2005 Modelling of vertical drains with smear installed in

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2005

Modelling of vertical drains with smear installed insoft clayIyathurai SathananthanUniversity of Wollongong

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Recommended CitationSathananthan, Iyathurai, Modelling of vertical drains with smear installed in soft clay, Doctor of Philosophy thesis, School of Civil,Mining and Environmental Engineering, University of Wollongong, 2005. http://ro.uow.edu.au/theses/1924

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MODELLING OF VERTICAL DRAINS WITH SMEAR

INSTALLED IN SOFT CLAY

A thesis submitted in fulfilment of the requirements

for the Award of the Degree of

Doctor of Philosophy

from

UNIVERSITY OF WOLLONGONG

by

IYATHURAI SATHANANTHAN, BSc Eng (Hons)

DEPARTMENT OF CIVIL, MINING AND ENVIRONMENTAL ENGINEERING

UNIVERSITY OF WOLLONGONG, AUSTRALIA.

2005

AFFIRMATION

I, Iyathurai Sathananthan, declare that I am the sole author of this dissertation. The work

presented in this thesis is original unless otherwise referenced or acknowledged and has

been carried out in the Department of Civil Engineering of the University of

Wollongong. The document has not been submitted for any other degree at any other

academic institution.

I authorise the University of Wollongong to lend this thesis to other institutions or

individuals for the purpose of scholarly research.

Iyathurai Sathananthan

September 2005

11

List of Publications

LIST OF PUBLICATIONS

The following publications were generated during my research period.

Book Chapter

Indraratna, B., Sathananthan, I., Bamunawita, C. and A.S. Balasubramaniam. (2005).

Theoretical and Numerical Perspectives and Field Observations for the Design

and Performance Evaluation of Embankments Constructed on Soft Marine Clay.

A Chapter in Ground Improvement - Case Histories Book (Volume 3), Edited by

Indraratna, B. and Chu, J., Elsevier, London, Chapter 2:61-106.

Journal Papers

Indraratna, B., Sathananthan, I., Rujikiatkamjorn, C. and A.S. Balasubramaniam.

(2005). Analytical and Numerical Modelling of Soft Soil Stabilized by

Prefabricated Vertical Drains Incorporating Vacuum Preloading. International

Journal of Geomechanics, ASCE, 5(2): 114-124.

Indraratna, B., Rujikiatkamjorn, C. and Sathananthan, I. (2005). Analytical and

numerical solutions for a single vertical drain including the effects of vacuum

preloading. Canadian Geotechnical Journal, 42(4):994-1014.

Sathananthan, I. and Indraratna, B. (2005). Laboratory Evaluation of Smear Zone and

Correlation between Permeability and Moisture Content. Submitted to Journal of

Geotechnical and Geoenvironmental Engineering, ASCE (in press).

iii

List of Publications

Indraratna, B., Rujikiatkamjorn, C. and Sathananthan, I. (2005). Radial Consolidation

of Clay using Compressibility Indices and Varying Horizontal Permeability.

Canadian Geotechnical Journal (in press).

Conference Papers

Indraratna, B. and Sathananthan, I. (2004). Numerical Prediction of Soft Clay

Consolidation with Geosynthetic Vertical Drains Using Plane Strain Solution.

Proceedings of the 9 Australia NewZealand Conference on Geomechanics.

Auckland, NewZealand. Vol.2. 633-639.

Indraratna, B. and Sathananthan, I. (2003). Comparison of Field Measurements and

Predicted Performance beneath Full Scale Embankments. Proceedings of the 6th

International Symposium on Field Measurements in Geomechanics. Oslo,

Norway. 17-27.

Indraratna, B., Rujikiatkamjorn, C. and Sathananthan, I. (2005). Analytical Modelling

and Field Assessment of Embankment Stabilized with Vertical Drains and

Vacuum Preloading. The 16th International Conference on Soil Mechanics and

Geotechnical Engineering. Osaka, Japan (in press)

Indraratna, B., Rujikiatkamjorn, C, Sathananthan, I., Shahin, M. and Khabbaz, H.

(2005). Analytical and Numerical Solution for Soft Clay Consolidation using

th

Geosynthetic Vertical Drains with Special Reference to Embankments. The 5

International Geotechnical Engineering Conference, Cairo. Egypt. 55-86.

IV

Abstract

ABSTRACT

In this research, analytical, experimental and numerical investigations were carried out

to pursue a better understanding of the consolidation of soft clay stabilized with

prefabricated vertical drains (PVD) subjected to preloading (with and without vacuum

pressure application). This investigation was carried out in fourfold. First, an analytical

solution based on Cylindrical Cavity Expansion analysis incorporating the Modified

Cam Clay theory has been formulated to estimate the extent of the smear zone. Second,

the smear zone characteristics were evaluated using a large-scale radial drainage

consolidometer and then compared with predicted values. Third, the existing

axisymmetric and plane strain theories of a unit cell were modified and incorporated

with a linearly distributed (trapezoidal) vacuum pressure for both Darcian and non-

Darcian flow. Finally, multi-drain plane strain analysis was conducted on a number of

case histories taken from Australia and Thailand to study the performance of the entire

embankment stabilised with vertical drains.

A series of large-scale model tests were conducted using a specially designed

consolidometer 650 mm diameter by 1040 mm high to study the characteristics of the

smear zone. A central vertical drain was installed in soil specimens (previously placed

inside a large cell) with a band shaped (rectangular, hollow) steel mandrel, during which

any variations in pore water pressure along the radial direction were recorded. Then, the

surcharge load was applied in stages up to 200 kPa and the clay then consolidated. At

the end of consolidation, soils samples were collected from several locations in order to

carry out a number of oedometer and triaxial tests to establish the variations of soil

properties. It was observed that drain installation disturbed the soil immediately

v

Abstract

adjacent to it, creating a "smear zone" in which a change in clay properties was caused

by reconsolidation due to dissipation of excess pore pressure, and remoulding due to

shear from the periphery of the mandrel. The smear zone was 2-3 times the equivalent

radius of the mandrel with horizontal permeability (in the smear zone) varying from

1.09 to 1.64, an average of 1.34 times smaller than that of the undisturbed zone. The

estimated extent of smear zone from the large-scale test was very close to the predicted

value, based on Cavity Expansion analysis.

The analytical solutions are incorporated in the finite element code (e.g. PLAXIS,

ABAQUS) employing the Modified Cam Clay theory. Selected numerical analysis

incorporating the proposed solution was carried out to study the behaviour of a number

of case histories, in view of various ground improvement schemes applied to stabilize

the soft clay foundation and the predictions are compared with available field data. A

good agreement between the finite element analysis and field data was found. Finally,

numerical analyses incorporating the proposed solution were conducted to study the

effect of embankment slope, construction rate, drain spacing, the characteristics of

smear zone, multi-stage loading and surface crust on the failure of the soft clay

foundation.

VI

Acknowledgments

ACKNOWLEDGMENTS

The author takes this opportunity to express his profound gratitude to Professor

Buddhima Indraratna for his enthusiastic guidance, invaluable suggestions, constructive

criticisms, and constant encouragement throughout this research project. His patience

and availability for any help whenever needed with his heavy workload is appreciated.

From Professor Indraratna, the author realized new ways of thinking about soil

mechanics, of blending complex mathematics with the principles of mechanics. This

guidance into new avenues of knowledge will be useful to author throughout his career.

I would like to express my gratitude to senior technicians Alan Grant for his

continuous help with the initial set-up of equipment and during the laboratory work. 1

appreciate Cholachat Rujikiatkamjorn, a fellow doctoral candidate for sharing views

and mutual help. A special note of sincere appreciation also extends to Dr. Hadi

Khabbaz, Professor A.S. Balasubramaniam, Dr. G.S.K. Fernando and Dr. Brett Lemas

for their continuing support and good wishes.

I would also like to extend thanks to fellow postgraduate friends for their

discussions, support, and social interaction outside the "geotechnical" world. M y

appreciation is also extended to all past and present members of the Department of

Civil, Mining and Environmental Engineering, University of Wollongong for their

warm-hearted cooperation. The author was financially supported by the Australian

Commonwealth (IPRS and U P A Scholarships) fund administered through the

University of Wollongong and this deserves special acknowledgment.

My special and sincere gratitude is offered to my beloved wife Ramani for her

constant love, prayers, encouragement and many sacrifices throughout the research

period; her constants affection and forbearance has been a source of strength.

Last but not the least, the author humbly dedicates this piece of work to his

beloved parents, wife, brothers and sisters, without whose sacrifice and understanding

the author could never have reached where he is today.

I. Sathananthan,

University of Wollongong, Australia.

