _E = X(t)E(t) dt 6W(r = 1)
u
3.2.5 Discussion
(3.41 )
(3.42)
A procedure was described for determining the excess porewater
pressure rise from conventional cyclic simple shear tests or cyclic
triaxial tests in terms of a variable that permits the use of these data
42
with irregular and random seismic loadings. The method circunvents the
need to calculate the equivalent uniform stress cycles in a total stress
analysis, as well as the need to measure volumetric strains and rebound
characteristics as required in a dynamic effective stress analysis.
The proposed method was developed using results of stress controlled
cyclic tests; however, an equivalent procedure may be developed using
results of constant strain cyclic tests. In the latter, it would be
necessary to know the number of cycles of constant amplitude shearing
strain N~ capable of causing an excess pore pressure ratio ru = 1.0, and
the variation of the excess pore pressure ratio r with the number of u
cycles of loading.
If the results of strain controlled tests are used, the quantity
~W(r = 1) is defined as u
6W(r u
1) (3.43)
where Ec(Y) is a function of the hysteretic energy dissipated by a cycle
of amplitude Y, and hey) is a weight function analogous to that 1n Eq.
3.9. ru = fy{ r N), as obtained from constant strain cylic loading
tests. Then, for a total stress analysis rN = r W' and ru may be
Let
calculated from
(3.44)
Finn and Bhatia (1981) suggest that the experimental scatter 1n
stress controlled test data 1S always greater than that for strain
controlled tests, primarily at higb excess pore pressure ratios,
43
exceeding 70 % of the effective overburden pressure. This would imply
that a technique to evaluate the excess pore pressure rise due to a
dynamic loading, using a continuous parameter such as aw, would be more
accurate if the results of strain controlled cyclic tests were used.
The shape of the hysteresis loops is much more stable throughout the
test for strain controlled cyclic loading test than for a stress
controlled cyclic loading test, implying that the deterioration of the
mechanical properties of the sand are more clearly defined for the
strain controlled tests.
If the stiffness deterioration, as given by Eq. 3.29, is taken into
account, then, in a manner similar to that described in Sect. 3.2.4, let
r 11 - r flEer ) NQ,Ec(Y)h(Y) f U
U dr U tfy(rN~ U
0
drN f (r ) ---JrN = Y U
(3.45)
and,
dr tfy(rN) . 1 U
drE drN
x
f-1 (r ) 11 - r ~rN = u
Y U
(3.46)
where r E =~E/AW(l). Eqs. 3.45 and 3.46 are equivalent to Eqs. 3.35 and
3.38, but ~E(ru) is always smaller than ~W(ru). To include the strength
deterioration it would be necessary to write
g(r ) u
(3.47)
44
where g(ru ) defines the variation of the energy dissipated by one loop
of shear strain amplitude, y, due to the effects of strength
deterioration alone. This function depends on the particular shearing
stress strain law, and on the mechanism of strength deterioration.
Usually the strength deteriorates according to
T (t) = T (1 - r ) m mo u (3.48)
which may be taken into account in defining the parameter 1) ~n Eq. 2.9;
e.g.
'J = ( 1 ) r 1 - r
u (3.49)
45
CHAPTER 4
SEISMIC RELIABILITY ANALYSIS
4.1 Introduction
The statistics of the seismic response of saturated sand deposits
are functions of the randomness in the frequency content and duration of
the strong ground motion, as well as of the uncertainties in the dynamic
soil properties. The randomness 1n the occurrence of the earthquake
loading is important for the lifetime reliability evaluation against
liquefaction.
The required dynamic soil properties may be obtained from field or
laboratory tests, or both (Richart, 1975; Woods, 1978; Marcuson and
Curro, 1981). Large uncertainties are unavoidable in the measurement of
some soil properties such as the relative density (Tavenas et aI, 1973).
Sample disturbance is a source of considerable uncertainty associated
with laboratory tests used to evaluate liquefaction potential, including
large shaking table tests, hollow cylinder torsional tests, cyclic
simple shear box tests, and cyclic triaxial tests (Marcuson and
Franklim, 1979; Fardis, 1979). The uncertainties in the dynamic soil
properties and their quantification have been the object of recent
research (Haldar, 1976; Haldar and Tang, 1979a, 1979b; Fardis, 1979, and
Fardis and Veneziano, 1981a, 1981b, 1982).
The methodology for the seismic reliability analysis against
liquefaction 1S formulated; the uncertainties 1n the dynamic soil
properties are identified and quantified based on the studies referred
46
to above; and, the uncertainties in the frequency content, intensity and
duration of the earthquake loading are subsequently discussed.
4.2 Reliability Evaluation
Given the occurrence of an earthquake with a given intensity,
Am = a, and a strong motion duration, TE = t, the performance function
against liquefaction at a given depth within the deposit is,
where
z 6W(r u 1) - 6W(a,t)
~W(ru = 1) = measure of resistance defined by Eq. 3.22;
~W(a, t) = damage parameter defined by Eq. 3.23.
(4.1)
Failure ~s then defined as,
z < 0 (4.2)
For the reliability evaluation, the statistics of ~W(r = 1) u and
~W(a, t) are necessary. The response of the deposit depends on
uncertain soil properties, e.g. shear modulus G , whose uncertainties m
are discussed in Sect. 4.3. Let the vector of such soil properties be
denoted by R ~
with mean and let GR. be the variance of the i-th ~
component of ~, and p .. be the correlation coefficient between the i-th ~J
and j-th component of!. Then, the quantity ~W(a, t) is also a function
of R, and failure is ~
z ~W(r = 1) - ~W(a,t,R) < 0 u
(4.3)
47
Using first order approximation (Ang and Tang, 1975) the mean and
variance of dW(a, t, R) can be calculated from
where
-
t t
f f t t
o 0
t
f x(s,a'~R)E[ET(s,a'~R)]dS t
o
(4.4)
(4.5)
(4.6)
(4.7)
These last two quantities are obtained from the random vibration
analysis, and the derivatives in Eq. 4.5 are evaluated by
finite-differences. An analytical technique to obtain the derivatives
~n Eq. 4.5 was recently proposed by Sues (1983).
The statistics of ~W(r = 1) depend on the uncertainties 1n the u
undrained resistance against liquefaction under uniform cyclic stress
loading N£, and on the uncertainties in the relative density of the sand
D • These uncertainties are discussed in Sect. 4.3. r
48
The lifetime reliability index ~, unconditional on Am and TE may be
calculated by
Am TEmax Q -- f maxf Q )' () d d ~ ~(a,t PA a,t a t
A T mTE m. E.
(4.8)
m~n m~n
where PA T (a, t)dadt is the joint probability that a < Am < a+dt and m E
t < TE < t+dt at the site. So far, available seismic hazard models deal
only with the earthquake intensity A , and not with its duration TE• . m
For the purpose of this study the statistics of TE will be conditional
on the value of Am' and the results suggested by Lai (1980) will be
used. The uncertainties ~n the duration and intensity of the earthquake
loading are discussed in Sect. 4.4. After including the effect of
duration uncertainties, the lifetime reliability is calculated as,
s m L 8(a
i=l i~a)Prob[i~a - 6a < A < i6a + ~a] (4.9)
where the probability that a-~a < a < a+~a ~s obtained with the
fault-rupture seismic hazard model of Der Kiureghian and Ang (1977).
The measure of earthquake intensity used in this study ~s the peak
ground acceleration. The statistics of TE conditional on Am' and the
randomness associated with the frequency content of the earthquake
ground motion are examined in Sect. 4.4.
49
4.3 Uncertainties in Soil Properties
4.3.1 Undrained Resistance to Liquefaction Under Uniform Cyclic Stress
Loading
The uncertainties in the undrained resistance to liquefaction under
uniform cyclic stress loading have been the object of recent study by
Haldar (1976), Ha1dar and Tang (1979), Fardis (1979), and Fardis and
. Veneziano (1981b). Fardis and Veneziano proposed a probabilistic model
for liquefaction under uniform cyclic stress loading that is well suited
for this study. The principal results of their study of interest for
this research are briefly summarized in the following.
The resistance of the sand against liquefaction is defined by the
number of cycles of constant amplitude cyclic shear stress loading, N£ '
capable of causing the critical excess pore pressure ratio
( excess pore pressure I initial effective vertical stress ) of one.
The uncertainty analysis is based on available laboratory data on cyclic
simple shear tests (Peacock and Seed, 1968; Yoshimi and Oh-Oka, 1973,
1975; Yoshimi and Tokimatsu, 1977; Ishihara and Yasuda, 1975; Ishibashi
and Sheriff, 1974; Sherif and aI, 1977; Finn et aI, 1977; Finn et aI,
1970; Finn et aI, 1971; Finn and Vaid, 1977) and large shaking table
tests (DeAlba et aI, 1975; Mori et aI, 1977). The mean-grain size,
D50 for the sands used in those tests is between 0.4 and 0.65 mm. The
analysis of those data lead to the following model for liquefaction in
the laboratory:
where,
(4.10)
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Urbana ~ Illinois 6180J..l
50
't = 't the shear stress ratio • (1"
co G' = co initial effective confining pressure;
0" ref an initial effective confining pressure of
reference equal to 4.8 psi;
Dr = relative density expressed as a fraction;
aI' a2 , a3 , a4 = jointly normal coefficients whose mean vector
and covariance matrix are shown in Table 4.1;
€L = a normal random variable with zero mean and
standard deviation 0.20, which accounts for
scatter about the regression equation.
Using first-order approximation (Ang and Tang, 1975) the c.o.v. of
the number of uniform stress cycles that causes liquefaction In the
laboratory, N£, can be calculated, in approximation, as,
(4.11)
Typical values of oN obtained with Eq. 4.11 are shown in Table 4.2 for >1-
several values of the relative density of the sand.
The c.o.v. of the shear stress ratio :; = T I err co capable of causing
liquefaction in the laboratory in a given number of cycles can be'
calculated, also In apprOxlmatlon, as,
1 2
11a 2
51
a' 2 2 2 2 co 2 + cr in CD ) + cr in (-, -) + a a3 r a4 0ref sL
(4.12)
Typical values of the c.o.v. of r/u' , calculated with Eq. 4.12, are co
shown in Table 4.2 for several values of the relative density.
Physically, the uncertainties in Nt are the result of uncertainties
1n the maximum and minimum dry densities of the sands used in the tests,
and the uncertainties 1n the corrections of the laboratory data for the
method of sample preparation, system compliance and stress
nonuniformities.
