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Soil Mechanics for Unsaturated Soils

D. G. Fredlund, Ph.D.

Professar of Civil Engineering University of Saskatchewan Saskatoon, Saskatchewan

H. Rahardjo, Ph.D.

Senior Lecturer School of Civil and Structural Engineering

Nanyang Technological University

A Wi1ey-Interscience Pub1ication

JOHN WILEY & SONS, JNC.

New York • Chichester • Brisbane • Toronto • Singapore

CHAPTER l

CHAPTER 2

CONTENTS

Introduction to Unsaturated Soil Mechanics l. l Role of Climate

l . 2 Types of Problems 1.2.1 Construction and Operation of a Dam

1.2.2 Natura! Slopes Subjected to Environmental Changes

1.2.3 Mounding Below Waste Retention Ponds

1.2.4 Stability of Vertical or Near Vertica1 Excavations

1.2.5 Lateral Earth Pressures

1.2.6 Bearing Capacity for Shallow Foundations

1.2.7 Ground Movements Involving Expansive Soils

1.2.8 Collapsing Soils

1.2.9 Summary of Unsaturated Soils Examples

1.3 Typical Profiles of Unsaturated Soils

1.3.1 Typical Tropical Residua! Soil Profile

1.3.2 Typical Expansive Soils Profile

1.4 Need for Unsaturated Soil Mechanics l. 5 Scope of the Book

1.6 Phases of an Unsaturated Soil 1.6.1 Definition of a Phasc

l .6.2 Air- Water Interface or Contractile Skin

l. 7 Terminology and Definitions

1.8 Historical Developments

Phase Properties and Relations 2.1 Properties of the Individuai Phases

2.1.1 Density and Specific Volume

Soil panicles

Water phase

Air phase

2. 1.2 Viscosity

2.1.3 Surface Tension

2.2 Interaction of Air and Water 2.2.1 Solid, Liquid, and Vapor States of Water

2.2.2 Water Vapor

2.2.3 Air Dissolving in Water

Solubility of Air in Water

Di.ffùsion of Gases Through Water

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XIV CONTéNTS

2.3 Volume-Mass Relations 29

2.3.1 Porosity 29 2.3.2 Void Ratio 30 2.3.3 Degree of Saturation 30 2.3.4 Water Conteni 31 2.3.5 Soil Densiry 32 2.3.6 Basic Volume- Mass Relationship 32 2.3.7 Changes in Volume-Mass Properties 33 2.3.8 Density of Mixtures Subjected to Compression of 34

the Air Phase

Piston-porous stone analogy 34 Conservation of mass applied to a mixture 36 Soil particles-water-air mixture 37 Air-water mixture 37

CHAPTER 3 Stress State Variables 38 3. 1 History of the Description of the Stress State 38

3. 1.1 Effective Stress Concept for a Saturated Soil 38 3.1.2 Proposed Effective Stress Equation for an 39

Unsaturated Soil

3.2 Stress State Variables for Unsaturated Soils 42 l 3.2.1 Equilibrium Analysis for Unsaturated Soils 42 Normal and shear stresses on a soil element 42 Equilibrium equations 43

1 3.2.2 Stress State Variables 43 Other combinations of stress state variables 44

i 3.2.3 Saturated Soils as a Special Case of Unsaturated 45 Soils

3.2.4 Dry Soils 45 t 3.3 Limit ing Stress State Conditions 46 .

3.4 Experimental Testing of the Stress State Variables 47

3.4.1 The Concept of Axis Translation 47 3.4.2 Null Tests to Test Stress State Variables 48 3.4.3 Other Experimental Evidence in Support of the 48

Proposed Stress State Variables

3.5 Stress Analysis 49

3.5.1 In Situ Stress State Component Profiles 49 Coefficient of latera l eanh pressure 52 Matric suction profile 53

Ground surface condition 53 .l Environmental conditions 53

. Vegetation 53 Water table 54 Permeability of the soil projìle 54

3.5.2 Extended Mohr Diagram 54

Equation of Mohr circles 55 Constructìon of Mohr circles 56

3.5.3 Stress Invariants 58

3.5.4 Stress Points 59

3.5.5 Stress Paths 59

3.6 Role of Osmotic Suction 63

CO]';TENTS xv

CHAPTER 4 Measurements of Soil Suction 64 4.1 Theory of Soil Suction 64

4.1.1 Componcnts of So il Suction 64 4. 1.2 Typical Suction Valucs and Thcir Mcasuring 66

Deviccs

4.2 Capillari ty 67 4.2.1 Capillary Height 67 4.2.2 Capillary Pressure 68 4.2.3 Height of Capillary Rise and Radius Effects 69