vii

Table of Contents

TABLE OF CONTENTS

Affirmation ii

List of Publications iii

Abstract v

Acknowledgments vii

Table of Contents viii

List of Figures xix

List of Tables xxviii

List of Symbols xxix

1 INTRODUCTION 1

1.1 General 1

1.2 Use of Vertical Drains 3

1.3 Methods of Deformation Analysis 5

1.3.1 Analytical Method 5

1.3.2 Numerical Analysis • 6

1.3.3 Observational Methods 7

1.4 Scope and Objective of Study 7

viii

Table of Contents

1.5 Organization of the Dissertation 9

2 LITERATURE REVIEW 12

2.1 History and Development of Vertical Drains 12

2.2 Types of Vertical Drains 13

2.2.1 Sand Drains 13

2.2.2 Prefabricated Vertical Drains 14

2.3 Installation and Monitoring of Vertical Drains 16

2.3.1 Inclinometers 18

2.3.2 Settlement Indicators 19

2.3.3 Piezometers 19

2.4 Drain Properties 19

2.4.1 Diameter of Influence Zone 19

2.4.2 Equivalent Drain Diameter of Band Shaped Vertical Drain 20

2.4.3 Filter and Pore Size 23

2.4.4 Discharge Capacity 24

2.5 Factors Influencing the Vertical Drain Efficiency 29

2.5.1 Smear Zone 29

ix

Table of Contents

2.5.1.1 Soil Macro Fabric 33

2.5.1.2 Size and Shape of the Mandrel 34

2.5.1.3 Installation Procedure 35

2.5.2 Effect of Sand Mat 36

2.5.3 Well Resistance 37

6 Development of Consolidation Theories 39

2.6.1 One-Dimensional Consolidation 39

2.6.2 Coupled Consolidation Theory 41

2.6.3 Development of Vertical Drain Theory 42

2.6.3.1 Rendulic and Carillo Diffusion theory 43

2.6.3.2 Barron's (1948) Suggestion - Equal Strain Hypothesis 44

2.6.3.3 Rigorous Solution (Yoshikuni and Nakanode, 1974) 47

2.6.3.4 Hansbo (1981)-Analysis with Smear and Well Resistance 48

2.6.3.5 X Method (Hansbo, 1979 and 1997) 49

2.6.4 2-D Modelling of Vertical Drains 50

2.6.4.1 Shinsha et al. (1982)-Permeability Transformation 52

2.6.4.2 Hird et al. (1992)-Geometry and Permeability Matching 52

x

Table of Contents

2.6.4.3 Bergado and Long (1994) -Equal Discharge Concept 53

2.6.4.4 Chaietal. (1995)-Well Resistance Matching 53

2.6.4.5 Lee etal. (1997)-Time Factor Analysis 54

2.6.4.6 Indraratna and Redana (1997) - Rigorous Solution for Parallel Drain

Wall 55

2.6.5 Simple Method of Modelling (1-D) 58

2.7 Evaluation of Design Parameters 59

2.7.1 Vertical Coefficient of Consolidation and Permeability 60

2.7.2 Horizontal Coefficient of Consolidation and Permeability 62

2.7.3 Coefficient of Consolidation with Radial Drainage 63

2.7.3.1 Log U vs t Approach 63

2.7.3.2 Plotting Settlement Data (Asaoka, 1978; Magnan et al., 1980) 64

2.8 Constitutive Models for Soils 67

2.8.1 Linear Elastic Model 67

2.8.2 Elastic-Perfectly Plastic Model 67

2.8.3 Critical State Models 67

2.8.3.1 Cam-Clay Model 72

xi

3 0009 03366597 2

Table of Contents

2.8.3.2 Modified C a m Clay Model 74

2.9 Salient Aspects of Numerical Modelling 75

2.9.1 Drain Efficiency by Pore Pressure Dissipation 76

2.9.2 Deformation as a Stability Indicator 77

2.10 Summary 79

3 THEORETICAL BACKGROUND 82

3.1 Prediction of Smear Zone Caused by Mandrel Driven Vertical Drains using

Cavity Expansion Analysis 82

3.1.1 General 82

3.1.2 Basic Assumptions and Definition of the Problem 83

3.1.3 Elastic Analysis 84

3.1.4 Plastic Analysis 86

3.1.4.1 Stress at Elastic-Plastic Boundary 86

3.1.4.2 Strain in Plastic Zone 87

3.1.4.3 Effective Stress in the Plastic Zone 89

3.1.4.4 Total Stress in the Plastic Zone 90

3.1.4.5 Pore Water Pressure in the Plastic Zone 90

xii

Table of Contents

3.1.4.6 Prediction of Smear Zone 91

3.1.4.7 Solution Procedure 91

3.1.4.8 Illustrated Application 91

3.2 Analytical Solution for Vertical Drain with Vacuum Preloading 95

3.2.1 General 95

3.2.2 Modelling of Axisymmetric Solution with Applied Vacuum Pressure.. 97

3.2.2.1 Excess Pore Water Pressure 107

3.2.2.2 Hydraulic Gradient 107

3.2.3 Modelling of Plane Strain Solution with Applied Vacuum Pressure.... 108



3.2.4 Comparison of Axisymmetric vs Plane Strain Conditions 114

3.2.5 Matching Approach and Theoretical Considerations 116

3.3 Plane Strain Consolidation Equation for a Single Drain under Non-Darcian

Flow 121

3.3.1 General 121

3.3.2 Proposed Plane Strain Solution 122

xiii

Table of Contents



3.3.3 Matching with Axisymmetric Consolidation 131

3.4 Summary 136

4 LABORATORY TESTING AND ANALYSIS OF RESULTS 138

4.1 General 138

4.2 Experimental Set-up and Testing Procedure 140

4.2.1 Apparatus 140

4.2.2 Testing Procedure 141

4.2.2.1 Preparation of Reconstituted Clay 142

4.2.2.2 Preparation of the Apparatus 143

4.2.2.3 Preparation of the Vertical Drain 144

4.2.2.4 Testing Procedure 144

4.3 Presentation of Results 146

4.3.1 Evaluation of Compressibility Indices 146

4.3.2 Pore Pressure Variation during Mandrel Installation 147

4.3.3 Permeability Tests 152

xiv

Table of Contents

4.3.4 Variations of Water Content 155

4.3.5 Comparison of Surface Settlement 160

4.3.6 Comparison of Excess Pore Water Pressures during Consolidation 161

4.4 Summary 162

5 CASE STUDY 1: SUNSHINE MOTORWAY (QLD, AUSTRALIA) 163

5.1 General 163

5.2 Finite Element Analysis 166

5.2.1 Element Types used in PLAXIS 167

5.2.2 Types of Material Models used in PLAXIS 169

5.2.2.1 Mohr-Coulomb Model 169

5.2.2.2 Soft Soil Model 170

5.2.3 Variation of Extent of Smear Zone 170

5.2.4 Plane Strain Permeability 171

5.2.5 Numerical Calculations and Comparison with Field Observations 172

5.2.5.1 Displacement Boundary 173

5.2.5.2 Drainage and Loading Boundary 173

5.2.5.3 Comparison of Centreline Settlement 174

xv

Table of Contents

5.2.5.4 Comparison of Lateral Displacements 177

5.2.5.5 Comparison of Excess Pore Pressure Variation 179

5.3 Summary 183

6 CASE STUDY 2: SECOND BANGKOK INTERNATIONAL AIRPORT

(SUVARNABHUMI, THAILAND) 185

6.1 General 185

6.2 Finite Element Analysis 189

6.2.1 Element Types used in ABAQUS 190

6.2.2 Variation of Extent of Smear Zone 191

6.2.3 Plane Strain Permeability 192

6.2.4 Numerical Predictions and Comparison with Field Data 194

6.3 Summary 2^0

7 NUMERICAL MODELLING OF VERTICAL DRAINS AND DESIGN

IMPLICATIONS 202

7.1 General.. 202

7.2 Design of Embankment Constructed on Soft Clay without Vertical Drains 203

7.2.1 Effect of Embankment Slope on Foundation Failure 205

7.2.1.1 Surface Settlements and Displacement 205

xvi

Table of Contents

7.2.1.2 Lateral Displacement 208


7.2.2 Effect of Loading Rate on Foundation Failure 210

7.3 Design of Embankment Constructed on Soft Clay with Vertical Drains 213

7.3.1 Effect of Drain Spacing 214

7.3.2 Effect of the Extent of Smear Zone 216

7.3.3 Effect of Smear Zone Permeability 219

7.3.4 Effect of Stage Loading 221

7.3.5 Effect of Surface Crust 224

7.3.6 Identification of the Critical Location 226

7.4 Summary 233

8 CONCLUSIONS AND RECOMMENDATIONS 235

8.1 General 235

8.2 Specific Observations 236

8.2.1 Mathematical Formulations and Modifications to the Existing Theories...

236

8.2.2 Laboratory Program 237

xvii

Table of Contents

8.2.3 Case History Analysis 239

8.2.4 Application of Finite Element Modelling for General Design 241

8.3 Suggestions for Future Research 242

BIBLIOGRAPHY 246

Appendix 1: Relationship between Isotropic and Conventional Overconsolidation Ratio

262

xvm

— List of Figures

LIST OF FIGURES

Figure 1.1 Potential benefit of vertical drains (adapted from Lau et al.2000) 4

Figure 2.1 Typical types of PVD (company brochure) 15

Figure 2.2 Typical installation rig (Source: Colbond bv, The Netherlands) 16

Figure 2.3 Basic instrumentation for a highway embankment (After Rixner, 1986) 17

Figure 2.4 Schematic of vacuum preloading consolidation (Tang et al., 2000) 18

Figure 2.5 Typical drain installation patterns and the equivalent diameters 20

Figure 2.6 Conceptual drawing of a PVD and equivalent diameter well 21

Figure 2.7 Equivalent diameters of band-shaped vertical drains 22

Figure 2.8 Possible deformation modes of PVD (adapted after Holtz, et al. 1991) 25

Figure 2.9 Typical values of discharge capacity (data from Rixner et al., 1986) 27

Figure 2.10 a) Schematic section of the test equipment showing the central drain and

associated smear zone; and b) locations of small specimens obtained to determine

the consolidation and permeability characteristic (Indraratna and Redana, 1998) .32

Figure 2.11 Ratio of kyfkv along the radial distance from the central drain (modified after

Indraratna and Redana, 1998) 33

Figure 2.12 Approximation of the disturbed zone around the mandrel (Rixnet et al).... 36

Figure 2.13 Minimum discharge capacity required (based on Eqn. 2.20) 38

xix

List of Figures

Figure 2.14 Unit cell model of a drain surrounding by soil cylinder 42

Figure 2.15 Schematic of soil cylinder with vertical drain (after Hansbo, 1979) 46

Figure 2.16 Average consolidation rates a) for vertical flow, b) for radial flow 47

Figure 2.17 Conversion of an axisymmetric unit cell into plane strain condition 51

Figure 2.18 Comparison of average degree of consolidation (data from Chai et al.,

1995) 54

Figure 2.19 Comparison of excess pore pressure variation (data from Chai et al., 1995)

55

Figure 2.20 Average degree of consolidation (modified after, Indraratna et al., 2000).. 57

Figure 2.21 Comparison of the average surface settlement (Indraratna et al., 2000) 58

Figure 2.22 Comparison of the excess pore pressure (Indrarataa et al., 2000) 58

Figure 2.23 Effect of Cdon degree of consolidation (Chai et al. 2001) 60

Figure 2.24 Aboshi and Monden (1963) method for determining Ch 64

Figure 2.25 Asaoka (1978) method to determine Ch 65

Figure 2.26 Isotropic normal consolidation line plot in critical state theory 69

Figure 2.27 Position of the critical state line 70

Figure 2.28 Position of the initial specific volume 71

Figure 2.29 Yield locus of Cam Clay and Modified Cam Clay model 73

xx

. List of Figures

Figure 2.30 Percentage of undissipated excess pore pressure at drain-soil interfaces

(Indraratna et al., 1994) 77

Figure 3.1 Expansion of a cavity 83

Figure 3.2 Normalized pore water pressure variation with radius 94

Figure 3.3 Consolidation process (a) conventional loading (b) vacuum preloading 96

Figure 3.4 Schematic of soil cylinder with vertical drain (adapted from Hansbo, 1979)

98

Figure 3.5 Average excess pore water pressure distribution with different vacuum

pressure distribution 104

Figure 3.6 Average excess pore water pressure distribution with different drain spacing

105

Figure 3.7 Average excess pore water pressure distribution with different smear zone

parameters 106

Figure 3.8 Plane strain unit cell 108

Figure 3.9 Comparison of average excess pore water pressure distribution 115

Figure 3.10 Conversion of an axisymmetric unit cell into plane strain condition (adapted

from, Hird et al., 1992; Indraratna and Redana, 1997) 117

Figure 3.11 Ratio between coefficients of permeability of undisturbed zone of

equivalent plane strain cell to those of axisymmetric cell as a function of n 119

xxi

List of Figures

Figure 3.12 Ratio between smear zone permeability to undisturbed zone permeability of

equivalent plane strain cell as a function of n, s and kh/kh 120

Figure 3.13 Exponential correlation (modified after Hansbo, 2001) 121

Figure 3.14 The variation of function gp (n, y) withy for selected n 124

Figure 3.15 The variation of function/^, (n, y) withy for selected n 126

Figure 3.16 Variation of average degree of consolidation with exponent n 129

Figure 3.17 Variation of normalized excess pore pressure with exponent n 130

Figure 3.18 Variation of normalized hydraulic gradient with exponent n 131

Figure 3.19 Equivalent plane strain ap value as a function of Blbw, bjbw and KhJKs of

axisymmetric cell for different n values 133

Figure 3.20 Equivalent plane strain J3p value as a function of B/bw, bslbw and KH/KS of

axisymmetric for different n values 134

Figure 3.21 Ratio between coefficients of consolidation of undisturbed zone of

equivalent plane strain cell to those of axisymmetric cell as a function of B/bw ..135

Figure 3.22 Ratio between undisturbed zone permeability to smear zone permeability of

equivalent plane strain cell as a function of Blbw, bjbw and KH/KS of axisymmetric

cell for different n values 136

Figure 4.1 Large-scale radial drainage consolidometer 141

xxii

List of Figures

Figure 4.2 Location of pore pressure transducers and cored samples 142

Figure 4.3 Mandrel, Guider, PPT and prepared sample 144

Figure 4.4 Variation of void ratio with consolidation pressure 146

Figure 4.5 Pore pressure variation during mandrel installation for different initial

surcharge pressure (preconsolidation pressure) 149

Figure 4.6 Normalized pore water pressure variation with distance 152

Figure 4.7 Variation of (a) horizontal permeability, (b) vertical permeability, (c)

permeability ratio, and (d) the normalized permeability, with radial distance .... 154

Figure 4.8 Variation of (a) water content and (b) normalized water content reduction,

with radial distance at a depth of 0.5 m 157

Figure 4.9 Variation of water content with depth and radial distance for an applied

pressure of 200 kPa 158

Figure 4.10 Correlation between the reduction of permeability and the water content

within the smear zone 159

Figure 4.11 (a) Comparison of surface settlement, and (b) FE mesh used in Plaxis.... 160

Figure 4.12 Comparison of excess pore water pressure 161

Figure 5.1 Map of Australia showing the location of the study area 164

Figure 5.2 Profile of the Geotechnical characteristics (Sunshine Motorway Stage 2

Interim Report, 1992) 164

xxiii

List of Figures

Figure 5.3 Plan view of Trial Embankment 165

Figure 5.4 Typical cross-section of embankment with selected instrumentation points

