IV
IlqA
1504CIVIL
UILU-ENG-83-2005
ENGINEERING STUDIES
. " , STRUCTURAL RESEARCH SERIES NO. 504
ell ••
.e~z Reference Room
University of Illinois
BIOG NeEL
208 N. Romine street
Urbana, Illinois 61801
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
i
50272 -101
REPORT DOCUMENTATION
1.1.. ~:REPORT
PAGE
NO.
UILU-ENG-83-2005
So Recipient'. AccHalon No.
I~
4-Title and Subtitle
5. Report Dete
APRIL 1983
STOCHASTIC ANALYSIS OF SEISMIC SAFETY AGAINST LIQUEFACTION
~-----------------------.--------------------.--------------------~---7. Author(s)
t---------------------------4
8. Performlne Oraanlzatlon Rept. No.
J. A. Pires, Y. K. Wen, A. H-S. Ang
9.
SRS No. 504
Performing Organization Name and Address
10. Project/T.sk/Work Unit No.
Department of Civil Engineering
University of Illinois
208 N. Romine Street
Urbana, IL 61801
11. Contract(C) or Grant(G) No.
(C)
NSF CEE 80-02584
NSF CEE 82-13729
(G)
12. Sponsoring Organization Name and Addr•••
13. Type of Report & Period Covered
National Science Foundation
Washington, D.C.
r---------------------------~
14.
15. Supplemflntary Notes
i-----------------------------· -------.
i-16.
Abstract (Limit: 200 words)
-------.----------.- ... -
. - ..-- ------------------------1
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
Descriptors
8_
Earthquake engineering
Hysteretic system
Liquefaction
Random vibration
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ACKNOWLEDGMENTS
Tois report is based on
submitted
in
partial
the
doctoral
fulfillment of the
thesis
of
require~ents
Dr. J. A. Pires,
for the Ph.D. in
Civil Engineering at the University of Illinois.
The study is part of a broader progran
of
structures
to
seismic
haza'rd,
Foundation under grants CEE 80-02584
03
the safety and reliability
supported
and
this study are gratefully acknowledged.
by the National
GEE 82-13729.
Scien~e
Supports
for
iii
TABLE OF CONTENTS
CHAPTER
1
Page
INTRODUCTION ••••••••••••••••••••••••••••••••••••••••.••••• 1
1.1
1.2
1.3
1.4
1.5
2
3
1
2
5
6
Notation ............................•.................. 7
THE RANDOM VIBRATION MODEL
10
2.1
2.2
2.3
2.4
2.5
10
11
14
Introduction ••••••••••••••••••••••••••••••••••••••••
The Smooth Hysteretic Model •••••••••••••••••••••••••
The Equivalent Linearization ••••••••••••••••••••••••
Energy Dissipation Statistics •••••••••••••••••••••••
DOF Reduction Technique •••••••••••••••••••••••••••••
SOIL CHARACTERIZATION AND LIQUEFACTION MODEL
3.1
3.2
4
Introductory Remarks •••••••••••••••••••••••••••••••••
Review of Procedures for Stochastic Analysis of
Liquefaction •••••••••••••••••••••••••••••••••••••••••
Purpose and Scope ••••••••••••••••••••••••••••••••••••
Organization •••••••••••••••••••••••••••••••••••••••••
4.2
4.3
4.4
23
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 Introduction •••••••••••••••••••••••••••••••••
3.2.2 Uniform Cyclic Loading •••••••••••••••••••••••
3.2.3 Irregular and Random Dynamic Loading •••••••••
3.2.4 Stiffness Deterioration •••••• '••••••••••••••••
3.2.5 Discussion •••••••••••••••••••••••••••••••••••
28
33
36
41
...............................
45
SEISMIC RELIABILITY ANALYSIS
4.1
17
20
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
45
46
49
49
53
55
55
57
iv
Page
5
....................................
........................................
ILLUSTRATIVE EXAMPLES
60
5.1
5.2
60
61
61
61
63
64
64
66
5.3
5.4
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 Introduction
5.3.2 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
.................................
67
.. ............................. . 67
"
5.5
6
SUMMARY AND CONCLUSIONS
6.1
6.2
68
69
..................................
..................................
70
70
72
Summary ..•••.•••...•••••.••••.•.•..••.•.•..••.••.•.• 72
Conclusions •••••••••••••••••••••••••••••• ~ •••••••••• 73
.............................................................
FIGURES ............................................................
TABLES
76
90
APPENDIX
A
EQUIVALENT LINEAR COEFFICIENTS
124
B
ENERGY DISSIPATION STATISTICS
125
REFERENCES
128
v
LIST OF TABLES
Page
Table
3.1
Function h(~) for a Sand with Dr = 0.54 •••.•••••••••••••• 77
3.2
Excess Pore Pressure with Cycles of Loading •••••••••••••• 77
3.3
Function h(i) for a Sand with D = 0.45 •••••••••••••••••• 78
3.4
Excess Pore Pressure with Cycles of Loading •••••••••••••• 78
4.1
Statistics of Parameters Defining Nt
4.2
c.o.v.
4.3
c.o.v.
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
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
r
•.•.•.•••••.•••.•••• 79
of the Undrained Resistance to Liquefaction
in Laboratory (a~oo = 4.8 psi) •••.••••••••••••••..••••.••• 79
of the Undrained Resistance to Liquefaction
f
in the Field (aco
= 4.8 ps')
80
].. · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
m
(seconds) .... '. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . 83
5.4
5.5a
J1. T And aT for a Mod ulaOt ed Load •••••••••••.••••••••••••• 84
L
L
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
5.7b
and a for Several wB ••••••••••••••••••••••••••••••• 87
L
't
Average ~ and G with Normal and Gamma
L
't
PDF's for w B ••••••••••••••••••••.•••••••••••••••.••••••• 88
5.8
Lifetime Reliability for Layered Deposit ••••••••••••••••• 88
5.9
Historical Data on Liquefaction (Partial Data) ••••••••••• 89
vi
LIST OF FIGURES
Page
Figure
2.1
Possible Hysteretic Shapes ••••••••••••••••••••••••••••••• 91
2.2
Lumped Mass Model •••••••••••••••••••••••••••••••••••••••• 92
2.3a
....................................
O'E and BE for a SDF ....................................
T
T
3 Degree-of-Freedom System .....................
for
a
SE
T
O'E for a 3 Degree-of-Freedom System .....................
T
Statistics of ET for a Nonstationary Load ................
2.3b
2.4a
2.4b
2.5
O'E
T
and &E
for a SDF
93
T
93
94
94
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/Y for Sands •••••••••••.• 98
r
3.5
Model and Empirical GIGm with Y for Sands •••••••••••••••• 99
3.6
Model and Empirical GIGm. with Y for Sands •.•••••••••••••• 99
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
Model and Empirical Skeleton Curves for Clays
3.11
Model and Empirical GIGm with Y for Clays ••••••••••••••• 102
3.12
Model and Empirical D with Y for Clays •.•••••••••••••••• 102
3.13a
NQ.
3.13b
Weight Function, h(i) ••••••••••••••••••••••••••••••••••• 103
3.13c
Excess Pore Pressure for Uniform Loading •••••••••••.•••• 103
vs
't
•••••••••••••••••••••••••••••••••••••••••••••••••
101
103
vii
Page
t
..................................................
3.14a
Nt vs
3.14b
Excess Pore Pressure with Cycles of Loading (Table 3.2) •• 104
3.15
Sand Stiffness Deterioration ••••••••••••••••••••••.•••••• 105
3.16
Damage Parameters r and rW with r
E
u
3.17
Excess Pore Pressure with Cycles of
3.18
Excess Pore Pressure with Cycles of Loading (Table 3.4)
4.1
PSD Function for "Rock" Sites
4.2
Kanai-Tajimi PSD
5.1
Homogeneous Sand Deposit
5.2
Cyclic Resistance Curves for Examp Ie I ..•..••...•........ 109
5.3
Statistics of Time until Lique£ act ion •••••••••••••••••••• 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 I I •............•.......... 113
5.10a
Reliability Against Liquefaction (IS-foot)
Funct~on
104
.....................
Loading ..............
106
..
106
............................
107
for Several w
B
105
107
108
115
5.11
...............
Reliability Against Liquefaction (25-foot) ...............
Seismic Hazard For Eureka (California) ...................
5.12
Sensitivity of Reliability to ~D
117
S.10b
5.13
5.14
Sensitivity of Reliability to &s
•••••••••••••••• e , •••••
r
.......................
114
116
117
u
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
The Committee on
Division
of
the
Introductory Remarks
Soil
Dynamics
A.S.C.E.
of
(1978)
the
Geotechnical
defined liquefaction of a saturated
cohesionless soil as the transfo·rmation of the soil from a
to
Engineering
solid
state
a liquid state as a result of the increase in porewater pressure and
attendant decrease in the effective
stress.
caused
shear
by
loadings,
monotonic
and
shock
changes
waves
explosions or blast loads.
~n
such
as
This
phenomenon
stresses,
cyclic
those
caused
by
may
be
vibratory
earthquakes,
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.
dilatancy, dense or medium sands harden
shear
deformations;
the
or
and
loss
of
strength
total stress conditions.
caused
by
the
flow
during
undrained
loss of strength is temporary and the flow of
strains is also limited after some time.
strains
solidify
Because of
In
In loose sands,
the
flow
of
will continue under undrained constant
either
case,
the
surface
deformations
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,
Mihailov,
1980).
Liquefaction
Petrovski
and
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
1964
(Seed,
1968 and 1979a).
the
Alaska
as
of
Seismically induced liquefaction is also
an important factor in the reliability assessment of
such
earthquake
lifeline
systems,
pipelines and highways (Sazaki and Taniguchi, 1981), and is an
important factor in
structures
(Seed,
the
aseismic
1979a).
A
design
survey
of
of
bridges
numerous
and
case histories of
liquefaction failures was done by Gilbert (1976), and a
history
of
earthquake-induced
liquefaction
1n
waterfront
Japan
reV1ew
of
the
was
done
by
Kuribayashi and Tatsuoka (1977).
According to Seed (1979a) there are
evaluating
the
liquefaction
basically
potential
of
two
saturated
approaches
sand
for
deposits:
methods based on observations of the performance of sand deposits during
past
earthquakes;
and
those
response of the deposits, using
based
on
the evaluation of the dynamic
laboratory
data
for
determining
the
conditions conducing to liquefaction.
1.2
Review
Attempts
deposits
decade.
1n
a
of Procedures
to
under
quantify
earthquake
for Stochastic
the
Analysis of
probability
loadings
have
of
Liquefaction
liquefaction
been
made
of
sand
during the past
A summary of these procedures was presented by Christian (1980)
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
methods,
Davis
and
Berrill
(1982).
In
these
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.
and was chosen to
uncertainties
This is a technique of multivariate statistics,
minimize
the
overall
probability
error.
The
caused by the scarcity and unreliability of the data used
in the analysis were not accounted for in the model.
(1978)
of
prepared
maps
of
ground
failure
Youd
potential,
1n
and
Perkins
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
BI06 NCEL
208 N. Romine Street
Urbana~ Illinois 61801
4
determination
of the conditions causing liquefaction of the soils.
The
deterministic simplified approach of Seed and Idriss (1971) was extended
to
include
properties.
assessment
the
uncertainties
~n
the
liquefaction
using
a
approach of Seed and Idriss (1971).
c~rthq~ake ind~c~d she~r
Faccioli (1973),
applied
a
loading
McGuire, Tatsuoka, et. a1 (1978) performed a
of
and in the
earthquake
Donovan
cumulative
damage
method
based
and
probabilistic
on the simplified
Uncertainties in the soil
~ere
stresses
(1971), and
theory
strength
included.
Donovan
to
and
evaluate
the
Singh
(1978),
potential for
liquefaction using laboratory data and dynamic analysis of the
In
both
cases
considered.
us~ng
the
only
uncertainty
in
Miner's
rule
of
the
peak
deposito
the input ground motion was
Donovan (1971) calculated the safety
density function of
parameters
the
soil
factor
for
deposits
fatigue, using an approximate probability
stresses
(Rayleigh
distribution);
the
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,
1982)
1981b,
incorporated the uncertainties in the soil parameters and in the spatial
variation of the deposito
behavior
The
nonlinear-hysteretic
and
deteriorating
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.
previous
Fardis
and
Veneziano,
1981),
all
the
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
In this study
pore
pressure
Purpose and Scope
a procedure
rise
~n
excitations in terms of
is developed that
saturated
a
sand
continuous
represents
the
excess
deposits under random seismic
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
the
amplitude
and number of loading cycles.
both
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
and
the
mean
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
zero
effect
under
repeated
alternate
shearing.
The
deterioration on the dynamic response of the soil
deteriorated
is
of
included
to
stiffness
1n
the
analysis. but the effect of drainage during the pore pressure buildup is
not taken into account.
The random vibration results together
uncertainty
analysis
of
with
the
results
from
the
the soil properties are used to calculate the
reliability of the deposits against liquefaction
under
random
seismic
6
excitations
with
given
durations
and intensities.
reliability indices, and the probabilities of
all
These conditional
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
degree
of
sophistication, are tipically deterministic.
safety are chosen such that, to the designer's opinion,
degree
their
The factors of
an
appropriate
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
actual
risk
~n
implicit
the
~s
design
the
not
design
known.
process,
The
the
proposed
probabilistic approach yields a measure of the likelihood of undesirable
performance such that the risk implicit
and the
relative
compared~
risk
between
various
the design can be calculated,
design
Chapter
2
statistics of the
the
can
be
a
model
resp~nse
Organization
for
calculating
model
~s
the
probabilities
and
of nonlinear-hysteretic and degrading systems
under random seismic loadings is summarized.
of
alternatives
Then, decisions concerning the design selection may be made.
1.4
In
~n
presented
An additional
development
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.
data
from
conventional
tests in the study
of
The technique
cyclic
deposits
allows
simple-shear
under
the
utilization
of
tests or cyclic triaxial
irregular
and
random
dynamic
evaluation
against
loadings.
The methodology
liquefaction
1S
for
the
formulated
se1sm1C
reliability
in Chapter 4.
The
rel~ability
defined and the techniques for their calculations
uncertainties
in
the
soil
properties
as
are
well
indices are
described.
The
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
are
compared
with the observed field behavior during past earthquakes,
namely: three locations in the city
Tokachioki
city of
model
of
earthquake of May 16, 1968;
Niigata
(Japan)
that
showed
Hachinohe ,(Japan)
and~
during
the
at several locations in the
evidence
of
liquefaction
and
no-liquefaction during past earthquakes.
1.5
Notation
Throughout the text, the time derivative of
denoted with a dot over the symbol.
a
a
=
m
= peak
input base acceleration.
ground acceleration.
any
quantity
will
be
8
A, a, ~, 0,
r
= parameters
controlling
the
hysteretic
stress
strain relation.
Am
[ B ]
intensity measure.
= covariance matrix of the random seismic loading.
c
= coefficient
of viscous damping.
D
= equivalent
viscous damping ratio.
D
r
e
E
= seismic
c
=
relative density.
= void
ratio.
= hystereti~
=
energy dissipated per cycle.
total hysteretic energy dissipated.
small strain shear modulus.
K
m
stiffness of each element of the system.
= mass of each element of the lumped mass system.
number of uniform cycles of loading.