4 .3 Measurements of Total Suction 70 4.3.1 Psychrometers 70

Seebeck effecrs 70 Peltier effects 70 Peltier psychrometer 71 Psychrometer calibration 73 Psychrometer performance 74

4.3.2 Filter paper 77 Principle of measurement (filter paper method) 77 Measurement and calibration techniques (jilter 77

paper method)

l The use of the filter paper merhod in practice 79 4.4 Measurements of Matric Suction 80

., 4.4.1 High Air Entcy Disks 81 4.4.2 Direct measurements 82

Tensiometers 83

i Servicing the tensiometer prior to installation 84 Servicing the tensiometer after insrallarion 86 Jet fili tensiometers 86 Small tip tensiometer 86 Quick Draw tensiometers 88 Tensiometer performance for field 88

measurements

Osmotic tensiometers 90 Axis-translation rechnique 91

4.4.3 Indirect Measurements 93 Thermal conductiviry sensors 95 Theory of operation 97 Calibrarion of sensors / 97 Typical results of matric suction measurements 99 The MCS 6000 sensors 99 The AGWA-11 sensors 100

4.5 Measurements of Osmotic Suction 104 4.5. 1 Squeezing technique 105

CHAPTER 5 Flow Laws 107 5.1 Flow of Water 107

5.1.1 Driving Potential for Water Phase 108 5.1.2 Darcy's Law for Unsaturated Soils 110 5.1.3 Coefficient of Permeability with Respect to the 110

Water Phase

Fluid .a1ul porous medium components IlO

xvi CONTENTS

Relationship between permeabìlity {md volume- I l i mass propenies

Ejfect of variatiom in degree of sarurarion 0 11 I l i permeabiliry

Relationship between coejjicient of permeability I l i

l and degree of saturarion

Relationship between water coejjicient of 113 permeability and matric suction

l Relationship between water coejjiciem of 113 permeability and volumetric water conteni

Hysteresis of the permeability function 116

Il )l( 5.2 Flow o f Air 117

5.2.1 Driving PotentiaJ for Air Phase 117 5.2.2 Fick's Law for Air Phase 117 Il 5.2.3 Coefficient of Penneability with Respect to Air 119

Phase

Relationship between air coejjicient of 120 11 permeabiliry and degree of saturation

Relationship between air coejjicient of 120 permeability and matric suction ~ 5.3 Diffusion 121

5.3. 1 Air Ditfusion Through Water 121

~ 5.3.2 Chemical Ditfusion Through Water 123

5.4 Summary of F low Laws 123

CHAPTER 6 Measurement of Permeability 124

6. 1 Measurement of Water Coefficient of Permeability 124 ~ 6.1.1 Direct Methods to Measure Water Coefficient of 124 Penneability

~ Laboratory test methods 124 Steady-state method 124 Apparatus for steady-state method 125

Il Computations using steady-state method 126 Presentation ofwater coejjiciems of 126

permeability

Il Dijficulties with the steady-state method 127 Instantaneous profile method 127 Instantaneous profile method proposed by 128 l Hamilton et al.

Cornputations for the instantaneous profile 129 method l In situ field methods 130

In situ instantaneous profile method 130 Computations f or the in si tu insrantaneous 131

profile method

6.1.2 Indirect Methods to Compute Water Coefficient of 133

li Penneability

Tempe pressure cell apparatus and test 133 procedure

lf Volumetric pressure p/ate extractor apparatus 134

and test procedure

Test procedure f or the volumetric pressure 135

tl plate extractor

Drying portion of soil- water characteristic 136 curve

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CHAPTER 8

COI':TENTS xvii

Wetting ponion of rh e .wil- warer characrerisric 136 curve

Cornpuration of k.,, using rhe soil- water characteristic curve

6.2 Mcasurement of Air Coefficient of Penneability

Triaxial permeamerer ce/l for the measurement of air permeability

Triaxial penneameter celi for air and water penneabi/iry measurements

6.3 Measurement of Diffusion

6.3. 1 Mechanism of Air Diffusion Through High Air Entry Disks

6.3.2 Measurements of the Cocfficient of Diffusion

Procedure for computing diffusion properties

6.3.3 Diffused Air Volume Indicators

Bubble pump to measure diffused air volume

Diffused air volume indicator (DAVI)

Procedure for measuring diffused air volume

Compuration of diffused air volume

Accuracy of the diffused air volume indicator

Steady-State Flow 7 .l Steady-State Water Flow

7. 1. 1 Variation of Coefficient of Permeability with Space for an Unsaturated Soil