166

Figure 5.5 Types of element used in PLAXIS (Version 8) 168

Figure 5.6 Distribution of nodes and stress points in interface elements and their

connection to soil elements 168

Figure 5.7 Normalised pore water pressure variation with radial distance 171

Figure 5.8 Variation of extent of smear zone with depth 171

Figure 5.9 Finite element mesh used to analyse Section B 173

Figure 5.10 Construction History of the Trial Embankment 174

Figure 5.11 Centreline settlement of trial embankment sections 175

Figure 5.12 Settlement contours at the end of construction 176

Figure 5.13 Settlement contours at the end of 100 days 176

Figure 5.14 Lateral displacement profile at the middle of the main batter 177

Figure 5.15 Lateral displacement profile at the toe of the berm 178

Figure 5.16 Lateral displacement contours at the end of construction 180

Figure 5.17 Lateral displacement contours at the end of 100 days 180

Figure 5.18 Excess pore pressure variation with time beneath the middle of the berm 181

xxiv

List of Figures

Figure 5.19 Variation of excess pore pressure below the embankment centreline 182

Figure 6.1 Location of SBIA site (after Moh and Lin, 1997) 186

Figure 6.2 General soil properties at SBIA site (modified after Sangamala, 1997) 186

Figure 6.3 Compressibility parameters at SBIA site (adopted from Sangmala, 1997). 187

Figure 6.4 Cross section of embankments with key instrumentation at SBIA (modified

after Indraratna et al., 2005) 188

Figure 6.5 Construction loading history 189

Figure 6.6 Types of elements used in ABAQUS (Hibbitt et al., (2004) 191

Figure 6.7 Normalized pore water pressure variation with radial distance 192

Figure 6.8 Variation of extent of smear zone with depth 193

Figure 6.9 Finite element mesh used in the analyses (Model 2) 194

Figure 6.10 Measured total pore pressure and simulated vacuum pressure at surface. 195

Figure 6.11 Settlement variation with depth for embankments (a) TV1 and (b) TV2.. 197

Figure 6.12 Variation of excess pore water pressure at 3m depth below ground level,

0.5m away from the centreline for embankments (a) TV1 and (b) TV2 198

Figure 6.13 Lateral displacement profiles (after 150days) through the toe of the

embankments (a) TV1 and (b) TV2 199

Figure 7.1 Finite element mesh (consists of 15-node elements) used in this analysis..204

xxv

List of Figures

Figure 7.2 Surface settlement at the embankment centre with fill height 206

Figure 7.3 Surface heave at embankment toe with fill height 206

Figure 7.4 Displacement contours for different slopes 207

Figure 7.5 Lateral displacement contours for different slopes 208

Figure 7.6 Excess pore pressure contours for different slopes 209

Figure 7.7 Surface settlement at embankment centreline with fill height 210

Figure 7.8 Displacement (heave) at embankment toe with fill height 211

Figure 7.9 Contour plots for a loading rate of 0.2 m/week (the height and the slope of

the embankment are 1.5m and 3:1, respectively) 212

Figure 7.10 Surface settlement at embankment centreline for different drain spacing 215

Figure 7.11 Displacement at embankment toe for different drain spacing 215

Figure 7.12 Lateral displacement contours for different drain spacing 216

Figure 7.13 Surface settlement at embankment centreline for two different smear zones

218

Figure 7.14 Displacement at embankment toe for two different smear zone 218

Figure 7.15 Surface settlement at embankment centreline for different permeability

ratios 220

Figure 7.16 Displacement at embankment toe for different permeability ratios 220

xxvi

. List of Figures

Figure 7.17 Construction loading history 222

Figure 7.18 Surface settlement at embankment centreline with (a) time, (b) fill height

222

Figure 7.19 Displacement contours when the fill height is 1.6 m 223

Figure 7.20 Lateral displacement contours when the fill height is 1.6 m 223

Figure 7.21 Effect of surface crust 224

Figure 7.22 Increment contours when the embankment height increases from 1.8 to 2.0

m (a) vertical displacement, (b) lateral displacement, and (c) shear strain 225

Figure 7.23 Measured settlement of Muar test embankment, Malaysia (after Indraratna

etal., 1992) 227

Figure 7.24 Typical embankment 227

Figure 7.25 Variation of induced vertical stress and shear stress 229

Figure 7.26 Variation of shear stress with horizontal direction at different depths 230

Figure 7.27 Shear stress distribution with normalized depth under embankments 231

Figure 7.28 Variation of lateral displacement at different cross section 232

Figure Al Variation of the isotropic and conventional overconsolidation ratio with slope

of critical state line 264

xxvn

List of Tables

LIST OF TABLES

Table 2.1 Percentage of discharge capacity of deformed drain condition 27

Table 2.2 Short term discharge capacity, in m3/year (Hansbo, 1981) 28

Table 2.3 Current recommended values for specification of discharge capacity 29

Table 2.4 Proposed smear zone parameters (After Xiao, 2001) 34

Table 2.5 Summary of proposed well resistance indexes 39

Table 2.6 Normalized deformation factors (modified after Indraratna et al. 1997) 79

Table 3.1 Input MCC parameters and calculation sheet 92

Table 4.1 Engineering properties of selected sample 142

Table 4.2 Modified Cam Clay Parameters 147

Table 5.1 Modified Cam-clay parameters used in the finite element analysis 167

Table 5.2 Mohr-Coulomb parameters of the sand layer 167

Table 5.3 Equivalent Plane Strain permeabilities of embankment sections 172

Table 6.1 Modified Cam-clay parameters of SBIA site (Indraratna et al., 2005) 190

Table 6.2 Axisymmetric and Plane Strain permeabilities for both embankments 193

Table 7.1 Soil parameters used in finite element analysis 204

Table 7.2 Equivalent permeability 214



xxviii

. — List of Notations

LIST OF SYMBOLS

A Cross sectional area

a Width of band drain

Radius of cavity after time t

av Coefficient of compressibility

B Equivalent half width of the plane strain cell

b Thickness of band drain

bs Equivalent half width of smear zone in plane strain

bw Equivalent half width of drain (well) in plane strain

c Vacuum propagation factor

Cc Compression index

Cf Ratio of field and laboratory coefficient of permeability

Ch Coefficient of horizontal consolidation

Ck Permeability change index

Cr Recompression index

c„ Undrained shear strength

cv Coefficient of vertical consolidation

Ca Secondary compression index

De Diameter of effective influence zone of drain

dm Diameter of mandrel

xxix

ds Diameter of smear zone

dw Equivalent diameter of band drain

D15 Diameter of clay particles corresponding to 15% passing



E Young's modulus

Ei Shearing strain in conventional triaxial tests

£_ Strain from pressure meter tests

Ei Strain from simple shear tests

Eu Young's modulus for undrained shear

e Void ratio

ecs Void ratio on the critical state line for value of p'=l

e0 Initial void ratio

Ft Influence factor of drain due to time

Fc Influence factor of drain due to drain deformation

Ffc Influence factor of drain due to clogging

F(n) Drain spacing factor

G Shear modulus

Well resistance factor

Gs Specific gravity

xxx

H Soil thickness

Hd The longest drainage path

H0 Initial thickness of compressible soil

Ip Plasticity index

i Hydraulic gradient

i0 Threshold hydraulic gradient

ii Limiting hydraulic gradient

k Permeability

kax Axisymmetric permeability

kfliter Permeability of filter

kh Horizontal coefficient of permeability

k'h Horizontal coefficient of permeability in smear zone

khp Equivalent undisturbed zone permeability in plane strain

k'hp Equivalent smear zone permeability in plane strain

kpi Plane strain permeability

ks Smear zone permeability

ksoii Permeability of soil

kv Vertical coefficient of permeability

kve Equivalent vertical coefficient of permeability

kw Coefficient of permeability of drain

xxxi

L Well resistance factor

/ Length of drain

/ Maximum discharge length

LL Liquid limit

M Oedometer modulus

Slope of critical state line

mv Coefficient of volume change

N Volume on the normal consolidation line corresponds to p'=l

n Spacing ratio

Exponent factor (non-Darcian flow)

O15 Size of particle which is larger than 15% of the fabric pores

O50 Size of particle which is larger than 50% of the fabric pores

O95 Apparent opening size

OCR Conventional (1-D) overconsolidation ratio

p Total mean stress

p' Effective mean stress

p c Stress representing the reference size of yield locus

p 'c0 Maximum stress on yield locus

p o In situ mean effective stress

PI Plasticity index

xxxii

PL Plasticity Limit

Q Discharge capacity

q Deviator stress

qp Deviator stress at elasto-plastic boundary

qreq Theoretical discharge capacity calculated from Barron's theory

qw Axisymmetric discharge capacity

qW(min) Minimum discharge capacity

qz Plane strain discharge capacity

R Radius of axisymmetric unit cell

Well resistance factor

Isotropic overconsolidation ratio

r Radius

R\ Index of well resistance

/-„ Permeability anisotropy (permeability ratio)

rm Radius of mandrel

rp Radius of plastic zone

rs Radius of smeared zone

rw Radius of vertical drain (well)

S Field spacing of drains

s Smear ratio

xxxiii

t Thickness of the drain walls in 2-D model

Time

Ty, Time factor for horizontal drainage

Thp Time factor for horizontal drainage in plane strain

Tv Time factor for vertical drainage

Tr5o Time factor for 50% consolidation of radial flow

Tso Time factor for 50% consolidation of laminar flow

Tr9o Time factor for 90% consolidation of radial flow

Tgo Time factor for 90% consolidation of laminar flow

u Pore water pressure

ux Displacement along the x\ direction

u0 Initial pore water pressure

Up Average degree of pore pressure dissipation

ur Excess pore water pressure due to radial flow only

Us Degree of consolidation settlement

usur Applied surcharge pressure

uvac Applied vacuum pressure

uw Pore water pressure

uz Excess pore water pressure due to vertical flow only

Uax Degree of consolidation in axisymmetric

xxxiv

List of Notations

Upi Degree of consolidation in plane strain

Uio 10 % degree of saturation

U Average degree of consolidation

u Average excess pore pressure throughout the soil mass

Ur Average degree of consolidation for radial flow only

ur Average excess pore pressure throughout the soil mass due to radial flow

uvac Average applied vacuum pressure

Uz Average degree of consolidation for vertical flow only

uz Average excess pore pressure throughout the soil mass due to vertical flow

V Volume

Specific volume

v Specific volume

v Smear factor

Velocity of flow

W Well resistance factor

w Water content

Width of band drain

WL Liquid limit

wp Plastic limit

XXXV

List of Notations

x Horizontal distance from centre of drain

x\ Cartesian coordinates

z Depth (thickness) of soil layer

Greek Letters

a Geometric parameter representing smear in plane strain under Darcian flow

A parameter representing the smear and well resistance in axisymmetric

consolidation under non-Darcian flow

Ratio of maximum lateral displacement to settlement

/? Geometric parameter representing smear in plane strain

A parameter representing the smear and well resistance in axisymmetric

consolidation under non-Darcian flow

Bj Ratio of maximum lateral displacement to corresponding fill height

B2 Ratio of maximum settlement to corresponding fills height

s Strain

sj Final settlement of the soft soil equivalent to 25% of the drain length installed

sr Radial strain

£0 Circumferential strain

§ Consolidation scalar potential

xxxvi

— — List of Notations

</) Effective friction angle

r Volume on the critical state line corresponds to p-1; r=ecs+\

yoc1 Octahedral strain

ys Saturated unit weight of soil

Yw Unit weight of water

n. Stress ratio

K Slope of unloading-reloading line in v-lnp space

Coefficient of permeability in non- Darcian flow

A Plastic volumetric strain ratio

X Slope of normal compression line in v-lnp space

Coefficient of consolidation in non- Darcian flow

fi Smear and well resistance factor in axisymmetric under Darcian flow

JUP Smear and well resistance factor in plane strain under Darcian flow

v Poisson's ratio

9 Geometric parameter representing well resistance in plane strain

p Settlement

o Total stress

xxxvii

a Effective stress

c_ Internal cavity pressure at time t

cr'ho In situ effective horizontal stress

o~rp Stress at elasto-plastic boundary

CvmaxMaximum effective vertical stress (past)

crr Radial stress

o~z Vertical stress at depth z

G'V Effective vertical stress

cr™ In situ effective vertical stress

o~e Circumferential stress

o~o Initial internal cavity pressure

o~i Axial stress or Major principle stress

(72 Confining stress or Intermediate principle stress

0-3 Confining stress or Minor Principle stress

r Shear stress

to Radial speed of soil element

E, Displacement

xxxviii

Subscripts

ax Axisymmetric

h Horizontal or Undisturbed zone

p Plane strain

r Radial

s Smear zone or Shear

v Vertical or Volumetric

0 Initial

0 Circumferential

Superscript

/ Effective parameter or Smear zone parameter

p Plastic

xxxix

Chapter 1 Introduction

1 INTRODUCTION

1.1 General

Rapid development and associated urbanization have compelled engineers to construct

earth structures including major highways over soft clay deposits of low bearing

capacity coupled with excessive settlement characteristics. In the coastal regions of

Australia and Southeast Asia, soft clays are widespread and extensive, often in the

vicinity of capital cities. These grounds have low shear strength and bearing capacity as

well as high compressibility and excessive settlements over long periods of time. When

these areas are selected for development work, the ground level must be raised above

the flood level with fill, but unacceptable differential settlement due to the heterogeneity

of the fill and compressibility of the underlying soft soils may cause the structures to be

damaged. Therefore, it is essential to stabilize the existing soft clay foundations prior to

construction in order to avoid excessive and differential settlement.