N
N~
= number
of uniform cycles of loading
capable
of
causing liquefaction.
total restoring force
q
r
u
=
~n
each element.
the excess pore pressure.
damage parameter ratio.
so
= 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
liquefaction occurs.
ground
motion
until
9
u
a::
relative displacement" of two consecutive
masses
or,
the
deformation
of
deformation
lumped
of the hysteretic
spring.
u
s
z:
the
skeleton
curve
of
the
of the displacement.
The
hysteretic spring.
z
= hysteretic
component
hysteretic restoring force is Kz.
= reliabil1ty
index
against
seismically induced
liqu,efaction.
= shear
==
strain.
parameters of the Kanai
Tajimi
power
density function.
,
Uco
==
effective confining stress.
= vertical
't
= shear
effective stress.
stress.
= normalized
shear stress.
= standard deviation of variable X.
= coefficient
of variation of variable X.
spectral
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
well in the nonlinear inelastic range.
material
is
hysteretic
and
the
In addition, the behavior of the
stiffness
or strength are likely to
deteriorate with the number of oscillations.
of
the
seismic
response
model capable of including
of
~s
Thus, an accurate analysis
such soil deposits requires a structural
the
nonlinear,
inelastic,
hysteretic
and
deteriorating behavior.
A number of models capable of reproducing this
proposed:
namely,
Hardin
and
Drnevich
behavior
have
been
(1972a, 1972b); Finn, Lee and
Martin (1977); Martin (1975); Martin and Seed (1978, 1979); Pyke (1979);
Bazant
and
Krizek
(1976);
(1971); Katsikas and
(1977).
Faccioli
horizontally
stress-strain
Wylie
and
layered
Richart
(1982);
(1975);
and
Liou,
Newmark
Streeter
and
soil
behavior
deposits
using
a
with
random
a
Ramberg-Osgood
technique.
Gazetas,
shearing
vibration model, in which the
by
a
harmonic
Debchaudhury and Gasparini (1981),
obtained the statistics of the response of earth dams excited by
motions
Richart
Ramirez (1976) analysed the seismic response of
parameters of an equivalent linear system were obtained
linearization
and Rosenblueth
strong
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
performed,
which can be very costly.
simulation,
must
be
Recently, Wen (1976, 1980), Baber
and Wen (1979, 1981) proposed a .hysteretic restoring force model, and an
analytical
method
for
the
solution
of
the
probabilities without statistical simulation.
will
briefly
describe
this
model
including
required statistics and
The
following
sections
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)
where (l-a)Kz
1S
aKu + (1 - a) Az
the hysteretic restoring force represented by:
.
(2.3)
z
in which
u = the relative displacement of the mass
z = the hysteretic component of the displacement.
m
(2.2)
= the
mass
12
a = the
base acceleration
c
=
K
= initial
coefficient of viscous damping
stiffness, and
= parameters
a,A,/3,o,r
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
the
inclusion
of
the
z
shapes
A,
/3
can
~s
the solution of the nonlinear
A
large
term, which
differential equation, i.e. Eq.
2.3.
of
number
of
hysteresis
be described by varying the parameters A, . /3, 0 and r, where
and 0 must satisfy certain criteria to assure that the total energy
dissipated
through
a
cycle
positive.
~s
combinations are shown in Fig. 2.1.
appropriate
combinations
of a, A,
If
Some
different
/3, 0
and r,
of
the
springs,
are
and/or in parallel, additional hysteresis shapes such
possible
each
with
combined in series
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:
_ _.....;;.d..::....~_ _
A - (0 + S)sr
+
fz
(2.4)
0
The incremental work done by hysteretic action is,
(1 - a.) Kzdu
and, the energy dissipated per cycle of amplitude z, Ec(z), is
(2.5)
g~ven
by
13
(Baber and and Wen ,1979),
E (z) = 2(1 - a)
c
K[foZ
A _
?:;d?:;
(8 + o)~r
- (
The ultimate hysteretic restoring force
0
?:;d?:;
A -
(0 - 8)sr
(l-~Kzu'
J
(2.6)
defined
as
the
limiting value of (l-a}Kz as u approaches infinity is:
(1 - a) Kz
(2.7)
u
The total energy dissipated by hysteretic action, E , is
T
ET
=
f
t
(1 -
a)
(2.8)
K(zu)dt
o
Deterioration of the stiffness and/or strength of the
be
included
material
can
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(E T )
and TJ =1](E T)
deterioration, respectively.
function
of
the
total
account
for
the
stiffness
and strength
The parameter A may also be defined
energy
dissipated by hysteresis, ETe
form, monotonic reduction in A(E T) will represent degradations
the stiffness and strength.
as
a
In this
in
both
14
2.3
The Equivalent Linearization
The response statistics, which reflect
loading,
have
been
the
random
nature
of
the
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
~n
motion
close
form,
without
resorting
to
the
Krylov-Bogoliubov (KB) approximation.
Consider the mu1tidegree
represents
a
lumped
of
freedom
system
in
Fig.
2.2,
which
mass model of a horizontally layered soil deposit
under vertically propagating shear waves.
The equations of
motion
may
be writ ten as:
ul
qi-1
(1 - c5';1) - •
mi - l
-
mi + l
x -m.
= - (-
2~
~
qi
+-
mi
[1
m.
+ (1 - c5. ) -~-] - (I - c5. )
~l
~n
mi - l
2
w U - wBu B) c5 1..1
BBB
qi+l
mi + l
i=l,n
(2.10)
i=l,n
(2.11)
where:
C.u. + a.K. + (1 - a.)K.z.
~
~
~
~
~
~
i
~
l,n
(2.12)
and,
.
A.u. -
z.
~
~
(8.u.lz.1
~
~
~
r.-l
~
z.
~
+
c.u.lz.1
~
~
n.
~
~
r.
~)v.
~
i=l,n
(2.13)
~
in which:
U
B
= 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
stationary
envelope
excitation
that
if
modulates
the
nonstationarity
1S
desired
5.11 ' 5.1n = Kronecker deltas.
The base excitation may be a white noise, tB (Caughey, 1960),
filtered
white
n01se
(Amin
and
or
a
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.
1
].
(z. U. )
(2.14)
]. 1
].
Only Eq. 2.13 needs
to
be
linearized.
The
linearized
form
of
Eq. 2.13 was obtained by Baber and Wen (1979) as follows:
.
(2.15)
ce].].
.u. + KeJ.]'
.z.
z.].
The equations for C . and K . are given in Appendix A.
e1
e1
The linearized
motion,
Eqs.
set
of
the
governing
differential
equations
of
2.10, 2.11, and 2.15, may be represented by the following
system of first-order differential equations:
{Y} + [G]{y}
(2.16)
16
in which:
(2.17)
[0 1 0 0 • . . 0]
(2.18)
__ ..:I
dUU
{y. }
T
~
[u.U.z.]
~
~
(2.19)
1
with,
(2.20)
The
matrix
[G]
~s
the matrix of the system coefficients, including the
Ce1. and Ke1'.
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
=
(2.21)
[B]
in which,
[Set)]
= E[{y(t)}{y(t)}T]
(2.22)
and,
(2.23)
[B]
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
The
domain.
[8]
=
stationary
solution
for
time
nondeteriorating systems, i.e.
0, may be obtained iteratively using
the
algorithm
reported
by
Bartels and Stewart (1972).
2.4
The expected rate
spring, ~E
T.
Energy Dissipation Statistics
of
hysteretic
energy
dissipated
by
the
i-th
(t) is given by:
1
ll·
E.r. (t)
=
(2.24)
(1 - et.)K.E[u.z.]
1.
1
1
1
1
(in
the
following the index i will be suppressed for simplicity).
value of E[uz] is an element of the
[Set)],
defined
in
Eq.2.22.
zero
time
lag
covariance
The
matrix
The mean square value of ET(t) is given
by:
(2.25)
Assuming that z and
u are
jointly Gaussian, the expected value of
may be calculated by:
=
2 2
222
a.
uz z u
(1 - a) K (1 + 2p. )0
The coefficient of variation of ET(t) is:
(2.26)
18
(2.27)
The
mean
total hysteretic energy dissipated,
~E
(t), from time to to t
T
is
~E (t)
T
t
= (1 - a)Kf E[z(T)U(T)]dT
t
(2.28)
o
whereas the corresponding mean square of the total energy dissipated is:
2
E[~(T)] = (1 -
a)
f
El!o!o
2 2
K
t
t
I
.
[z(s)u(s)z(v)u(v)]dsdv
(2.29)
Evaluation of the right-hand side of Eq. 2.29 requires the knowledge
the two time joint probability distribution of
calculation of a
six
fold
integral.
u and
However,
of
z, and involves the
if
jointly
Gaussian
behavior is assumed for the two time joint probability distribution of
and z, calculations may be performed without
stationary and nonstationary cases.
t
(1 - a)2K2j
f
t t
o
+
difficulty
for
both
u
the
With this latter assumption,
t
{E[z(s)u(s)]E[z(v)u(v)] +
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.
to,
For the stationary case, the above expression is
simplified
19
E[~(t)]
(1 - a)2 K2\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
o
(2.31)
(t)
T
The two-time covariance matrix is obtained from:
d
for s>v
ds[S(s,v)] =-[G(s)][S(s,v)]
(2.32)
= [S(v,v)].
s:v
Alternatively, the matrix [S(s,v)] may be determined from,
with the initial conditions -[S(s,v)]
[S(s,v)]
=
where the matrix
[¢(s)][¢(v)]
[~(t)]
is
-1
the
for s>v
x [S(v,v)]
solution
of
the
matrix
(2.33)
differential
equation,
[ ¢ (t) ] =
with
-[ G (t)
] [¢ ( t ) ]
the initial conditions [~(to)]
matrix.
The
derivation
of
Eqs.
=
(2.34)
[I], in which [I] is the identity
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
presented was compared with the results of simulations.
investigation are summarized in Figs. 2.3a and 2.3b.
variation
DE
decreases
previously
Results of this
The coefficient of
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 5 E for each element of a three-degree of freedom system are
T
1n Figs.
2.4a and 2.4b. The behavior is similar to that of the
T
shown
single degree of freedom system.
coefficients
It is interesting to observe that
the
variation,
of
almost equal.
BE ' for all elements of the system are
T
This is because the only source of uncertainty is in the
loading" which
1.S
through
2.36e.
the same for all elements of the system, see Eqs. 2.35
Similar
behavior
-of
observed
for
equation,
Eq.
nonstationary loadings (see Fig. 2.5).
2.5
DOF Reduction Technique
The numerical integration of the matrix differential
2.21,
in
toe
nonstationary
case,
or - the
iterative solution of the
remaining Liapunov matrix equation for the stationary case,
easy
task
for
a
system
unknowns in the zero time
(3n+2)(3n+3)/2
for
a
with many degrees of freedom.
lag
system
covariance
matrix,
[8],
is
not
an
The number of
is
equal
to
with n degrees of freedom subjected to a
filtered Gaussian shot noise excitation.
The number of unknowns in the
recognized
that
the
motion
of
problem
the
may
be
system may
reduced
if
it
be described with a
combination of a few modes of vibration, usually the first or the
and
second
modes.
Makdisi
first
It has been shown that the response of embankments
under earthquake loadings
vibration,
is
1S
primarily
and Seed (1979).
layered soil deposits that
do
not
in
the
first
few
modes
of
The same is true for horizontally
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.
of
saturated
loose
For deteriorating systems, such
for
the
deposits
sands under random seismic excitations, a frequent
updating of the modal shape of vibration is necessary.
motion
as
The equations of
multidegree of freedom inelastic, nonlinear-hysteretic
system are written as:
[M]{v} +
{Q({v},{v},t)}
(2.35)
P (t)
where:
i
I
v.
l
Q.l
j=l
= q.l
(2.36a)
i=l,n
ll.
1.
(2.36b)
i=l,n
- q.l - 1
(2.36c)
{p (t) }
M ..
lJ
m.o
..
l lJ
j=l,n
i=l,n
1]
[1 1
(2.36d)
(2.36e)
with all the other quantities as previously defined.
If only one mode is used, the displacement can be expressed as
(2.37)
{v}
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}
•
(2.38)
(-2s W u B B B
=
{W1}T[M]{Wl}
and
c'.
(v.~
e~
- v.~-l ) + K'e~.z.~
i=l,n
The unknowns are reduced to u '
.
B' u B ' xl' xl'
number
of
(n+4)(n+5)/2.
to
unknowns
ln
the
zero
time
lag
Z i'
= 1 ,n •
covariance
If two modes are used, the number of
(n+6)(n+7)/2.
.
1
(2.39)
unknowns
The
total
matrix
is
increases
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
variance of the strain energy dissipated by hysteresis.
calculate
the
23
CHAPTER 3
SOIL CHARACTERIZATION AND LIQUEFACTION MODEL
3.1
3.1.1
Dynamic Shearing Stress-Strain Relation for Soils
Introduction
The factors that control the basic shearing stress-strain curve
shown
ln
Fig.
are
At zero shearing strain the tangent to the curve
3.1.
establishes the maximum value of the
shear
modulus,
Gm •
The
secant
shear modulus corresponding ·to any intermediate point on the curve, such
as point P in Fig. 3.1, is denoted by
The
G.
shearing stress obtained in a static test is T •
m
maximum
skeleton
curve
the
ln
Fig.
3.2.
shown by the dashed line in Fig. 3.2 describes the
variation of the secant
respective
of
Cyclic torsional tests
of soils produce hysteresis loops similar to those shown
The
value
shearing
shear
modulus,
y.
strain,
G,
of
the
loops,
with
the
The loop shape and width describe the
increase in damping with the shearing strain, y.
It is often convenient to approximate the strain softening
of
soils,
as
shown
in
Figs. 3.1 and 3.2, by analytical expressions.
Hardin and Drnevich (1972b), have adopted modified hyperbolic
as those shown in Fig. 3.1.
other
expressions
have
been
(1977) proposed a model based on
behavior.
the
smooth
characterize
In
the
relations
Richart (1975), Streeter, Wylie and Richart
(1974), have used the equations proposed by Ramberg and
and
behavior
the
Osgood
(1943),
suggested by Martin (1976).
critical
state
theory
Pender
of
soil
present study the differential equations describing
hysteretic
model
presented
in
Chapter
2
are
used
to
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
Gaddah
(1977),
determine the
ratio,
D,
Stokoe
ve~ieticn
and
(1978),
and
__ ..1
of the ratio
Q.L1.U
others,
were able to
-_ .. .!_--,-_ . . --:---.. -
C~ULVQ.LCU~
VLC~VUC
defined in Fig. 3.3, with the shearing strain, y, for a wide
number of sands and clays.
the
Lodde
and
In the next two sections the
parameters
of
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,
1
G
(3.1)
where,
(3.2)
in
which
empirical
parameters.
a
constants
Values
reference
that
of
strain defined as TmIGm, and a and bare
represent
a
and
the
influence
of
various
test
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/Y
sands
subjected
r
for
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
s = ~,
is shown by the solid line.
shape
of
loop
the
and
the
The values of
S
= 0.50, A = 1.0, and
and ~
strain
reference
between
the
model
the
These curves are
replotted in Fig. 3.4b to compare the variation of the
Y/Yr
control
ratio
GIG with
m
and the empirical data of Hardin and Drnevich
(1972).
The variation of GIGm with the shearing strain
is
model
Yr
shown
= 4x10- 4
1n
Fig.