The model is then modified to predict the resistance of the sand 1n
the field under uniform cyclic stress loading. This is accomplished
adding two normal random variables, Ao and ao ' to the right-hand side of
Eq. 4.10. The mean and variance. of these random variables are shown in
Table 4.1. Ao accounts for differences between the laboratory and
in-place structure of the soil, and for site-to-site variations; and,
ao accounts for the effect of multidirectional motions. The c.o.v. of
the number of constant amplitude shear stress cycles that cause
liquefaction 1n the field is then calculated, in approximation, as,
a'
0.2 2 2 2 2 2 )
2 i 2(~) exp{O A + a +0 + a in (l1D
+ 0a n a' Ni a a
1 a 3 r 4 ref 0 0
11a 2 (4.13) 2 (_3)2 + 20 a p in(l1
D)} - 1 + a + aD
SL 11D r a
1 a3 .al a 3 r r
52
where the in-situ value of the relative density is also a random
variable. Typical values for this coefficient of variation are shown in
Table 4.3 for several values of the mean and c.o.v. of the in-situ
relative density of the sand. The c.o.v.'s of the shear stress ratio
T =7/~' capable of causing liquefaction in the field in a given number co
of cycles, are also shown ~n Table 4.3.
The statistics of the undrained resistance to liquefaction under
uniform cyclic stress loading, Ni
, are necessary to obtain the
statistics of the quantity ~W(ru = 1) in Eq. 4.1.
defined in Sect. 3.2.2, is
-where 'T ~s
~W(r = 1) ~s u -
b.W(r u
1)
the shear stress ratio
a constant for any pair
'T / a:.' co
of Ni
and h(7) is
and 'T. For
This quantity,
(4.14)
chosen such that
a given set of
I T, D and (J this quantity has a lognormal distribution with mean r co
(4.15)
and standard deviation
(4.16)
The c.o.v. of ~W(ru = 1) ~s the same as the c.o.v. of Ni
. Typical
values of these coefficients of variation are shown 1n Table 4.3 for
several values of the mean and c.o.v. of the in-situ relative density of
the sand.
53
4.3.2 Additional Soil Properties
The statistics of the undrained resistance to liquefaction are
strongly dependent on the uncertainties in the in-situ relative density
of the sand, D (see Table 4.3). r Furthermore, the response of sand
deposits under random seismic loads depends on the uncertainties in the
small strain shear modulus, G , and the static shearing strength of the m
sands, Tm. In the following, the c.o.v ..... s of the in-situ relative
density, small strain shear modulus, and static shear strength will be
discussed.
Relative Density The in-place values of the relative density, D , r
are usually related to the results of SPT-N tests. A few in-situ joint
measurements of SPT-N and D are necessary to estimate the statistical r
parameters of the relationship between SPT-N and the relative density
for a given soil. The spatial variability within the profile 1S taken
into account with measurements 6f the SPT-N along several borings.
Fardis (1979) developed a probabilistic model for estimating the SPT-N
and D relationship. r
An uncertainty analysis of the the relative density was also done by
Haldar (1976) and Haldar and Tang (1979b), and the results used in a
probabilistic evaluation of liquefaction potential (Haldar and Tang,
1979a). The in-situ relative density of the sand can be determined by
either direct methods (laboratory determination), or indirect methods
that relate the in-situ value of the relative density to the SPT
blowcount. Typical values of the c.o.v ..... s of the in-situ relative
density of the sand determined with either of the above mentioned
methods are shown in Table 4.4 (Haldar and Tang, 1979b).
54
The relationship between the mean and c.o.v. of the in-situ relative
density of the sand can be approximated (Haldar, 1976) as
0.258 - 0.138 ~D r
0.633 - 1.475 ~D r
2 + 1.192 ~D
r
~D >0.60 r
~D ~0.60 r
(4.17a)
(4.17b)
when the in-situ relative density is determined as suggested by Gibbs
and Holz (1957).
Small Strain Shear Modulus In the present study, the small strain
shear modulus of the sand ~s calculated with the following equation
(Martin and Seed, 1982):
G m
5,000(1 -D - 75
r ) 0cr' 100 m
(psi) (4.18)
The form of Eq. 4.18 suggests that the uncertainties in the small strain
shear modulus are strongly dependent on the uncertainties ~n the
relative density of the sand. The c.o.v.'s of G for several values of m
the mean and c.o.v. of the in-situ relative density of the sand are
shown in Table 4.5. Errors in the predicition of G 'th E m w~ q. 4.18
should also be considered. An additional c.o.v. of 0.12 is used to
account for the differences between the laboratory and in-situ values of
the shear modulus Gm• This value is based on the study by Fardis (1979)
for the equation proposed by Hardin and Drnevich (1972b). The total
c.o.v. of the small strain shear modulus is shown in Table 4.5 for
several values of the in-situ relative density of the sand.
55
Static Shear Strength -- The c.o.v. of the static shear strength of
the sand ~s related to the uncertainties in the friction angle of the
sand. Meyerhoff (1982) suggested that the c.o.v. of the static shear
strength of the sand is between 0.10 and 0.20. The value of 0.10 was
used in this study. The c.o.v. of the undrained shear strength of the
clay is between 0.20 and 0.40 (Meyerhoff, 1983); the value of 0.20 was
used in this study.
4.4 Earthquake Loading
4.4.1 Ground Motion Model
For the purposes of the this study the earthquake induced strong
ground motion should be specified in terms of its amplitude frequency
content and duration. The intensity of the earthquake loading ~s
characterized by its peak ground acceleration, a and the frequency max'
content by its power spectral density function (PSD function). The
Kanai-Tajimi PSD function
Sew) 2S o
(4.19)
is used to model the frequency content of the earthquake loading. The
parameter So is the intensity scale of the PSD function; and, wB and
~B shape parameters.
The peak ground acceleration a has been related to the root mean max
square ground acceleration arms' and excellent correlations have been
found by Lai (1980), Vanmarcke and Lai (1980), Moayyad and Mohraz
(1982), and Hanks and McGuire (1981). In particular, Vanmarcke and Lai
56
(1980) suggested the following relationship:
Where:
a max
a !2£nC 2TE) rms T
o (4.20)
TE = duration of the strong-motion phase of the ground
excitation;
To = predominant period of the ground motion.
The predominant period of the earthquake motion is defined as
T o
2'IT C1}
C
where the quantity W is calculated from c
(4.21)
(4.22)
The duration of the strong-phase of motion is determined from Eq. 4.20
and the condition that the total energy of the ground motion is
retained, i.e. with
I = a2
T o rms E
(4.23)
where 10 is the Arias intensity (time integral of the squared
accelerations). In this manner, the expected peak ground acceleration
and the energy of the recorded accelerograms (that provided the data)
are reproduced by the model.
When the parameters of the Kanai-Tajimi PSD function are known, the
root mean square ground acceleration is calculated by
a rms (4.24)
In general,
a max
57
= (PF)a rms (4.25)
where (PF) 1S a peak factor, defined by Eq. 4.20; this peak factor is
very insensitive to the values of the duration and predominant period of
the ground motion, and is usually between 2.5 and 3.0.
4.4.2 Uncertainties in Earthquake Load Parameters
PSD Function -- Housner and Jennings (1964) have initially suggested
the values of wB = 511" orad/sec and ~B = 0.64 for "firm" ground
conditions; whereas based on 140 horizontal accelerograms Lai (1980)
found wB to vary from 5.7 rad/sec to 51.7 rad/sec, and ~B between 0.10
and 0.90. F 22 tr k" . t d 6 / or roc S1 e recor s wE had a mean of 2 .7 rad sec and
c.o.v. of 0.40, and ~B had a mean of 0.35 and c.o.v. of 0.36. For
"soft" site records the corresponding means and c.o.v.'s are 19 rad/sec
and 0.43 for wB and 0.32 and 0.36 for 'B. Moayyad and Mohraz (1982) obtained average shapes of the PSD
function for vertical and horizontal accelerations. When a Kanai-Tajimi
PSD function, Eq. 4.19, is fitted to the average shape for "rock" sites
proposed by Moayyad and Mohraz (1982), wB and 'B were found to be
wB = 16.9 rad/sec and ~B = 0.94 (Sues, 1983) • The Kanai-Tajimi PSD
function with these values of wB and ~B is used in this study to model
the earthquake ground motion except if otherwise stated.
The Kanai-Tajimi PSD function with wB = 16.9 rad/sec and 'B = 0.94,
and the Housner and Jennings PSD function are shown in Fig. 4.1. The
average of PSD density function for "rock" sites obtained using the
statistics and distributions for wB and 'B proposed by Lai (1980) is
58
also shown in Fig. 4.1.
The PSD function with wB = 16.9 rad/sec and 'B = 0.94, and the
average PSD function obtained with Lai's data are similar, implying that
the response statistics calculated with either of them will be similar.
However, the study by Lai (1980) proves that the large c.o.v.'s of
wB and 'B ( ..... 0.40) imply high likelyhoods of occurrence of earthquakes
with very different frequency content (see PSD functions in Fig. 4.2).
The effect of these differences on the seismic response of a saturated
sand deposits was calculated in Sect. 5.3.1; a c.o.v. of the time to
liquefaction of the order of 0.50 was obtained.
Peak Ground Acceleration -- The probabilities of exceedance of all
significant se~sm~c intensities at the site during the lifetime of the
project are calculated with the fault-rupture seismic hazard model of
Der Kiureghian and Ang (1977). The probabilities calculated with this
model will depend on the physical relations assumed in the model (e.g.
the intensity attenuation equation, and the slip-length magnitude
relation) as well as the values of parameters such as the slope of the
"magnitude recurrence" curve (Der Kiureghian and Ang, 1977). The effect
of these uncertainties on the calculated probabilities can be
systematically evaluated with the above referred seismic hazard model.
In general, the uncertainties in the intensity attenuation equation will
tend to dominate. The c.o.v. of the peak ground acceleration obtained
with the attenuation equation may be as high as 0.70 (Der Kiureghian and
Ang, 1977).
Duration of Strong-Motion Phase Lai (1980) suggested the
following relationship between the expected peak ground acceleration and
59
the mean duration of the Btrong-motion phase:
11T (a ) E max
30 exp(-3.254 aO. 35 ) max
The c.o.v. of TE conditional on the value of
(4.26)
ST = 0.804, E
and
the gamma distribution is considered appropriate for TE• For "rock"
site conditions the strong motion durations are slightly shorter
(~foayyad and Mohraz, 1982; Lai, 1980). However, because of scarcity of
the data for "rock" sites Eq. 4.26 is retained.