Even though there are a variety of soil improvement techniques available to

stabilize the soft ground, the application of preloading with prefabricated vertical drains

is still regarded as one of the classical and popular methods in practice. It involves the

loading of the ground surface to induce a greater part of the ultimate settlement of the

underlying soft strata. In other words, a surcharge load equal to or greater than the

expected foundation loading is applied to accelerate consolidation by rapid pore

pressure dissipation via vertical drains. Application of vacuum pressure with surcharge

loading can further accelerate consolidation while reducing the required surcharge fill

material without any adverse effects on the stability of an embankment built on soft

clays. The rate of consolidation attributed to vacuum-assisted preloading is greater than

the conventional method because of an increase in the lateral hydraulic gradient.

1

. Chapter 1 Introduction

The major use of preloading and vertical drains is that preloading on its own can

reduce total and differential settlement facilitating the choice of foundations, whereas

when vertical drains are used with preloading, the settlement process can be accelerated

considerably. Soil consolidation is the process of decreasing the volume in saturated

soils by expelling the water that occupies inside the pores between the solid soil

particles. Therefore, the rate of consolidation is governed by the compressibility,

permeability and length of the drainage path. The amount of settlement is directly

related to the change in void ratio, which in turn is directly proportional to the amount

of dissipation of excess pore water pressure.

A number of researchers (e.g. Barron, 1948; Yoshikuni and Nakanodo, 1974;

Hansbo, 1981; Onoue, 1988; Hird and Russell, 1992; Indraratna and Redana, 1998)

have investigated the factors affecting the progress of consolidation with vertical drains.

The influence of smear on consolidation and the discharge capacity of the drain have

been the main focus of such investigations. The term 'smear zone' is generally referred

to as the disturbance that occurs when installing a vertical drain. This causes a

substantial reduction in soil permeability around the drain, which in mm retards the rate

of consolidation. In reality, the discharge capacity of a drain can be reduced due to well

resistance as a result of kinking, folding etc and also due to clogging of drain filters

during installation. The difficulties and uncertainties that prevent a proper assessment of

radial consolidation around vertical drains arise from the following aspects:

(i) Obtaining accurate and representative soil parameters, especially hydraulic

properties, from laboratory and/or field tests;

(ii) Estimating the smear effect due to insufficient knowledge of smear zone

characteristics;

(iii) Incorporating variations of soil properties with consolidation;

2

.— . Chapter 1 Introduction

(iv) Oversimplifying the loading history and stress condition;

(v) Inability of the existing theories to completely model the consolidation;

(vi) Evaluating the effectiveness of vertical drains and discharge capacity; and

(vii) Difficulties in incorporating installation details.

Some of these uncertainties and difficulties are due to a lack of understanding,

which could be overcome through research such as this, and indeed, the main focus of

this thesis is the effect of smear, which is a significant and unavoidable problem.

In this study, the effectiveness of vertical drains was evaluated by modelling its

behaviour in large-scale laboratory consolidation tests. Special attention was given to

modelling the smear zone developed during installation. The extent of the smear zone

was determined in the laboratory utilising the large-scale consolidometer apparatus and

verified with proposed analytical smear zone. A plane strain model including smear

effect is introduced by modifying Hansbo's axisymmetric solution of vertical drains.

The sub-soil properties were modelled according to the Modified Cam-clay theory, and

the finite element technique was adopted as the main tool for numerical analysis.

1.2 Use of Vertical Drains

Vertical drains are artificially created drainage often used in soft deposits to accelerate

primary consolidation by shortening the drainage path such that dissipation of pore

water pressure can occur radially rather than the vertically. The main advantages of

vertical drains are as follows:

(i) Increase the shear strength of soil by decreasing the void ratio;

(ii) Decrease the preloading time to minimize the same level of post-construction

settlement;

3

Chapter 1 introduction

(iii) Reduce differential settlements during the primary consolidation stage; and

(iv) Curtail the height of surcharge fill required to achieve the desired

precompression.

The potential benefit of vertical drains is shown in Fig. 1.1, where a faster rate of

foundation settlement is obtained compared to those without them. Vertical drains can

also be used as pressure relief wells to reduce pore pressure due to seepage, for instance,

below natural slopes, and to improve the effectiveness of natural drainage layers below

loaded areas.

Time (month)

Figure 1.1 Potential benefit of vertical drains (adapted from Lau et al.2000)

4


1.3 Methods of Deformation Analysis

1.3.1 Analytical Method

A stress increase caused by construction of embarikments or other compresses soil

layers. This compression is caused by (i) deformation of soil particles, (ii) relocation of

soil particles and (iii) expulsion of water or air from the void spaces. In general,

embankments and subsoil deformation may be classified into three components;

immediate or distortion settlement, consolidation or primary settlement, and creep or

secondary compression settlement. Immediate or undrained deformation caused by the

elastic deformation of moist, dry, and saturated soils without any change in the moisture

content, might be calculated using various solutions, hence these settlement calculations

are generally based on equations derived from the theory of elasticity (Giroud and

Rebatel, 1971; Poulos and Davis, 1974). These elastic solutions have been popular due

to their simplicity, although the sub-soil is often non-elastic.

Consolidation settlements, the result of a volume change in saturated cohesive

soils due to the expulsion of wafer that occupies in the void spaces, are predicted by

evaluating one dimensional compression characteristics of an undisturbed sample in the

laboratory oedometer test. The rate of settlement can be predicted using Terzaghi's one-

dimensional consolidation theory (Terzaghi and Peck, 1967; Holtz and Kovacs, 1981).

For embankments stabilized with vertical drains, the rate of settlement can be predicted

using Barron's (1948) or Hansbo's (1981) analytical solutions.

Secondary compression is a continuation of volume change beyond primary

consolidation, and usually occurs at a much slower rate. Secondary compression is

different from primary consolidation in that it takes place under constant effective

5

. Chapter 1 Introduction

stress, i.e. after the entire excess pore pressure has dissipated. Secondary compression

might be predicted using the relationships given by Raymond and Wahls (1976), and

Mesri and Godlewski (1977).

1.3.2 Numerical Analysis

There are many practical engineering problems for which one cannot obtain exact

solutions. This may be attributed to either the complex nature of governing differential

equations or the difficulties that arise from dealing with boundary and initial conditions.

To deal with this problem one needs to resort to numerical approximations. In contrast

to analytical solutions, which show the exact behaviour of a system at any point within

the system, numerical solutions approximate exact solutions only at discrete points,

called nodes. The first step of any numerical procedure is discretisation, which divides

the medium of interest into a number of small sub-regions and nodes. There are two

common classes of numerical methods: (i) finite difference and (ii) finite element

methods. With finite difference methods, the differential equation is written for each

node, and the derivatives are replaced by finite difference equations. This approach

results in a set of simultaneous linear equations. Although finite difference methods are

easy to understand and employ in simple problems, they become difficult to apply to

problems with complex geometries or boundary conditions. In contrast, the finite

element method uses integral formulations rather that finite difference equations to

create a system of algebraic equations. Moreover, an approximate continuous function

is assumed to represent the solution for each element. The complete solution is then

generated by connecting or assembling the individual solutions, allowing for continuity

at the inter-elemental boundaries. The deformation response of each element is defined

6


by element shape, the displacement variation within each element, and the constitutive

model (stress-strain behaviour) employed to represent element behaviour.

Numerical techniques in geotechnical engineering have been used increasingly

because of advances in computer applications. The main advantages of numerical

analysis are that the settlement and stresses within the soil are coupled, and therefore

more realistic soil behaviour can be simulated. Finite difference methods for vertical

drains have been developed by several researchers (Brenner and Prebaharan, 1983 and

Onoue, 1988b). Of these, two techniques are generally known in the finite difference

approach as "explicit" and "implicit" solutions. A comparative study between them

concluded that the implicit method provides better numerical stability, although a set of

simultaneous equations needs to be solved at each time step (Desai and Christian,

1977).

1.3.3 Observational Methods

Due to a variety of uncertainties in theoretical procedures, final settlement may not be

correct, and therefore observational support is required to justify the predictions. There

are several observational methods available whereby using the initial settlement

observations, the final consolidation settlement could be predicted (Asaoka, 1978,

Magnan and Deroy, 1980).

1.4 Scope and Objective of Study

In order to increase embankments stability on soft, clayey foundations, it has become

common practise to construct vertical drains underneath an embankment to accelerate

consolidation, for which engineers estimate the time-settlement relations using Barron's

7


solution (which assumes a simple case of no peripheral smear, no well resistance, and

no well rigidity). However, remarkable differences in the consolidation process have

been often found, which may be caused by the aforementioned factors. M a n y

researchers and engineers in soil mechanics have been interested in both peripheral

smear and well resistance. Nowadays, most of the prefabricated drains used in practice

have negligible well resistance. Therefore, the accuracy of the predictions using the

consolidation theories depends on the correct assessment of the extent of smear zone

and its permeability. But, these are often difficult to quantify and determine from

laboratory tests. So far, there is no comprehensive or standard method for measuring

them.

In this study, an attempt is made to introduce an analytical method to determine

the extent of smear zone using cylindrical cavity expansion analysis incorporating the

Modified Cam-Clay model. This predicted extent of smear zone was compared with the

experimentally evaluated smear zone propagation using a large scale, radial drainage

consolidometer. Then, the existing axisymmetric (3-D) and plane strain (2-D) theories

were modified incorporating the vacuum pressure. The settlements observed in

laboratory tests were compared with the predicted value, using the finite element

computer programmes incorporating the critical state theory in soil mechanics. Various

case studies on vertical drains were also analysed to verify the analytical model. This

study includes the performance of vertical drains installed underneath embankments

built on soft clay foundations in Australia and Thailand.

Various laboratory tests have been conducted to model the smear effect around

vertical drains. Based on those findings, the smear zone can be quantified and verified

with the proposed analytical solution and then incorporated in the finite element

analysis. Compressibility and permeability tests were also conducted to determine the

8


variation of compressibility, and vertical and horizontal permeability of the soil close to,

and away from the central drain, in a large, radial drainage consolidometer. The

apparatus can accommodate 650 mm diameter samples, upto 950 mm in height. The

permeability was indirectly measured using a conventional (50 mm diameter)

oedometer apparatus on samples recovered from the large-scale consolidometer, and

subsequently analysed using Terzaghi's one-dimensional theory. By measuring the

samples recovered at a known radial distance surrounding the centrally inserted drain in

the large-scale consolidometer, the extent and the variation of the permeability inside

and outside the smear zone were determined. The soil sample (450 mm diameter x 950

mm height) placed in the large-scale consolidometer was loaded to measure the

settlement and pore water pressure response of the soft clay.

The model proposed in this study has been applied to several embankments built

on soft clay stabilized with vertical drains founded in Australia and Thailand. The

settlements and pore water pressure responses of these soft clay foundations were

determined and compared with the field measurements. Finally, the knowledge gained

from the analysis of consolidometer cell and real embankments was used to demonstrate

how the plane strain analysis can be employed to predict the stability of embankments

under different conditions such as, with and without vertical drains, preloading,

different embankment geometries, and different drain spacing.

