a biot formulation for geotechnical earthquake engineering applications

A numerical study of sand liquefaction is performed and compared with the centrifuge experimental results to show the capabilities of the proposed formulation on pore water pressure gene

A BIOT FORMULATION FOR GEOTECHNICAL EARTHQUAKE ENGINEERING APPLICATIONS

of the requirement for the degree of

Doctor of Philosophy Department of Civil, Environmental, and Architectural Engineering

2006

For Tote #:

Vol.Issue: 67-05

School Code: 0051 E COL-BOUL-E

See :

CAO 2006

USE THESE INSTRUCTIONS FOR BEPRESS SUBMISSIONS Do not photo lib pg do not use acct

Normal size (8 1/2 X 11) Camera and EBeam

This thesis entitled:

A Biot Formulation for Geotechnical Earthquake Engineering Applications

written by Yu Bao has been approved for the Department of Civil, Environmental, and

A Biot Formulation for Geotechnical Earthquake Engineering Applications

Thesis directed by Professor Stein Sture

The mechanical behavior of saturated soil is mainly governed by the interaction between the soil skeleton and the pore fluid, and this interaction may lead

to significant loss of strength known as liquefaction under seismic loadings The main objective of this thesis is to develop and implement a cyclic constitutive model capable of modeling soil skeleton dilatancy during earthquake excitation The constitutive model is based on the fuzzy-set plasticity theory and enhancement is made on the description of dilatancy behavior under cyclic loading A robust Biot formulation, in which the governing equations of motion of the soil mixture are coupled with the global mass balance equations, is developed to describe the realistic behavior of saturated soil The finite element discretization is established without neglecting the convective terms An unconditionally stable implicit time integration scheme, Hilber-Hughes-Taylor α method is used and an iterative algorithm based on Newton-Raphson method is developed to solve the nonlinear time-discretized problem A numerical study of sand liquefaction is performed and compared with the centrifuge experimental results to show the capabilities of the proposed formulation

on pore water pressure generation and strength loss occurred in loose granular soil deposit under cyclic loading The computed results show good agreement with the experimental data The capability of the enhanced fuzzy-set model in simulating cyclic soil behaviors including liquefaction is validated It is concluded that the developed Biot formulation and computational procedure are an effective means to assess liquefaction potential and liquefaction-related deformations

Dedication

To My Family For your love and encouragement

Acknowledgments

I would like to express my most sincere appreciation to my advisor Professor Stein Sture for his invaluable guidance, encouragement and support throughout this study

I would like to thank Professor Hon-Yim Ko for his friendship, and his guidance and support in centrifuge modeling I am also very grateful to Professor Richard Regueiro for his sharing of knowledge and support

I am grateful to Professor Tad Pfeffer and Professor Carlos Felippa for serving as

my defense committee I would like to thank Professor Ronald Pak, who was my academic advisor during my first year of Ph.D study

My appreciation goes to Dr Yu-Ning Ge for his help in understanding the set plasticity constitutive model and Dr Sung Ryul Kim for his help in the centrifuge experiments

My thanks go to my parents, my husband Miao and my son Dylan for their love, encouragement and support