60
CHAPTER 5
ILLUSTRATIVE EXAMPLES
5.1 Introduction
The principles and general procedures described 1n the earlier
chapters are illustrated herein for specific soil deposits. The first
example illustrates the calculation of the mean and standard deviation
of the duration of strong motion, with a specified intensity, necessary
to cause liquefaction failure. An homogeneous deposit with low relative
density is considered. The differences in the results of a total stress
analysis and an effective stress analysis are presented.
In the second example, the seismic safety of a nonhomogeneous
layered soil deposit is examined. The corresponding prototype problem
would be a deposit in a reclaimed area; the top layer may be a hydraulic
fill and the other layers are of medium clay and sand with higher
relative densities.
The probabilities of liquefaction predicted with the proposed
methodology are compared with the performance of sand deposits during
past earthquakes. Examples of these are: three locations in an area of
sandy soil 1n the city of Hachinohe (Japan) which showed a variety of
damage features during the Tokachioki earthquake of May 16, 1968; and
several locations 1n the city of Niigata (Japan) which showed evidence
of liquefaction or no-liquefaction during past earthquakes.
61
5.2 Homogeneous Saturated Sand Deposit
5.2.1 Problem Description
A homogeneous sand deposit is considered. The deposit is idealized
as consisting of several layers to account for changes ~n soil
properties with depth as shown in Fig. 5.1. The soil properties used in
the analysis are specified in Table 5.1. The deposit was discretized
into 10 elements of equal thickness. The small strain shear modulus r-~m'
the shear strength Tm, the stiffness K, as well as the parameters 5 and
fi for all the elements in the profile are shown ~n Table 5.2. The
hysteretic model parameters were chosen as A = 1.0 and r = 0.50 for all
the elements.
The undrained cyclic resistance curves against liquefaction for this
sand are shown in Fig. 5.2 for three different values of the vertical
effective stress. The values of T/~~O for the same Nt increase for
decreasing values of U~o. Accordingly, the cyclic resistance curves for
other a~o between those ~n Fig. 5.2 were obtained by linear
interpolation.
5.2.2 Total Stress Analysis
The statistics of the time till liquefaction, TL, for several
stationary loadings with different peak accelerations a , and with max
WB = 5rr rad/sec and 'B = 0.64, are given in Table 5.2.
~ were calculated by L
The values of
~T L
~W(r u
1) = ----------
X~· ET
(5.1)
where all the quantities. are defined in Chs. 2 and 3, X being the
62
equivalent amplitude for random loading defined by Eq. 3.24. The
denominator of Eq. 5.1 ~s obtained from the random vibration analysis.
The standard deviation of TL is calculated by u T =~T 8T where L L L
the same as 8E at T
time t = ~T ~ L
The validity of this technique for
calculating uT was verified with Monte-Carlo simulations. L
The variation of ~T at 12.5-foot depth with the intensity of the L
earthquake loading ~s shown in Fig. 5.3. The mean plus one standard
deviation and the mean minus one standard deviation of TL are also shown
~n Fig. 5.3.
The coefficient of variation of the duration of strong motion
necessary for liquefaction decreases as the mean value of that duration
increases. A similar behavior was found with the coefficient of
variation of the total hysteretic energy dissipated, i.e. the c.o.v. of
the hysteretic energy decreases as the duration of the load increases.
For long durations of the loading the c.o.v. of the aforementioned time
to liquefaction decreases because of the ergodic nature of the ground
excitation for such long durations.
These seismic resistance curves for the deposit may be obtained at
several depths within the soil deposit, and used to calculate factors of
safety against liquefaction for a specified seismic loading, as well as
the associated reliability levels. Suppose that the intensity of the
loading is given as amax = 0.05 g, and the strong-phase duration
TE = 5.0 seconds. According to Fig. 5.3, the deposit can withstand an
intensity of 0.067g for a duration of 5.0 seconds prior to liquefaction.
The factor of safety therefore, ~s F = 0.067/0.05 = 1.34, with a a
coefficient of variation of 0.14. The associated reliability is
63
P[F > 1.0] = 0.98. a
This information can not be obtained with any other method currently
available (Finn, et aI, 1977; Martin and Seed, 1979; Katsikas and Wylie,
1982) without recourse to statistical simulations. Current methods for
evaluating liquefaction potential are limited to deterministic ground
excitations; therefore, the factor of safety only applies to a
particular accelerogram.
The profile of the total accelerations in the soil deposit is shown
~n Fig. 5.4 for A = 0.20, 0.10, and 0.05 g. The nonlinear behavior max
~s apparent in the variation of the ratio of the standard deviation of
the total acceleration at the surface and the root mean square base
acceleration, arms.
5.2.3 Stiffness Deterioration
Martin and Seed (1979) analyzed this deposit under the 1940 El
Centro earthquake scaled to a maximum acceleration of 0.10 g. When a
total stress analysis was used and undrained conditions assumed, the
time until liquefaction was estimated to be 3.0 seconds. The effective
stress analysis gave a value of 8.0 seconds. Under the same conditions,
Katsikas and Wylie (1982) calculated this duration as 4.0 second~,
whereas Zienckiewicz et al (1982) obtained 2.8 seconds. The statistics
of TL under a load with an expected peak ground acceleration
amax = 0.10 g, wB = ~ rad/sec, and 'B = 0.64, with the modulating
function shown in Fig. 5.4 may be seen in Table 5.4. In particular, at
12.5 feet it is ~T = 3.2 seconds and uT = 1.7 seconds. L L
The effective stress analysis was carried out for this load and the
statistics of TL at 12.5 feet were calculated as ~T = 3.6 seconds and L
64
U T = 1.8 seconds. The expected pore pressure ratio rise and the L
stiffness deterioration are shown ~n Figs. 5.6a and 5.6b, respectively,
at several depths within the deposit. This analysis shows that the mean
values of TL calculated with the total stress analysis and the effective
stress analysis differ by less than 15 % for the homogeneous soil
deposit. Martin and Seed (1979) concluded, based on the results of
several deterministic analysis, that the stresses in the soil calculated
by an effective stress analysis and a total stress analysis are almost
identical until the onset of liquefaction.
5.3 Reclaimed Fill
5.3.1 Introduction
The profile of the deposit representing a reclaimed area is shown in
Fig. 5.7. The seismic reliability evaluation of this reclaimed fill is
performed, and its sensitivity to the uncertainties in some dynamic soil
properties is examined. The seismic response of the deposit as obtained
from the random vibration analysis depends on the method of
discretization of the continuum (Martin and Seed, 1978; Seed and Idriss,
1968). In Tables 5.5a, 5.5b and 5.5c, some results of the stationary
random vibration analysis are shown for two levels of intensity of the
loading and three schemes of discretization (shown in Fig. 5.9). The
9-element model seems appropriate for this problem. The statistics of
the shear strains and stresses converge faster than the statistics of
the hysteretic shear strain energy dissipated. However, for the purpose
of this study, the seven element model may be sufficient.
65
The average dynamic properties of the elements of the lumped mass
model are shown in Table 5.6. The undrained cyclic resistance curves
are shown in Fig. 5.8. These curves were obtained assuming that this
sand can be described by Eq. 4.10, and the statistics shown in Table
4.1.
The effect of the randomness in the frequency content of the ground
excitation on the statistics of the response of this deposit was
quantified. The deposit was analyzed for arms = 0.08 g and 'B = 0.4,
but with five different values of wB- With the statistics of wB for
"rock" site conditions suggested by Lai (1980), the expected response
statistics of the deposit were calculated assuming both normal and gamma
distributions for wB• The results of this analysis are summarized 1n
Tables s.7a and s.7b.
The response statistics for a load with the same root-mean-square
ground acceleration but with a different frequency content i.e.
W B = 16.9 rad/seco and 'B o = 0.94 are also shown 1n Table S.7b. These
latter statistics are similar to the average statistics obtained with
the earlier analysis. The reasons for this good agreement are
summarized 1n Sect. 4.4.2; the particular natural frequency of the
deposit also contributes to this agreement, as shown in Fig_ 4.1.
The results thus show that a c.o.v. of the time to liquefaction of
the order of 0.50, can be attributed to the randomness of the frequency
content of the ground motion energy. The corresponding coefficient of
variation of the damage parameter ~W(a, t), is also of the order of
0.50. This randomness should be included in the reliability analysis.
66
The statistics of TL were obtained with an effective stress analysis
for a loading with intensity amax = 0.15 g, and the cyclic resistance
curves in Fig. 5.8 marked 8N = 8n = 0.0. At a depth of 15 feet, these l r
statistics are ~T = 3.5 seconds and ~T = 1.2 seconds; whereas the same L . L
statistics obtained with a total stress analysis are ~T = 3.1 seconds L
and seconds. These results agree with the statements of
Martin and Seed (1979).
5.3.2 Reliability Evaluatio~
The reliability indices conditional on the intensity and duration of
the earthquake loading were calculated at depths of 15 and 25 ft for
several levels of the intensity of the load and a wide range of the
duration of the strong-phase motions. These indices are shown in Figs.
5.l0a and S.lOb. A value of 0.20 for the coefficient of variation of
the undrained shear strength of the clay was used, as suggested by
Meyerhoff (1982). The coefficient of variation of the relative density
1S taken as 0.15 following Haldar and Tang (1979b). The statistics and
probabilities of the strong-phase motion given in Sect. 4.4 are combined
with the results 1n Figs. S.10a and 5.10b, to obtain unconditional
reliability measures. These unconditional values are shown by the
dashed lines in Figs. 5.10a and S.10b.
The fault rupture model of Der Kiureghian and Ang (1977) is used to
calculate the probabilities of all significant loadings at the site.
Using the data reported by Kiremidjian and Shah (1975), the predicted
probabilities for Eureka (California) are shown
4.24 and the information in Figs. 5.10 and 5.11 are used to calculate
the lifetime reliability against liquefaction. The calculated
67
reliabilities are summarized in Table 5.8.
The sensitivity of the reliability to the uncertainties in some soil
properties can be seen in Figs. 5.12 to 5.14. In Fig. 5.12, values of
aD = 0.0 and 0.15 are assumed. In Fig. 5.13 the sensitivity of the r
reliability to the uncertainties in the undrained shear strength of the
clay layer are examined. Finally, in Fig. 5.14, all the soil properties
were considered to be exactly known.
5.4 Case Studies
5.4.1 Introduction
The probabilities of liquefaction predicted with the proposed
methodology are compared with the field performance of sand deposits
during past earthquakes. Usually, the historical data for the in-situ
resistance of the soil are plotted graphically against the intensity of
the earthquake load, and a boundary separating the data between
liquefaction and no-liquefaction is determined (Seed and Idriss, 1981).