1.5 Organization of the Dissertation

This introductory Chapter highlights the aim and scope of the present research, while

Chapter 2 is devoted to a detailed literature review associated with the present work. In

fact, the available publications on vertical drains, plane strain modelling, and the

9

.—. Chapter 1 Introduction

relevant laboratory tests to measure the coefficient of consolidation and permeability

inside and outside smear zone, and the analysis of embankments stabilized with vertical

drains, are reviewed in detail.

Chapter 3 presents the mathematical formulation for the present research. First, an

analytical solution is described to evaluate the extent of the smear zone caused by

mandrel driven vertical drains using the cavity expansion theory incorporating Modified

Cam Clay model. Then, a modified consolidation theory incorporating vacuum pressure

for axisymmetric and plane strain conditions, with a linearly distributed (trapezoidal)

vacuum pressure was discussed. Finally, a new plane strain consolidation equation

(incorporating smear) for non-Darcian flow are presented in detail.

In Chapter 4, the experimental set up in the laboratory to estimate the extent of

smear zone, to monitor settlement and pore pressure was discussed. The settlement

response of the soil with vertical drains (tested in the large-scale equipment) is predicted

using the proposed plane strain analysis as previously mentioned in Chapter 3.

Chapters 5 and 6 present the application of the proposed model to several

embankments stabilized with vertical drains in Australia and Thailand. Multi-drain

analysis was carried out and the predictions were compared with available field data.

Chapter 7 discusses the use of 2-D plane strain numerical analysis for soft clay

foundations to predict the failure height under different conditions such as, with and

without vertical drains, preloading, different embankment geometries, and different

drain spacing. Comparisons between those conditions are elucidated.

10


Concluding remarks and recommendations for further studies are summarized in

Chapter 8, followed by References and Appendices. The letter symbols (notation) in

this thesis are defined where they first appear, either in the text, or by diagrams.

11

. Chapter 2 Literature Review

2 LITERATURE REVIEW

2.1 History and Development of Vertical Drains

To improve the soft ground different types of vertical drainage system (e.g. sand

compaction piles, prefabricated vertical drains) have been used extensively over the past

few decades. This concept was first proposed around 1920's and patented in 1926 by

Daniel.J.Moran, an American engineer. Moran suggested the first practical application

of sand drains to stabilize the mud soil beneath the roadway approach to the San

Francisco Oakland Bay Bridge (Johnson, 1970).

This solution led to comprehensive laboratory and field-testing by the California

Division of Highways in 1933 on the effectiveness of sand drains on the rate of

consolidation, and in 1936 Porter described these successful trials and contributed to the

further study and development. After World War Two the application of sand drains has

undergone enormous development, largely due to better methods of installation and

greater knowledge of the principle controlling their performances in different types of

soft clays (Jamiolkowski et al., 1983). In Japan, during 1940's, vertical sand drain

behaviour was not understood very well because the foundation bearing capacity was

considered sufficient for a full load immediately after installation which resulted in

frequent foundation failure (Aboshi, 1992).

Walter Kjellman installed the first prefabricated drain system in a field test in

1937 using tubes made from a wood/fibre material but after realizing this material was

inappropriate and too expensive, Kjellman invented and patented a band shaped

cardboard drain in 1939, and a method for driving it into the ground. This cardboard

drain consisted of two cardboard sheets glued together with an external cross-section

100 mm wide by 3 mm thick, with ten 3 mm wide by 1 mm thick longitudinal internal

12

— Chapter 2 Literature Review

channels. The efficiency of cardboard wicks was first investigated in Sweden at Lilla

Mellosa in a full-scale test, after which several types of prefabricated band drains such

as Geodrain (Sweden), Alidrain (England), Mebradrain (Netherlands), etc, were

developed. Basically, prefabricated band drains have a rectangular cross-section

consisting of filter fabric sleeve or jacket surrounding a plastic core. The sleeve acts as a

physical barrier separating the core and the surrounding soil but permits pore water to

enter the drain. It is made from non-woven polyester, polypropylene geotextile, or

synthetic paper. The plastic core has grooved channels which act as flow paths and

supports for the filter sleeve (Bergado et al. 1996).

2.2 Types of Vertical Drains

There are many types of vertical drains used for the ground improvement, the first were

cylindrical columns of sand constructed using conventional piling equipment and were

placed at fairly large centres to avoid disturbing the ground. It was soon realized that

their performance depended on the spacing and surface area of the drain. To reduce the

spacing a sand drain called "sandwicks" were used where the sand was contained within

a 50 m m diameter geotextile sock and installed using a vibrated or jetted mandrel. At

this time Kjellman was experimenting with a narrow cardboard drain, all of which had

major drawbacks so in 1970 Akzo Research, in cooperation with the Royal Adrian

Volker Group, developed the first synthetic drain using a non-woven polyester fabric.

2.2.1 Sand Drains

Sand drains collect water expelled from the soil and transport it out of the ground. The

discharge capacity of sand drain is governed by their permeability, which in turn is

controlled by the size of the grain. The discharge capacity however, can be reduced and

13

Chapter 2 Literature Review

ultimately clogged, by the intrusion of fine soil particles. Generally, sand drains were

relatively large (30-70 cm) but later on smaller diameter drains (sand wick - 50 mm,

sand pack -120 mm) were used. In these drains, the sand is packed into a synthetic fibre

net tube which prevents the drains from necking. Sand compaction pile is also a sand

drain which is compacted during its installation and can carry more loads and operate as

a drain well. Some disadvantages of sand drains are:

(i) The sand must have adequate drainage properties and is seldom found near the

construction site;

(ii) The drains may become discontinuous due to careless installation or excessive

lateral soil displacements during consolidation;

(iii) Sand bulking during placement may lead to formation of cavities and subsequent

collapse on flooding;

(iv) The large diameter required for the sand drain may pose a construction problem

and/or budgetary burden;

(v) Disturbing the soil around each drain during installation may reduce its hydraulic

conductivity, flow, and efficiency of the system; and

(vi) The reinforcing effect may reduce the efficiency of the surcharge loading in

consolidating the subsoil.

2.2.2 Prefabricated Vertical Drains

The development of various types of prefabricated band shaped drains that can

overcome most of the shortcomings of sand drains are being marketed under different

trade names (Rixner et al. 1986). The number of commercially available band drains has

increased rapidly since the first cardboard wick type drain used by Kjellman in 1948.

With the development of the first band drain by Akzo in 1970, and the rapid

14


development of geotextile and geomembrance technology in recent years, the sleeves

(or filter jacket) of today's wick drains are usually made from synthetic geotextile

which provides higher tensile strength and ensures consistent performance and quality.

Several types of PVD are available, such as Geodrain, Mebra drain, Audrain, Colbond

and Flodrain. PVD is more efficient than a sand drain due to its rapid installation rate

which saves time and money. PVD also creates fewer disturbances to the soil during

installation and has a greater resistance (stiffness) to lateral ground movement. Typical

types of PVDs such as Mebra and Colbond drain are shown in Figure 2.1. The most

common band shaped drains are 100 mm x 4 mm.

Band drains are generally placed within a steel mandrel by displacement methods

using static pull down or vibratory techniques. Static pushing is preferable for driving

the mandrel into the ground; whereas the dynamic methods seem to create a higher

excess pore water pressure and a greater soil disturbance during installation. The

installation rig was shown in Figure 2.2, where the vertical drain is protected during the

installation by a mandrel.

Figure 2.1 Typical types of P V D (company brochure)

15


Figure 2.2 Typical installation rig (Source: Colbond bv, The Netherlands)

2.3 Installation and Monitoring of Vertical Drains

The site must be prepared before installing a vertical drain, this may involve removing

surface vegetation and debris and grading the site for a sand blanket to act as a medium

for expelling water from the drains, and as an appropriate working mat. Vertical drains

can be installed by the washing jet method, the static method, or the dynamic method.

The washing jet method is primarily used when installing large diameter sand drains,

where sand is washed in through the jet pipe.

Prefabricated vertical drains (PVDs) are usually installed by the static or dynamic

method. In the latter, the mandrel is driven into the ground with a vibrating or drop

hammer, but in the former, the mandrel is pushed in by a static load. The static method

usually causes less ground disturbances and is much preferred for more sensitive soils.

Although faster, the dynamic methods generate higher excess pore pressures and more

soil around the mandrel.

16


Surcharge fill

Deep settlement points

Permanent fill

Piezometers

n

D u m m y Piezometers

Settlement plate

Berm

Inclinometer

Figure 2.3 Basic instrumentation for a highway embankment (After Rixner, 1986)

O n major projects instrumentation is essential for verifying performance and

observing design amendments, as warranted, to prevent unacceptable displacement.

Figure 2.3 shows a typical scheme of instruments required to monitor the soft clay

foundation beneath an embankment containing PVD. The most commonly used

instruments are inclinometers, settlement indicators, and piezometers.

A diagram of the vacuum preloading method is shown in Figure 2.4 (Shang et al.,

1998). The working platform consists of a sand layer through which vertical drains are

placed into the soil. The treatment area is sealed by a flexible membrane which is keyed

into an anchor trench surrounding the area and a perforated pipe system is placed

beneath the liner to collect water. It is essential that the site be securely sealed and

isolated from any surrounding permeable soils to avoid leaks in the membrane and loss

of vacuum. To obtain and sustain a high vacuum, the membrane is covered with water.

17


Finally the vacuum pumps with sufficient capacity to generate a vacuum in the soil and

capable of pumping water and air are connected to the collection system.

Water Vacuum pump Election pipe Impervious

membrane liner Clay revetment

Sand layer

Figure 2.4 Schematic of vacuum preloading consolidation (Tang et al., 2000)

2.3.1 Inclinometers

These instruments are used to monitor the lateral (transverse) movements of natural

slopes or embankments. An inclinometer casing has a grooved metal or plastic pipe that

is placed into a borehole (Durmicliff, 1988). The space between the borehole and the

casing is backfilled with a sand or gravel grout. The bottom of the pipe must rest on a

firm base to achieve a stable point of fixity. To monitor the embankment inclinometers

are normally placed at or near the toe where excessive lateral movement is usually of

some concern.

18


2.3.2 Settlement Indicators

Settlement plates or points are commonly installed where significant settlement is

predicted (Dunnicliff, 1988) to record the magnitude and rate of settlement under a load.

Settlement plates are steel plate which should be placed immediately after installing the

vertical drains and before constructing the embankment. Typically, a reference rod and

protecting pipe are attached to the settlement-monitoring platform and settlement is

often evaluated periodically until the surcharge embankment is completed, then at a

reduced frequency, measuring the elevation at the top of the reference rod. Benchmarks

used for reference datum must be stable and remote from all vertical movements

2.3.3 Piezometers

Piezometers should be installed at the bottom of the sand blanket and at various

intermediate depths within the compressible layer. A dummy piezometer is usually

installed a sufficient distance away from the embankment to record the natural ground

water level and excess pore water pressure at a given location is determined by

comparison with the 'dummy' level.

2.4 Drain Properties

2.4.1 Diameter of Influence Zone

Vertical drains are generally installed in either a square or equilateral triangular pattern,

as illustrated in Figure 2.5. The related consolidation problem is normally simplified to

an axisymmetric unit cell in most vertical drain consolidation theories where the drain

well and its influence are assumed to be cylindrical. The influence area of each drain is

usually approximated into a cylinder with an equivalent cross sectional area. As shown

19


in Figure 2.5, the equivalent diameter of the influence zone (De) can be found in terms

of the drain spacing (S) as follows (Hansbo, 1981):

De-1.128 S for drains installed in a square pattern, and (2.1)

De-1.05 S for drains installed in an equilateral triangular pattern (2.2)

Drains in a square pattern may be easier to lay out and control during installation

in the field but a triangular pattern usually provides a more uniform consolidation

between them.