CONTENTS

CHAPTER

1 INTRODUCTION………1

1.1 Motivation……… 1

1.2 Research Focus……… 4

1.3 Scope of Work and Layout………5

2 BACKGROUND AND LITERATURE REVIEW……… 7

2.1 Laboratory Tests………8

2.2 Centrifuge Modeling………15

2.3 Constitutive Modeling……….16

2.4 The Theory of Mixtures……… 24

2.5 Finite Element Implementation………26

2.6 Time Integration Scheme……….27

3 ENHANCED FUZZY-SET PLASTICITY CONSTITUTIVE MODEL… 29

3.1 Fuzzy Sets………30

3.2 Classical Plasticity Theory……… 31

3.3 Fuzzy-Set Plasticity Theory……….35

3.4 Stress Control Formulation in p-q Space……….38

3.5 Stress Control Formulation in Cartesian Stress Space……….46

3.6 Strain Control Formulation……… 50

3.7 2-D Plane Strain Formulation……… 52

3.8 Kinematic Mechanism of Deviatoric Membership Function γd………54

3.9 Kinematic Mechanism of Locking Membership Function γl…………62

3.10 Model Parameters………67

3.11 Model Capabilities on Cyclic Mobility………67

3.12 Model Responses……….71

3.13 Dilatancy Parameters……… 91

3.14 Model Calibration………93

3.14.1 Unconstrained Numerical Optimization……… 93

3.14.2 Numerical Optimization in Constitutive Model Calibration 96

4 THEORY OF MIXTURES FOR FLUID-SATURATED POROUS MEDIA……… 98

4.1 Average Quantities……… 99

4.2 Laws of Balance……….103

4.2.1 Balance of Mass……….104

4.2.2 Balance of Linear Momentum……… 107

4.2.3 Balance of Angular Momentum……….110

4.2.4 First Law of Thermodynamics (Balance of Energy)……….113

4.2.5 Second Law of Thermodynamics (Entropy Inequality)…….113

4.3 Field Equations for Saturated Soil……….114

5 FINITE ELEMENT FORMULATION….……… 119

5.1 The Finite Element Method (FEM)……… 119

5.2 Matrix Form of the Field Equations for Saturated Soil……….122

5.3 FEM Form of the Balance of Linear Momentum in the Solid Phase…125 5.4 FEM Form of the Balance of Linear Momentum in the Fluid Phase…129 5.5 FEM Form of Conservation of Mass for the Mixture………132

5.6 Combination of the Discretized Governing Equations……… 134

6 COUPLED FINITE ELEMENT – INFINITE ELEMENT MODEL…… 137

6.1 Mixed Displacement – Pore Pressure Element……… 138

6.1.1 2-D 9-node Lagrangian Isoparametric Quadrilateral Element…… 139

6.1.2 4-node Lagrangian Isoparametric Quadrilateral Element…………145

6.2 2-D 6-Node Infinite Element………147

6.2.1 1-D 2-Node Infinite Element………148

6.2.2 2-D 6-Node Infinite Element……… 150

6.3 2-D Coupled Finite Element - Infinite Element Numerical Model… 158

7 TIME INTEGRATION AND NONLINEAR ANALYSIS……… 159

7.1 Direct Time Integration Techniques……… 160

7.1.1 Newmark Method………161

7.1.2 Hilber-Hughes-Taylor α-Method……….163

7.2 Implementation of Hilber-Hughes-Taylor α-Method……… 165

7.3 Newton-Raphson Method: Nonlinear Analysis……….167

7.4 Calculation Procedure……… 169

8 VERIFICATION OF NUMERICAL SIMULATION……… 172

8.1 Centrifuge Modeling of Soil Liquefaction……….173

8.1.1 Centrifuge Model Test on a Layer of Liquefiable Sand Deposit… 173

8.1.2 Experimental Results………176

8.2 Fully Coupled FEM Code “DYNSOILS”……… 187

8.2.1 Pre-Processing Module……… 187

8.2.2 Post-Processing Module……….188

8.2.3 Analysis Module………188 8.3 Case 1: A Liquefiable Layer of Sand Deposit

– Comparison of Numerical Simulation with Centrifuge Experiment 190

8.4 Case 2: A Footing Subject to Vertical Sinusoidal Loading………… 206

8.5 Summary………215

9 SUMMARY, CONCLUSION AND RECOMMENDATION FOR FUTURE WORK……… 216

9.1 Summary and Conclusions………216

9.2 Recommendation for Future Work………218

BIBLIOGRAPHY………219

TABLE

Table

2.1 Table of Scaling Relations (Centrifuge Modeling at N*g)……… 15 3.1 Fuzzy-Set Model Parameters……… …72 3.2 Dilatancy Parameters……… 91

FIGURES

Figure

2.1 Excess Pore-Water- Pressure Response Behavior and Deviatoric Stress – Strain Curve during an Undrained Stress-Controlled Cyclic Triaxial Test of Nevada Sand (Dr = 40%) (Arulmoli et al., 1992)……….9 2.2 Stress-Strain and Excess Pore Water Pressure Histories during an Undrained Stress-Controlled Cyclic Triaxial Test of Bonnie Silt (Arulmoli et al., 1992) 10 2.3 Variations of Shear Stress with Axial Strain (Lee and Schofield, 1988)………11 2.4 Stress-strain Curve and Stress-path for Nevada Sand of Dr = 60% Obtained from Cyclic Undrained Simple Shear Test (Arulmoli et al., 1992)……….12 2.5 Recorded Time Histories of Acceleration and Pore Water Pressure (Koga and Matsuo, 1990)……….13 2.6 Time Records of Acceleration and Lateral Displacement Responses (Ishihara et al., 1991)……….14 2.7 Iai’s Model Performance (Iai 1991)……… 17 2.8 Schematic Illustration of Multi-Surface Model……… 19 2.9 Schematic Illustration of the Bounding Surface in Uniaxial Stress-Strain Space (reproduced from Dafalias and Popov, 1975)……….20 2.10 Schematic Representation of a Loading and a Bounding Surface (reproduced from Dafalias and Popov, 1975)……….21 2.11 Schematic Illustration of Fuzzy-Set Model……….23 3.1 “Fuzzy” Yield Surface Specified by a Given Constant Value of the Membership Function (Klisinski, 1987)……… 31 3.2 Lode’s Angle in Deviatoric Plane (William and Warnke, 1973)………40