= 7.5x10-4 •
and Y
r
'3.5
for
r
for the
= 0.50,
hysteretic
A = 1.00,
The range of experimental data
= ~,
8
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.
that
the
variation
On these bases, it
may
be
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,
shearing
3.7,
Y.r
=
the
7.5x10
-4
model
•
with
r=0.50,
per
cycle;
this
The model appears to
reported
by
dissipate
more
could be because of the shape of the loop, or
r
Tatsuoka
et
a1.
The
(1979)
comparison
for
drained
saturated and dry sands, at confining pressures of 2,020 psf,
in Fig. 3.8.
the
Yr = 4x10 -4 and
A=1.00,
because of different reference strains, Y •
results
with
The experimental data in Seed and Idriss (1970) is also
shown in Fig. 3.7 for comparison.
energy
~.
amplitude, y, as defined in Fig. 3.3, is shown in Fig.
strain
for
concluded
with
the
tests of
is
shown
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)
shown
with the dashed line in Fig. 3.9.
The corresponding variation of
D with Y/Yr for the hysteretic model using r
shown
solid lines in Fig. 3.9.
with
is
= 0.50,
A
= 1.00, S = fi,
is
If a linear spring with stiffness
m, where a = 2.5 %,. is added in parallel with a hysteretic spring, the
aG
values
of
Fig. 3.9.
A
= a,
D decrease
noticeably for high values of y/y , as shown in
r
A similar effect may be obtained if a nonlinear
and
~
= 0,
1S
spring
with
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
result
in
y>O.l %.
better
a
a,~,
Other values of r,
characterization
stress-strain relation for a particular sand.
of
the
5 and
A,
dynamic
A system
may
shearing
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
cycles,
for
the
for
two
respectively.
and Y/Y
r'
at
1,000
Hardin and Drnevich equations, is shown by the dashed
line in Fig. 3.10.
lines
m
~
The same relationship is
springs
with A
= 1.0,
show~
o={3, and r
in
= 0.25
3.11 by solid
and r
The decrease of the ratio G/Gm with the shearing
= 0.20,
strain,
27
is
)',
~n
shown
and r
= 0.20.
and
the
Fig. 3.11 for A = 1.0, Y = 4x10- 4 , S ={3, and r = 0.25
r
(1970),
Experimental results reported by Seed and Idriss
curve
of
Hardin
and
Drnevich
1,000
for
cycles
and
~'r = 4xlO- 4 are also shown in Fig. 3.11.
In Fig. 3.12 the variation of
A
= 1.0,
S={3,
and
D with
= 4x10 -4
)' r
~s
)'
and
Yr
=
shown
3x10
-3
•
r
The
range
experimental data given in Seed and Idriss (1970), and the
results
reported
= 0.25,
for
of
experimental
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
3.2.1
Pore Water Pressure and Stiffness Deterioration
Introduction
For the purpose of
porewater pressure rise
with the random
mechanisms
this
~n
study,
a
technique
for
predicting
the
the sand caused by ground shaking, compatible
vibration
analysis,
~s
necessary.
The
fundamental
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
deposits
systems.
as
a
problem
of
random
vibration
sand
of nonlinear-hysteretic
28
3.2.2
Uniform Cyclic Loading
Based on energy
developed
soils.
an
considerations,
approach
for
The amount of energy
Nemat-Nasser
modeling
r~quired
and
Shokooh
(1979)
the liquefaction of cohesionless
to change the void ratio from e to
(e+de) is defined as:
de
- v f(l + r )g(e - e )
dW
u
(3.3)
m
where
dW
= work
=
jj
performed in rearranging the particles
parameter
that
depend
may
on
the
effective
confining
pressure, (J"c'o) but not on the void ratio, e.
em
minimum void ratio for the sand.
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:
f(l)
1
~> 0
and
dr u
The dimensions of dW and v
g (0)
0
are the same, and all
in Eq. 3.4 are nondimensional.
1lw
the
bulk
modulus
effective confining pressure.
other
(3.4)
quantities
For the saturated undrained case,
eO'
de = - __
c dr
where ~W ~s
~ > 0
de -
(3.5)
u
of
the
water, and a~o ~s the initial
Then Eq. 3.3 becomes:
29
(3.6)
The onset of liquefaction occurs
when
ru
= 1.
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
A
uW
= \)
* _____e
0____
fr
U
gee. 0 -em) 0
dr'U
(3.7)
f(l+r ')
u
or by the differential equation
dr
U
d/:).W
gee - e )
__~o_*___m=- x f(l + r )
\) e
.
(3.8)
u
o
- , ITJ •
the initial void ratio, and v * = vcr
co W
The value of ~W is related to the shear strain energy that the
h
were
eo
1.S
dissipates
by
hysteresis,
neglecting
the
work
done
volumetric
by
changes, which could be several orders of magnitude smaller.
Using
data reported by De Alba et al (1975) it was shown that the value of
corresponding to the onset of liquefaction, i.e. r
u
=
1
'
is
sand
a
the
~W
constant
for a given initial state of the sand.
Let the energy dissipated by hysteresis in one
T
=
rIa'co be
denoted
as
Ec(T).
The
value
of
cycle
~W
amplitude
after N cycles of
constant amplitude
may be considered proportional
cycles
if the amplitude of shearing is large (Nemat-Nasser
of
loading
and Shokooh, 1979); and thus,
to
of
the
number
of
30
= h(=t)NEc (=()
~w
(3.9)
where he:;:) ~s a function of the shearing stress amplitude.
Equation 3.9
then becomes
\) * e
=
gee
f
0
o
- e )
m
r
d'r
U
(3.10)
u
f(l + r')
u
0
using the following simple forms-for f(l+r ) and gee - e )
u
g (e
o
- e )
m
=
(e
0
- e m)
n
£(1 + r )
,n > 1
m'
0
(1
+
U
r
)5
,
5
> 0 (3.11)
U
Eq. 3.10 yields,
\) *e
=
gee
0
o
- e ) x (s
1
1)
m
For initial liquefaction, r
u
[1
\) * e
(5 - 1) (e
of
(1
-
r ) 1-5 ]
U
_
(3.12)
= 1,
_____
0 ___
where Nt is the number
-
cycles
o
-
of
(1 _ 21-
5 )
(3.13)
e )n
m
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
31
r )l-s
1 + (1 +
where rN
=
u
(3.14)
N/N£.
Seed, Martin and Lysmer (1976) have suggested that
rN and
ru might
be related by
rN =
in
which
value of
8
nru
(3.15)
<-2-)
an empirical parameter that varies from 0.50 to 0.9; a
1S
e = 0.70
Lysmer, 1976).
. 26
Sl.n
valid for a large range of data (Seed,
1S
Martin
and
Equation 3.15 would be obtained if f(l + ru) in Eq. 3.10
is replaced by
1
f(l + r')
u
en sin
'
29-16 1Tru
nru,
(3.16)
(2)cos (-2-)
If the functions in Eqs. 3.11 and 3.16 are used for fC1+r ) in Eq.
u
3.8,
the following respective incremental relations are obtained:
dr
u
-- =
dr
W
(1 + r )s
u
s - 1
(1 _ 21 - s )
(3.17)
and,
dr
u
-- =
dr
W
1
28_18nr u
nru
TIe sin
(-2-) cos ( 2 )
(3.18)
32
where,
(3.19)
1)
N~
Consider the T vs.
resistance
~s
curve
= 8.0
psi, and was reported
= 45 % at a
by
De
Alba
all pairs of
T
u
= 1) = Nnh(T)E c (7)
N
al
is a constant
and N)1. on the cyclic resistance curve in Fig. 3.13a,
are enough to determine h(T) and the variation of r u with r W·
of
et
The knowledge of the cyclic shearing stress-strain relation for
this sand and the assumption that ~W(r
for
This cyclic
for a sand with a relative density D
r
confining pressure u~o
(1975).
relationship shown in Fig. 3.13a.
The value
Ec(r) is calculated for the initial state of the sand and the cyclic
shearing stress-strain relation is represented with the hysteretic model
~n
proposed
Chapter 2 with r
= 0.50"
A
= 1.0,
the values of N)1. and their corresponding stress
and 6 =~.
ratios
are
In Table 3.1
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
~n
shown
column
4
of
=3
are
Table 3.1, and the values of h(T) are shown in
column 5, where,
3E
C
(:r 3 )
N)1.Ec CENt)
and
The
T
is the stress ratio corresponding to liquefaction in
function
h(T)
is
shown
in
Fig.
(3.20)
Nt cycles ..
3.13b and, using Eq. 3.18, the
33
variation of the porewater pressure with the number of cycles
is
shown
present
value
in Fig. 3.13c.
Irregular and Random Dynamic Loading
3.2.3
Assuming that future values of ru depend only on its
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
given
in
Fig.
'3.13a,
and
as
those
with the shearing stress-strain
relation, calculate the function h(T) and the value of ~W for
ru
(ii)
=
1 as Nth(T)Ec(T).
For the i-th cycle of loading, calculate the value of
the ratio rW
= ~W/~W(l),
~W,
and
using Ec (T) and h(T), where
i
6W.
~
and T.
J
(iii)
~s
I E (T.)h(T.)
. 1 c J
J
(3.21)
J=
the amplitude of the j-th cycle of loading.
The value of rW together with
Eq.
are
3.18
sufficient
to
calculate the excess pore pressure ratio r u·
It is tacitly assumed that the sand does not deteriorate, therefore,
E (To)
c
J
18
always calculated for the initial state of the soil.
The case
of a deteriorating material is considered subsequently.
The cyclic resistance curve of
Dr = 45 %,
at
a
confining
a
pressure
sand
with
by
Martin,
relative
density,
of u~o= 4000 psf, as reported by
Martin, et a1. (1975), is shown in Fig. 3.14a.
described
a
Using
the
methodology
Finn and Seed (1975) the porewater pressure rise
for the nonuniform cyclic loading of Table 3.2 was calculated
for
this
34
sand;
the
3.14b.
results are shown in Table 3.2 and by the solid line in Fig.
The porewater pressure, r u ' was also
with 8=1.30;
Fig. 3.14h.
0.50,
using
Eq.
3.18
the corresponding results are shown by the dashed line in
The stress-strain curve for the sand was modeled with
8 = f3.
= 1.0,
A
computed
It should be mentioned that the stress-strain
curve of the sand in question is not accurately modeled
for
this type of
ex~mple~
~n
the
with
and all the others in
the results shown by a dashed line in Fig. 3.14b are
differences
r
not
r
t:hiA
-
-- -
-
= 0.50.
r.h~nt.:),._
- -- -.- JI
sensitive
to
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
the
same
quite
function in Table 3.1 decreases rapidly as
is, of course, the result of different slopes in the
curves.
curve
~n
Fig.
3.13a.
(1981a,1981b),
have
shown
resistance
curves
Fardis
that
obtained
in
the
the
slope in the curve of Fig. 3.13a.
amplitude
of
the
loading
shearing strain energy that
volume
whereas
decreases.
cyclic
This
resistance
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
the
uniform,
changes
of
the
It
cycles,
~s
sand
(1979)
and
average
3.1
corresponds
Fardis and Veneziano
slope
of
the
cyclic
laboratory is almost equal to the
appears
the
that
the
resulting
greater
the
greater the percentage of the
capable of contributing to
approach suggested by Martin, Finn and
to
from
Seed
slip
(1975)
the
apparent
deformation.
implies
that
The
the
35
apparent reduction in the volume of the sand due to slip deformation for
ru
=
1 ,
E
approach
vd ' is a constant for a given initial state of the sand.
also
shows
that
if
the
This
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
~.e.
r
~
=
I.Oc
This
ratio,
threshold value for the sand of Table 3.3 with a
confining pressure of 4,000 psf would be approximately 130 psf.
me~hodo1ogy
The extension of the
described above to random
loadings
may be described as follows:
(i)
From the cyclic resistance curves for sand such as
Figs.
3.13a
and
3.14a
relation of the soil
(ii)
U
=
under
cyclic
loading,
At time t, the value of
~W(t)
t
ft
~W(t)
where
t
0
designates
u
= 1
'
calculate
the
as
= Nnh(T)E
(T)
N
c
1)
~n
and from the shearing stress-strain
function h(T) and the value of ~W for r
~W(r
those
(3.22)
is calculated as,
.
(3.23)
X(s)E(s)ds
0
the
starting
time
of the excitation,
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;
T
f
X(t)
o
max h(T')E (T')p-(T' ,0-,0.,t)dT'
c
T
T T
(3.24)
E (T')p-(T' 0c
o~
T ' T' T'
t)dT'
Me~z Reference Room
University of IllinOis
Bl06 NCEL
208 N. Romine Street
Urbana, IllinOis 61801
36
where PTC • ) is the
peaks
of
the
probability
normalized
density
function
of
the
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:
dr w
1
dt = b.W(r u
=
1)
X(t)E(t)
(3.25)
and,
dr u
1
--=-x
dt
1T0'.
.
X(t)E(t)
x b.W(r = 1)
1
TIr u
cos (-2-) sin
20'.- 1
TIr u
(3.26)
u
(2)
Steps (ii) and (iii) continue until liquefaction occurs or until the
excitation
stops.
deterioration
pressure
of
The
the
increase,
above
material
procedure
properties
does
caused
and
[T(t)
by
include
the
the
porewater
and is appropriate for a total stress analysis.
the case of a stationary excitation, implementation
X(t)
not
are
constants
for
is
any given loading.
simple
In
because
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
vibratory
at low
pressure
development
in
saturated
sands
under
loading leads to a decrease in the value of the shear modulus
strains,
Gm'
as
defined
earlier.
When
the
pore
pressure
37
approaches the initial effective confining stress, the stress path is in
the failure
envelope
stress-strain
for
behavior
a
large
becomes
portion
very complex.
tilting of the hysteresis loops caused by
will
be
considered,
sand
at
high
pore
the
of
the
cycle,
and
the
In this study, only the
stiffness
deterioration
because the general stress-strain relation of the
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
this basis (Martin and Seed, 1979).
be
obtained
on
The value of Gm at time t after the
start of the excitation is related to G and to the square root of
mo
the
effective confining stress, as follows:
(3 .27)
Therefore, the shear modulus at low strains, Gm(t), is given by;
= 11 -
G (t)
m
Let
~ 1n
Eq. 2.9 be defined as:
n
1
11 -
Then, the energy dissipated by a cycle
amplitude
-
T,
(3.28)
r (t) G
u
mo
when
(3.29)
r
u
of
normalized
shearing
the porewater pressure has risen to r '
u
given by (Baber and Wen, 1979),
stress
38
E (T,n) = nE (T)
C
(3.30)
C
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
shearing
stress
submitted
1S
amplitude,
to
the
cycles
N
of
loading
of
constant
total energy dissipated by hysteresis
would be,
(3.31)
In order to apply the
problems
with
same
procedure
deterioration,
described
in
Sect.
3.2.1
for
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
u
= 1iT
arcsin(rNI/Z6)
(3.32)
1:
and
~
dr
The equation for
~,
= eITsin
N
~o-
1
'ITr
U
'ITr
u
(3.33)
(-Z-)cos(--2-)dr u
obtained from substituting Eq. 3.32 into
Eq.
3.29,
1S
1
2
.