Drains

De=1.128S

Square pattern De=1.05S

Triangular pattern

Figure 2.5 Typical drain installation patterns and the equivalent diameters

2.4.2 Equivalent Drain Diameter of Band Shaped Vertical Drain

The radius of sand drains, or their modern derivatives such as sand wicks or plastic tube

drains, can easily be determine from the size of the mandrel, which is usually circular in

cross section. However, most prefabricated drains have rectangular cross section (band

20


shaped, Figure 2.6), but for design purposes, the rectangular (width-a and thickness-,))

section has to be converted into an equivalent circle (Figure 2.6) with a diameter of dM

because the conventional theory of radial consolidation assumes that drains are circular.

band-shaped

cross section

Polypropylene core

dw=f(a,b)

Geotextile filter

equivalent circular

cross-section

Figure 2.6 Conceptual drawing of a P V D and equivalent diameter well

Kjellman (1948) first suggested that, "the draining effect of a drain depends to a

great extent upon the circumference of its cross-section, but very little upon its cross-

sectional area". Based on finite element analysis Hansbo (1979) verified Kjellman

suggestion and thus band shaped and circular drains lead to practically the same degree

of consolidation, provided their circumferences are equal. Accordingly, the equivalent

diameter dw of a band shaped drain with width a and thickness b can be expressed as

(using "perimeter equivalence"):

21


, 2(a + b) dw=— " (2.3)

7t

Atkinson and Eldred (1981) proposed that a reduction factor of K/4 should be

applied to Eqn. (2.3) to take account of the corner effect where the flow lines converge

rapidly. This was subsequently confirmed with finite element studies performed by

Rixner et al. (1986) and Hansbo (1987). This leads to:

' „ - % - * (2-4)

Pradhan et al. (1993) suggested that the equivalent diameter of band-shaped drains

should be estimated by considering the flow net around the soil cylinder of diameter de

(Figure 2.7). The mean square distance of their flow net is calculated as:

-2 1 2 1 2 2® s =~di+ — a' jde (2.5)

71 4e 12

Then, dw=de-2A\s2 ]+b (2.6)

dw=0.5(a+b) Rixner etal.( 1986)

dw=0.5a+0.7b <ông&Covo(1994)

\ dw=2(a+b)/7t

™ ', Hansbo (1979)

*b!

' Assumed water yi—iflow net

Pradhan etal. (1993) e.. - "

Figure 2.7 Equivalent diameters of band-shaped vertical drains

22


More recently, using an electrical analogue field plotter, Long and Covo (1994)

found that the equivalent diameter dw should be computed using:

dw= 0.5a + 0.7b (2.7)

No definitive research exists that clearly shows that one of these formulations is

superior to the others. Various research efforts have supported each of the equations

presented above.

2.4.3 Filter and Pore Size

The pore or apparent opening size (AOS) of the filter should meet its design criteria. On

the one hand, the AOS has to be small enough to prevent fine particles of the soil

entering the filter and the drain but on the other hand it cannot be too small because it

must provide sufficient permeability. The two key parameters that indicate filter quality

are AOS and permeability. Generally, filter permeability (&/;.._,-) must be at least one

order of magnitude higher than the soil (ksou), i.e.:

kfllter > 10 ksoil (2-8)

Several researchers have proposed some criteria for AOS; a commonly used

criterion is given by Carroll (1983):

°2L*(2-3) (2-9) D85

and 5___(70-72) (2-10) D50

23


where, O95 is the AOS (the size of largest particle that would effectively pass through),

O50 is the size which is larger than 50% of the fabric pores, and Dgs and D50 refer to the

sizes for 85% and 50% of passing soil particles by weight. The apparent opening size

(AOS) of a prefabricated vertical drain is usually taken to be less than 0.09 mm. Filter

material can also become clogged if the soil particles become trapped but this can be

prevented by ensuring that (Christopher and Holtz, 1985):

D15

o15

>3

= (2-S)

(2.11)

(2.12)

2.4.4 Discharge Capacity

The discharge capacity is the most important parameter that controls the performance of

prefabricated vertical drains. Only PVDs having sufficient discharge capacity can

function well. There are two major uncertainties related to the discharge capacity of a

vertical drain. The first one is the determination of the required discharge capacity to be

used in design (Holtz et al., 1991) and the second one is the measurement of the

discharge capacity of the drain in the laboratory and the field. To measure discharge

capacity it is necessary to simulate field conditions as closely as possible. According to

Holtz et al. (1991), the discharge capacity depends primarily on the following factors:

(i) The area of the drain core available for flow (free volume);

(ii) The effect of lateral earth pressure;

(iii) Possible folding, bending, and crimping of the drain (Figure 2.8); and

(iv) Infiltration of fine soil particles through the filter.

24


Relatively uniform soil

mass

a) uniform bending b) sinusoidal bending

Weak zones

1 1

AH

>=•

Weak V"

zones \

•

A H i - i

* <

\

v \ p> >

i c) local bending d) local kinking e) multiple kinking

Figure 2.8 Possible deformation modes of P V D (adapted after Holtz, et al. 1991)

By incorporating the above factors the actual discharge capacity qw, is then given by:

<lW=(FtXFcXFfc}lreq

(2.12)

where, F,, Fc and Ffc are the reduction factors due to time, folding, or drain condition,

and filtration and clogging, respectively. The term qreq is the theoretical discharge

capacity calculated from Barron's theory of consolidation, which is given by:

Hreq

€fU10l7B2h

4T, (2.13)

where, sf is the final settlement of the soft soil equivalent to 2 5 % of the length of the

drain installed into soft ground, U10 is the 10% degree of consolidation, / is the depth of

25


the vertical drain, ch is horizontal coefficient of consolidation and Th is the time factor

for horizontal consolidation.

The reduction factor due to time Ft, has been estimated from laboratory tests to be

between 1.03 and 1.48, with an average of 1.25 (Bergado et al., 1996). The percentage

reduction of discharge capacity under the worst conditions of bending, folding, and

twisting has been tabulated in Table 2.1, from which the average reduction can be taken

as 48%), which gives the reduction factor of the deformed drain condition Fc about 2.

The filtration tests show that the trapped fine soil particles decrease the permeability of

the PVD and its discharge capacity. This deterioration is complicated by the biological

and chemical growth in the geotextile filter. From filtration tests the value of Ffc is

suggested to vary between 2.8 and 4.2 with an average of about 3.5. After considering

all the worst conditions that may occur in the field the discharge capacity qw could 500-

800 m /year but reduce to 100-300 m /year where the hydraulic gradient is unity under

elevated lateral pressure (Rixner et al., 1986). The discharge capacity of various types of

drains is shown in Figure 2.9 where the discharge capacity is influenced by lateral

confining pressure.

In lieu of laboratory test data, it is also suggested that the discharge capacity can

conservatively be assumed as 100 m3/year. Based on laboratory test results Hansbo

(1981) suggested a much smaller discharge capacity of drains, as summarised in Table

2.2.

26


Table 2.1 Percentage of discharge capacity of deformed drain condition

Deformed Condition % Reduction in qy

1 0 % Bent 26%

20% Bent

90° Twist

180° Twist

32%

33%

43%

One-clamp 20% Bent 48%

Two-clamp 30% Bent 78%

Average 48%

1200

m

a

fr

a 03

U <_>

c_ _3 a 173

1000

800

600

400

Q 200

Hydraulic Gradient = 1

Mebradrain MD7407 (4)

Castel Drain Board (4)

CX-l 000(4)

Colbond CX-l 000(4)

200 300 400 500

Lateral confining pressure (kPa) 600 700

Figure 2.9 Typical values of discharge capacity (data from Rixner et al., 1986)

27


Table 2.2 Short term discharge capacity, in m3/year (Hansbo, 1981)

Drain type

Geodrain

Other drain types

40

26

21

24

15

10

21

—

19

80

20

20

22

14

05

19

17

17

Lateral pressure

250

20

18

14

14

01

17

13

09

inkPa

500

16

10

12

12

Clogged

15

12

04

Kremer et al. (1982) stated that the minimum vertical discharge capacity must be

160 m3/year under a hydraulic gradient of 0.625 applied across a 40 cm drain length

subjected to a 100 kPa confining pressure. Based on laboratory data and their

experience Jamiolkowski et al. (1983) concluded that for an acceptable quality of drain

qw should be at least 10-15 m3/year at a lateral stress range of 300-500 kPa, and for

drains that may be 20 m long. Hansbo (1987) specified that qw becomes a critical

property for long drains if its capacity is less than 50-100 nrVyear. Holtz et al. (1991)

reported that the qw of PVD could vary from 100-800 nrVyear. For certain types of PVD

affected by significant vertical compression and high lateral pressure, qw values may be

reduced to 25-100 nrVyear (Holtz et al., 1991). The current recommended values are

given in Table 2.3.

28


Table 2.3 Current recommended values for specification of discharge capacity

Source

den Hoedt (1981)

Kremer et al. (1982)

Kremeretal. (1983)

Jamiolkowski et al. (1983)

Koda et al. (1989)

Rixner etal. (1986)

Van Zanten (1986)

Hansbo (1987)

Lawrence and Koerner (1988)

Holtz etal. (1989)

de Jager and Oostveen (1990)

Value

95

256

790

10-15

100

100

790-1580

50-100

150

100-150

315-1580

Lateral stress (kPa)

50-300

100

15

300-500

50

Not given

150-350

Not given

Not given

300-500

150-300

2.5 Factors Influencing the Vertical Drain Efficiency

2.5.1 Smear Zone

Vertical drains are installed in the field using a steel mandrel which is pushed into the

ground statically or dynamically and then withdrawn leaving the drain in the subsoil.

This process causes significant remoulding of the subsoil, especially in the immediate

vicinity of the mandrel. The resulting smear zone will have reduced lateral permeability

which adversely affects consolidation. Barron (1948) stated that if drain wells were

installed by driving cased holes which are back filling as the casing is withdrawn,

driving and pulling the casing would distort and remould the adjacent soil. The finer and

more impervious layers in varved soils will be dragged down and smeared over the

29


more pervious layers, resulting in a zone of reduced permeability in the soil adjacent to

the well periphery.

The combined effect of permeability and compressibility within the smear zone

causes a different behaviour from the undisturbed soil. Predicting soil behaviour

surrounding the drain requires an accurate estimation of the smear zone properties. In

many classical solutions (Barron, 1948; Hansbo, 1981, Indraratna et al., 1997),

considered the influence of the smear zone with an idealized two-zone model where the

smear zone is the disturbed region in the immediate vicinity of the drain and the other

zone is the intact or undisturbed region outside the smear zone. Onoue et al., (1991);

Madhav et al., 1993; Bergado et al., 1996 introduced a three zone hypothesis defined

by:

(i) An inner smear zone in the immediate vicinity of the drain, where the soil is

highly remoulded during installation;

(ii) An outer smear zone where permeability is moderately reduced as a result of the

initial reduction of void ratio during installation; and

(iii) An undisturbed zone where the soil is not affected by installation.

The use of constant but different values of permeability for the smear zone and

undisturbed soil helps to obtain closed form solutions for consolidation with vertical

drains but due to the complex variation of permeability in the radial direction the

solution for the three-zone approach is difficult. For practical purposes the two-zone

approach is generally sufficient but two parameters are necessary to characterize smear

effect, namely, the smear zone diameter [ds) and the permeability ratio (Mk), i-e., the

value in the undisturbed zone (h) over the smear zone (ks). Both the smear zone

diameter and its permeability are difficult to quantify and determine from laboratory

30


tests and so far there is no comprehensive or standard method for measuring them. The

extent of the smear zone and its permeability vary with the installation procedure, size

and shape of the mandrel, and the type and sensitivity of soil (macro fabric). Field and

laboratory observations (Onoue et al., 1991; Madhav et al., 1993; Bergado et al., 1996;

Indraratna and Redana., 1998) indicated a continuous variation of soil permeability with

the radial distance away from the drain centreline. Also, the smear zone diameter (ds)

has been the subject of much discussion in literature dealing with PVD.

Investigations by Holtz and Holm (1973) and Akagi (1977) indicate that:

d,=2dm (2.14)

where, dm is the diameter of the circle with an area equal to the cross sectional area of

the mandrel. Jamiolkowski et al., 1981, proposed that:

da=&°&dm (2-15) 2

Hansbo (1981, 1997) proposed another relationship as follows:

ds=(l.5-3.0)dw (2-16)

where, dw is the equivalent drain diameter. Based on a laboratory study and back

analyses, Bergado et al. (1991) proposed that the following relationship could be

assumed:

d =2d (2-17)

31


a) Settlement erne >ouc

transducer

permeable.

U M

Til

specimen

TI

smear zone vertical drain

T5-L. £j~

Pore water T f pressure transducer.

L o a d

1,1 1 1 __ .

-eg

impermeable

k'

/ JU.