3.3 Effective Stress Path under Undrained Cyclic Loading……… 42 3.4 Deviatoric Stress – Strain Response Curve of Two Unloading-Reloading Cycles……… 55 3.5 (a) Pictorial Illustration of Kinematic Mechanism of Deviatoric Membership Function γdfrom Point “a” to Point “b”……….56 (b) Pictorial Illustration of Kinematic Mechanism of Deviatoric Membership Function γdfrom Point “b” to Point “c”……… 57 (c) Pictorial Illustration of Kinematic Mechanism of Deviatoric Membership Function γdfrom Point “c” to Point “d”……….58 (d) Pictorial Illustration of Kinematic Mechanism of Deviatoric Membership Function γdfrom Point “d” to Point “e”……….59 (e) Pictorial Illustration of Kinematic Mechanism of Deviatoric Membership Function γdfrom Point “e” to Point “f”……….60 3.6 Mean Stress – Strain Curve of Two Unloading – Reloading Cycles………….63 3.7 (a) Pictorial Illustration of Kinematic mechanism of locking membership Function γlfrom Point “a” to Point “b”……….63 (b) Pictorial Illustration of Kinematic mechanism of locking membership Function γlfrom Point “b” to Point “c”……… ….64 (c) Pictorial Illustration of Kinematic mechanism of locking membership Function γlfrom Point “c” to Point “d”………64 (d) Pictorial Illustration of Kinematic mechanism of locking membership Function γlfrom Point “d” to Point “e”……… 65 (e) Pictorial Illustration of Kinematic mechanism of locking membership Function γlfrom Point “e” to Point “f”………65 3.8 Schematic Cross section of Sand Particles’ Deformation under Cyclic Loading (Youd, 1977)……….……… 68 3.9 Stress – Strain Curve under Cyclic Loading……… 69