1/29
/ 1 - TI arcsl.n(r
N )
Now, define the following quantity,
(3.34)
39
(3.35)
flEer )
u
after
N<
NQ, cycles of loading with uniform shearing stress amplitude,
where rN is related to ru through Eq. 3.32.
value
This equation
defines
of ~W(r u ) when the deterioration of G is taken into account; its
m
value is always larger
than
~W(r
)
u
for
the
Dividing both sides·of Eq. 3.35 by
ratio.
'ITr
flEer )
u
t:.W(r
Define r
E
u
= 1)
ru
f
o
9'ITCOS
u
C-2-) sin
same
~W(r
26 - 1
u
'ITr
r
pressure
= 1),
u
(-2-)
------~~--------~~
II -
porewater
(3.36)
dr
U
u
as
flEer )
u
flW(r
The
the
differential
equation
u
(3.37)
= 1)
relating
r E to ru may be derived from Eqs.
3.36 and 3.37; thus,
dr
~
dr E
11 -
r
= _____________u__________
nru
9'ITcos (-2-) sin
(3.38)
29-1 nru
(-2-)
The solution of Eq. 3.38 is shown by the solid line in Fig. 3.16 and
may
line.
be
compared with the solution of Eq. 3.18 which is shown in dashed
For the early stages, the curves are
almost
coincident
because
40
very
little
stiffness
deterioration
has
yet
occurred,
1.e. ~l{r )
u
and ~E(r u ) are very similar.
In the derivation of Eq. 3.38 it was
cycles
of
N,
loading,
right-hand side of Eq.
is
3.38
assumed
continuous
a
drE/d N,
by
that
the
parameter.
the
number
of
Multiplying the
equation
governing
the
porewater pressure increase under nonuniform loading may be writen as,
dr
/1 -
u
ru
1
---=-=-1-) h
1Tr u
29 - I 1Tr u 6W(r u
e1Tcos(--2--)sin
(--2-)·
dN
(T (N) )
(T (N) )
dE
C
(3.39)
dN
where,
T(N) =
the shearing stress amplitude of the N-th cycle.
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;
compared with the solid line of Fig. 3.14b.
the
results
may
The same problem was solved
again, this time with the cycles of loading applied in a reverse
The
results
are
shown
1n
Table
The
order.
3.4 for the fundamental approach of
Martin, Finn and Seed (1975), and plotted in
line.
be
Fig.
3.17
with
a
solid
solution of Eq. 3.39 for this loading is shown in Fig. 3.18
in dashed line.
theoretical
The results still
ones;
the
compare
differences
are
reasonably
mainly
the
well
with
the
of
the
result
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
analysis,
buildup
may
be
under
random
extended
to
seismic
loading, using a total stress
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)
f
~E(t)
t
However,
ET(t)
is calculated by
t
(3.40)
X(s)E(s)ds
o
not· the same as in Eq. 3.23 because the
1S
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
Eq. 3.29.
defined
by
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
II -
u
"dt=
r
rrr u
X(t)E(t)
= 1)
u
~W(r
2e 1 Trr
rrecos( 2 )sin
- ( 2u)
(3.41 )
u
Eq. 3.41 1S obtained with Eq. 3.38 and
dr
1
=
dt
6W(r
_E
u
3.2.5
X(t)E(t)
(3.42)
Discussion
A procedure was
pressure
= 1)
rise
from
described
for
conventional
determining
cyclic
the
excess
porewater
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
In the
results of constant strain cyclic tests.
necessary
strain
the
N~
to
know
latter,
would
it
the number of cycles of constant amplitude shearing
= 1.0,
capable of causing an excess pore pressure ratio ru
variation
of
be
the excess pore pressure ratio r
u
and
with the number of
cycles of loading.
If the results of strain controlled tests
~W(r u
=
are
used,
the
quantity
1) is defined as
6W(r
(3.43)
1)
u
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
3.9.
Let
tests.
ru
=
1n
Eq.
fy{ r N), as obtained from constant strain cylic loading
Then, for
a
total
stress
analysis
rN
= r W'
and
ru may
be
calculated from
(3.44)
Finn and Bhatia (1981) suggest
stress
controlled
controlled
tests,
test
data
primarily
1S
at
that
the
always
higb
experimental
scatter
1n
greater than that for strain
excess
pore
pressure
ratios,
43
70 % of the effective overburden pressure.
exceeding
that a technique to evaluate the excess pore
dynamic
pressure
loading, using a continuous parameter such as
accurate if the results of strain controlled
The
shape
of
test for strain
controlled
the
cyclic
rise
aw,
due
a
to
would be more
tests
were
used.
hysteresis loops is much more stable throughout the
controlled
cyclic
This would imply
cyclic
loading
test
than
for
a
stress
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
flEer U )
NQ,Ec(Y)h(Y)
f
11 -
r
U
0
r
U
dr
tfy(rN~
dr
N
U
(3.45)
---JrN = f Y(r U )
and,
dr
dr
where r E
U
E
=~E/AW(l).
tfy(rN
).
dr
N
~rN
x
f- 1 (r )
= Y
1
11 -
(3.46)
r
u
U
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
of
g(r u )
shear
defines the variation of the energy dissipated by one loop
strain
deterioration
y,
amplitude,
alone.
stress strain law, and
due
to
the
effects
of
strength
This function depends on the particular shearing
on
the
mechanism
of
strength
deterioration.
Usually the strength deteriorates according to
T (t)
m
which
= Tmo (1
(3.48)
- r )
u
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
loading is important for the
1n the occurrence of the earthquake
lifetime
reliability
evaluation
against
liquefaction.
The required dynamic soil properties may be obtained from
laboratory
tests,
Curro, 1981).
or
both
(Richart,
field
or
1975; Woods, 1978; Marcuson and
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
simple
shear
box
tests,
tests,
hollow
and
Franklim, 1979; Fardis, 1979).
properties
and
their
cylinder
cyclic
torsional
triaxial
tests
The uncertainties in
quantification
have
been
the
tests,
cyclic
(Marcuson
and
dynamic
soil
the object of recent
research (Haldar, 1976; Haldar and Tang, 1979a, 1979b; Fardis, 1979, and
Fardis and Veneziano, 1981a, 1981b, 1982).
The
methodology
liquefaction
1S
for
the
formulated;
seismic
the
reliability
uncertainties
1n
properties are identified and quantified based on the
analysis
the
against
dynamic soil
studies
referred
46
to above; and, the uncertainties in the frequency content, intensity and
duration of the earthquake loading are subsequently discussed.
4.2
Given the occurrence
= a,
Am
and
Reliability Evaluation
of
an
earthquake
a strong motion duration, TE
with
= t,
a
given
intensity,
the performance function
against liquefaction at a given depth within the deposit is,
z
6W(r
(4.1)
1) - 6W(a,t)
u
where
~W(ru =
~W(a,
Failure
1) = measure of resistance defined by Eq. 3.22;
t)
= damage
parameter defined by Eq. 3.23.
then defined as,
~s
z
For
~W(a,
the
t)
reliability
are
uncertain
evaluation,
necessary.
soil
The
properties,
are discussed in Sect. 4.3.
denoted
by
R with
~
(4.2)
< 0
the
response
statistics
of
the
of
deposit
u
= 1)
and
depends
on
e.g. shear modulus Gm, whose uncertainties
Let the vector of such soil
mean
~W(r
properties
be
and let GR. be the variance of the i-th
~
component of
~,
and p .. be the correlation coefficient between the
~J
and j-th component of!.
i-th
Then, the quantity ~W(a, t) is also a function
of R, and failure is
~
z
~W(r
u
= 1) - ~W(a,t,R) <
0
(4.3)
47
Using first order approximation
(Ang
and
1975)
Tang,
the
mean
and
-
variance of dW(a, t, R) can be calculated from
(4.4)
(4.5)
where
f
t
t
x(s,a'~R)E[ET(s,a'~R)]dS
(4.6)
o
t
t
ff
t t
o
0
(4.7)
These
last
analysis,
two
and
quantities
the
finite-differences.
~n
are
derivatives
An
obtained
in
Eq.
from
the
4.5
are
random vibration
evaluated
by
analytical technique to obtain the derivatives
Eq. 4.5 was recently proposed by Sues (1983).
The statistics of ~W(r u
undrained
resistance
= 1)
against
depend
on
the
uncertainties
1n
the
liquefaction under uniform cyclic stress
loading N£, and on the uncertainties in the relative density of the sand
Dr •
These uncertainties are discussed in Sect. 4.3.
48
The lifetime reliability index
~,
unconditional on Am and TE may
be
calculated by
~
Q
f
Am
TEmax
maxf
A
T
--
m.
m~n
~(a,t )' P
Q
E.
PA T (a, t)dadt is
m E
t < TE < t+dt at the site.
with
the
(4.8)
a,t d a d t
()
m~n
where
only
AmTE
the joint probability that a < Am < a+dt and
So far, available seismic hazard models deal
earthquake
intensity A , and not with its duration TE•
.
m
For the purpose of this study the statistics of TE will
on
the
used.
loading
value
of
Am'
discussed
conditional
and the results suggested by Lai (1980) will be
The uncertainties
are
be
~n
the duration and intensity of the earthquake
in
Sect.
4.4.
After including the effect of
duration uncertainties, the lifetime reliability is calculated as,
m
s
L
8(a
i~a)Prob[i~a
- 6a < A < i6a +
(4.9)
~a]
i=l
where
the
probability
fault-rupture
seismic
a-~a
that
hazard
< a <
model
acceleration.
The
randomness associated with
~s
obtained
with
the
of Der Kiureghian and Ang (1977).
The measure of earthquake intensity used
ground
a+~a
in
this
study
~s
the
peak
statistics of TE conditional on Am' and the
the
frequency
ground motion are examined in Sect. 4.4.
content
of
the
earthquake
49
4.3
Uncertainties in Soil Properties
Undrained Resistance to Liquefaction Under Uniform Cyclic Stress
4.3.1
Loading
The uncertainties in the undrained resistance to liquefaction
uniform
cyclic
stress
loading have been the object of recent study by
Haldar (1976), Ha1dar and Tang (1979), Fardis
. Veneziano
(1981b).
under
(1979),
and
Fardis
and
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
( excess pore pressure I initial effective vertical stress )
ratio
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;
and
Sheriff,
1974;
Sherif and aI, 1977; Finn et aI, 1977; Finn et aI,
1970; Finn et aI, 1971; Finn and Vaid, 1977)
tests
(DeAlba
et
aI,
1975;
and
Mori et aI, 1977).
large
shaking
of
table
The mean-grain size,
D50 for the sands used in those tests is between 0.4 and 0.65
analysis
Ishibashi
mm.
The
those data lead to the following model for liquefaction in
the laboratory:
(4.10)
where,
Me~z ReIerence Room
University of Illinois
B106 NeEL
_ 208 N. Romine Street
Urbana ~ Illinois 6180J..l
50
't
G'
co
= the shear stress ratio
=
•
co
initial effective confining pressure;
an
0"
ref
't
initial
(1"
effective
confining
pressure
of
reference equal to 4.8 psi;
Dr
aI' a 2 , a 3 , a4
= relative density expressed as a fraction;
= jointly normal coefficients whose mean vector
and covariance matrix are shown in Table 4.1;
€L
=a
normal random
standard
variable
deviation
0.20,
with
which
zero
mean
and
accounts
for
scatter about the regression equation.
Using first-order approximation (Ang and Tang, 1975) the
the
number
of
uniform
stress
c.o.v.
of
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
liquefaction
in
the
laboratory
in
calculated, also In apprOxlmatlon, as,
a
:; =T I errco capable of causing
given
number of cycles can be'
51
1
2
11a
+ cr
2
a
in
2
CD ) + cr
r
3
a'
2
2 co
2
in (-,-) + a
a4
0ref
sL
2
(4.12)
Typical values of the c.o.v. of
r/u'co ,
calculated
Eq.
with
4.12,
are
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
method
of
sample
preparation,
system
compliance
and
the
stress
nonuniformities.
The model is then modified to predict the resistance of the sand
the
field
under
uniform
cyclic stress loading.
1n
This is accomplished
adding two normal random variables, Ao and a o ' to the right-hand side of
Eq.
4.10.
The mean and variance. of these random variables are shown in
Table 4.1.
in-place
Ao accounts
structure
for
of
differences
between
the
number
of
and
the soil, and for site-to-site variations; and,
a o accounts for the effect of multidirectional motions.
the
laboratory
constant
amplitude
shear
stress
The
cycles
c.o.v.
that
of
cause
liquefaction 1n the field is then calculated, in approximation, as,
2
0.
Ni
2
2
exp{O A + a a
0
2
+0
a
0
a'
2 i 2(~)
2
2
+ a in (l1 D ) + 0a n a'
a3
r
ref
4
1
11a
2
(_3)2
a p
in(l1 )} - 1
+ a
+
aD + 20 a a .a a
Dr
SL
11D
1 3 l 3
r
r
2
(4.13)
52
where the in-situ value
variable.
of
the
relative
T
=7/~'
co
density
of
the sand.
also
a
random
mean
and
c.o.v.
of
the
in-situ
The c.o.v.'s of the shear stress ratio
capable of causing liquefaction in the field in a given number
~n
of cycles, are also shown
Table 4.3.
The statistics of the undrained
uniform
is
Typical values for this coefficient of variation are shown in
Table 4.3 for several values of the
relative
density
cyclic
statistics of the
stress
loading,
resistance
N ,
i
quantity ~W(ru = 1)
are
in
to
liquefaction
necessary
Eq.
4.1.
to
under
obtain
This
the
quantity,
defined in Sect. 3.2.2, is
b.W(r
where
~W(r
-
u
-
'T
=
T, D
r and
1)
(4.14)
the shear stress ratio
~s
1)
u
~s
'T /
a:.'co and h(7) is chosen such that
a constant for any pair of Ni and
'T.
For a given
set
of
I
co this quantity has a lognormal distribution with mean
(J
(4.15)
and standard deviation
(4.16)
The
c.o.v.
of ~W(ru
= 1)
~s
the
same as the c.o.v. of N .
values of these coefficients of variation are shown
i
1n
Table
Typical
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
strongly
the
undrained
resistance
to
liquefaction
dependent on the uncertainties in the in-situ relative density
of the sand, D
(see Table 4.3).
r
deposits
Furthermore,
the
response
of
sand
under random seismic loads depends on the uncertainties in the
small strain shear modulus, Gm, and the static shearing strength of
sands,
are
Tm.
In
the
following,
the
the
c.o.v ..... s of the in-situ relative
density, small strain shear modulus, and static shear strength
will
be
discussed.
Relative Density
are
The in-place values of the relative density, D ,
r
usually related to the results of SPT-N tests.
A few in-situ joint
measurements of SPT-N and Dr are necessary to estimate
parameters
of
the
for a given soil.
into
account
the
statistical
relationship between SPT-N and the relative density
The spatial variability within the profile
with
measurements
6f
the
1S
taken
SPT-N along several borings.
Fardis (1979) developed a probabilistic model for estimating
the
SPT-N
and Dr relationship.
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
The
1979a).
potential
relate
blowcount.
density
of
and
Tang,
in-situ relative density of the sand can be determined by
either direct methods (laboratory determination),
that
(Haldar
the
in-situ
value
Typical values of
the
sand
the
determined
of
the
relative
c.o.v ..... s
with
or
of
either
the
indirect
methods
density to the SPT
in-situ
relative
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
~D
0.258 - 0.138
~D
r
0.633 - 1.475
+ 1.192
~D
(4.17a)
>0.60
r
2
~D ~0.60
r
~D
r
r
when the in-situ relative density is determined as
suggested
(4.17b)
by
Gibbs
and Holz (1957).