T2 w

T4 "

T6"

23 cm

24 cm

24 cm

24 cm

D = 45 c m

vertical specimen

smear zone

horizontal specimen

Figure 2.10 a) Schematic section of the test equipment showing the central drain and

associated smear zone; and b) locations of small specimens obtained to determine the

consolidation and permeability characteristic (Indraratna and Redana, 1998)

Indraratna and Redana (1998) proposed that the estimated smear zone could be as

large as (4-5)_4, which was verified using a specially designed large-scale

consolidometer (Indraratna and Redana, 1995). The schematic section of the

consolidometer and the location of the recovered specimen are shown in Figures 2.10(a)

and 2.10(b). Figure 2.11 shows the variation of kyjkv ratio along the radial distance from

the central drain. According to Hansbo (1987) and Bergado et al. (1991), the /QA ratio

32


was found to be close to unity in the smear zone, which agrees with the study by

Indraratna and Redana (1998). The studies of Bo et al. (2000) and Xiao (2001) indicate

that the smear zone can be 4 times of the size of the mandrel or 5-8 times the equivalent

drain diameter. The recommended smear zone parameters by different researchers have

been listed in Table 2.4.

-^ 2.00

%

c_ s -

_

u

1.50

1.00

| 0.50

_

o N

"S 0.00

drain Smear zone

expected trend

Mean Consolidation

Pressure: — e — 6.5 kPa

16.5 kPa 32.5 kPa 64.5 kPa 129.5 kPa 260 kPa i o

X 5 10

Radial distance, R (cm) 15 20

Figure 2.11 Ratio of ki/kv along the radial distance from the central drain (modified after

Indraratna and Redana, 1998)

2.5.1.1 Soil Macro Fabric

For soil with a pronounced macro fabric, the ratio of horizontal permeability to vertical

permeability (Mfcv) can be very high, whereas the kh/kv ratio becomes unity within the

disturbed (smear) zone. Vertical drains are very efficient when the clay layers contain a

lot of thin horizontal sand or silt lenses (micro layers) but if they are continuous in the

horizontal direction, vertical drains may not be effective because the rapid drainage of

pore water may occur irrespective of whether they are installed or not.

33


Table 2.4 Proposed smear zone parameters (After Xiao, 2001)

Source Extent Permeability Remarks

Barron (1948)

Hansbo (1979)

Hansbo (1981)

Bergado etal. (1991)

Onoue(1991)

rs=1.6rm kh/ks=3

rs—1.5~3rm Open

rs=1.5rm kf/ks=3

rs=2r„

r*=L6rn

kf/kv=l

kf/ks=3

Almeida etal. (1993) rs=1.5~2rm kh/ks=3~6

Indraratna et al. (1998) rs=4~5rw kh/kv=1.15

Assumed

Based on available literature at that time

Assumed in case study

Laboratory investigation and back analysis for Bangkok soft clay

From test interpretation

Based on experiences

Laboratory investigation (For Sydney clay)

Chai & Miura (1999) rs=2~3rn kh/ks=C/(kh/ks) Qthe ratio between lab and field values

Hird et al. (2000)

Xiao (2000)

r.=1.6r„

r,-4r„

kt/ks=3

k}/ks=1.3

Recommend for design

Laboratory investigation (For Kaolin clay)

rs: radius of smear zone, ks: smear zone permeability, and kv: vertical permeability.

2.5.1.2 Size and Shape of the Mandrel

Disturbance generally increases with the total cross sectional area of the mandrel so it

should be as close as possible to the drain to minimize displacement. Akagi (1977,

1981) observed while studying the effect of mandrel driven drains in soft clays that

when a closed-end mandrel is driven into saturated clay there was a great deal of excess

pore water pressure due to heave and lateral displacement which caused a decrease in

the strength and coefficient of consolidation of the surrounding soil. However, this

34

— . Chapter 2 Literature Review

excess pore pressure dissipated rapidly followed by consolidation after the mandrel was

installation or before the fill was placed. Bergado et al. (1991) reported from a case

study of a Bangkok clay embankment stabilized with vertical drains that settlement was

faster where the drains were installed using a mandrel with a smaller cross sectional

area, rather than a larger one. This verifies that a smaller smear zone was developed in

the former.

2.5.1.3 Installation Procedure

Evaluating the effect of installation on the degree of disturbance is a very complex

matter in soil mechanics. Baligh (1985) developed a strain path method to estimate the

disturbance caused by the installation of various rigid objects into the ground. The

strain state during undrained axisymmetric penetration of closed end piles has three

deviatoric strain components, namely, __./, is. and E3. Ei is the shearing strain in a

conventional triaxial test, __._ is the strain from pressure-metre tests and E3 is the strain

from simple shear tests. The second deviatoric strain invariant, the octahedral strain,

Yoct, is then given by:

Figure 2.12 shows the theoretical distribution of octahedral shear strain (;&_,) with radial

distance from a circular mandrel.

35


w

A/2

r fflZZZZZZZZZZZZZZZL .•;"/ 6 ___ Q/2 '<_/:..

r - * -/5ro6,

w/s/f//w/r. PV drain

Mandrel

m •71 wl

Undisturbed soil

Figure 2.12 Approximation of the disturbed zone around the mandrel (Rixnet et al)

2.5.2 Effect of Sand Mat

Part or all water ingress into drains will flow to the ground first and then exit through

the sand mat. Since the hydraulic conductivity of sand is considerably higher than clay

it can usually be assumed there is no hydraulic resistance in the sand mat but in some

cases, depending on local materials, lower quality clayey sand may be used. In these

instances the hydraulic resistance in the sand mat may influence the rate of

consolidation of the clay sub-soil, the amount of which is a function of its hydraulic

conductivity and embankment geometry.

36


2.5.3 Well Resistance

Well resistance refers to the finite permeability of the vertical drain with respect to soil.

Head loss occurs when water flows along the drain and delays radial consolidation to a

certain extent. It should be pointed out that well resistance is controlled by the drain's

discharge capacity qw, and also by the soil's permeability fa, the maximum discharge

length /-, and any geometric deficiencies (bending, kinks, etc) in the drains.

Mesri and Lo (1991) proposed the governing equation for vertical flow within the

vertical drain in terms of excess pore water pressure at the soil-drain interface. Based on

k Mesri's equation, a well resistance factor is defined as K—-

V2 and is an index for

the magnitude of pore water pressure. Eventually, the well resistance factor (R) can be

expressed as follows:

„=-^V (2-19) k I2 Khlm

Analysis of the field performance of vertical drains in soft clay deposits indicated

that well resistance is negligible when R is greater than 5, i.e., the minimum discharge

capacity qw(mi„), of vertical drains required for negligible well resistance may be

determined from:

_... = 5k J2 (2-20)

The above relationship is illustrated in Figure 2.13 for most typical values of fa

and lm. The most typical value of qw(mi„) can range from 2 to 80 m3/year (Lin et

al.,2000). Table 2.5 summarizes the well resistance indices proposed by various

37


investigators to evaluate the influence of finite discharge capacity of vertical drains.

Note that the proposed indices are also converted to the well resistance factor (R)

proposed by Mesri and Lo (1991). It can be seen that all these indices depend only on R,

except for the expression proposed by Aboshi and Yoshikuni (1967) and Stamatopoulos

and Kotzias (1985) which also includes the drain spacing.

5 10 15 20 25 30

Drain length (m)

Figure 2.13 Minimum discharge capacity required (based on Eqn. 2.20)

Laboratory and field data generally indicate that the discharge capacities of most

commercial PVDs have almost no influence on the consolidation rate of clay, especially

drains that are not too long (Indraratna et al., 1994). For values of qw greater than 100-

150 m3/year (in the field) and where drains are shorter than 30 m, there should be no

significant increase in the consolidation time. Given these circumstances it may be

claimed that for commercial PVDs, well resistance is usually negligible in most

38


practical situations unless the drains are excessively long and geometric deficiencies

occur during installation (bending, kinks, etc). In most soft clays, well resistance can be

ignored for PVD less than 15m long.

Table 2.5 Summary of proposed well resistance indexes

Source Index of well resistance

' 4F(n)n2 k. Aboshi and Yoshikuni (1967) /_,. =

Yoshikuni and Nakanodo (1974) and

Onoue (1988)

Hansbo (1981) W= 2^-

jn2-l) kh(lm)2 _7t{n2-l)l

w\rwj 4F(n)n2 R

I- 8k" fl \

n2K VI' V w J

fl \

wVwJ

________

n R

2n— R

Stamatopoulos and Kotzias (1985) /?, = 1 ku

F{n)kw

fi >

V^y ________

F(n)R

ZengandXie, 1989

Mesri andLo, 1991

G = ^ -fl \

4 k wVwj

R=7r^-f \ r

_______

4 R

k^m

n: spacing ratio = De/d^

2.6 Development of Consolidation Theories

2.6.1 One-Dimensional Consolidation

Deformation problems have occupied civil engineers for over a century but a rational,

quantitative approach has only been available since the introduction of Terzaghi's

39


(1923) one-dimensional theory, considered by many to be the birth of modern soil

mechanics. Since then a great number of contributions have been made to improve the

capability of predictions. The one-dimensional governing equation for the dissipation of

excess pore pressure was proposed by Terzaghi as follows:

du d2u .--.. - = c v T T (2-21) dt dz

where, u = excess pore pressure, t = time, z = coordinate and cv - vertical coefficient of

consolidation, can be expressed as :

cv=K/Ywmv (2-22)

where, kv = coefficient of vertical permeability, YW ~ urut weight of water and mv =

coefficient of volume change.

In deriving the above equation (2.21), the following assumptions were adopted:

The soil is fully saturated and homogeneous;

Water and soil particles are incompressible;

Darcy's linear flow law is valid;

Compression and flow are one-dimensional and in vertical;

Soil skeleton follows time independent constitutive law;

External loading is applied suddenly and remains constant;

Soil deformations are small;

The permeability is constant throughout the layer and the process; and

The soil skeleton follows isotropic linear elastic law.

40


2.6.2 Coupled Consolidation Theory

The application of one-dimensional consolidation theory to practical situations is very

limited but they can be analysed through a consolidation theory which accounts for

three-dimensional drainage and strain conditions. There are two basic approaches for

two or three-dimensional consolidation problems, the first being the direct extension of

Terzaghi's one-dimensional diffusion type theory proposed by Rendulic (1936) which

assumes that total stress remains constant during consolidation, i.e. the internal

volumetric components of total stress are assumed to have the same time history of

behaviour as the applied volumetric stress. This pseudo-theory does not provide for a

coupling between the magnitude and the progress of settlement, it only assumes that the

dissipation of excess pore pressure assists settlement.

The second theory was derived directly from the theory of elasticity by Biot

(1941), and is commonly known as the Biot's theory. This theory is mathematically

much more complex and provides for a coupling between magnitude and progress of

displacement i.e., at any point in the consolidating soil continuum, there is continuous

interaction between dissipating excess pore water pressure and changing total stress.

The final form of Biot's equations in terms of excess pore water pressure is as follows:

__________ V^ (2-24)

41


in which x, = Cartesian coordinates, k = permeability constant, w, = displacement along

f _2 A the xi direction, G = shear modulus, V = Laplacian operator = V

I dx2 and s =

/

volumetric strain.

It should be pointed out that in two and three-dimensional problems the degree of

consolidation settlement Us, is no longer equal to the average degree of pore pressure

dissipation Up, since there will always be some stress redistribution.

2.6.3 Development of Vertical Drain Theory

•f •9 i 'ed

Q o tgi —.

_

Vertical Drain

-Smear Zone

"Undisturbed Clay

(a) Ideal Drain (a) Real Drain

Figure 2.14 Unit cell model of a drain surrounding by soil cylinder

Analytical solutions already developed for consolidation of ground improved with

vertical drains invariably employs the "unit cell" model as illustrated in Figure 2.14.

This is appropriate and reasonable considering that the flow of water into a single drain

is axisymmetric in nature. Theory for radial drainage consolidation has been addressed

42


by many researchers (Rendulic, 1936; Carrillo, 1942; Barron, 1948; Yoshikuni and

Nakanode, 1974; Hansbo, 1981,1997; Onoue, 1988a, 1988b; andZeng and Xie, 1989)

2.6.3.1 Rendulic and Carillo Diffusion theory

Rendulic (1936) formulated and solved the differential equation for one-dimensional

vertical compression by radial flows:

du

~d~t = ch

d2u 1 du)

Kdr2 r dr j

(2.25)

where r=coordinate, c/,=horizontal coefficient of consolidation (kh//wmv).