3.10 Pore-Water-Pressure vs Deviatoric Strain Response under Undrained Cyclic

Loading………70

3.11 Stress Path Showing Cyclic Mobility……… 71

3.12 Loading History of Deviatoric Stress……… 73

3.13 (a) Deviatoric Strain History……….74

(b) Deviatoric Stress – Deviatoric Strain Curve………75

(c) Pore Water Pressure – Deviatoric Strain Response Curve……… 76

(d) Pore Water Pressure Buildup History……… 77

(e) Effective Stress Path Approaching Liquefaction……….78

3.14 Loading History of Deviatoric Stress with Increasing Amplitude……… 79

3.15 (a) Deviatoric Strain History……….80

(b) Deviatoric Stress – Deviatoric Strain Curve………81

(c) Pore Water Pressure – Deviatoric Strain Response Curve……… 82

(d) Pore Water Pressure Buildup History……… 83

3.15 (e) Effective Stress Path Approaching Liquefaction……….84

3.16 Loading History of Deviatoric Stress with Decreasing Amplitude………… 85

3.17 (a) Deviatoric Strain History……… 86

(b) Deviatoric Stress – Deviatoric Strain Curve………87

(c) Pore Water Pressure – Deviatoric Strain Response Curve……… 88

(d) Pore Water Pressure Buildup History……… 89

(e) Effective Stress Path Approaching Liquefaction……….90

3.18 Effect of Change of Dilatancy Parameters on Pore Water Pressure Buildup…92 4.1 Typical Averaging Volume………100

6.1 Mixed Displacement-Pore Pressure Element……….138

6.2 2-D 9-node Isoparametric Quadrilateral Element……… 139

6.3 2-D 4-node Isoparametric Quadrilateral Element……… 145

6.4 1-D infinite element……… 148

6.5 2-D 6-node infinite element……… 150

7.1 Stability of Newmark Time Integration Scheme……… 163

8.1 Cross-Section of the Model and Instrumentation Layout……… 175

8.2 Particle Gradation Curve for Nevada Sand #120……… 176

8.3 Base Input Motion: Horizontal Acceleration……… 177

8.4 Measured Horizontal Acceleration at the Surface……….178

8.5 Measured Horizontal Acceleration at the Depth of 2 m……… 179

8.6 Measured Horizontal Acceleration at the Depth of 4 m………180

8.7 Measured Horizontal Acceleration at the Depth of 6 m………181

8.8 Measured Horizontal Acceleration at the Depth of 8 m………182

8.9 Measured Excess Pore Water Pressure at Different Depths……… 183

8.10 Short-Term Excess Pore Water Pressure History…… ……….184

8.11 Measured Surface Settlement……… 185

8.12 Short-Term Settlement Record………186

8.13 Flowchart of Computing Procedure……….189

8.14 Flowchart for Fuzzy-Set Plasticity Constitutive Model……… 190

8.15 2-D FEM Mesh………192

8.16 Base Horizontal Acceleration……… 193

8.17 Short-Term Pore Water Pressure at the Depth of 9.6 m……… 194

8.18 Short-Term Pore Water Pressure at the Depth of 8 m……….195

8.19 Short-Term Pore Water Pressure at the Depth of 6 m……….196

8.20 Short-Term Pore Water Pressure at the Depth of 4 m……….197

8.21 Short-Term Pore Water Pressure at the Depth of 2 m……….198

8.22 Short-Term Settlement……….199

8.23 Long-Term Pore Water Pressure at the Depth of 9.6 m……… 200

8.24 Long-Term Pore Water Pressure at the Depth of 8 m……….201

8.25 Long-Term Pore Water Pressure at the Depth of 6 m……….202

8.26 Long-Term Pore Water Pressure at the Depth of 4 m……….203

8.27 Long-Term Pore Water Pressure at the Depth of 2 m……….204

8.28 Long-Term Settlement……….205

8.29 Finite Element Mesh with Boundary Conditions……….208

8.30 Input Distribution Force……… 209

8.31 Settlement……….210

8.32 Excess Pore Water Pressure at 2 m……… 211

8.33 Excess Water Pressure at 4 m……… 212

8.34 Excess Water Pressure at 8 m……… 213

8.35 Deformed Mesh……… 214

CHAPTER 1 INTRODUCTION

1.1 Motivation

Liquefaction is a phenomenon in which the strength and stiffness of a soil is reduced by strong ground motion or earthquake shaking or other rapid cyclic loading Liquefaction has historically been responsible for tremendous amounts of damage including landslides, differential settlements, lateral spreading, structural and earth system failures throughout the world Cyclic mobility is a liquefaction phenomenon, triggered by cyclic loading, occurring in soil deposits with static shear stresses lower than the soil strength Deformations due to cyclic mobility develop incrementally, because of static and dynamic stresses that develop during an earthquake Lateral spreading, a common result of cyclic mobility, can occur on gently sloping and on relatively flat ground close to rivers and lakes Liquefaction problems have received

a great deal of attention, and great efforts have been made to understand the basic mechanism and phenomenon However, due to the complexity of liquefaction problems, such as nonlinearity of responses, sudden phase transition from solid to liquid behavior, material instability, interaction and relative movement between the porous soil skeleton and interstitial water, limitations on experimental and numerical techniques, material model formulation and so on, reliable and accurate predictive methods have yet to be developed

Since 1960’s, significant progress has been made in understanding the liquefaction phenomena There are three main approaches: (1) field observations before, during and/or after earthquakes, (2) laboratory experiments, and (3) numerical simulations Among the theoretical studies, three different methods have been developed: (1) total stress approaches, (2) quasi-effective stress approaches, and (3) effective stress – based techniques, which are also referred to as fully coupled methods

Strong ground motion induces a tendency for volume change and the skeleton dilation/contraction effects are of great importance to cause a progressive pore water pressure build-up as well as cyclic pore pressure variations depending on the drainage condition and soil permeability Even if liquefaction were not to take place, the development of excess pore pressures may lead to excessive soil softening, weakening or to partial loss of stability and even to bearing capacity failures Rational analysis of the development of earthquake – induced pore pressures requires

soil-a fundsoil-amentsoil-al description of the soil’s constitutive relsoil-ations Therefore, the soil-ability of the constitutive model to predict permanent volume changes during cyclic loading becomes a major factor in seismic analysis