Small Strain Shear Modulus
shear
modulus
of
the
sand
In the present study, the small strain
~s
calculated with the following equation
(Martin and Seed, 1982):
D - 75
G
m
5,000(1 -
r
)
100
0cr'
(4.18)
(psi)
m
The form of Eq. 4.18 suggests that the uncertainties in the small strain
shear
modulus
are
strongly
relative density of the sand.
the
mean
and
c.o.v.
shown in Table 4.5.
should
also
be
of
dependent
on
the
uncertainties
~n
the
The c.o.v.'s of Gm for several values
of
the in-situ relative density of the sand are
Errors in
considered.
the
An
predicition
of
Gm
'th
w~
Eq.
4.18
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
c.o.v.
of
the
small
strain
shear
Drnevich
(1972b).
The
total
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
the
sand
sand.
~s
related to the uncertainties in the friction angle of the
Meyerhoff (1982) suggested that the c.o.v. of
strength
of
the
sand is between 0.10 and 0.20.
used in this study.
clay
of
the
static
shear
The value of 0.10 was
The c.o.v. of the undrained shear strength
is between 0.20 and 0.40 (Meyerhoff, 1983);
of
the
the value of 0.20 was
used in this study.
4.4
4.4.1
Earthquake Loading
Ground Motion Model
For the purposes of the this study
ground
motion
should
content and duration.
characterized
by
the
earthquake
induced
strong
be specified in terms of its amplitude frequency
The
intensity
of
the
earthquake
loading
~s
its peak ground acceleration, a max' and the frequency
content by its power spectral
density
function
(PSD
function).
The
Kanai-Tajimi PSD function
Sew)
is
used
2S
(4.19)
o
to model the frequency content of the earthquake loading.
parameter So is the intensity scale of the
~B
PSD
function;
and,
The
wB and
shape parameters.
The peak ground acceleration amax has been related to the root
square
ground
acceleration
found by Lai (1980),
(1982),
mean
arms' and excellent correlations have been
Vanmarcke
and
and Hanks and McGuire (1981).
Lai
(1980),
Moayyad
and
Mohraz
In particular, Vanmarcke and Lai
56
(1980) suggested the following relationship:
a
a !2£nC
max
rms
2TE)
T
(4.20)
o
Where:
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
2'IT
T
o
(4.21)
C1}
C
where the quantity W c is calculated from
(4.22)
The duration of the strong-phase of motion is determined from
and
the
condition
that
the
total
energy
of
the
Eq.
4.20
ground motion is
retained, i.e. with
I
where
10 is
the
accelerations).
Arias
In
o
2
T
= a rms
E
intensity
(4.23)
(time
integral
of
the
squared
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
(4.24)
rms
57
In general,
a
where
(PF)
max
=
(4.25)
(PF)a rms
a peak factor, defined by Eq. 4.20; this peak factor is
1S
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
conditions; whereas based on 140
found
c.o.v.
F or 22
of
horizontal
= 0.64
for
"firm"
accelerograms
Lai
ground
(1980)
vary from 5.7 rad/sec to 51.7 rad/sec, and ~B between 0.10
wB to
and 0.90.
~B
and
0.40,
tr
roc k"
and
S1. t
e recor d s wE had a mean of 2 6 .7 rad / sec
~B had
a
mean of 0.35 and c.o.v. of 0.36.
"soft" site records the corresponding means and c.o.v.'s are 19
Mohraz
(1982)
obtained
average
shapes
function for vertical and horizontal accelerations.
rad/sec
of
the
by
Moayyad
and
Mohraz
wB = 16.9 rad/sec and ~B = 0.94
function
with
(1982),
(Sues,
these values of w and
B
w and
B
1983) •
~B
PSD
When a Kanai-Tajimi
PSD function, Eq. 4.19, is fitted to the average shape for "rock"
proposed
For
'B.
and 0.43 for wB and 0.32 and 0.36 for
Moayyad and
and
sites
'B were found to be
The
Kanai-Tajimi
PSD
is used in this study to model
the earthquake ground motion except if otherwise stated.
The Kanai-Tajimi PSD function with wB
and
the
Housner
= 16.9
'B = 0.94,
and Jennings PSD function are shown in Fig. 4.1.
average of PSD density function for
"rock"
statistics
wB and
and
rad/sec and
distributions
for
sites
obtained
'B proposed by Lai
using
The
the
(1980) is
58
also shown in Fig. 4.1.
The PSD function with w B
= 16.9
rad/sec
and
= 0.94,
'B
and
the
average PSD function obtained with Lai's data are similar, implying that
the response statistics calculated with either of them will be
However,
the
study
by
Lai
(1980)
proves that the large c.o.v.'s of
w B and 'B ( ..... 0.40) imply high likelyhoods of occurrence
with
very
of
earthquakes
different frequency content (see PSD functions in Fig. 4.2).
The effect of these differences on the seismic response of
sand
similar.
deposits
was
calculated
a
saturated
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
significant
se~sm~c
Kiureghian
and Ang (1977).
hazard
intensity
attenuation
model
equation,
and
the
model
slip-length
"magnitude recurrence" curve (Der Kiureghian and Ang, 1977).
these
uncertainties
systematically
evaluated
on
the
calculated
probabilities
(e.g.
magnitude
relation) as well as the values of parameters such as the slope
of
of
The probabilities calculated with this
model will depend on the physical relations assumed in the
the
all
intensities at the site during the lifetime of the
project are calculated with the fault-rupture seismic
Der
of
of
the
The effect
can
be
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:
30 exp(-3.254 aO. 35 )
max
11T (a
)
E max
(4.26)
ST = 0.804,
The c.o.v. of TE conditional on the value of
and
E
the
site
gamma
distribution
conditions
(~foayyad
the
is
strong
considered appropriate for T •
E
motion
and Mohraz, 1982; Lai, 1980).
durations
are
For "rock"
slightly
shorter
However, because of scarcity of
the data for "rock" sites Eq. 4.26 is retained.
60
CHAPTER 5
ILLUSTRATIVE EXAMPLES
5.1
The principles and
chapters
are
Introduction
general
procedures
described
the
illustrated herein for specific soil deposits.
example illustrates the calculation of the mean and
of
1n
standard
earlier
The first
deviation
the duration of strong motion, with a specified intensity, necessary
to cause liquefaction failure.
density is considered.
An homogeneous deposit with low relative
The differences in the results of a total stress
analysis and an effective stress analysis are presented.
In the second
layered
soil
example,
the
seismic
deposit is examined.
safety
of
a
nonhomogeneous
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
methodology
are
compared
past earthquakes.
sandy
soil
1n
of
liquefaction
predicted
with
proposed
with the performance of sand deposits during
Examples of these are: three locations in an area
of
the city of Hachinohe (Japan) which showed a variety of
damage features during the Tokachioki earthquake of May
several
the
locations
16,
1968;
and
1n the city of Niigata (Japan) which showed evidence
of liquefaction or no-liquefaction during past earthquakes.
61
5.2
5.2.1
Homogeneous Saturated Sand Deposit
Problem Description
A homogeneous sand deposit is considered.
as
consisting
of
several
layers
to
The deposit is
account
for
idealized
~n
changes
soil
properties with depth as shown in Fig. 5.1.
The soil properties used in
the
The deposit was discretized
analysis
are
specified in Table 5.1.
into 10 elements of equal thickness.
r-
~m'
shear strength Tm, the stiffness K, as well as the parameters 5 and
the
fi
The small strain shear modulus
for all the elements in the profile
hysteretic
are
~n
shown
model parameters were chosen as A
= 1.0
Table
and r
5.2.
= 0.50
The
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
The values of T/~~O for
effective stress.
decreasing values of U~o.
other
a~o between
the
same
Nt
increase
for
Accordingly, the cyclic resistance curves for
~n
those
Fig.
5.2
were
obtained
by
linear
T ,
L
for
several
,
and with
interpolation.
5.2.2
Total Stress Analysis
The statistics of
stationary
W
B
~
= 5rr
loadings
rad/sec and 'B
the
with
time
till
different
= 0.64,
liquefaction,
peak accelerations a
are given in Table 5.2.
max
The
values
of
were calculated by
L
~W(r
~T
where
all
the
L
1)
u
(5.1)
= ---------X~·
quantities. are
ET
defined
in
Chs. 2 and 3, X being the
62
equivalent amplitude for
denominator
of
random
~s
Eq. 5.1
loading
defined
by
Eq.
3.24.
The
obtained from the random vibration analysis.
=~T 8
where
T
T
L
L L
The validity of this technique for
The standard deviation of TL is calculated by u
the
same
8 E at
as
= ~T
time t
T
~
L
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
necessary
variation
of
the
duration
of
motion
for liquefaction decreases as the mean value of that duration
increases.
A
variation
similar
behavior
was
found
with
the
coefficient
of
of the total hysteretic energy dissipated, i.e. the c.o.v. of
the hysteretic energy decreases as the duration of the
For
strong
load
increases.
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
the
associated
loading
TE = 5.0
is
reliability
given
seconds.
as
levels.
amax = 0.05 g,
as
Suppose that the intensity of the
and
the
strong-phase
duration
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,
coefficient
of
variation
of
0.14.
~s
F
The
a
= 0.067/0.05 = 1.34,
associated
with a
reliability
is
63
P[F a > 1.0] = 0.98.
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.
evaluating
liquefaction
excitations;
potential
therefore,
the
Current methods
for
are limited to deterministic ground
factor
of
safety
only
applies
to
a
particular accelerogram.
The profile of the total accelerations in the soil deposit is
~s
5.4 for Amax = 0.20, 0.10, and 0.05 g.
Fig.
~n
The nonlinear behavior
apparent in the variation of the ratio of the standard
the
total
acceleration
at
the
shown
deviation
of
surface and the root mean square base
acceleration, arms.
Stiffness Deterioration
5.2.3
Martin and Seed (1979) analyzed
Centro
earthquake
scaled
deposit
under
undrained
conditions
stress analysis gave a value of 8.0 seconds.
and
(1982)
Wylie
calculated
this
TL under
a max
= 0.10
function
a
g, w B
load
=~
with
rad/sec,
an
expected
and
'B
The effective
duration as 4.0
peak
= 0.64,
The
ground
with
shown in Fig. 5.4 may be seen in Table 5.4.
the
second~,
statistics
acceleration
modulating
In particular, at
= 3.2 seconds and u T = 1.7 seconds.
L
L
The effective stress analysis was carried out for this load and
12.5 feet it is
statistics
of
the
Under the same conditions,
whereas Zienckiewicz et al (1982) obtained 2.8 seconds.
of
El
When a
assumed,
until liquefaction was estimated to be 3.0 seconds.
Katsikas
1940
the
to a maximum acceleration of 0.10 g.
total stress analysis was used and
time
this
~T
TL at 12.5 feet were calculated as
~T
L
=
the
3.6 seconds and
64
= 1.8 seconds. The expected pore pressure ratio rise and the
L
stiffness deterioration are shown ~n Figs. 5.6a and 5.6b, respectively,
UT
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
deposit.
Martin
and
less
Seed
than
(1979)
15 % for
the
homogeneous
soil
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
5.3.1
Reclaimed Fill
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.
from
the
random
The seismic response of the deposit as obtained
vibration
analysis
depends
the
on
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
random
vibration
of
the
stationary
analysis are shown for two levels of intensity of the
loading and three schemes of discretization (shown in
Fig.
9-element
The statistics of
model
seems appropriate for this problem.
the shear strains and stresses converge faster than
the hysteretic shear strain energy dissipated.
the
5.9).
statistics
The
of
However, for the purpose
of this study, the seven element model may be sufficient.
65
The average dynamic properties of the elements of
model
are
shown
in Table 5.6.
are shown in Fig. 5.8.
sand
can
be
the
lumped
mass
The undrained cyclic resistance curves
These curves were obtained
assuming
that
this
by Eq. 4.10, and the statistics shown in Table
described
4.1.
The effect of the randomness in the frequency content of the
excitation
on
quantified.
but
with
the
statistics
of
the
response
The deposit was analyzed for
five
different
values of wB-
=
arms
of
ground
this deposit was
0.08 g
and
'B
=
0.4,
With the statistics of w for
B
"rock" site conditions suggested by Lai (1980),
the
expected
response
statistics of the deposit were calculated assuming both normal and gamma
distributions for w B•
The results of this analysis
are
summarized
1n
Tables s.7a and s.7b.
The response statistics for a load with
ground
W
acceleration
but
with
a
the
different
B = 16.9 rad/sec and 'B = 0.94 are also shown
o
o
latter
the
statistics
earlier
summarized
are
Sect.
root-mean-square
frequency
content
1n
S.7b.
Table
i.e.
These
similar to the average statistics obtained with
analysis.
1n
same
The
4.4.2;
reasons
the
for
this
particular
good
agreement
are
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
the
of
order of 0.50, can be attributed to the randomness of the frequency
content of the ground motion energy.
variation
0.50.
liquefaction
of
the
damage
parameter
The corresponding
~W(a,
coefficient
of
t), is also of the order of
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 a max
= 8n
curves in Fig. 5.8 marked 8N
l
statistics are ~T
= 3.5
L
= 0.15
g, and the cyclic resistance
0.0.
At a depth of 15 feet, these
=
r
= 1.2
seconds and ~T
.
L
seconds; whereas the same
= 3.1
~T
statistics obtained with a total stress analysis are
seconds
L
and
seconds.
These
results
agree
with the statements of
Martin and Seed (1979).
Reliability
5.3.2
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
duration
of the strong-phase motions.
5.l0a and S.lOb.
the
undrained
wide
shear
strength
of
range
of
the
These indices are shown in Figs.
A value of 0.20 for the coefficient
Meyerhoff (1982).
1S
a
the
of
variation
of
clay was used, as suggested by
The coefficient of variation of the relative
taken as 0.15 following Haldar and Tang (1979b).
density
The statistics and
probabilities of the strong-phase motion given in Sect. 4.4 are combined
with
the
results
1n
reliability measures.
Figs.
These
S.10a
and 5.10b, to obtain unconditional
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
calculate
the
probabilities
of
to
all significant loadings at the site.
Using the data reported by Kiremidjian and Shah
(1975),
the
predicted
to
calculate
probabilities for Eureka (California) are shown
4.24 and the information in Figs. 5.10 and 5.11 are
the
lifetime
reliability
against
used
liquefaction.
The
calculated
67
reliabilities are summarized in Table 5.8.
The sensitivity of the reliability to the uncertainties in some soil
properties
aD = 0.0
can
be seen in Figs. 5.12 to 5.14.
and 0.15 are assumed.
In Fig. 5.13
In Fig. 5.12, values of
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
Introduction
5.4.1
The
probabilities
methodology
are
resistance
of
liquefaction
compared
during past earthquakes.
the
Case Studies
with
predicted
with
earthquake
the field performance of sand deposits
Usually, the historical data for
load,
and
a
boundary
separating
the
in-situ
the
data
between
and no-liquefaction is determined (Seed and Idriss, 1981).
A convenient parameter to represent the intensity of
ratio
proposed
of the soil are plotted graphically against the intensity of
liquefaction
the
the
of
the
average
surfaces of the sand to
the
shear
an
earthquake
is
stress Tave developed on horizontal
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
of
the
a
function
standard penetration resistance (SPT), N , corrected to a value
1
of a~o equal to 1 ton/sq ft (Seed and Idriss, 1981).