Heterogeneous, isotropic steady-state seepage

Heterogeneous, anisotropie sready-state seepage

7.1.2 One-Dimensional Flow

Formulation for one-dimensional ftow

Solution far one-dimensional flow

Finite difference method

Head boundary condition

Flux boundary condition

7.1.3 Two-Dimensional Flow

Formulation for two-dimensional flow

Solutions far two-dimensional jfow

Seepage analysis IISing the finite element method ~

Examples of two-dimensional problems

Infinite slope

7. 1.4 Three-Dimensional Flow

7.2 Steady-State Air Flow

7 .2.1 One-Dimensional Flow

7.2.2 Two-Dimensional Flow

7.3 Steady-State Air Diffusion Through Water

Pore Pressure Parameters 8.1 Compressibility of Pore Fluids

8.1 . l Air Compressibiliry

8.1 .2 Water Compressibility

8.1.3 Compressibility of Air- Watcr Mixtures

The use of pore pressure parameters in the compressibili/)' equation

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XVIII CONTI:::NTS

CHAPTER 9

8.1.4 Components of Compressibi1ity of an Air- Watcr Mixture

Effects of free air on the compressibility of the mixture

Effec:ts of dissolved air 011 t h e compressibility of the mixture

8.1.5 Other Re1ations for Compressibi1ity of Air- Water Mixrure

Limitation of Kelvin 's equation in formulatin.g the compressibility equation

8.2 Derivations of Pore Pressure Parameters

8.2.1 Tangent and Secant Pore Pressure Paramcters

8.2.2 Summary of Necessary Constitutivc Relations

8.2.3 Drained and Undrained Loading

8.2.4 Tota1 Stress and Soil Anisotropy

8.2.5 Ko-Loading

8.2.6 Hilf's Analysis

8.2.7 Isotropic Loading

8.2.8 Uniaxial Loading

8.2.9 Triaxial Loading

8.2.10 Three-Dimensional Loading

8.2.11 ex Parameters

8.3 Solutions of the Pore Pressure Equations and Comparisons with Experimental Results

8.3.1 Secant B;, Pore Pressure Parameter Derived from Hilf's Analysis

8.3.2 Graphica1 Procedure for Hi1f's Ana1ysis

8.3.3 Experimental Results of Tangent B Pore Pressure Parameters for Isotropic Loading

8.3.4 Theoretical Prediction of B Pore Pressure Parameters for Isotropic Loading

8.3.5 Experimental Results of Tangent Band A Parameters for Triaxia1 Loading

8.3.6 Experimental Measurements of the a Parameter

Shear Strength Theory 9.1 History of Shear Strength

9.1.1 Data Associated with Incomplete Stress Variab1e Measurements

9.2 Failure Envelope for Unsaturated Soils

9 .2.1 Failure Criteria

9.2.2 Shear Strength Equation

9.2.3 Extended Mohr-Coulomb Failure Enve1ope

9.2.4 Use of (a - u..,) and (u0 - u,.,) to Define Shear Strength

9.2.5 Mohr-Coulomb and Stress Point Envelopes

9.3 Triaxial Tests on Unsaturated Soils

9.3.1 Consolidate<! Drained Test

9.3.2 Constant Water Content Test

9.3.3 Consolidated Undrained Test with Pore Pressure Measurements

9.3.4 Undrained Test

9.3.5 Unconfined Compression Test

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CHAPTER IO

CHAPTER 11

CONTENTS xix

9.4 Direct Shear Tests on Unsaturcd Soils 247

9 .5 Selection of Strain Rate 248

9.5.1 Background on Strain Rates for Triaxial Testing 248

9.5.2 Strain Rates for Triaxial Tests 250

9.5.3 Displacement Rate for Direct Shear Tests