Although some geotechnical engineering problems may be idealized as static analysis or dynamic analysis in one-phase media, due to the coupled effects of the solid and the fluid, it is difficult to predict the soil dynamic behavior by using simplified procedures A saturated soil behaves as a two-phase system and any comprehensive analysis should take into consideration the interaction between the soil skeleton and the interstitial fluid The mixture theory, characterized by the concept of volume fractions, perhaps yields the most consistently developed framework for the treatment of liquid-saturated porous solids

The finite element method (FEM) is considered as a powerful numerical technique in solving geotechnical engineering problems However, every finite

element model has to be terminated at some finite boundary For wave propagation problems, the usual finite boundary of the finite element model will cause the seismic waves to be reflected and superimposed on the progressing waves Therefore, the boundary conditions involved in simulating the dynamic behavior of semi-infinite soil media are a major challenge in FEM The desired boundary should ideally be radiating to outgoing waves and transparent to incoming waves In addition, the boundary should be located as close to the finite structure as possible for computational efficiency The infinite element is a recently developed technique to deal with the infinite media in the finite element analysis The basic idea is to place elements with a special shape function for the geometry at the infinite boundary The accuracy of numerical analysis needs to be validated However, due to the difficulty of predicting when and where a major earthquake will occur and the general random nature of these events, most field liquefaction failures have occurred

at sites which were not instrumented The advent of centrifuge modeling that incorporates scaled dynamic events brought light on the development of proper numerical techniques to simulate the consequences of soil liquefaction Centrifuge testing creates stress conditions in the model that closely simulate those in the full-scale prototype, so that the behavior of the model can approximate that of the prototype Since 1988, the University of Colorado – Boulder (CU) has operated a

400 g-ton centrifuge, which among the ones having highest capacity in the world A servo-controlled electro-hydraulic shake table is mounted on the swing platform of the centrifuge, which could be operated inflight to produce earthquake-like motions The centrifuge is currently used to study the liquefaction mechanisms related to earth systems, as well as to validate numerical simulations A large body of knowledge is related to centrifuge experiments using a wide range of soils and medium-structure configurations The Nevada Sand, which was used in the VELACS project, is specific to this research

1.2 Research Focus

This thesis concentrates on developing and implementing an efficient, transparent and accurate constitutive model based on the fuzzy set plasticity theory to describe the nonlinear volume changes for granular soils and Biot field equations used in overall analysis Traditional concepts of critical state soil mechanics, state parameter and phase transformation surface are introduced in the enhanced model The new development of a strongly dilative/contractive phase beneath the failure surface is added and new analytical developments for the dilatancy parameter are presented The numerical optimization procedures for model calibration are investigated

In this thesis, the governing equations of motion of the soil mixture are coupled with the global mass balance equations, and necessary assumptions are made to obtain the equivalent Biot’s equations from the general balance equations The

x , u and x denote solid skeleton displacement, pore water pressure and fluid ( f)

displacement, respectively The convective terms in the general Biot equations are not neglected during the discretization and the effect of convective terms on the finite element formulation is investigated The three-parameter time integration scheme Hilber-Hughes-Taylor α -method is used to integrate the spatially discrete finite element equations

The developed fuzzy set plasticity constitutive model is implemented in the originally – developed fully coupled finite element code Several examples will be analyzed to show the capabilities of the finite element formulation Centrifuge experiments on soil liquefaction are conducted The experimental results are used to verify the elasto-plastic numerical solutions by comparing the numerical simulation results with the data obtained from centrifuge modeling tests

1.3 Scope of Work and Layout

The main purposes of the thesis are:

(1) To enhance the performance of the fuzzy-set plasticity model proposed by Klisinski (1988) and make the model more versatile and perform realistic analysis under a wider range of cyclic loading conditions during liquefaction

(2) To develop the dilatancy parameters in the enhanced constitutive model to simulate the shear-induced soil dilation/contraction along loading, unloading and reloading cycles

(3) To use the theory of mixtures to describe the behavior of saturated soil by using the volume fraction concepts

(4) To establish the coupled finite element – infinite element numerical model

to realistically simulate the boundary conditions of semi-infinite soil body under dynamic loadings

(5) To include the nonlinear convective terms, which in previous formulations have frequently been excluded for the sake of simplicity, in the finite element formulation and to develop the corresponding fully coupled FE analysis code

(6) To use the enhanced fuzzy-set constitutive model and the FE procedure to study pore water pressure development, the liquefaction phenomenon, and the consequences of liquefaction