A convenient parameter to represent the intensity of an earthquake is
the ratio of the average shear stress T developed on horizontal ave
surfaces of the sand to the initial effective vertical stress a' . vo
Values of the stress ratios known to be associated with some evidence of
liquefaction or no-liquefaction in the field are plotted as a function
of the standard penetration resistance (SPT), N1 , corrected to a value
of a~o equal to 1 ton/sq ft (Seed and Idriss, 1981).
A graphical representation of some of the historical data available
is shown 1n Fig. 5.15a, where the boundary separating the cases of
liquefaction and no-liquefaction (Seed and Idriss, 1981) is shown by a
Me-cz Reference Roonr University of Illinois
BIOS NCEL 208 N. Romine 2trs8t
t';~"b ar~:~ ~ I], J. i :~ ~-- .: ~~ r ~.: :~" ':.} ..
68
solid line. The same data is plotted in Fig. 5.15b using the estimated
value of the in-situ relative density of the sand as the measure of the
resistance of the soil. Information about these data points can be
found in Seed, Arango and Chan (1975) and in Table 5.9.
The proposed method for evaluating the reliability of sand deposits
during earthquakes is used to calculate the probability of liquefaction
for somE of the data points in Fig. 5.15a, namely: (i) at three
locations in the city of Hachinohe (Japan) during the Tokachioki
earthquake of May 16, 1968 (points 5, 6 and 9 ~n Fig. 5.15a); (ii) at
three locations ~n the city of Niigata (Japan) during the Niigata
earthquake of 1964 (points 1, 3 and 4 in Fig. 5.15a); and, (iii) at one
location ~n the city of Niigata (Japan) for two historical earthquakes
of magnitude 6.1 and 6.6 (points 10 and 11 in Fig. 5.15a).
In the following a brief description of the modeling of the soil
profiles for each case study as well as the assumptions made in the
analyses are presented. The results are summarized in Fig. 5.16 and
discussed in Sect. 5.4.4.
5.4.2 Hachinohe (Japan)
The locations studied correspond to the borings designated P6, PI
and P2 by Ohsaki (1970). The soil profiles at each location were
idealized as shown ~n Figs. 5.l7a, 5.17b and 5.17c, corresponding to
points 5, 6 and 9 ~n Fig. 5.15a. The characteristics of the Tokachioki
earthquake of May 16, 1968 are shown in Table 5.9. The expected peak
ground acceleration of the base excitation .. .. . 1·· usen ~n ~ne anaLys~s ~s
approximately 0.09 g. The statistics of the strong-motion duration
TE as defined ~n this study are obtained with the correlations with
69
magnitude and epicentral distance proposed by Lai (1980) and Shinozuka,
Kameda and Koike (1983), and are also shown in Table 5.9. The gamma
distribution was considered appropriate for TE•
The probabilities of liquefaction were calculated for these three
locations using the c.o.v.'s in Figs. 5.17a, 5.17b and 5.17c, and Ch. 4
( Table 4.3); and, then for a to~al c.o.v. of 0.30 for the shear stress
ratio T/U~o that causes liquefaction in a given number of uniform
loading cycles. These probabilities are summarized in Fig. 5.16.
5.4.3 Niigata (Japan)
The soil profiles corresponding to three locations in the city. of
Niigata (Japan) were idealized as shown in Figs. 5.18a, 5.18b and 5.18c,
corresponding to points 1, 3 (also 10 and 11) and 4 in Fig. 5.15a (Seed
and Idriss, 1981). The characteristics of the Niigata earthquakes of
1964, 1887 and 1802 are summarized in Table 5.9, as reported by Kawasumi
(1968) and Seed and Idriss (1967). The data concerning the earthquakes
of 1887 and 1802 are only approximate. The expected peak ground
accelerations of the base excitations are approximately 0.07 g, 0.05 g
and 0.025 g for the earthquakes of 1964, 1802, and 1887, respectively.
The statistics of the strong-motion duration TE were obtained from the
correlations with magnitude and epicentral distance proposed by Lai
(1980) and Shinozuka, Kameda and Koike (1983), and are shown ~n Table
5.9. The gamma distribution was considered appropriate for TE•
The probabilities of liquefaction during the Niigata earthquake of
1964 were calculated for three locations (namelv. nointg I. 3 and 4 In ---------- --- ------ ------~-- ... ------.17 r------ -7 - ---- . ---
Fig. 5.15a); and, for the location in Fig. 5.1Sb during the earthquakes
of 1802 and lS87 (respectively points 10 and 11 in Fig. 5.15a). As
70
before, these probabilities were calculated for the c.o.v.'s shown in
Figs. 5.18a, S.18b and S.18c, and Ch. 4 ( Table 4.3); and, for a total
c.o.v. of 0.30 for the shear stress ratio T/U' that causes liquefaction vo
in a given number of uniform loading cycles. The results are summarized
in Fig. 5.16.
5.4.4 Main Observations
The probabilities· of liquefaction obtained with the proposed method
of analysis appear to be consistent with the observed results. It is
expected that for the data points close to the boundary separating the
cases of liquefaction and no-liquefaction the predicted probabilities of
liquefaction will be higher than for those points further below this
line. For the same standard penetration resistance (as measured by N1)
the ratios of the shear stress on the boundary and the earthquake
induced shear stress are also shown in Fig. 5.16. Usually, these ratios
will increase as the probabilities of liquefaction decrease.
5.5. Summary of Results
The first example shows how seismic resistance curves against
liquefaction can be obtained for a specific soil deposit using the
results from the random vibration analysis. The results of this
analysis showed that the c.o.v. of the duration of strong ground motion
necessary for liquefaction to occur decreases as the mean value of this
duration increases. This means that the reliability against seismically
induced liquefaction becomes less sensitive to the ground motion
randomness as the expected time to liquefaction increases. This
71
analysis also shows that the statistics of the time to liquefaction
obtained with a total stress analysis differ only slightly from the same
statistics obtained with an analysis that considers the deterioration of
the sand stiffness.
The second example illustrates the procedure proposed for the
lifetime reliability evaluation against liquefaction. The results show
(see Fig. 5.14) that the probabilities of liquefaction conditional on
the duration and intensity of the load are very sensitive to the
uncertainties in the undrained resistance against liquefaction for a
given relative density, and the in-situ value of the relative density of
the sand. This example clearly indicates that the reliability against
liquefaction ~s more sensitive to the intensity of the seismic
excitation than to the duration of the strong-motion phase.
Finally, the comparison with historical data shows that the proposed
methodology appears to be a viable procedure for predicting the seismic
reliability of sand deposits against liquefaction, and for assessing the
relative reliability of design alternatives.
72
CRAPTER 6
SUMMARY AND CONCLUSIONS
6.1 Summary
A procedure was developed that represents the excess pore pressure
rise in conventional uniform-stress cyclic shear tests in terms of a
continuous damage parameter, which permits the study of liquefaction of
sand deposits as a problem of random vibration of nonlinear-hysteretic
systems. This parameter is a function of the hysteretic shear-strain
energy dissipated by the soil and of the amplitude of the hysteretic
restoring shear stress, thus measuring both the number and amplitude of
loading cycles.
The mean and variance of the strong-motion duration until
liquefaction ~n the deposit were calculated with the random vibration
analysis. These satistics can be used to define the seismic resistance
curves against liquefaction in the deposit. On this basis, factors of
safety and the associated reliability levels can be obtained for any
seismic loading.
Alternatively, the results of the random vibration analysis can be
used to calculate the reliabili~y against liquefaction of a saturated
sand deposit under a random seismic load with a prescribed duration and
intensity. With this approach, the uncertainties in the resistant
properties of the soil can also be systematically included ~n the
reliability evaluation.
73
The reliability against liquefaction conditional on the intensity
and duration of the loading was obtained using the results from the
random vibration analysis together with the results from the uncertainty
analysis of the soil properties. The sensitivity of the reliability
measures to the the duration and intensity of the strong-motion, as well
as to the uncertainties in the soil properties was examined. The
randomness in the intensity and duration of the earthquake load were
included in the lifetime reliability evaluatfon.
The probabilities of liquefaction predicted with the proposed method
were compared with field observations of some saturated sand deposits
during past earthquakes.
6.2 Conclusions
The main results and conclusions of this study are summarized in the
following:
I - The proposed methodology, that evaluates the seismic reliability
of sand deposits conditional on the intensity and duration of the
earthquake loading, is necessary and useful for a risk-based design
against liquefaction. The comparison with the historical data shows
that the method is an effective tool for determining the seismic
reliability of saturated sand deposits against liquefaction, and for
assessing the relative risks between design alternatives (see Sect.
5.4).
2 - The reliability against liquefaction conditional on. the
intensity and duration of the loading is sensitive to the uncertainties
74
1n the in-situ relative density of the sand, and the uncertainties in
the undrained resistance to liquefaction for a given relative density.
In fact, the uncertainties in these soil properties dominate the
reliability against liquefaction for given intensity and duration of the
loading.
3 - The conditional reliability against liquefaction is sensitive to
the strong-motion duration, and is even more sensitive to the intensity
of the seismic excitation (as measured by its peak ground acceleration).
This implies that the probabilities of occurrence of all significant
loads at the site have to be considered 1n the design process, as
opposed to the deterministic methods that postulate the lifetime
earthquake load.
4 - The technique that was developed to represent the excess pore
pressure generation 1n saturated sand deposits under random seismic
excitations has the following advantages: (i) it uses a continuous
damage parameter formulation, thus removing the need for calculating the
equivalent number of loading cycles; (ii) does not require the
measurement of volumetric strains or the rebound characteristics of the
sand under cyclic loading, whose measurement requires sophisticated
equipment and is time consuming; and, (iii) can be used with the results
of constant stress or constant strain cyclic loading tests.
5 - The statistics of the strong-motion duration that will cause
liquefaction can be well predicted with the total stress analysis. The
stiffness deterioration leads to slightly longer expected times for
liquefaction,
accelerations.
and changes the frequency content of the surface
75
6 - The probability of occurrence of earthquake loadings with very
different frequency content can result in very different expected times
till liquefaction. The c.o.v. of the time to liquefaction associated
with the uncertainties in the Kanai-Tajimi filter parameters is of the
order of 0.50.
7 - The shear-strain energy dissipated by hysteresis in saturated
sand deposits ~s a good indicator of the damage and deterioration of
such deposits under random seismic loadings.