A graphical representation of some of the historical data
is
shown
1n
Fig.
5.15a,
where
available
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
resistance
of
the
soil.
Information
the
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
during
earthquakes is used to calculate the probability of liquefaction
for somE of the
locations
in
data
the
points
city
of
in
Fig.
Hachinohe
5.15a,
locations
~n
the
city
of
namely:
(Japan)
earthquake of May 16, 1968 (points 5, 6 and 9
three
deposits
Niigata
(i)
during
~n
at
three
the Tokachioki
Fig. 5.15a);
(ii)
at
(Japan) during the Niigata
earthquake of 1964 (points 1, 3 and 4 in Fig. 5.15a); and, (iii) at
location
~n
one
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
profiles
for
each
case
analyses are presented.
study
modeling
of
the
soil
as well as the assumptions made in the
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
and
P2
by
Ohsaki
(1970).
The
soil
idealized as shown
~n
Figs. 5.l7a, 5.17b
points
~n
Fig. 5.15a.
5, 6 and 9
acceleration
of
approximately 0.09 g.
TE as
defined
~n
The
this
the
and
5.17c,
corresponding
to
The characteristics of the Tokachioki
base
excitation
statistics
study
PI
P6,
profiles at each location were
earthquake of May 16, 1968 are shown in Table 5.9.
ground
designated
of
the
The
expected
..
usen. .~n ~ne
peak
1 · ·~s
anaLys~s
strong-motion
duration
are obtained with the correlations with
69
magnitude and epicentral distance proposed by Lai (1980) and
Kameda
and
(1983),
Koike
Shinozuka,
and are also shown in Table 5.9.
The gamma
distribution was considered appropriate for T •
E
The probabilities of liquefaction were calculated
locations
these
three
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
T/U~o that
ratio
for
loading cycles.
causes
to~al
c.o.v. of 0.30 for the shear
liquefaction
in
a
given
stress
number of uniform
These probabilities are summarized in Fig. 5.16.
Niigata (Japan)
5.4.3
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
and
Idriss,
1981).
(Seed
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).
of 1887 and
1802
accelerations
are
of
only
The data concerning the earthquakes
approximate.
The
statistics
5.9.
and
1887,
Shinozuka,
epicentral
distance
respectively.
proposed
Kameda and Koike (1983), and are shown
~n
Lai
Table
were
earthquake
1802
of
calculated
for three
locations (namelv.
nointg
I. 3 and 4 In
----------------... - - - - - - . 1 7
r-----------~--
Fig. 5.15a); and, for the location in Fig. 5.1Sb during the
of
by
The gamma distribution was considered appropriate for T •
E
The probabilities of liquefaction during the Niigata
1964
ground
of the strong-motion duration TE were obtained from the
correlations with magnitude and
(1980)
peak
the base excitations are approximately 0.07 g, 0.05 g
and 0.025 g for the earthquakes of 1964, 1802, and
The
expected
and
lS87
(respectively
-7
-
----
.
---
earthquakes
points 10 and 11 in Fig. 5.15a).
As
70
before, these probabilities were calculated for the
Figs.
5.18a,
c.o.v.'s
shown
in
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'vo that causes liquefaction
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
of
analysis
appear
method
to be consistent with the observed results.
expected that for the data points close to the boundary
It is
separating
the
cases of liquefaction and no-liquefaction the predicted probabilities of
liquefaction will be higher than for those
line.
points
further
below
this
For the same standard penetration resistance (as measured by N )
1
the ratios of the shear
stress
on
the
boundary
induced shear stress are also shown in Fig. 5.16.
and
the
earthquake
Usually, these ratios
will increase as the probabilities of liquefaction decrease.
5.5.
The first
example
shows
liquefaction
can
be
results from
the
random
analysis
Summary of Results
obtained
how
for
vibration
seismic
a
resistance
curves
against
specific soil deposit using the
analysis.
The
results
of
showed that the c.o.v. of the duration of strong ground motion
necessary for liquefaction to occur decreases as the mean value of
duration increases.
induced
this
This means that the reliability against seismically
liquefaction
randomness
as
the
this
becomes
expected
less
time
sensitive
to
to
liquefaction
the
ground
increases.
motion
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
lifetime
example
illustrates
the
proposed
reliability evaluation against liquefaction.
(see Fig. 5.14) that the probabilities of
the
procedure
duration
and
intensity
of
the
liquefaction
load
uncertainties in the undrained resistance
are
against
for
the
The results show
conditional
on
very sensitive to the
liquefaction
for
a
given relative density, and the in-situ value of the relative density of
the sand.
This example clearly indicates that the
liquefaction
~s
more
sensitive
to
the
reliability
intensity
of
the
against
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
rise
in
conventional
uniform-stress
pore
pressure
cyclic shear tests in terms of a
continuous damage parameter, which permits the study of liquefaction
sand
deposits
systems.
energy
of
as a problem of random vibration of nonlinear-hysteretic
This parameter is a function of
dissipated
by
the
the
hysteretic
shear-strain
soil and of the amplitude of the hysteretic
restoring shear stress, thus measuring both the number and amplitude
of
loading cycles.
The
mean
liquefaction
analysis.
curves
and
~n
variance
the
of
the
strong-motion
duration
deposit were calculated with the random vibration
These satistics can be used to define the seismic
against
until
liquefaction in the deposit.
resistance
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
used
to
calculate
the
reliabili~y
can
against liquefaction of a saturated
sand deposit under a random seismic load with a prescribed duration
intensity.
With
this
properties of the soil
reliability evaluation.
approach,
can
also
be
the
be
uncertainties
systematically
and
in the resistant
included
~n
the
73
The reliability against liquefaction conditional
and
duration
of
the
loading
on
the
intensity
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.
randomness in the intensity and duration of
the
earthquake
load
The
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
earthquake loading, is necessary and
against
liquefaction.
that the method
is
reliability
saturated
of
an
The
sand
assessing the relative risks
useful
comparison
effective
intensity
tool
for
and duration of the
a
risk-based
design
with the historical data shows
for
determining
the
seismic
deposits against liquefaction, and for
between
design
alternatives
(see
Sect.
5.4).
2 - The
intensity
reliability
against
liquefaction
conditional
on.
the
and duration of the loading is sensitive to the uncertainties
74
1n the in-situ relative density of the sand, and
the
undrained
In fact,
the
the
uncertainties
in
resistance to liquefaction for a given relative density.
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
opposed
to
the
be
considered
1n
methods
that
deterministic
the
design
postulate
process,
the
as
lifetime
earthquake load.
4 - The technique that was developed to represent
pressure
generation
1n
saturated
excitations has the following
sand
the
excess
pore
deposits under random seismic
advantages:
(i)
it
uses
a
continuous
damage parameter formulation, thus removing the need for calculating the
equivalent
number
measurement
of
loading
cycles;
(ii)
does
not
require
the
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
liquefaction
accelerations.
that
will
cause
can be well predicted with the total stress analysis.
stiffness deterioration leads to
liquefaction,
duration
and
changes
the
slightly
longer
frequency
expected
content
of
times
the
The
for
surface
75
6 - The probability of occurrence of earthquake loadings
different
the
very
frequency content can result in very different expected times
till liquefaction.
with
with
The c.o.v. of the time
to
liquefaction
associated
uncertainties in the Kanai-Tajimi filter parameters is of the
order of 0.50.
7 sand
The shear-strain energy dissipated by hysteresis
deposits
~s
a
in
saturated
good indicator of the damage and deterioration of
such deposits under random seismic loadings.
76
TABLES
77
Function h(T) for a Sand with Dr =0.54·
Table 3.1
N.e,
E (z)
c
z
T
0'
vo
z
=3
[Ec(z)]N
u
Q,
h(a; )
va
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
CYCLES
Excess Pore Pressure with Cycles of Loading
T
(psf)
r
T
u
0'
vo
NQ,
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
0'
va
E
c
z
T
NQ,
z
c-n
E (0.113)
c
u
T
h(G')
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
CYCLES
Excess Pore Pressure with Cycles of Loading
T
(psi)
r
T
u
0'
NQ,
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
79
Table 4.1
Mean
Vector
a
a
a
a
a
l
2
3
4
0
A
0
-0.66
-5.17
7.8
Statistics of Parameters Defining N£
Covariance Matrix
a
l
0.982
a
a
2
0.013
0.015
synunetric
-0.72
3
a
a
4
-
-0.02
-
-
-
0.36
-
-
-
-
-
0.36
0.59
Table 4.2 C.O.V. of the Undrained Resistance to
Liquefaction in Laboratory (a' = 4.8 psi)
co
r
0
-
2.58
l1D
A
-
0.0864
0.59
-0.44
0
oN
i
r
°Tla co
50%
1.6
0.25
60%
1.55
0.24
75%
1.5
0.23
80
Table 4.3
c.o.v.
Liquefact~on
J.l.D
fiu r
r
50%
60%
75%
Table 4.4
Method
of
Determination
of the Undrained Resistance to
in the Field (cr' = 4.B psi)
co
ON
i
.
!lr Icr ,
co
0.0
2.2
0.26
0.15
4.7
0.34
0.20
B.l
0.40
0.0
2.1
0.25
0.15
4.6
0.34
O.lB
6.2
0.37
0.0
2.1
0.25
0.15
4.5
0.34
0.18
6.1
0.37
C.O.V. of the In-Situ Relative Density
Mean Relative Density
30%
40%
50%
60%
70%
0.18-0.36
0.14-0.27
0.13-0.22
0.12-0.20
0.11-0.18
Indirect Method
(all data)
0.27
0.27
0.27
0.27
0.27
Indirect Method
(Gibbs & Holz
data)
0.30
0.23
0.21
0.19
0.18
Direct Method
(£1
=
0.01) *
Ys
*C.O.V.
of the in-situ dry density
81
Table 4.5
c.o.v.
of the Small Strain Shear Modulus
G of Sands
m
r'b
l1
n
r
50%
60%
75%
Table 5.1
°Gm
st
r
0.0
0.0
0.12
0.15
0.10
0.16
0.20
0.13
0.18
0.0
0.0
0.12
0.15
0.11
0.16
0.20
0.14
0.19
0.0
0.0
0.12
0.15
0.11
0.16
0.20
0.15
0.19
K
m
Properties of Sand in Example I
PARA.l1ET ER
cp
G
VALUE
36.8
0
0.85
Y
l20pcf
n
r
0.45
max
0.973
min
0.636
e
e
D50
0.65
Me~z Rererence Room
University of Illinois
BI06 NCEL
208 N. Romine Street
Urbana, Illinois 61801
82
Table 5.2
0'
Lumped Hass Model for Example I
G
't
K
U
2
ELEHENT
va
(psf)
ma
(psf)
mo
(psf)
(~b)
1
3,150
2,200,000
1,670
254
0.0108
2.3420
2
2,850
2,090,000
1,510
242
0.0108
2.4016
3
2,550
1,980,000
1,350
229
0.0108
2.4695
4
2,250
1,860,000
1,190
215
0.0108
2.5484
5
1,950
1,730,000
1,030
200
0.0180
2.6402
6
1,650
1,600,000
870
184
0.0108
2.7533
7
1,350
1,440,000
715
166
0.0108
2.8949
8
1,050
1,270,000
560
147
0.0108
3.0813
9
750
1,070,000
400
124
0.0108
3.3407
10
300
680,000
160
78
0.0108
4.2104
1.n
(lb sec )
in
o,
S
I
Statistics of TI. for Severnl Load Intensities (seconds)
Table 5.3
--
..-.".. --.~--.------
a
DEPTII
(ft)
J-l T
cO.20g
__
.--
a
max
J-l
01'
L
L
47.5
0.51
42.5
r
=O.lSg
aT
a
max
J-l T
=0.10g
°T
a
max
l-IT
L
L
0.47
0.90
0.6
2.1
1.0
4.0
0.48
0.44
0.B5
0.6
2.0
1.0
37.5
0.46
0.42
0.B3
0.6
1.9
32.5
0.44
0.40
0.79
0.6
27.5
0.42
0.38
0.76
22.5
0.40
0.35
17.5
0.38
12.5
7.5
-
max
-_.
---------~----
L
L
.
L
=.075g
°T
L
a
max
l-IT
=0.05g
aT
L
L
1.6
10.5
2.6
3.7
1.5
9.B
2.5
1.0
3.6
1.5
9.4
2.4
1.9
1.0
3.5
1.5
9.1
2.4
0.6
1.8
0.9
3.4
1.4
8.7
2.4
0.73
0.6
1.7
0.9
3.2
1.4
B.4
2.3
0.33
0.70
0.5
1.7
0.9
3.2
1.4
8.2
2.3
0.38
0.31
0.69
0.5
1.6
0.8
3.1
1.3
8.2
2.2
0.40
0.31
0.73
0.5
1.7
0.8
3.3
1.3
8.6
2.2
----
C'J
{J)
Table 5.4
~T
and crT
L
ELEMENT
for a Modulated Load
L
1-1T
O'T
L
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
a
max
llE
ELEMENT
a
1
(psi)
Response Statistics with 5 Elements
=0.20g
a
max
llE
T
3
(Psi in )
sec
llT
1
-m
a
L
(sec)
a
1
(psi)
T
=0.40g
T
3
(psi in )
sec
1-1T
T
-m
a
L
(sec)
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
a
a
1
(psi)
a
max=0.20g
llE
ELEMENT
Response Statist1cs with 7 Elements
max
1-1E
T
3
llT
l'
(pSi in )
sec
-m
al'
L
(sec)
a
l'
(psi)
=0.40g
T
3
llT
l'
sec
m
a
(~si in )
L
(sec)
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
a
max
~E
ELEMENT
a
T
(psi)
Response Statistics with 9 Elements
=0.20g
a
max
~E
T
3
(psi in )
sec
T
-
~T
m
aT
L
(sec)
a
T
(psi)
=0.40g
T
3
~T
T
(psi in )
sec
-m
L
(sec)
aT
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
ELEMENT
DEPTH
(ft)
Lumped Mass Model for Example II (Mean Values)
height
(ft)
T
mo
(psi)
mo
(psi)
(~b)
G
M
K
2
(lb ~ec )
~n
<5
'.
S
~n
1
135
30
32
30,100
83.6
0.0648
0.808
2
105
30
25
26,900
74.6
0.0648
0.864
3
75
30
8
16,400
45.6
0.0648
0.772
4
45
30
8
16,400
45.6
0.0648
0.772
5
25
10
7.5
11,700
97.5
0.0216
1.803
6
15
10
5.4
9,900
82.5
0.0216
1.954
7
5
10
2.2
5,420
45.2
0.0216
2.812
Table 5.7a
~T
L
S
PMF
w
B
and aT for Several w
B
0
~T
11T
L5
(sec)
L6
(sec)
T5
(psi)
T6
(psi)
2.41
4.00
2.2
1.5
a
a
GAMMA
NORMA.L
2
(~)
3
sec
5.
0.0323
0.0664
25.37
15.
0.3141
0.2404
8.457
1.63
2.00
2.1
1.5
25.
0.3895
0.3833
5.074
2.46
1.96
1.8
1.4
35.
0.1999
0.2465
3.625
5.75
3.50
1.5
1.2
45.