(7) To conduct centrifuge experiments to validate the numerical procedure and compare physical and simulated phenomena

This thesis consists of 9 chapters A literature review of liquefaction research related to cyclic mobility is included in Chapter 2 Chapter 3 presents the formulation and improvement of the fuzzy-set plasticity model, along with the necessary material parameters calibration and model responses Chapter 4 addresses the modern theory

of mixtures and its application to modeling two-phase saturated soil The governing

equations of motion of saturated soil are also presented in Chapter 4 In Chapter 5, the finite element semi-discretization of the governing equations is established without neglecting the convective terms by means of a xs) −u−x(f) weighted residual Galerkin method The coupled finite element – infinite element numerical model is presented in Chapter 6 Time integration of the semi-discretized governing equations using the Hilber-Hughes-Taylor α-method and its implementation in the nonlinear problem is presented in Chapter 7 Several numerical examples, which include various geotechnical structures subjected to sinusoidal loadings, are presented

in Chapter 8 Centrifuge tests on liquefiable soils are also described in Chapter 8 Some of the numerical solutions are compared with the data obtained from centrifuge experiments Finally, Chapter 9 summarizes the thesis work that has been presented,

as well as conclusions and recommendations for future research work

CHAPTER 2 BACKGRPOUND AND LITERATURE REVIEW

The background and the literature review of liquefaction research related to cyclic mobility, including cyclic laboratory sample tests, centrifuge experiments, recent development of soil constitutive modeling, Biot-type fully coupled analysis and mixture theory, finite element implementation and time integration scheme, are described in this Chapter The seismic behaviors of soils have in many instances been recorded at specific locations and the liquefaction-induced shear-deformation mechanism has been observed These ground acceleration records show a strong influence of soil dilation or contraction at large cyclic shear strain excursions Dilation phases can cause significant regain in shear stiffness and strength and lead to

a strong restraining effect on the magnitude of cyclic and accumulated permanent shear strains, while a contractive soil skeleton typically leads to immediate liquefaction Under cyclic loading, a granular soil will show strongly coupled shear-dilation behavior In a saturated soil, the volume change caused by dilation or contraction is achieved by the migration of fluid into or from the pore-space This migration process will be retarded by the relatively low permeability of soil or the relatively fast rate of loading and result in pore water pressure generation and associated change of effective confinement Pioneering and traditional liquefaction studies related to the overall phenomenon and cyclic mobility include the work by Seed and Lee (1966), Casagrande (1975), Castro (1975), Castro and Poulos (1977) and Seed (1979)

2.1 Laboratory Tests

A thorough review of laboratory tests and case studies of sand liquefaction was conducted by Ishihara (1993) The triaxial test has been widely used for laboratory testing granular soils under both monotonic and cyclic loading conditions

to obtain constitutive parameters The cyclic triaxial test requires that the apparatus should be capable of applying extensional as well as compression loads to the soil specimen so that the cyclic stresses can reverse between triaxial compression and extension state Fig 2.1 Shows the result of a typical triaxial test when a cyclically imposed shear stress is present (Arulmoli et al 1992) Fig 2.2 (Arulmoli et al 1992) shows the stress-strain and excess pore water pressure response histories during an undrained stress-controlled cyclic triaxial test of Bonnie silt Fig 2.3 (Lee and Schofield, 1988) shows the sand response during one cycle in triaxial test investigations

Fig.2.1 Excess Pore-Water- Pressure Response Behavior and Deviatoric Stress – Strain Curve during an Undrained Stress-Controlled Cyclic Triaxial Test

of Nevada Sand (Dr = 40%) (Arulmoli et al., 1992)

Fig 2.2 Stress-Strain and Excess Pore Water Pressure Histories during an Undrained Stress-Controlled Cyclic Triaxial Test of Bonnie Silt (Arulmoli et al., 1992)

Fig 2.3 Variations of Shear Stress with Axial Strain (Lee and Schofield, 1988)

Any type of simple shear test apparatus can be used to test soil specimen under cyclic loading conditions if it is connected to a cyclic loader One of the representative responses of cyclic simple shear tests is illustrated by Fig 2.4 (Arulmoli et al 1992)

Laboratory shake table tests have been frequently used to study soil cyclic behaviors Most of these studies were conducted in Japan A number of tests were conducted by using Toyoura sand with relative density ranging from 68% to 79% (Koga and Matsuo, 1990) Fig 2.5 shows a representative of the recorded accelerations and pore water pressures Fig 2.6 shows the time histories of

acceleration and lateral displacement in a shake table test conducted by Ishihara et al (1991) Sasaki et al (1991, 1992) reported a series of tests on a liquefiable layer of loose sand prepared by an underwater drop method