76
TABLES
77
Table 3.1 Function h(T) for a Sand with D =0.54· r
T z E (z)
h(a; ) N.e, c
0' z [Ec(z)]N =3 vo u va Q,
3 0.190 0.311 1.000 1.000
5 0.172 0.282 0.749 0.801
10 0.149 0.244 0.493 0.609
20 0.130 0.21:3 0.385 0.448
30 0.120 0.197 0.268 0.373
60 0.104 0.170 0.178 0.281
100 0.094 0.154 0.135 0.224
200 0.082 0.134 0.092 0.163
Table 3.2 Excess Pore Pressure with Cycles of Loading
CYCLES T T NQ, (psf)
r 0' u vo
1 450. 0.349 0.113 4.0
2 416. 0.531 0.104 5.0
3 384. 0.657 0.096 6.0
4 355. 0.755 0.089 7.5
5 328. 0.837 0.082 9.2
6 303. 0.914 0.076 11.4
7 280. 1.000 0.070 14.2
78
Table 3.3 Function h(T) for a Sand with D =0.45 r
E c-n T
NQ, z c T
0' z E (0.113) h(G') va u c va
0.113 4.0 0.206 1.000 1.000
0.104 5.0 0.190 0.797 1.004
0.096 6.0 0.175 0.634 1.052
0.089 7.5 0.163 0.520 1.025
0.082 9.2 0.150 0.414 1.051
0.076 11.4 0.139 0.336 1.045
0.070 14.2 0.128 0.268 1.050
Table 3.4 Excess Pore Pressure with Cycles of Loading
CYCLES T T
NQ, (psi) r 0' u
va
1 280. 0.178 0.070 12.7
2 303. 0.279 0.076 10.0
3 328. 0.410 0.082 8.4
4 355. 0.521 0.089 6.7
5 384. 0.642 0.096 6.0
6 416. 0.790 0.104 5.0
7 450. 1.000 0.113 4.0
al
a2
a3
a4
a 0
A 0
79
Table 4.1 Statistics of Parameters Defining N£
Mean Covariance Matrix
Vector a
l a
2 a3 a
4 a 0
-0.66 0.982 0.013 0.0864 - -
-5.17 0.015 -0.02 - -7.8 synunetric 0.36 - -
-0.72 0.59 -
-0.44 0.36
2.58
Table 4.2 C.O.V. of the Undrained Resistance to Liquefaction in Laboratory (a' = 4.8 psi)
co
l1D oN °Tla r r i co
50% 1.6 0.25
60% 1.55 0.24
75% 1.5 0.23
A 0
-
-
-
-
-
0.59
80
Table 4.3 c.o.v. of the Undrained Resistance to Liquefact~on in the Field (cr' = 4.B psi)
co
J.l.D fiu ON !lr Icr , r r i . co
0.0 2.2 0.26
50% 0.15 4.7 0.34
0.20 B.l 0.40
0.0 2.1 0.25
60% 0.15 4.6 0.34
O.lB 6.2 0.37
0.0 2.1 0.25
75% 0.15 4.5 0.34
0.18 6.1 0.37
Table 4.4 C.O.V. of the In-Situ Relative Density
Method Mean Relative Density of
Determination 30% 40% 50% 60%
Direct Method
* 0.18-0.36 0.14-0.27 0.13-0.22 0.12-0.20 (£1 = 0.01)
Ys
Indirect Method 0.27 0.27 0.27 0.27 (all data)
Indirect Method (Gibbs & Holz 0.30 0.23 0.21 0.19 data)
* C.O.V. of the in-situ dry density
70%
0.11-0.18
0.27
0.18
81
Table 4.5 c.o.v. of the Small Strain Shear Modulus G of Sands
m
l1n r'b °G stG r r m m
0.0 0.0 0.12
50% 0.15 0.10 0.16
0.20 0.13 0.18
0.0 0.0 0.12
60% 0.15 0.11 0.16
0.20 0.14 0.19
0.0 0.0 0.12
75% 0.15 0.11 0.16
0.20 0.15 0.19
Table 5.1 Properties of Sand in Example I
P ARA.l1ET ER
cp
K 0
Y
n r
e max
e min
D50
VALUE
36.8
0.85
l20pcf
0.45
0.973
0.636
0.65
Me~z Rererence Room University of Illinois
BI06 NCEL 208 N. Romine Street
Urbana, Illinois 61801
82
Table 5.2 Lumped Hass Model for Example I
0' va
ELEHENT (psf)
1 3,150
2 2,850
3 2,550
4 2,250
5 1,950
6 1,650
7 1,350
8 1,050
9 750
10 300
G ma
(psf)
2,200,000
2,090,000
1,980,000
1,860,000
1,730,000
1,600,000
1,440,000
1,270,000
1,070,000
680,000
't mo
(psf)
1,670
1,510
1,350
1,190
1,030
870
715
560
400
160
K
(~b) 1.n
254
242
229
215
200
184
166
147
124
78
U 2 (lb sec )
in
0.0108
0.0108
0.0108
0.0108
0.0180
0.0108
0.0108
0.0108
0.0108
0.0108
o , S
2.3420
2.4016
2.4695
2.5484 I 2.6402
2.7533
2.8949
3.0813
3.3407
4.2104
Table 5.3 Statistics of TI. for Severnl Load Intensities (seconds)
-- .. -." .. --.~--.------ -_. __ .--a cO.20g a =O.lSg a =0.10g a =.075g
DEPTII max max max max
(ft) J-l T 01' J-l r aT J-l T °T l-IT °T
L L L L L L . L L
47.5 0.51 0.47 0.90 0.6 2.1 1.0 4.0 1.6
42.5 0.48 0.44 0.B5 0.6 2.0 1.0 3.7 1.5
37.5 0.46 0.42 0.B3 0.6 1.9 1.0 3.6 1.5
32.5 0.44 0.40 0.79 0.6 1.9 1.0 3.5 1.5
27.5 0.42 0.38 0.76 0.6 1.8 0.9 3.4 1.4
22.5 0.40 0.35 0.73 0.6 1.7 0.9 3.2 1.4
17.5 0.38 0.33 0.70 0.5 1.7 0.9 3.2 1.4
12.5 0.38 0.31 0.69 0.5 1.6 0.8 3.1 1.3
7.5 0.40 0.31 0.73 0.5 1.7 0.8 3.3 1.3 - ---------~---- ----
a =0.05g max
l-IT L
aT L
10.5 2.6
9.B 2.5
9.4 2.4
9.1 2.4
8.7 2.4
B.4 2.3
8.2 2.3
8.2 2.2
8.6 2.2
C'J {J)
Table 5.4 ~T and crT for a Modulated Load L L
ELEMENT 1-1T
L O'T
L (sec) (sec)
1 3.5 1.8
2 3.4 1.8
3 3.3 1.8
4 3.3 1.8
5 3.3 1.8
6 3.2 1.8
7 3.2 1.7
8 3.1 1.7
9 3.4 1.7
85
Table 5.5a Response Statistics with 5 Elements
a =0.20g max
a =0.40g max
ELEMENT llE llE T llT a T 1-1T a 3 1 3 1 L 1 T L
(psi) (Psi in ) m (sec) (psi) (psi in ) m (sec) - -sec a sec a
T T
1 5.6 2.6 5.0 9.4 13.5 3.0
2 3.7 7.1 2.4 5.2 28.2 1.7
3 1.4 0.06 5.4 15.6 2.0 0.14 3.8 3.2
4 1.0 0.04 5.4 16.8 1.5 0.08 3.7 4.5
5 0.5 0.03 4.3 0.7 0.06 3.1
Table 5.sb Response Statist1cs with 7 Elements
a max=0.20g a =0.40g max
ELEMENT llE 1-1E a T llT a T llT
1 3 l' L l' 3 l' L (psi) (pSi in ) m (sec) (psi) (~si in ) m
(sec) - -sec a sec a l' l'
1 5.2 1.2 6.8 10.0 6.6 3.5
2 4.6 1.0 6.0 7.5 4.4 3.7
3 3.7 4.2 2.4 5.3 20.1 1.7
4 3.0 2.6 2.9 4.3 8.1 1.7
5 1.9 0.22 4.0 2.3 2.6 0.54 2.9 0.58
6 1.4 0.18 3.9 2.2 1.9 0.40 2.8 0.80
. 7 0.8 0.20 2.7 1.1 0.51 2.0
86
Table s.sc Response Statistics with 9 Elements
a =0.20g max a =0.40g max
ELEMENT ~E ~E T ~T T ~T a
3 a 3 T T L T T L (psi) (psi in ) m
(sec) (psi) (psi in ) m (sec) - -sec a sec aT T
1 5.6 0.84 5.8 9.7 4.8 3.4
2 5.0 0.75 5.7 8.3 3.8 3.4
3 4.3 0.63 5.6 6.8 2.7 3.5
4 3.7 3.2 2.4 5.4 15.6 1.6
5 3.2 2.0 2.8 4.7 7.3 1.9
6 2.7 1.3 3.3 3.9 4.0 2.3
7 2.0 0.24 3.8 1.9 2.7 0.65 2.8 0.46
8 1.5 0.20 3.6 1.8 2.0 0.50 2.7 0.63
9 0.9 0.26 2.5 1.2 0.82 1.9
87
Table 5.6 Lumped Mass Model for Example II (Mean Values)
DEPTH ELEMENT (ft)
1 135
2 105
3 75
4 45
5 25
6 15
7 5
wB GAMMA
5. 0.0323
15. 0.3141
25. 0.3895
35. 0.1999
45. 0.0642
height T G K mo mo (ft) (psi) (psi) (~b)
~n
30 32 30,100 83.6
30 25 26,900 74.6
30 8 16,400 45.6
30 8 16,400 45.6
10 7.5 11,700 97.5
10 5.4 9,900 82.5
10 2.2 5,420 45.2
Table 5.7a ~T and aT for Several wB
L
PMF S ~T 11T 0
2 L5 L6 NORMA.L (~) (sec) (sec) 3 sec
0.0664 25.37 2.41 4.00
0.2404 8.457 1.63 2.00
0.3833 5.074 2.46 1.96
0.2465 3.625 5.75 3.50
0.0634 2.819 11.80 6.14
M 2 <5 '. S (lb ~ec )
~n
0.0648 0.808
0.0648 0.864
0.0648 0.772
0.0648 0.772
0.0216 1.803
0.0216 1.954
0.0216 2.812
a a T5 T6
(psi) (psi)
2.2 1.5
2.1 1.5
1.8 1.4
1.5 1.2
1.3 1.1
88
Table S.7b Average ~T and crT with Normal and Gamma PDF's for wB L
E[IlT ] e.O.'\[ [ llT ] E[crt
] e.QV.[cr ] L L T
PHF
TL TL TL T 5 6 5 L6 T5 T6 T5 T6
(sec) (sec) (sec) (sec) (psi) (psi) (psi) (psi)
Gamma 3.45 2.62 0.77 0.43 1.82 1.38 0.14 0.09
Normal 3.66 2.75 0.72 0.42 1.79 1.36 0.15 0.10
wB=16.9 3.20 2.80 1.77 1.34
sB=O.94
Table 5.8 Lifetime Reliability for Layered Deposit
15 ft - Depth 25 ft - Depth
Lifetime (years) 10 25 50 10 25 50
L ~
Reliability Indices 0.45 -Q.04 -0.37 0.40 -0.02 -0.34 I \
Probability 0.68 0.52 0.43 of 0.66 0.50 0.41
No-Liquefaction
Table 5.9 Historical Data on Liquefaction (Partial Data)
Distance TE c.o.v. Depth of D 0' ,.