0.0642
0.0634
2.819
11.80
6.14
1.3
1.1
88
Table S.7b
Average
and crT with Normal and Gamma PDF's for w
B
~T
L
E[cr
e.O.'\[ [ llT ]
E[IlT ]
L
L
t
e.QV.[cr ]
T
]
PHF
TL
TL
T
TL
(sec)
6
(sec)
5
(sec)
L6
(sec)
T5
(psi)
T6
(psi)
T5
(psi)
T6
(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
3.20
2.80
1.77
1.34
w =16.9
B
5
sB=O.94
Table 5.8
Lifetime Reliability for Layered Deposit
25 ft - Depth
15 ft - Depth
Lifetime (years)
10
25
0.45
-Q.04
I
Probability
of
No-Liquefaction
0.66
10
25
50
-0.37
0.40
-0.02
-0.34
0.41
0.68
0.52
0.43
~
L
Reliability Indices
50
\
0.50
Table 5.9
Case
History
Date
Site
M
Distance
(miles)
Historical Data on Liquefaction (Partial Data)
TE
max
(g)
Depth of
Water Table
(ft)
D
(ft)
vo
(psff
N-SPT
TE
c.o.v.
(sec)
a
0'
N1
D
,. ave
r
--orvo
Field
Behavior
Reference
1964
Niigata
7.5
32
15
0.8
0.17
3
20
1200
6
8
53
0.195
Liq.
Seed Hnd
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)
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)
1968
lIakodate
7.8
100
15
0.8
0.21
15
1000
6
9
55
0.205
Liq.
Kishida (1970)
1968
Hachinohe
7.8
45-100
15
0.8
0.21
47
2900
25
21
75
0.19
No-Liq.
Ohsaki (1970)
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)
2
8
9
8
9
T
ldri~~
(lY71)
00
\0
90
FIGURES
"91
z
z
______
u
(a)
6+0>0, 0-6<0
(b)
~~~~------.u
6+0>0, 0-6
=0
________~~~------__u
"(c)
6+0>0-6>0
(d)
6+0
u
(e)
Fig. 2.1
0>8+0>0-6
Possible Hysteretic Shapes
0, 0-8<0
92
=l
m
I
/
kn
/
I
I
Cn
j
~
=i
/
k3
I
I
~
I
~+ 1 kit1l1.1 +(1..a."1)~ + 1 ~.1
~I
m
I ;.1 9i.1
/~
:=j
k2
C2
3
m1
. I
I
I -- -
----fmjXJ
I
Cj
I
l1
I
'W
k,
u2
C1
a i kj 4+(1-'1)~ zt
x.=± u.+~8
J.
J: 1
58
U
1
Fig. 2. 2
/
Lumped Mass Model
J
I
I
93
A=1.Q
6=~=1.0
CI=0.05
T=0.28sec
~ r=1.0
Filtered White Noise
So =50 in2/sec3
n
WS= 15.6 rad/sec
SB=0.64
00 0
Simulation (n::: 60)
--Analytical Results
°E
_-.:.T_ (in2)
(1- a) K
1.4
~/6-Q)K
1.2
6E
1.0
T
.8
.6
.4
.2
~--------~-----%
2
4
6
e
Fig. 2.3a
10
12
0
and 0E
E
T
14 16
18 20
TIME, t (seconds)
for a SDF
T
So =2.0 ir:f/se2
o 0 0 Simulation
Analytical Results
°E
_---.,;T_ x 20 (jrf )
(1-
a
)K
1.4
1.2
1.0
.8
.6
.4
.2
2
4
6
Fig. 2.3b
e
0
10 12
ET
14
and 0E
16 18 20
TiME, t (seconds)
for a SDF
T
t
94
White f\oise
BE T-
I
.8
50 = 50 irf/se2
.7
r, =.63, ~=.23, "I;=.16
.6
(sec)
.5
.4
.3
.2
.1
.0
t
2
8
10
12
Fig. 2. 4a
0E
for a 3 Degree-of-Freedom Sys.tem
4
6
14
16
18 20 22 24 26 28 30 32
TIME, t (seconds)
34
T
4.4
aE
4.0
T·
I
(1- C4)K j
(irf) 3.6
3.2
2.8
2.4
2.0
1.6
~_-2
12
.8
.4
.0
2
4
6
8
10
12
14
16
18 20 22 24
TIME, t
Fig. 2.4b
3
t
26 28 30 32
(seconds)
34
for a 3 Degree-of Freedom System
0E
T
95
GE
(1-
T
a) K
(jrf ).
1.6
1, =.45
1.4
1.2
BET
------,
1.0
4>1
\
\
\
\
.8
.6
.4
,,
'~1
" ",
O'E 2
I
.2
I
I
O.
O.
/
10.
20.
TIME,
~ET
(1 - Ci) K
t
(seconds)
8
2
(in )
7
6
5
4
3
2
o~~~~~
o
__ __
~
~~
__
~~
~
~
20.
10.
TIME,
Fig. 2.5
__ __ _____
t
(seconds)
Statistics of ET for a Nonstationary Load
Me~z
Reierence Room
University
o~
Illinois
BI06 NeEL
208 N. Romine Street
Urbana, Illinois 61801
96
sand
til
til
hyperbolic
---
W
a::
I-
til
SHEAR
STRAIN.
Y
I-'
til
I/)
W
a::
lI/)
a::
<{
w
:::c
I/)
SHEAR STRAIN.
Fig. 3.1
Y
Skeleton Curve of the Dynamic Shear Stress-Strain
Relation for Soils
97
Y
Fig. 3.2
Cyclic Shearing'Stress-Strain Loops for Soils
t
G
,
f:lW
4'TT
.l..Yata
D=-
Fig. 3.3
2
Equivalent Viscous Damping Ratio D
98
- - - Hardin and Drnevich (1000. cycles)
1 .25
--model
- -
1.
-
--
E
"
!-J
.75
r=O.5
A=1.0
V1
V1
0=8
w .5
a=O.O
a:
lV1
a:
<i
ii
V
.25
w
I
U1
O·
O·
Fig. 3.4a
I
1.
2.
3.
I
I
4.
5.
SHEAR STRAIN, Y/Yr
6
Model and Empirical Skeleton Curves for Sands
- - - Hardin and Drnevich (1000 cycles)
---model
.5
r =0.50
A =1.0
5=(3
Ct=O.O
10°
SHEAR
Fig. 3.4b
Model and Empirical GIG
m
10 2
10'
STRAIN,
with y/y
r
Y/Y r
for Sands
99
7 LLt.seed and Idrlss (1970-range ot data)
1.
- - y =7.5x1cJ'
r=0.5
A =1.0
.5
5=(3
Ct =0.0
O.
10- 4
10-3
SHEAR STRAIN, Y
Fig. 3.5
Hodel and Empirical GIG
with y for Sands
m
,.
-
-
- Tatsuo ka et al (1979)
--Y.r :7.5x1cr 4
-..,.(--- Yr: 4.0x1c)'
<.
r =0.5
A :1.0
5=(3
til
a:O.O
0:
.5
...w
....
Z
<.
u
w
til
O.
10- 6
10-5
10- 4
SHEAR
Fig. 3.6
10-3
STRAIN,
Hodel and Empirical GIG
m
10-2
Y
with y for Sands
100
."
Wseed and Idriss (1970 - range of data)
y r= 7.5.x10-~
:::J
0
a
~
>
o·
~
t-
ee:
w
\!)
u
z
...J
4:
>
:::J
0
w
Y,. =4. Ox1 0-"
*
(/)
r =0.5
A=1.0
.5
5=(3
4:
a=O.O
lIZ!:
Z
a..
~
a
Q
10-6
10-5
10-3
10~
SHEAR
Fig. 3.7
-
1.
STRAIN,
10- 2
Y
Model and Empirical D with y for Sands
- - Tatsuoka et a/ (1979)
y.,.=7.5x1(r4
)(
(/)
:::J
0
u
a
(/)
6
>
~
~
ee:
~
t9
Z
>
a..
z
4:
4:
~
2:
w
Cl
0
\::4.0xKf'
r =0.5
A =1.0
C =(3
o. =0.0
.5
-
<{
O.
10- 6
,o-~
SHEAR
Fig. 3.8
10-2
10-3
10- 4
STRAIN,
Y
Model and Empirical D with y for Sands
101
'1.
- - - Ramberg - OsgOOd (Richart. 1975)
en
::>
0
u
en
*
c
(1=
0.00 }
Model
--(1::0.025
o·
> i=
~
l-
e::
w
...J
c.!)
z
r =0.50
A =1.0
.5
Z
~
> a:::
:5 ~
0 ~
w a
5=a
10'
10°
SHEAR STRAIN.
Fig. 3.9
Y/Yr ·
Ramberg-Osgood' s (Richart, 1975) and
D with y for Sands
~ven'
s Model
1•
A=1.0
E
5=~
(1=0.0
·8
j-I
""'~
vi
en
w
e::
I-
en
.6
- - - - - Hardin and Drnevich
~r=0.25
.4
--r=0.20
a:
~
w
~
.2
O·
0,
1.
2.
3.
SHEAR STRAIN,
Fig. 3.10
4.
5.
Y/Yr
Model and Empirical Skeleton Curve for Clays
6.
102
- - - Seed and Idriss (1970)
1.
~
(!)
- * -Hardin and
-"*'
X
.8
Drnevich
r = 0.25
---r=0.20
vf
:::)
...J
:J
Q
.6
0
2:
0::
<:
.4
A =10
5 =s
a =0.0
~
(/'J
.....
z
.2
4:
U
~
O.
10~
10~
10- 4
SHEAR
Fig. 3.11
1.
IJ')
:J
0
U
IJ')
.
Model and Empirical GIG
with y for Clays
!lI.t.Seed and Idriss (1970 ..
Range 01
m
- *-
!
a
10-3
STRAIN, Y
Oat a)
Tsai, L.m and Martin (1980)
~Yr=4.0X10~
-_.'f',.= 3. Ox 10-3
0
;; .....
..... 4:
0::
z
w (!)
r =0.25
A :11.0
.5
5 :IS
...J
« z
a=O.O
> a:::
:; 2:
«
(3
UJ Q
O.
10- 6
.
10-5
10- 4
10-3
SHEAR STRAIN,
Fig. 3.12
10-2
Y
Hodel and Empirical D with y for Clays
103
.20
~ ~ I-J.1 5
a::
\I
t/)
(I-J
l-
0:
or = 0.5 ~
.1 0
0'
<i
W
I-
VJ
0:
I
<i
.05
O.
2
3
10
5
20 30 50
100
CYCLES TO LIQUEFACTION,
Fig. 3.13a
200
N{
Nt vs T (DeAlba et aI, 1975)
~1J 1.0
:c
.So
.60
D
.40
r
= 0.54
l-
I
(!)
.20
o.
.05
.10
.15
.20
.1 5
SHEAR STRESS RATIO, 1:
1:
=-d
vo
Fig. 3.13b
Weight Function, h(T)
L:J
6
1.0
Dr
I-
<i-
Q:
VJ
VJ
=0.54
.8
w
0:
w ::>
VJ
u t/)
x w
w a:
.6
w
a:
0
a..
.2
.4
0-
10
100
CYCLES OF LOADING,
Fig. 3.13c
N
Excess Pore Pressure for Uniform Loading
104
~ h:?20
D
o~
....<{ .15
r
= 0.45
a::
til
til
W
.10
a::
....
til
a::
.05
4:
~ O.
10
100
CYCLES TO LIQUEFACTION, N~
Fig. 3.14a
0
w
=>
c.n
-
L
(Uartin, Finn and Seed, 1975)
1.0
....<t:
a::
NQ, vs
D
r
= 0.45
0.8
a::
til
W
a::
0.6
~
w
a:
Martin et aJ (1975 )
0.4
0
- - -This Study
~
r.n
tf)
w
u
x
w
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)
'105
z/z u
.3
1
.2
2
3
4
,
.
Y/Y
____--~~--__- -__~~4_~~__--~~--__-r
.6
4
.5
3
.2
.3
.4
.3
.4
.5
.2
2 1
.3
Fig. 3.15
ru
1.0
Sand Stiffness Deterioration
,
---rw
O.B
/
r
I
E
/
/
/
0.6
/'
./
./
04
./.
0.2
O.
.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
106
1.0
w
a:
0
a.
to?
~
0.6
w
0.4
c.n
c.n
a::
~
w
a:::
w
0.8
0
Martin
:::)
c.n
c.n
w
a:::
a.
et a/ (1975)
- - - This Study
0.2
o.
2
3
5
4
6
7
CYCLES OF LOADING, N
Fig. 3.17
c...:l
w
a:::
0
a.
c.n
c.n
w
U
X
W
Q-
1.0
Excess Pore Pressure with
Cycles of Loading
Martin et al (1975)
- - - This Study
0.8
~
«
a:::
0.6
w
a:
:::)
If)
If)
w
a:::
a..
0.4
0.2
o.
2
3
CYCLES
4
5
6
7
OF LOADING, N
Fig. 3.18 Excess Pore Pressure with
Cycles of Loading (Table 3.4)
'107
C"lu
o
Q)
Housner and Jennings
0
tn
-...
Ne
..........
9
W : 1-6.9 radlsec, ~= 0.94
B
.......... 20
'-'
=0.08
arm s
ROOT MEAN SQUARE ACC ELERATION:
Lai's
data average
10
3
'-'
(/)
0
Wo
2TT
0
Brr
6rr
4TT
FREQUENCY,
Fig. 4.1
12TT
1OTT
w
14rr
(rad / sec)
PSD Function for "Rock rr Sites
..........
{\
I\
arms =0.08 9
I I
I
I
I
I
We =STT radl sec
~ ~B=O.4
I
I
1
I
I
\
W
/
\
X
./"
./'
\
~ ~
B
"e
=1SlT radlsec
=0.4
Wa= 25rr rad/se~
S = 0.4
\ ,.."
/\.................
W =35lT rad/sec
8
B
/ ' . . . . . . .~ - -cc:-.------- r':J 8=0.4
A""'"
-,-.........
--...--
-- ---.:::::- --" --- --_--------........
........
o--~~~~-----~~--~--~~----~-~~-~~~~--~
14TT
16IT
18TT
err
10fT
12TT
4lT
o
FREQUENCY,
Fig. 4.2
W
(rad/sec)
Kanai-Tajimi PSD Function for Several wE
108
Gmo
Lmo
1
I
J
10
9
-
2
1
I
I
I
I
~
....
I-.
3
-
7
5
2
I
"-
8
6
I
10 pst
5
10 pSf
I-
....
....
..,
0
0
-
I-
"-
I
ltl
....
)(
0
I.
~
4
~
1-0
3
....
2
I.
l-
1
I
I
~ig.
5.1
I
,
I
I
Homogeneous Sand Deposit
I
I
I
109
- - - - Finn et aI (1978)
~
-
o
-
o>
"
.2
-
-
This Study
I
1C'P-----Ovo=782
pst
I
~~___----Ovo: 1720
.1
pst
<0=2973 pst
o.
10
1
100
1000
CYCLES TO UQUEFACTION,
Fig. 5.2
N£
Cyclic Resistance Curves for Example I
.2
en
"'-
Z
a
I-
<t:
a:
w
'"
.1
...J
a
w
u
u
max
~TL+aTL
" "
OT.
L
=0.07 9
«
-.....;:
0max =005 9
~
<t:
.........
---= ----=~--
W
a..
O.
O.
10.
1.