Fig.2.4 Stress-strain Curve and Stress-path for Nevada Sand of Dr = 60% Obtained

from Cyclic Undrained Simple Shear Test (Arulmoli et al., 1992)

Fig 2.5 Recorded Time Histories of Acceleration and Pore Water Pressure

(Koga and Matsuo, 1990)

Fig 2.6 Time Records of Acceleration and Lateral Displacement Responses

(Ishihara et al 1991)

The laboratory testing results show that:

1) At low confinement, sands undergo volume changes when subject to shear loadings This process is known as the coupled shear-dilation

2) In a saturated soil, under relatively fast rate of loading, the dilation induced volume increase will result in an immediate change of the pore pressure and a consequent change of effective confinement

3) Due to the change of effective confinement, the soil stiffness instantaneously changes

4) For the same level of applied shear stress, increasing larger strain excursions show a cycle-by-cycle degradation of strength

2.2 Centrifuge Modeling

The basic principle of centrifuge small-scale modeling of earth structure systems is to create stress conditions in the model that closely simulate those in the full-scale prototype together with uniform and measurable soil properties and desired boundary and input conditions, so that the behavior of the model can approximate that

of the prototype Various scaling relations can be developed to enable the model measurements to be extrapolated to the corresponding prototype quantities The scaling relations for achieving similarity in geotechnical centrifuge modeling are presented in Table 2.1 (Ko, 1988)

Quantity Prototype Model

A large number of centrifuge tests using Nevada Sand have shown clear evidence in cyclic shear-dilatancy Many of these experiments were described in the VELACS (Verification of Liquefaction Analysis by Centrifuge Studies) project (Arulanandan and Scott, 1994) Relative density of the Nevada Sand varied from 40%

to 75%

2.3 Constitutive Modeling

Unlike most engineering materials, the constitutive relationships of soils are highly nonlinear from the very beginning of loading Deformation and strength characteristics of soils are greatly affected by such factors as soil structure, loading rate, stress history, strain level and current stress state In practice, it is necessary to idealize the soil behavior in order to develop usable constitutive models For example, under short term loading, the soil behavior can be simplified as time – independent

A number of isotropic and kinematic plasticity models have been developed

to simulate the processes associated with sand dilation during liquefaction (Prevost

1981, 1985, Sture, Mould and Ko 1982, Klisinski 1988, 1991, Iai 1991, Bardet et al

1993, Manzari and Dafalias 1997, Yang and Elgamal 2000) and many essential features of cyclic mobility have been successfully modeled Fig 2.7 shows the model responses of Iai’s constitutive model (1991) There are mainly three categories among these widely used plasticity models: multi-surface models, bounding surface model and fuzzy-set model

Fig 2.7 Iai’s Model Performance (Iai 1991)

The concept of multiple yield functions was first proposed by Koiter (1953) Iwan (1967) extended Koiter’s work by adding kinematic hardening to each yield surface An anisotropic hardening model for metals was developed by Mroz (1967) Based on Mroz’s work, Prevost (1975, 1977, 1979a, 1979b), and Prevost et

al (1981) study the behavior of clays under monotonic and cyclic loading conditions Mroz, Norris, and Zienkiewicz (1978) developed an anisotropic hardening model for soils to simulate both drained and undrained soil behaviors under cyclic loading Sture, Ko and Mould (1982) described the analytical procedure and application of a multi-surface anisotropic hardening model in detail and derived the explicit stress-strain relations Vermeer (1982) proposed an isotropic hardening model for both cone and cap yield surfaces

For multi-surface plasticity models, a collection of nesting surfaces is used The innermost surface bounds a region in which the stress-strain relation is assumed

to be linearly elastic and the outermost yield surface acts as the ultimate strength limit The strain increment associated with a stress increment is calculated by the magnitude of surface translation and the yield surface size change The surfaces translate by the stress point without intersecting each other during the movement The yield surfaces cannot overlap, and if they contact at the stress point, they translate together as a bundle and share the common tangent plane at the contact point The stress point and the collected nesting surfaces cannot move outside the outermost yield surface The schematic illustration of multi-surface model is shown in Fig 2.8