Field Case a ave Date Site M max Water Table vo N-SPT N1 D --or-History (miles) (sec) TE (g) (ft) (ft) (psff
r Behavior Reference vo
1964 Niigata 7.5 32 15 0.8 0.17 3 20 1200 6 8 53 0.195 Liq. Seed Hnd ldri~~ (lY71)
2 1964 Niigata 7.5 32 15 0.8 0.17 3 25 1500 15 18 64 0.195 Liq. Kishida (1966)
1964 :-'iigata 7.5 32 15 0.8 0.17 3 20 1200 12 16 64 0.195 Nu-Liq. Seed and Idriss (1971) 00 \0
1964 Niigata 7.5 32 15 0.8 0.17 12 25 2000 6 6 53 0.12 No-Liq. Seed and Idriss (1971)
5 1968 lIachinohe 7.8 45-100 15 0.8 0.21 3 12 800 14 21 78 0.23 No-Liq. Ohsaki (1970)
6 1968 Hachinohe 7.8 45-100 15 0.8 0.21 3 12 800 <4 <6 -45 0.23 Liq. Ohsaki (1970)
1968 Hachinohe 7.8 45-100 15 0.8 0.21 5 10 800 15 23 80 0.1£15 No-Liq. Ohsaki (1970)
8 1968 lIakodate 7.8 100 15 0.8 0.21 15 1000 6 9 55 0.205 Liq. Kishida (1970)
98 1968 Hachinohe 7.8 45-100 15 0.8 0.21 47 2900 25 21 75 0.19 No-Liq. Ohsaki (1970)
9T 1968 Hachinohe 7.8 45-100 15 0.8 0.21 9 850 15 22 80 0.16 No-Liq. Ohsaki (1970)
10 1802 Niigata 6.6 24 10 0.8 0.12 3 20 1200 12 16 64 0.135 No-Liq. Seed and Idriss (1967)
11 1887 Niigata 6.1 29 8 0.8 0.08 3 20 1200 12 1fi 64 0.09 No-Liq. Seed and Idriss (1967)
90
FIGURES
"91
z z
u ______ ~~~~------.u
(a) 6+0>0, 0-6<0 (b) 6+0>0, 0-6 = 0
________ ~~~------__ u
"(c) 6+0>0-6>0 (d) 6+0 0, 0-8<0
u
(e) 0>8+0>0-6
Fig. 2.1 Possible Hysteretic Shapes
92
m =l I
kn /
/ Cn
I I
j
~
=i k3 I
/
~ I I ~+ 1 kit1l1.1 +(1..a."1)~ + 1 ~.1
~ m
I ;.1 9i.1 I / k2
/~ I . I
I -- - ----fmjXJ C2 :=j 3 I I
m1 I C j l1 I I
k, 'W a i kj 4+(1-'1)~ zt C1 u2 x.=± u.+~
J. J 8 J: 1
58 U1
Fig. 2. 2 Lumped Mass Model
°E _-.:.T_ (in2)
(1- a) K 1.4
1.2
6E
1.0 T
.8
.6
.4
.2
°E _---.,;T_ x 20 (jrf ) 1.4
(1- a )K 1.2
1.0
.8
.6
.4
.2
93
A=1.Q 6=~=1.0 CI=0.05 n T=0.28sec
~ r=1.0
Filtered White Noise So = 50 in2/sec3
WS= 15.6 rad/sec
S =0.64 B
00 0 Simulation (n::: 60)
--Analytical Results
~/6-Q)K
~--------~-----%
2 4 6 e 10 12 14 16 18 20 TIME, t (seconds)
Fig. 2.3a 0 E and 0E for a SDF T T
So = 2.0 ir:f/ se2 o 0 0 Simulation - Analytical Results
2 4 6 e 10 12 14 16 18 20
TiME, t (seconds)
Fig. 2.3b 0 E and 0E for a SDF T T
t
BE .8
T- .7 I
.6
.5
.4
.3
.2
.1
.0
4.4
a 4.0 E T·
(irf) I 3.6 (1- C4)K j
3.2
2.8
2.4
2.0
1.6
12
.8
.4
.0
94
White f\oise
50 = 50 irf/se2
r, =.63, ~=.23, "I;=.16 (sec)
t 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
TIME, t (seconds)
Fig. 2. 4a 0E for a 3 Degree-of-Freedom Sys.tem T
~_-2
3 t
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 TIME, t (seconds)
Fig. 2.4b 0E for a 3 Degree-of Freedom System T
GE T (jrf ).
(1- a) K
BET
4>1
~ET (in2 )
(1 - Ci) K
1.6
1.4
1.2
1.0
.8
.6
.4
.2 I I
O. /
O.
8
7
6
5
4
3
2
I
95
1, =.45
------, \
\ \
\ , , '~1
" ", O'E
2
10. 20.
TIME, t (seconds)
o~~~~~ __ ~ __ ~~ __ ~~ __ ~ __ ~ __ ___ o 10. 20.
TIME, t (seconds)
Fig. 2.5 Statistics of ET for a Nonstationary Load
Me~z Reierence Room University o~ Illinois
BI06 NeEL 208 N. Romine Street
Urbana, Illinois 61801
til til W a:: Itil
I-'
til I/)
W a:: lI/)
a:: <{ w :::c I/)
96
sand
hyperbolic ---
SHEAR STRAIN. Y
SHEAR STRAIN. Y
Fig. 3.1 Skeleton Curve of the Dynamic Shear Stress-Strain Relation for Soils
97
Fig. 3.2 Cyclic Shearing'Stress-Strain Loops for Soils
t
, f:lW D=-
4'TT .l..Yata 2
G
Fig. 3.3 Equivalent Viscous Damping Ratio D
Y
E
" !-J
V1 V1 w a: l-V1
a: <i w I U1
1 .25
1 .
.75
.5
ii
.25 V O· O· 1 .
98
- - - Hardin and Drnevich (1000. cycles)
--model
2.
- - - --
r=O.5 A=1.0 0=8 a=O.O
I I I
3. 4. 5. SHEAR STRAIN, Y/Yr
6
Fig. 3.4a Model and Empirical Skeleton Curves for Sands
.5
- - - Hardin and Drnevich (1000 cycles)
---model
r =0.50 A =1.0 5=(3 Ct=O.O
10° 10' 102
SHEAR STRAIN, Y/Yr
Fig. 3.4b Model and Empirical GIG with y/y for Sands m r
0: <. w ... til
.... Z <. u w til
1.
.5
O.
, .
.5
O.
99
7 LLt.seed and Idrlss (1970-range ot data)
r=0.5 A =1.0 5=(3 Ct =0.0
--y = 7.5x1cJ'
10-4 10-3
SHEAR STRAIN, Y
Fig. 3.5 Hodel and Empirical GIG with y for Sands m
r =0.5 A :1.0 5=(3 a:O.O
- - - Tatsuo ka et al (1979)
--Y. :7.5x1cr4
r
-..,.(---Y r: 4.0x 1c)'
10-6 10-5 10- 4 10-3 10-2
SHEAR STRAIN, Y
Fig. 3.6 Hodel and Empirical GIG with y for Sands m
100
." Wseed and Idriss (1970 - range of data)
y -~ r= 7.5.x10
* Y,. =4. Ox1 0-" (/)
:::J a 0 u r =0.5 ~ o· .5 A=1.0 > ~
4: 5=(3 t- ee: a=O.O z w \!)
lIZ!: ...J 4: Z > a.. :::J ~ 0 w a
Q 10-6 10-5 10~ 10-3 10-2
SHEAR STRAIN, Y
Fig. 3.7 Model and Empirical D with y for Sands
1. - - - Tatsuoka et a/ (1979)
y. = 7.5x1(r4 ,.
)( \::4.0xKf' (/)
:::J a 0 u 6 (/)
> ~ .5 r =0.5 4: A =1.0
~ ee: z C =(3 -~ t9 o. =0.0 4: Z
> a.. ~ 2: 0 <{
w Cl
O. 10-6 ,o-~ 10-4 10-3 10-2
SHEAR STRAIN, Y Fig. 3.8 Model and Empirical D with y for Sands
en ::> 0 u en > l-z w ...J ~ > :5 0 w
'1.
c
o· i= ~ e::
.5 c.!) Z a::: ~ ~ a
101
- - - Ramberg - OsgOOd (Richart. 1975)
* (1= 0.00 } Model
--(1::0.025
r =0.50 A =1.0 5=a
10° SHEAR STRAIN. Y/Yr ·
10'
Fig. 3.9 Ramberg-Osgood' s (Richart, 1975) and ~ven' s Model D with y for Sands
E j-I
""'-~ vi en w e:: I-en
a: ~ w ~
1 •
·8
.6
.4
.2
O· 0,
A=1.0
5=~ (1=0.0
1.
- - - - - Hardin and Drnevich
~r=0.25
--r=0.20
2. 3. 4.
SHEAR STRAIN, Y/Yr
5.
Fig. 3.10 Model and Empirical Skeleton Curve for Clays
6.
~ (!)
vf :::) ...J :J Q
0 2: 0:: <: ~ (/'J
..... z 4: U ~
IJ')
:J a 0 U . IJ') 0 ;; ..... ..... 4:
0:: z w (!) ...J « z > a::: :; 2: (3 « UJ Q
1. -"*'
.8
.6
.4 A =10 5 =s a =0.0
.2
O. 10~ 10~
102
- - - Seed and Idriss (1970)
- * -Hardin and Drnevich
X r = 0.25
---r=0.20
10-4 10-3
SHEAR STRAIN, Y
Fig. 3.11 Model and Empirical GIG with y for Clays m
!lI.t.Seed and Idriss (1970 .. Range 01 Oat a) 1.