TIME
Fig. 5.3
TO LIQUEFACTION,
TL (seconds)
Statistics of Time until Liquefaction
100.
110
10
9
a
8
a0..,0
-2.1
Co
7
6
(J)
I-
z
W
~
5
4
W
...J
w 3
2
1
0
o.
0.10
0.05
ROOT MEAN SQUARE .ACCELERATION,
Fig. 5.4
0.15
Uo T
Profile of Total Accelerations
1
o
2
4
TIME,
Fig. 5.5
t
6
8
(seconds)
~1odulating Function of the Base Excitation
( 9 )
III
1.
'-.:J
0"
f=
<{
Ct:
W
Ct:
::J
(./)
(./)
w
a.
w
Ct:
.3
Ct:
0
a.
U1
U1
w
u
x
lLJ
.2
.1
O.
O.
1.
3.
2.
TIME,
Fig. 5.6a
t
4.
(seconds)
Expected Excess Pore Pressure Rise
200------~~----~--~----~--~~
100
vi
(/)
I.LJ
Z
IJ..
IJ..
~
(/)
t(sec)
o
o.
TIME,
Fig. 5.6b
3.
2.
t
4.
(seconds)
Stiffness Deterioration
5.
5.
112
===-______ } Dr: 50
~EDIUM
60'
%
D : 0.45
50
SAND
mm
MEDIUM CLAY
60'
DENSE SAND
Fig. 5.7
Layered Deposit
0.4
- - -
L
I
0.3
15 ft
- - 25
Ovo
tt
"&-::::::...~---- 6D
0.2
r
=0.0
0.1
0.0
1
10
CYCLES TO LlCUEFACTIO N,
Fig. 5.8
1000
100
N
J
Cyclic Resistance Curves for Example II
113
:3
2
5
4
3
-
10'
2d
"'0'
\
I
Gm
10
l'
4
.
20 30
I
tm
( psi)
(10 psi)
2
-
90'
1
150'
,
1
1
I
2
:3
10 20 30
Hi
7
't m
- 2a
r:I
6
I
1
4
(10 psi)
sf/
(psi)
5
(j
4
9d
3
0'
2
13d
1
150'
2
7
6
5
1d
-=--
2d
:"d
4
\
3
Gm
(104 pSI)
.
10 20 30
~
1ln
(psi)
60'
:3
~'
2
120'
1
Fig. 5.9
150'
----
f
Lumped Mass Hodels for Example II
Ms"tz Reference Room
University o~ Illinois
BI06 NCEL
208 N. Romine Street
Urbana~. Illinois
61801
114
i J
j
Ji
I
I III
JOo0001
r
o001
2.
0.01
.10 9
a.u..
z
o
0.1
1.
toU
<
lL.
W
::>
x
w
a
z
o
::i
0.5
Q
>-
>::i
I-
:J
co
<
lL.
o
to-
co
-1.
0.9
..J
W
0:
<
co
o
0.95 a:
a..
-2.
0.1
10.0
1.0
TIME,
Fig. S.lOa
t
(seconds)
Reliability Against Liquefaction (IS-foot)
115
0.001
z
2.
o
....
u
.~
LLJ
1.
::>
x'"
"
::J
LIJ
a
z
IJ...
o
0.
~
.....
-1
CD
-1.
"*'" - iF Co )
<t
aJ
-~(a)
0.95 ~
-
o
-2.
0.98
~--~~~~~~--~~~~~~--~~~~~0.99
0.1
10.0
1.0
TIME,
Fig. S.10b
t
100.0
(seco nds)
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
Peak base
Fig. 5.11
.25
.30
.35
acceleration,
.40
.45
c max
(g)
.50
Seismic Hazard for Eureka (California)
.55 .60
117
2.
x
w
o
z
1.
o.
>-
- - 5 Dr =0.0
- - 5 =0.15
~
--.J
aJ
-t
~
--1
W
0:::
,
Dr
-2.~~~~~~~~~~~~~--~~~~
.1
TIME,
Fig. 5.12
10.
1.
t
100.
(seconds)
Sensitivity of Reliability to
on
r
2.
- 60r=0.15
--6 0r=o.o
1.
c
=020 9
max
c::L
X
w
O.
0
z
>-
-, .
~
--1
C!J
<
-2.
--1
w
0:
.1
1.
TIME,
Fig. 5.13
t
10.
(seconds)
Sensitivity of Reliability to
100.
°
s
u
118
.0001
~
3-
\
2.
.001
\
\
\
.15 9
t
~
\.109
\.15 9
\
\
0.
.S
-t
- .-Ex.ct
-2.
.1
'"
Pro QeMles
ncertain
t
.9
,"-
~peMies
~ tCMd
10.
99
100.
Fig. 5.14 Sensitivity of Reliability to the
Uncertainties in Soil Properties
119
.. 0>
~
•
wO.3
>
<{
j-J
Q"
~
a:
en
(J)
o NO-LIQUEFACTION
6)
0.2
I(J)
(GOOD ACCELERATION DATA)
.-
NO-L1 QUEFACTION ( ESTIMATED ACCELERATIONS)
·6
5
8
cjB
1
4
1JJ
0:
L10UEFACTION (GOOD ACCELERATION DATA)
0
0
07
9,-
11 s
0.1
0:
<{
W
J:
U1
0.0
5
STANDARD
10
15
PENETRATION
( CORRECTED
TO
25
20
N1
RESISTANCE,
I
crvo
:1
t on I 5 q ft )
Fig. 5.l5a Historical Data of Liquefaction and
No-Liquefaction (Partial Data)
0
- >
~
UJ
>
0.3
<!
!--'
0'"
I-
•
UQUEFACTION (GOOD ACCELERATrON DATA)
o
NO LIQUEFACTION (GOOD ACCELERATION
~
NO LIQUEFACTION (ESTIMATED ACCELERATIONS)
6.
0.2
<!
c:::
tR
w
0:
l-
1"
a
DATA)
2
30
~
10
0.1
~11
V')
c:::
<!
w
J:
(/')
0.0
20
GIBBS AND
40
HOLTZ
60
80
RELATIVE DENSIT~
100
Dr
Fig. 5.lSb Historical Data of Liquefaction and
No-Liquefaction (Partial Data)
_
•
LIQUEFACTION
8
NO-UQUEFACTION (GOOD ACCELERATION DATA)
~ &
b
(GOOD ACCELERATION DATA)
> 0.4
<l:
6
0.3
0:
6.
W
0:
0.2
1.
~09B
O~
0:::
<!
w
40
I
Ul
0.611
0.34
0.42
0.14
0.0062
0.33
0.92
0.26
0.11
2
3
10
11
5
6
9a
9T
.....
Ul
0.1
/
0
w
~
0
Dt
O.37<Qf<O.40
1
(/)
=O.30
0.64
0.32
0.41
0.09
0.0023
0.30
0.94
0.23
O.OB
t
t
=--:1-
avo
9T
11 (9
0
z
O.OK
5
10
15
20
25
STANDARD PENETRATION RESISTANCE CORRECTED
AN EFECTIVE VERTIC AL PRESSURE OF 1 ton/sq ft.
Fig. 5.16
AVE/~O)BOUNDAFlY
HYSTORY
w
:n
PROBABILITIES OF LlOUEFAC TlON
NO-LIQUEFACTION (ESTIMATED ACCELERATIONS)
...............
~
(1:
CASE
30
TO
N1
Predicted Probabilities of Liquefaction for Some Historical Data
( 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;
flDr
ELEMENT
(f t)
3.3 WT
3.7
10.
1 3.3
nD
r
N-SPT
D50
[(mean) (mm)
_14
0.50
0.50
14
().~
15
0.50
16
K
(Ih/ln)
tm
(OSI )
11 2
180
2-1 1
238
4.3
103
6.
85
133
8.4
4
85
151
10.8
3
85
168
13.2
2
85
1 82
15.7
1
85
196
1 8.
10
9
8
7
78
78
78
78
0.15
0.15
0.15
0.15
G
80
0.15
5
19
1.
2.5
3.4
22.
31.
40.
49.
58.
67.
Fig. 5.l7a
Sand Deposit for Case History 5 (Hachinohe)
DEPTH
(f t )
3.3
6.7
~O
ELEMENT
WT
-
1 O.
1 3.3
Q
r
Or
N -SPT
0 50
( mean) (mm)
015
3
4
015
4
015
015
4
K
Clb/in)
78
1 22
lm
( psi)
10
9
B
7
45
45
45
45
0.19
0.19
0.19
0.19
G
70
0.15
5
85
133
8.4
4
85
151
10.8
3
85
168
13.2
2
85
182
15.7
1
85
196
1 8.1
14
1.
-2.5
144·
-i.A
163
4.3
93
6.
22.
3"
40.
49.
58.
67.
Fig. 5.l7b
Sand Deposit for Case History 6 (Hachinohe)
122
DEPTH
Q
~O
ELEMENT
r
( fi)
3.3
6.7
10.
WT
-
1 3.3
75
75
O
r
0.15
0.15
0.15
0.12
N-SPT
(mean)
0
50
K
1:
m
(mm)
(osj)
-
-
(I bll n)
16
25
0.05
0 ..05
11 2
195
243
300
1.
3.
4.5
5.4
-
10
9
8
7
90
6
90
27
135
7.
5
90
30
157
9.
4
90
40
181
11.5
3
95
60
272
13.
2
75
242
1 5.
1
85
105
1 8.
78
22.
30.
38.
44.
0.15
SO.
25
0.50
30
67.
Fig. S.17e
DEPTH
(t t)
3.
11.5
20.
Sand Deposit for Case History 9 (Haehinohe)
~O
ELEMENT
-
35.
50.
80.
r
!"y
0 50
Dr N-SPT (mm)
(mean)
K
(I blin)
tm
(psi)
0.4
67
68
0.4
0.4
0.4
83
65
80
0.4
60
0.7
2.3
4.
6.5
10.
14.4
90
57
22.
3
90
68
31.
2
90
77
40.
1
90
85
50.
10
9
B
7
6
5
65
60
53
60
60
0.16
0.18
0.16
0.16
80
0.15
4
6
7
7
12
25
30
123.
165.
208.
250.
Fig. S.18a
Sand Deposit for Case History 1 (Niigata)
123
DEPTH
(f tj
3.
11.5
20.
35.
50.
ELEMENf
WT
-
nD
65
0.15
0.16
0.16
0.16
r
N -SPT D50
(mea n) (mm)
K
(Ib/in>
lm
( ps i)
85
0.7
2.3
4.
6.5
10.
60
14.4
90
57
22.
3
90
68
31.
2
90
77
40.
1
90
85
50.
10
9
8
7
80.
~D
r
GO
6
64
64
64
5
80
4
0.15
6
9
12
15
25
0.4
0.4
0.4
0.4
>30
67
71
95
68
123.
165.
20B
250.
Fig. S.18b
DE PTH
(f t )
12.
20.5
29.
44.
59.
Sand Deposit for Case Histories 3, 10 and 11 (Niigata)
~D
ELEMENT
WT
-
r
nDr
N -SPT
DSn9
(meCiln) (mm
K
lm
(Ib/rn)
(DS i)
68.
2.8
6.5
8.3
11.
14.
1 9.
90
62.
26.
3
90
72.
35.
2
90
81.
44.
1
90
89.
53.
10
8
7
6
65
60
53
60
80
5
90
4
9
0.15
0.16
0.18
0.16
6
7
7
12
25
> 30
0.4
0.4
0.4
0.4
65.
11 4.
11 8.
83.
95.
89.
132.
1 74.
217.
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:
(A.l)
(A.2)
where
(A.3a)
(A.3b)
rO.O
r
u z
1T
(A.3c)
F4
r
=-
lIT
r-1
(A.3d)
p. 0.0
uz U z
e = arctan(
11 -
P
uz
P
uz )
(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 )}
the same order of [G],
be
the
=
solution
{c}.
of
Let the matrix [<p(t)J, of
the
matrix
differential
equation
.
[¢(t)] =-[G(t)][¢(t)]
with
toe
initial
(B.2)
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}
(B.3)
In particular
lS(t,t)]
+
[G(t)][S(t,t)]
+
[S(t,t)][G(t)]
T
2nS {F}{F}T
o
(B.4)
with
[Set ,t )]
o 0
(B.5)
126
Eq. B.3 may be used to calculate the
matrix
of
the
response.
nonzero
time
lag
covariance
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
s
= v.
If
both
sides
of
Eq.
B.6
are
[~(v)J-land
by
premultiplied by
making
[~(s)]
the
following equation results:
[¢(s)][¢(v)]
-1
[¢(s)]{ .•. }[4J(v)]T
[S(v,v)]
The right hand side of Eq. B.7
~s
(B.7)
precisely the right hand side
of
Eq.
B.3, then
[9(s)] [¢(v)]
[S(s,v)]
-1
which is the same as Eq. 2.37.
for s > v
[S(v,v)]
(B.B)
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
B.3 and B.B may
[~(t)]
not
covariance matrix.
can not be inverted with enough accuracy and Eqs.
be
used
to
calculate
In this case, for s
= v,
the
non
is
time
lag
it is true that
3[S(s.v)] _ 3[¢(s)]
- - ._-l~_,
,.,
d;'
- dS
x L¢(v)J - lStv,v)J
which
zero
(B.9)
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)
dS
t= _[G (S) ] [ S (S ,V) ]
for
S >
The solution of the differential equation, Eq. B.lO,
conditions
[S(s,v)] s
of the response.
Eq.
B.lO
=v =
v
(B.lO)
with
the
initial
[S(v,v)], is the two time covariance matrix
No numerical
~roblems
were found with the solution
of
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)]
(iii)
Because
the
values
LS
not known.
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
negligible.
to 8.
LS
T is the apparant period of vibration and n - 5
128
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2.
Amin, M., and Ang, A. H-S., "Nonstationary Stochastic Model for
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3.
Anderson, D. G., "Dynamic Modulus of Cohesive Soils," Thesis
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4.
Ang, A. H-S., and Tang, W. H., Probability Concepts in Engineering
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Baber, T. T., and Wen, Y. K., "Stochast ic Equivalent Linearization
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Hysteretic, Degrading, Multistory Structures," Structural
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of
Civil
Engineering,
University of Illinois, Urbana, Illinois, Dec., 1979.
6.
Baber, T. T., and Wen, Y. K., "Random Vibration of Hysteretic,
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7.
Bartels, R H., and Stewart, G. W., "Solution of the Matrix
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AX + XB = C," Algorithm 432, Communications of the
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8.
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9.
Christian, J. T., and Swiger, W. F., "Statistics of Liquefaction
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10.
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Clough, R. W., and Penzien, J., Dynamics of Structures, McGraw-Hill
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129
12.
Committee on Soil Dynamics
of
"Definition of Terms Related to
the
Geotechnical
Division,
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13.
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14.
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Donovan, N. C., and Singh, S., "Liquefaction Criteria for Trans
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18.
Faccioli, E., IfA Stochastic Model for Predicting Seismic Failure in
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19.
Faccioli, E., and Ramirez, J., "Earthquake Response of Nonlinear
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Fardis, M. N., "Probabilistic
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of
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During
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Massachusetts
Institute
of
Technology,
Cambridge,
Massachusetts, Mar., 1979.
21.
Fardis, M. N., and Veneziano, D., "Estimation of SPT-N and Relative
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130
23.
Fardis, M. N., and Veneziano, D., "Probabilistic Analysis of
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