Fig 2.8 Schematic Illustration of Multi-Surface Model

Bounding surface concept was originally proposed by Dafalias (1975) and was developed by the motivation that the results of some monotonic and cyclic uniaxial tests show the convergence of the stress strain response curve (Fig 2.9), i.e there are “bounds” in the stress strain space These bounds cannot be crossed, but can change position during the loading process In bounding surface models, the variation of the state of stress and plastic modulus are defined on the basis of a very

simple radial mapping rule For each actual stress point within or on the bounding surface, a corresponding “image” point on the surface is specified as the intersection

of the surface with the straight line connecting the origin with the current stress point The actual plastic contribution to the total deformation is then assumed to be a function of the plastic modulus on the bounding surface, at the “image” point, and the distance between the actual stress point and its “image” This feature enables the plastic deformation to occur when the stress state lies on or within the bounding surface, by allowing the plastic modulus to be a decreasing function of the distance of the stress state from a corresponding point on the bounding surface The bounding surface formulation closely follows that of classical plasticity, requiring the definition

of the elastic response, loading (bounding) surface, flow rule, and hardening In addition, it is necessary to define the direction and magnitude of plastic strain occurring within the bounding surface The schematic illustration of bounding-surface model is shown in Fig 2.10

Fig 2.9 Schematic Illustration of the Bounding Surface in Uniaxial Stress-Strain

Space (reproduced from Dafalias and Popov 1975)

Fig 2.10 Schematic Representation of a Loading and a Bounding Surface

(reproduced from Dafalias and Popov 1975)

Fuzzy-set plasticity theory was first proposed by Klisinski et al (1987) and later enhanced by Kisinski (1991), Ge (2003) and Bao (2005) Due to its transparency and simplicity, this concept has received increasing attention recently The model is capable of simulating all essential nonlinear characteristics of soils, including nonlinear stress-strain and volume change behavior during unloading and reloading cycles In fuzzy-set plasticity, it is assumed that there exists an ultimate yield surface where the material behavior is entirely plastic In addition, the material behavior inside an initial yield surface is purely elastic The main difference in the formulation is that the elasto-plastic response between the initial and the ultimate yield surfaces is not characterized in the conventional sense but by a fuzzy set representing conditions between the elastic and plastic states For the fuzzy set plasticity theory, a real number γ( ) on the interval [0, 1] is assigned to each point

in the region F<0( F is the yield function) If the point lies on the yield

surfaceF =0, the value of the membership function γ( )is equal to zero; while the point lies in the purely elastic region, the corresponding value of γ( )is equal to one Thus, the membership function γ( ) for the fuzzy set F ≤0 represents the

“membership degree” of to the set of purely elastic material behavior Instead of determining plastic moduli from classical plasticity theory, they are defined in terms

of the value of a membership functionγ( )∈(0,1), such that γ =1for the purely elastic behavior, whereas γ =0when the stress point is on the ultimate yield surface The schematic illustration of fuzzy-set model is shown in Fig 2.11

Fig 2.11 Schematic Illustration of Fuzzy-Set Model

2.4 The Theory of Mixtures

The response of fluid-saturated porous media which are subjected to dependent loads has been studied since the early 1950s However, reliable procedures to describe the pore pressure changes and the effective stress field in the solid phase have not been developed yet In a saturated porous medium, solid and fluid phases co-exist and are in relative motion Therefore, the equations of motion for individual constituents involve interaction terms and the stress depends on kinematics of both phases

Before 1975, dynamic analysis of geotechnical engineering problems was mainly based on total stresses because of the lack of practical models that were capable of predicting pore water pressures Then a model presented by Martin, Finn and Seed (1975) revealed the densification mechanism during liquefaction They proposed that for saturated sand, if drainage is not allowed during the loading sequence, the tendency for volume reduction during each cycle of loading leads to a corresponding progressive increase in pore water pressure In this model, the plastic volumetric strain that occurs during one cycle of uniform shear strain in an undrained simple shear test is assumed to be equal to the total volumetric strain in a drained simple shear test

A general theory of three-dimensional deformation of porous fluid-saturated media was first proposed by Biot (1941) Later, Biot (1956, 1961) extended this quasistatic theory to wave propagation in saturated geological media For an isotropic saturated porous medium, Biot (1956, 1961) proposed the kinetic energy of the mixture to be quadratic in velocities of the solid and fluid and included a coupling term The dissipative function was postulated to be quadratic in relative velocity For constitutive equations, Biot (1941, 1956, 1961) stated that there exists an energy function quadratic in solid strain and change of water content (fluid strain) The