9.6 Multistage Testing 9.7 Nonlinearity of Failure Envelope 9.8 Relationships Between q/> and x

Measurement of Shear Strength Parameters 10. 1 Special Design Considerations

10.1. 1 Axis-Translation Technique

10.1.2 Pore-Water Pressure Contro! or Measurement

Saturation procedure f or a high air entry disk

10.1.3 Pressure Response Below the Ceramic Disk

10.1.4 Pore-Air Pressure Contro! or Measurement

10.1.5 Water Volume Change Measurement

10.1.6 Air Volume Change Measurement

10.1. 7 Overall Volume Change Measurement

10. 1.8 Specimen Preparation

10.1. 9 Backpressuring to Produce Saturation

10.2 Test Procedures for Triaxial Tests

10.2.1 Consolidated Drained Test

10.2.2 Constant Water Content Test

10.3 10.4

10.2 .3 Consolidated Undrained Test with Pore Pressure Measurements

10.2.4 Undrained Test

10.2.5 Unconfined Compression Test

Test Procedures for Direct Shear Tests / ­

Typical Test Results

10.4. 1 Triaxial Test Results

Consolidated drained triaxial tests

Constant water content triaxial tests

Nonlinear shear strength versus matric sucrion

Undrained and unconjìned compression tests

10.4.2 Direct Shear Test Results

Plastic and Limit Equilibrium I l . l Earth Pressures

11.1. l At Rest Eanh Pressure Conditions

11. 1.2 Estimation of Depth of Cracking

11. 1.3 Extended Rankine Theory of Earth Pressures

Active earth pressure

Coefficient of active earth pressure

Active earth pressure distriburion (constan.t matric suction with depth)

Tension zone deprh

Active earth pressure distriburion (linear decrease in matric suction to the water table)

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XX CONTENTS

Acri ve earth pressurc distribution whcn rhe soil has tension cracks

Passive earth pressure

Coefficient of passive earth pressure

Passive earth pressure distribution (constanr matric sucrion wirh depth)

Passive earth pressure distribution (linear decrease in marric suction to the warer table)

Deformarions wirh active and passive states

I l . 1.4 Total Latera! Earth Force

Acri ve earth force

Passive earth force

Il. l .5 Effect of Changes in Matric Suction on thc Active and Passive Earth Pressure

Relationship berween swelling pressures and the earth pressures

I l . l. 6 Unsupported Excavations

Ejfect oftension cracks on the unsupported height

Il. 2 Bearing Capacìty ~o.' 11 ·"

11 .2.1 Tenaghi Bearihg Capacity Theory

11.2.2 Assessment of Shear Strength Parameters and a Design Matric Suction

Stress state variable approach

Total stress approach

11.2. 3 Bearing Capacity of Layered Systems

11.3 st'ope StabiJity

11 .3. 1 Location of the Criticai Slip Surface

11.3.2 Generai Limit &juil ibrium (GLE) Method

Shear force mobilized equarion

Normalforce equation

Factor of safery with respect t o moment equilibrium

Factor of safety with respect t o force equilibrium

lnterslice force function

Procedures for solving rhe factors of safety equation

Pore-water pressure designation

11.3.3 Other Limit Equilibrium Methods

11.3 .4 Numerica! Difficultics Associated with the Limit Equilibrium Method of Slices