! - *- Tsai, L.m and Martin (1980)
~Yr=4.0X10~
-_.'f',.= 3. Ox 10-3
.5 r =0.25 A :11.0
5 :IS a=O.O
O. 10-6 . 10-5 10- 4 10-3 10-2
SHEAR STRAIN, Y
Fig. 3.12 Hodel and Empirical D with y for Clays
.20
~ ~ I-J.1 5 a:: \I l-t/) (I-J .1 0 0: 0' <i W I- .05 I <i VJ 0:
O. 2 3
103
o = 0.5 ~ r
5 10 20 30 50 100 200 CYCLES TO LIQUEFACTION, N{
Fig. 3.13a Nt vs T (DeAlba et aI, 1975)
VJ VJ w u x w
~1J 1.0 :c
lI
.So
.60
.40
(!) .20
o.
D = 0.54 r
.05 .10 .15 .20 .15 1: SHEAR STRESS RATIO, 1: =-d
vo
Fig. 3.13b Weight Function, h(T)
L:J
6 1.0 I-<i-
Dr = 0.54 Q: .8 w 0: .6 ::> VJ t/)
w .4 a: 0-
w .2 a: 0 a..
10 100
CYCLES OF LOADING, N
Fig. 3.13c Excess Pore Pressure for Uniform Loading
104
~ h:?20
o~
.... <{ a:: til til W a:: .... til
a:: 4:
~
0 .... <t: a:: w a:: => c.n til W a:: ~
w a: 0 ~
r.n tf)
w u x w
D = 0.45 .15 r
.10
.05
O. 10 100
CYCLES TO LIQUEFACTION, N~
-Fig. 3.14a NQ, vs L (Uartin, Finn and Seed, 1975)
1.0 D = 0.45 r
0.8
0.6
Martin et aJ (1975 ) 0.4
- - -This Study
0.2
O. 1 2 3 4 5 6 7
CYCLE 5 OF LOADI NG. N
Fig. 3.14b Excess Pore Pressure with Cycles of Loading (Table 3.2)
r u
'105
.3
.2
. ,
z/z u
1 2 3 4
Y/Y ____ --~~--__ --__ ~~4_~~ __ --~~--__ -r
.6 .5 .4 .3 .2 .3 .4 .5
4 3 2 1 .2
.3
Fig. 3.15 Sand Stiffness Deterioration
1.0
O.B
0.6
04
0.2
O.
, ---rw /
/ rE I
/ /
/' ./
./ ./.
.2 .4 .6 .8 1.0 1.2 1.4 1.6
rE , rW
Fig. 3.16 Damage Parameters r E and rW with ru
to? w a: 0 0 a.
~ c.n a:: c.n w w ~
a::: :::)
w c.n c.n w a::: a.
c...:l w a::: Q-0 ~ a. «
a::: c.n c.n w w a: U :::) X If)
W If)
w a::: a..
106
1.0
0.8
0.6
0.4 Martin et a/ (1975)
0.2 - - - This Study
o. 2 3 4 5 6 7
CYCLES OF LOADING, N
Fig. 3.17 Excess Pore Pressure with Cycles of Loading
1.0 Martin et al (1975)
- -- This Study 0.8
0.6
0.4
0.2
o. 7 2 3 4 5 6
CYCLES OF LOADING, N
Fig. 3.18 Excess Pore Pressure with Cycles of Loading (Table 3.4)
.......... C"lu
Q) tn -...
Ne '-'
..........
3 '-'
(/)
..........
20
10
0 0
'107
ROOT MEAN SQUARE ACC ELERATION: arm s = 0.08 9 W
B: 1-6.9 radlsec, ~= 0.94
o 0
2TT Wo 4TT 6rr Brr
FREQUENCY,
Housner and Jennings
Lai's data average
1 OTT 12TT
w (rad / sec)
Fig. 4.1 PSD Function for "Rock rr Sites
a =0.08 9 rms
14rr
{\ I \ I I I I I We = STT radl sec
~ ~B=O.4 I
I I I 1
I \
WB
=1SlT radlsec
\ ~ ~ =0.4
/ "e Wa= 25rr rad/se~ X \ ,.." S = 0.4 W =35lT rad/sec
./' \ /\................. 8 B /'....... r 8=0.4 ./" ~ - -cc:-.------- ':J
A""'" -,-- ......... --...---- --" ........ _----- ---.:::::- ........ --------- --o--~~~~-----~~--~--~~----~-~~-~~~~--~ o 4lT err 10fT 12TT 14TT 16IT 18TT
FREQUENCY, W (rad/sec)
Fig. 4.2 Kanai-Tajimi PSD Function for Several wE
10
9 -
8
7
6
5
4
3
2
1 I
Lmo
I
"-
....
-....
.., 0 0 -I ltl )(
0 ~
I
108
1
J I
3 10 pst
I-
....
1-0
....
l-
I ,
2 I
I
Gmo 1
I I ~
I I
~ig. 5.1 Homogeneous Sand Deposit
2 I I
I-. 5 10 pSf
I-
....
"-
I.
~
I.
I I
o ~ - > o
en
Z a I-<t: a: w ...J w u u « ~
<t: W a..
.2
.1
o. 1
.2
.1
O. O.
"
109
---- Finn et aI (1978)
- - - This Study
I
1C'P-----Ovo= 782 pst
I ~~ ___ ----Ovo: 1720 pst
<0=2973 pst
10 100
CYCLES TO UQUEFACTION, N£
Fig. 5.2 Cyclic Resistance Curves for Example I
"'-'" ~TL+aTL
" " OT. L
a =0.07 9 max
-.....;: ......... 0max =005 9 ---= ----=~--
1. 10.
TIME TO LIQUEFACTION, TL (seconds)
Fig. 5.3 Statistics of Time until Liquefaction
1000
100.
10
9
8
7
6 (J)
I- 5 z W
4 ~ W ...J 3 w
2
1
0 o.
110
a a 0..,0 -2.1
Co
0.05 0.10 ROOT MEAN SQUARE .ACCELERATION, Uo
T
Fig. 5.4 Profile of Total Accelerations
1
o 2 4 6 8 TIME, t (seconds)
Fig. 5.5 ~1odulating Function of the Base Excitation
0.15 ( 9 )
'-.:J
0" f= <{ Ct:
W Ct: ::J (./) (./)
w Ct: a.
w Ct: 0 a.
U1 U1 w u x lLJ
III
1.
.3
.2
.1
O. O. 1. 2. 3. 4.
TIME, t (seconds)
Fig. 5.6a Expected Excess Pore Pressure Rise
vi (/)
I.LJ Z IJ.. IJ..
~ (/)
200------~~----~--~----~--~~
100
t(sec)
5. o o. 2. 3. 4.
TIME, t (seconds)
Fig. 5.6b Stiffness Deterioration
5.
112
===-______ } Dr: 50 %
D : 0.45 m m ~EDIUM SAND 50
60' MEDIUM CLAY
60' DENSE SAND
Fig. 5.7 Layered Deposit
0.4
L - - - 15 ft
I 0.3 -- 25 tt Ovo
0.2
"&-::::::...~---- 6D = 0.0 r
0.1
0.0 1 10 100 1000
CYCLES TO LlCUEFACTIO N, NJ
Fig. 5.8 Cyclic Resistance Curves for Example II
113
2 :3 10 20 30
5 10' \ I l' . I
4 - 2d Gm tm
3 "'0'
(104
psi) ( psi)
2
90' -1
150' , I 1 1 I
1 2 :3 10 20 30
Hi 't m - 2a 7 r:I 6 (10
4 psi) (psi)
sf/ 5
(j
4 9d
3 0'
2 13d
1 150'
2 3 10 20 30 7 1d \ Gm ~ 6 -=-- 2d 1ln 5 :"d
4 ( 4 . 10 pSI) (psi)
60'
:3 ~' ----
2 120' - -
1 150' f
Fig. 5.9 Lumped Mass Hodels for Example II
Ms"tz Reference Room University o~ Illinois
BI06 NCEL 208 N. Romine Street
Urbana~. Illinois 61801
x w a z
>I-:J co < ..J W 0:
114
i J j J i I
2. .109
1.
Q
-1.
-2.
0.1 1.0 10.0
TIME, t (seconds)
I III JOo0001
ro001
a.u.. 0.01
0.1
0.5
z o toU < lL. W ::> o ::i lL. o >to-::i co < 0.9 co o
0.95 a: a..
Fig. S.lOa Reliability Against Liquefaction (IS-foot)
x'" LIJ a z
2.
1.
0.
-1.
-2.
"*'" - iFCo )
- -~(a)
115
0.001
z o .... u .~ LLJ ::>
" ::J
IJ... o ~ ..... -1
CD <t aJ o
0.95 ~
0.98 ~--~~~~~~--~~~~~~--~~~~~0.99
0.1 1.0 10.0 100.0
TIME, t (seco nds)
Fig. S.10b Reliability Against Liquefaction (25-foot)
116
1.
10-1
(]) u c ro "tJ (]) (]) u X (])
0 10-2
>, ...... 'is ro
..0 0 L
CL
.05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60
Peak base acceleration, c max (g)
Fig. 5.11 Seismic Hazard for Eureka (California)
x w o z
>~
--.J
117
2.
1.
o.
, aJ -t
- - 5 =0.0 Dr
-- 5 =0.15 Dr ~
--1 W 0:::
c::L
X w 0 z
>-~
--1 C!J < --1 w 0:
-2.~~~~~~~~~~~~~--~~~~
.1 1. 10. 100.
TIME, t (seconds)
Fig. 5.12 Sensitivity of Reliability to on r
2.
1.
O.
-, .
-2.
. 1
Fig. 5.13
-60r
=0.15
--60r
=o.o
c = 020 9 max
1. 10. TIME, t (seconds)
Sensitivity of Reliability to ° s
u
100 .
118
3- ~ \.159
\.109 .0001
~ .001
\ \ 2. \ \ t
.159 \ \
0. .S
-t
'" .9 - .-Ex.ct Pro QeMles ,"-
-2. ncertain ~peMies ~ tCMd 99
.1 t 10. 100.
Fig. 5.14 Sensitivity of Reliability to the Uncertainties in Soil Properties
119
0 .. > ~ • L10UEFACTION (GOOD ACCELERATION DATA)
wO.3 o NO-LIQUEFACTION (GOOD ACCELERATION DATA) > <{
j-J 6) NO-L1 QUEFACTION ( ESTIMATED ACCELERATIONS)
·6 5
Q" 8 ~ 0.2 .- cjB 0 7 a: 1 en 0 (J) 4 9,-1JJ 0 0: I- 0.1 11 s (J)
0: <{ W J: U1 0.0
5 10 15 20 25 STANDARD PENETRATION RESISTANCE, N1
I ( CORRECTED TO cr : 1 t on I 5 q ft )
vo
Fig. 5.l5a Historical Data of Liquefaction and No-Liquefaction (Partial Data)
0 - > ~
UJ 0.3 > <!
!--'
0'" 0.2 I-<! c:::
tR w 0: 0.1 l-V')
c::: <! w J: (/') 0.0
• UQUEFACTION (GOOD ACCELERATrON DATA)
o NO LIQUEFACTION (GOOD ACCELERATION DATA)
~ NO LIQUEFACTION (ESTIMATED ACCELERATIONS)
6.
20 40
a 2 1" 30
~ 10 ~11
60 80 GIBBS AND HOLTZ RELATIVE DENSIT~ Dr
100
Fig. 5.lSb Historical Data of Liquefaction and No-Liquefaction (Partial Data)
• LIQUEFACTION (GOOD ACCELERATION DATA)
8 NO-UQUEFACTION (GOOD ACCELERATION DATA)
_ ~ & NO-LIQUEFACTION (ESTIMATED ACCELERATIONS)
b
CASE PROBABILITIES OF LlOUEFAC TlON
...............
w > 0.4 <l:
.....
6 ~ 0.3 0:
(/)
Ul W 0:
:n 0.2 0::: <! w I
Ul 0.1 0 w ~ 0 z
6. 1. ~09B O~ 0 9T
40 / 11 (9
HYSTORY
O.37<Qf<O.40 Dt =O.30
1 0.611 0.64 2 0.34 0.32 3 0.42 0.41
10 0.14 0.09 11 0.0062 0.0023 5 0.33 0.30 6 0.92 0.94 9a 0.26 0.23 9 T 0.11 O.OB
t t =--:1-
avo
5 20 O.OK 25 30 10 15 STANDARD PENETRATION RESISTANCE CORRECTED TO
AN EFECTIVE VERTIC AL PRESSURE OF 1 ton/sq ft. N1
Fig. 5.16 Predicted Probabilities of Liquefaction for Some Historical Data
(1: AVE/~O)BOUNDAFlY
( lAV J~olpo INT
0.5
0.9
0.6 1.3
2.0 1.0 0.3 1. 2 1.5
J-I N 0
121
PEPn; ELEMENT flD n
D N-SPT D50 K tm (f t) r r [(mean) (mm) (Ih/ln) (OSI )
3.3 WT 10 78 0.15 _14 0.50 11 2 1. 9 78 0.15 1 4 0.50 180 2.5 3.7 8 78 0.15 1 5 ().~ 2-1 1 3.4
10. 7 78 0.15 1 6 0.50 238 4.3 1 3.3
G 80 0.15 1 9 103 6. 22.
5 85 133 8.4 31.
4 85 151 10.8 40.
3 85 168 13.2 49.
2 85 1 82 15.7 58.
67. 1 85 196 1 8.
Fig. 5.l7a Sand Deposit for Case History 5 (Hachinohe)
DEPTH ~O Q
O N -SPT 0 50 K ELEMENT lm
( f t ) r r ( mean) (mm) Clb/in) ( psi)
3.3 WT 1 0 45 0.19 3 015 78 1.
6.7 - 9 45 0.19 4 015 1 22 -2.5 B 45 0.19 4 015 144· -i.A
1 O. 7 45 0.19 4 015 163 4.3
1 3.3 G 70 0.15 14 93 6.
22. 5 85 133 8.4
3" 4 85 151 10.8
40. 3 85 168 13.2
49. 2 85 182 15.7
58.
1 85 196 1 8.1 67.
Fig. 5.l7b Sand Deposit for Case History 6 (Hachinohe)
122
DEPTH
ELEMENT ~O QO N-SPT 0
50 K 1:
r r m ( fi) (mean) (mm) (I bll n) (osj)
3.3 10 75 0.15 - - 11 2 1.
6.7 WT 9 75 0.15 - - 195 3.
10. - 8 78 0.15 16 0.05 243 4.5
1 3.3 7 90 0.12 25 0 .. 05 300 5.4
6 90 27 135 7. 22.
5 90 30 157 9. 30.
4 90 40 181 11.5 38.
3 95 60 272 13. 44.
2 75 0.15 25 0.50 242 1 5. SO.
1 85 30 105 1 8.
67.
Fig. S.17e Sand Deposit for Case History 9 (Haehinohe)
DEPTH ~O !"y N-SPT 0 50 K tm ELEMENT Dr (t t) r (mean) (mm) (I blin) (psi)
3. 10 65 6 67 0.7
- 9 60 0.16 7 0.4 68 2.3 11.5
20. B 53 0.18 7 0.4 83 4.
35. 7 60 0.16 12 0.4 65 6.5
50. 6 60 0.16 25 0.4 80 10.
5 80 0.15 30 0.4 60 14.4 80.
4 90 57 22.
123.
3 90 68 31.
165.
2 90 77 40.
208.
1 90 85 50. 250.
Fig. S.18a Sand Deposit for Case History 1 (Niigata)
123
DEPTH ~D nD N -SPT D K lm ELEMENf 50 (f tj r r (mea n) (mm) (Ib/in> ( ps i)
3. WT 1 0 65 0.15 6 67 0.7
11.5 - 9 GO 0.16 9 0.4 71 2.3
20. 8 64 0.16 12 0.4 95 4.
35. 7 64 0.16 15 0.4 68 6.5 6 64 0.15 25 0.4 85 10.
50. 5 80 >30 60 14.4
80.
4 90 57 22. 123.
3 90 68 31. 165.
2 90 77 40.
20B
1 90 85 50.
250.
Fig. S.18b Sand Deposit for Case Histories 3, 10 and 11 (Niigata)
DE PTH ~D nD N -SPT DSn9
K lm ELEMENT (f t ) r r (meCiln) (mm (Ib/rn) (DS i)
12. WT 10 65 0.15 6 0.4 65. 2.8 - 9 60 0.16 7 0.4 11 4. 6.5
20.5
29. 8 53 0.18 7 0.4 11 8. 8.3
44. 7 60 0.16 12 0.4 83. 11.
59. 6 80 25 95. 14.
5 90 > 30 68. 1 9. 89.
4 90 62. 26.
132.
3 90 72. 35.
1 74.
2 90 81. 44.
217.
1 90 89. 53. 259.
Fig. S.18e Sand Deposit for Case History 4 (Niigata)
124
APPENDIX A
EQUIVALENT LINEAR COEFFICIENTS
For real r > 0 the coefficients in 2.13 are:
where
r rO.O
u z 1T
r r-1 F4 = - p. 0.0 lIT uz U z
11 - P uz e = arctan( ) Puz
(A.l)
(A.2)
(A.3a)
(A.3b)
(A.3c)
(A.3d)
(A.3e)
·125
APPENDIX B
ENERGY DISSIPATION STATISTICS
The linearized equations of motion are described by the system of
first order differential equations given by Eq. 2.20.
(B.l)
with the initial conditions {y(t o)} = {c}. Let the matrix [<p(t)J, of
the same order of [G], be the solution of the matrix differential
equation
. [¢(t)] =-[G(t)][¢(t)] (B.2)
with toe initial conditions, [<P(to)J = [I], where [I] 1S the identity
matrix.
Then, the two time covariance matrix of the response is
[S(s,v)] [¢(s)]{E[ (tc}
In particular
with
T lS(t,t)] + [G(t)][S(t,t)] + [S(t,t)][G(t)]
[Set ,t )] o 0
2nS {F}{F}T o
(B.3)
(B.4)
(B.5)
126
Eq. B.3 may be used to calculate the nonzero time lag covariance
matrix of the response. An alternative equation for [s(s,v)J may be
obtained if it is observed that for s>v it is
obtained from Eq. B.3 premultiplying both sides by [~(v)J-land making
s = v. If both sides of Eq. B.6 are premultiplied by [~(s)] the
following equation results:
-1 [¢(s)][¢(v)] [S(v,v)] [¢(s)]{ .•. }[4J(v)]T (B.7)
The right hand side of Eq. B.7 ~s precisely the right hand side of Eq.
B.3, then
[S(s,v)] -1
[9(s)] [¢(v)] [S(v,v)] for s > v (B.B)
which is the same as Eq. 2.37. In many cases, as when the excitation is
modeled with a Kanai-Tajimi PSD function with a value for 'B larger than
0.4, the matrix [~(t)] can not be inverted with enough accuracy and Eqs.
B.3 and B.B may not be used to calculate the non zero time lag
covariance matrix. In this case, for s = v, it is true that
3[S(s.v)] _ 3[¢(s)] - - ._-l~_, ,., d;' - dS x L¢(v)J - lStv,v)J (B.9)
which is the partial derivative of [S(s,v)] with respect to s obtained
from Eq. B.B. If Eq. B.2 is used,
i27
d [S (S , V) t= _ [G (S) ] [ S (S ,V) ] dS
for S > v (B.lO)
The solution of the differential equation, Eq. B.lO, with the initial
conditions [S(s,v)] s = v = [S(v,v)], is the two time covariance matrix
of the response. No numerical ~roblems were found with the solution of
Eq. B.lO even when the base excitation was modeled as Kanai-Tajimi PSD
function with wB = 0.94. Eq. B.lO is the same as Eq. 2.36.
Notes:
(i) It is only necessary to calculate S(s,v) for s > v or s <v,
because of the properties of the autocorrelation function.
(ii) It LS necessary to solve Eq. 2.25, before solving Eqs. 2.36
or 2.37, because the value of [G(s)] LS not known.
(iii) Because the values of [S(s,v)] decay very quickly with the
value Is-rl, it is not necessary to carry out the integration
indicated in Eq. 2.34 over the whole domain. For /s-r/ > n T
the contribution of [S(s,r)] for the integral in Eq. 2.34 LS
negligible. T is the apparant period of vibration and n - 5
to 8.
128
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"Dynamic Modulus of Cohesive Soils," Thesis University of Michigan, Ann Arbor, 1974, in of the Requirements for the Degree of Doctor of
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129
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130
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