11 .3.5 Etfects of Negative Pore-Water Pressure on Slope Stability

The " total cohesion " method

Two examples using the ''rotai cohesion · ' method

Example no. l

Example no. 2

The "extended shear strength" method

Genera l /ayout of problems and soil properties

lnitial condirfons for the seepage analysis

Seepage and slope stability resulrs under high-intensity rainfall conditions

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CHAPTER 12

CHAPTER 13

CONTENTS xxi

Volume Change Theory 12.1 Literature Rcvicw

12.2 Concepts of Volume Change and Deformation

12.2.1 Continuity Requiremcnts

12.2.2 Ovcrnll Volume Change

12.2.3 Water and Air Volume Changes

12.3 Constitutive Relations

12.3.1 Elasticity Form

Water phase constitutive relation

Change in the volume of air

lsotropic loading

Uniaxial loading

Triaxial loading

K0-loading

Piane strain loading

Piane stress loading

12.3.2 Compressibility Form

12.3.3 Volume- Mass Form (Soil Mechanics Terminology)

12 .3.4 Use of (u - uw) and (ua - uw) to Formulate Constitutive Relations

12.4 Experimental Yerifications fo r Uniqueness of Constitutive Surfaces

12.4.1 Sign Convention for Volumetric Deformation Properties

12.4.2 Verification of Uniqueness of the Constiturive Surfaces Using Small Stress Changes

12.4.3 Verification of the Constitutive Surfaces Using Large Stress State Variable Changes

12.5 Relationship Among Volumetric D eformation Coefficients ~

12.5.1 Relationship of Volumetric Deformation Coefficients fo r the Void Ratio and Water Content Surfaces

12.5.2 Relationship of Volumetric Deformation Coefficients for the Volume-Mass Form of the Constitutive_ Surfaces

12.5.3 Laboratory Tests Used for Obtaining Volumetric Deformation Coefficients

12.5.4 Relationship of Volumetric Deformation Coefficients for Unloading Surfaces

12.5.5 Re1ationship of Volumetric Deformation Coefficients for Loading and Unloading Surfaces

12.5.6 Constitutive Surfaces on a Semi-Logarithmic Plot

Measurements of Volume Change Indices l 3. l Literature Review

13.2 T est Procedures and Equipments

13.2.1 Loading Constitutivc Surfaces

Oedometer tests

Pressure plate drying tests

Shrinkage tesrs

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XXII CONTGNTS

CHAPTER 14

CHAPTER 15

Determination of volume change indices

Determinati an of volume clumge indice.\· associated with the /ransition piane

Typica/ resu/ts from pressure p/ate tests

Determinarion of in si tu stn•ss state using oedomeler test results

" Constant volume" cest

''Free-swell ' ' test

Correction for the compressibility of the apparatus

Correctionfor samp/ing disturbance

13 .2 .2 Unloading Constitutive Surfaces

Unloading tests after compression

Pressure plate weuing tests

Free-swell tests

Determination of volume change indices

Volume Change Predictions 14.1 Literature Review

14.1.1 Factors Affecting Tota1 Heave

14.2 Past , Present, and Future States of Stress

14.2.1 Stress State History

14.2.2 In Situ Stress State

14.2.3 Future Stress State and Ground Movements

14.3 Theory of Heave Predictions 14.3.1 Total Heave Formulations

14.3.2 Prediction of Fina! Pore- Water Pressures

14.3.3 Example of Heave Calculations

14.3.4 Case Histories

Slab-on-grade jloor, Regina, Saskatchewan

Esron school, Eston, Saskatchewan

14.4 Control Factors in Heave Prediction and Reduction

14.4. 1 Closed-Form Heave Equation when Swclling Pressure is Constant

14.4.2 Effect of Correcting the Swelling Pressure on the Prediction of Total Heave

14.4.3 Example with Wetting from the Top to a Specified Depth

14.4.4 Example with a Portion of the Profile Removed by Excavation and Backfìlled with a Nonexpansive Soil

14.5 Notes on Collapsible Soils

One-DimensionaJ Consolidation and Swelling 15. l Literature Review

15.2 Physical Relations Required for the Formulation

15.3 Derivation of Consolidation Equations 15.3. 1 Water Phase Partial Differential Equation

Saturaced condition

Dry soil condition

Special case of an unsaturated soil condition

15.3.2 Air Phase Partial Differentia1 Equation

Sacuraced soil condicion

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CONTENTS XXIJI

Dry soil condition 426 Special case of an unsaturared soi/

15.4 Solution of Consolidation Equations Us ing Finite Difference Technique

15.5 Typical Consolidation Test Results for Unsaturated Soils

15 .5. 1 Tcsts on Compacted Kao1in

Presentation of results

Theoretical ana/yses

15 .5.2 Tests on Silty Sand

Presentarion of results

Theoretical analysis

15.6 Dimensionless Consolidation Parameters

Two- and Three-Dimensional Unsteady-State Flow and Nonisothermal Analyses 16.1 Uncoupled Two-Dimensional Formulations

16.1.1 Unsteady-State Seepage in lsotropic Soil

Water phase panial differential equation

Air phase pania/ differential equation

16.1.2 Unsteady-State Seepage in an Anisotropie Soil

Anisotropy in permeability

Water phase panial differemial equation

.Seepage ana/ysis using the finite element method

Examples of two-dimensional problems and their solutions

Example of water flow rhrough an eanh dam

Example of groundwater seepage below a lagoon

Example of seepage within a layered h ili slope .r-

16.2 Coupled Formulations of Three-Dimensional Consolidation

16.2.1 Constitutive Relations

Soil structure

Water phase

Air phase

16.2.2 Coup1ed Consolidation Equations

Equilibrium equations

Water phase continuiry

Air phase continuiry

16.3 Nonisothermal Flow 16.3.1 Air Phase Partia\ Differential Equation

16.3.2 Fluid and Vapor Flow Equation for the Water P ha se

16.3.3 Heat Flow Equation

16.3.4 Atmospheric Boundary Conditions

Surface boundary conditions for air and fluid water flow

Surface boundary conditions for water vapor fio w

Surface boundary conditions for heat flow

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xxiv CONTENTS

APPENDIX A

APPENDIX B

Units and Symbols

Theoretical Justification for Stress State Variables B. l Equilibrium Equations for Unsaturated Soils B.2 Total or Overall Equilibrium

B.3 Independent Phase Equilibrium

8.3.1 Water Phase Equilibrium

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485 8 .3.2 Air Phase Equilibrium 485 8.3.3 Contractile Skin Equilibrium 485

B.4 Equilibrium of the Soil Structure (i. e., Arrangement of 488 Soil Particles)

B.5 Other Combinations of Stress State Variables 489

References 490

About the Authors 508

lndex 510