Carillo (1942) showed that for radial drainage and associated one-dimensional

consolidation the excess pore water pressure u,-,z is given by:

du '*>- '--^

8t = Ch

d u 1 du

ydr2 r dr j

«-._ = _______

u0

+ cv^ (2.26) dz

(2.27)

where, ur and uz are the excess pore water pressure due to radial flow and vertical flow

only, and wo=initial pore water pressure. By substituting the average excess pore water

pressure into Eqn. (2.27), the average degree of consolidation of the stratum can be

obtained by combining Uz and Ur which are evaluated separately from Terzaghi's and

Rendulic's solutions, respectively.

{l-u)={l-Uz){l-Ur) (2-28)

43


where, U = the average degree of consolidation of the clay at time / for combined

vertical and radial flow, Uz and Ur are respectively the average degree of

consolidation at time t for vertical flow and radial flow only. It should be noted that

both Rendulic's and Carill's solutions are for ideal drains (infinite discharge capacity

with no smear zone).

2.6.3.2 Barron's (1948) Suggestion - Equal Strain Hypothesis

Since perfect drains are impossible Barron (1948) addressed the smear and well

resistance effects that can decrease the performance of vertical drains. Barron (1948)

included the smear effect in his solutions for consolidation by vertical drains. He

presented closed form solutions for two extreme cases for radial drainage induced

consolidation which might occur in the clay layer, namely, 'free strain' and 'equal

strain' and showed that the average consolidation in both cases is almost the same. The

'free strain hypothesis' assumes that the load is uniform over a circular zone of

influence for each vertical drain, and that the differential settlements occurring over this

zone have no effect on the redistribution of stresses by arching of the fill load. The

'equal vertical strain hypothesis' on the other hand, assumes that arching occurs in the

upper layer during the consolidation process without any differential settlement in the

clay layer, which means the vertical strain is uniform in a horizontal section of the soil..

The difference in the predicted pore water pressures calculated using the free strain and

equal strain assumptions are shown to be very small. Therefore, the approximate

solution based on the 'equal strain hypothesis' gives satisfactory results compared to the

rigorous free strain hypothesis.

The Figure 2.15 shows the schematic illustration of a soil cylinder with a central

vertical drain where rw = the radius of the drain, rs = the radius of smear zone, R = the

44


radius of soil cylinder and / = the length of the drain installed into the soft ground. The

coefficient of permeability in the vertical and horizontal directions are kv and fa,

respectively and k'h is the coefficient permeability in the smear zone. Based on the same

assumptions as Terzaghi's theory, Barron (1948) showed the consolidation of a soil

cylinder with an ideal centre vertical drain and impervious boundary except at the drain,

is characterized by the Eqn. (2.25) and gave a solution of the excess pore pressure for

radial flow only, ur of Eqn. (2.25) incorporating the effect of smear is given by:

7

v Ur —Ur - In

(r\

\rs J

______) 4. *i

2R2 \'

(' J2 J n —s \ln(s)

h \ n (2.29)

where, u = u0 exp f 5.0

v. y J (2.30)

in which, u0: initial excess pore pressure, and the smear factor vis given by:

v = F\ n,s,kh,kh n 2 2

n -s

In fn\ 3 S Kf, ku\ n -s

VsJ 4 4n2 ln(s) (2.31)

where, n=R/rw is drain spacing ratio and s is the extent factor of the smear zone with

respect to the size of the drain and is given by: s = rs/rw.

The average degree of consolidation Ur, in the soil body is given by:

u Ur = 1 - = - = 1-exp Uo

'_*_T { y )

(2.32)

The time factor Th in the above equation is defined as:

h~D2 (2.33)

45

Chapter 2 Literature Re\ iev.

Drain Smear zone

Figure 2.15 Schematic of soil cylinder with vertical drain (after Hansbo, 1979)

The coefficient of radial drainage consolidation ch, is represented by:

ch = "v/w

(2.34)

where, av is the coefficient of compressibility of the soil, and e is the void ratio.

Curves of average degree of radial consolidation versus time factor Th for various values

of n are shown in Figure 2.16. The average degree of vertical consolidation versus time

factor Tv is also indicated.

46


0

-3

-. 20 K

~s 40

? s_ •2 *-_ ._ ._> _

o to

S,

a

60

80

100

fl . Vr. \d) V LI ru\ r>«. yuj iva.

tical Flow lial Flow

\ \°

sÔO

t6\

\ A<

.-,

s\ \ \

X ^ 0.005 0.01 0.02 0.05 0.1 0.2

Time Factor, Tv and Th

0.5

Figure 2.16 Average consolidation rates a) for vertical flow, b) for radial flow

2.6.3.3 Rigorous Solution (Yoshikuni and Nakanode, 1974)

Yoshikuni and Nakanodo (1974) developed a rigorous solution of the consolidation

with vertical drains based on the 'free strain hypothesis'. The development of the

solution is very long and therefore not included here. However, a summary of the

governing equations is given below. In this solution, only the effect of well resistance is

included but the effect of smear is not taken into consideration. The consolidation of the

soil cylinder with vertical drain at radius r = rw and at depth z is given by:

' O U

dz' + —

2 kh fdu^

Kdtj = 0 (2.35)

Also, first time he introduced the effect of well resistance on the analytical solution

by the following factor:

_.= «.*» it 2 fa

m 8khlj

71 °w

(2.36)

47


2.6.3.4 Hansbo (1981)-Analysis with Smear and Well Resistance

Hansbo (1981) presented an approximate solution for vertical drain by considering both

smear and well resistance based on the 'equal strain' hypothesis. The general concept of

this solution is the same as illustrated previously in Figure 2.15. The average degree of

consolidation at depth z by radial flowt/,., presented by Hansbo (1981) can be

expressed as:

Un =l-exp f 8T^

\ V J (2.37)

in which

f n M = ^ l

In

+

V

h i

fn\ / A

\khj

ln(s)-0.75 + „2-i V ^ 4n'

khn2~l

S-l 2 , - S+l

4n2 + nz(2l-z)^-\l~

(2.38)

Or in a simplified form (neglecting the minor significance terms):

p-ln

f \

ln(s)-0.75 + 7a(2l-z)-*-<lw

(2.39)

The average degree of consolidation l/-,_v of the whole layer can be obtained by

exchanging the value of /_ (Eqn 2.39) for:

p.=ln fn\

+ \s)

f \

ykhj

ln(s)-0.75 + 2khid

2

3<lw

(2.40)

48


2.6.3.5 X Method (Hansbo, 1979 and 1997)

Although the classical theory of consolidation of vertical drains (Barron, 1948) and its

later developments are all based on the validity of Darcy's law, in the consolidation

process permeability is subjected to a gradual reduction. According to laboratory

investigations carried out by Hansbo (1960) on Ska-Edeby clay, a deviation from

Darcy's law was observed at small hydraulic gradients. It was concluded that Darcy's

linear law v=ki should be replaced by exponential flow correlation as follows (n is

exponential correlation factor):

v = Ki" when i<i1 (2.41a)

v = mi"'1 (i - i0) when i > i} (2.41b)

where, i0 = i,{n -l)/n, is a threshold gradient, below which no flow will take place.

Hansbo (1979, 1997) proposed an alternative consolidation equation Based on the

exponential flow correlation which is supported by the full-scale field test at Skd-

Edeby, Sweden. The average degree of consolidation is related with time as follows:

Ur=l-, At 1+

f \n-l

u0 aD2 \Drwj

l/n-l

(2.42)

where, the coefficient of consolidation X is given by KhMJYw , M=l/mv=oedometer

modulus, D is diameter of the influence zone of the drain, a is n2nJ3" J4{n -1)"+ and

omitting the terms of minor significance

49


P = 1 n-1 (n-1)2

3n-l n(3n-l)(5n-l) 2n2(5n-l)(7n-l)

+ • 2n

V n \n-i K f £) Y«-i D

KdsJ s \uwJ

(2.42a)

+ -nj yDj DJ

Kh7JZ\2l - z)

2qM

The average degree of consolidation Ur,av of the whole layer can be obtained by

exchanging the last term in the above expression for:

(i-i) v. n)

J—

D

f j2\

I — Dz

Kh7d

3qv

(2.42b)

W h e n the exponent n-»l, Eqn (2.42) yields the same result as Eqn. (2.37) assuming

A, = ch and Kh/Ks=kh/ks.

2.6 A 2-D Modelling of Vertical Drains

Even though each vertical drain is axisymmetric, finite element analyses dealing with

multi drain embankments have commonly been conducted under 'plane strain'

conditions for optimising computational efficiency. Therefore, to employ a realistic 2-D

plane strain analysis for vertical drains, the appropriate equivalence between the plane

strain and axisymmetric analysis must be established. Figure 2.17 depicts the

conversion of an axisymmetric vertical drain into an equivalent drain wall. This can be

achieved in several ways (Hird et al. 1992, Indraratna and Redana, 1997), for example:

50


(i) Geometric matching - the drain spacing is matched while the same

permeability coefficient is maintained;

(ii) Permeability matching -coefficient of permeability is matched while the

same drain spacing is kept; and

(iii) Combination of (i) and (ii), with the plane strain permeability calculated for

a convenient drain spacing.

kh

Drain

Smear zone

__ _"

0

» *

> * — » * — » * » * • *

a) Axisymmetric b) Plane Strain

Figure 2.17 Conversion of an axisymmetric unit cell into plane strain condition

51


2.6.4.1 Shinsha et al. (1982)-Permeabil_ty Transformation

Shinsha et al. (1982) first proposed an acceptable matching criterion for converting the

permeability coefficients. The equivalent coefficient of permeability was calculated

based on the assumption that the required time for a 50% degree of consolidation in

both schemes was the same, giving:

kptftax^B/DjTw/Trso (2.43)

where, 7^0=0.197 is a dimensionless time factor at 5 0 % consolidation of laminar flow

and Trso is corresponding radial flow.

2.6.4.2 Hird et al. (1992)-Geometry and Permeability Matching

By adapting Hansbo's (1981) theory for the plane strain case Hird et al. showed that the

average degrees of consolidation U, at any depth and time in the two unit cells were

theoretically identical in the absence of well resistance if

V __ 2B'

3R' In (R\ (k

\rsJ + v ks

In f \ 3 K 3

V rW

(2.44)

where, subscripts ax and pi denote axisymmetric and plane strain conditions

respectively. Note that the geometric matching can be obtained by substituting

kp!^kax=fa in Eqn. (2.44), where as permeability matching can be achieved by

substituting B=R. In the event of significant well resistance, its effect can be matched

independently by ensuring that

Qjqw = 2B/itR: (2.45)

52


2.6.4.3 Bergado and Long (1994) -Equal Discharge Concept

The converted permeability, including smear effect, is introduced, based on an equal

discharge rate in both schemes, assuming the coefficient of permeability is independent

of the state of seepage flow.

V _ _ 7dO{l-as)

2S in ^

\ds )

fu \

+ Kks J

f J \ in \dwJ

(2.46)

where, as=t/D; t: thickness of the walls in 2-D model; D and S: the row spacing and

pile spacing of actual case respectively; a = De/D; S=D and a = 1.05 for square

pattern; S=0.866D and a = 1.13 for triangular pattern.

2.6.4.4 Chai et al. (1995) - Well Resistance Matching

Chai et al. (1995) successfully extended the analysis by Hird et al. (1992) to include the

effect of well resistance and clogging. In this approach, the discharge capacity of the

drain in plane strain (qwp) for matching the average degree of horizontal consolidation is

given by (geometry and permeability are same in both schemes):

q 4khV

wp

3B In ^

W + -ln(s) +

17 2l2i±h

12 3qy

(2-47)

The model developed in this study was refined using a single drain model of 5 m

long, and both elastic and elasto-plastic analyses were applied to predict its

performance. Excellent agreement was obtained between the axisymmetric and plane

strain models, especially with the varied discharge capacity qwp, as shown in Figures

53


2.18 and 2.19. There the varied discharge capacity yielded a more uniform and closer

match between the axisymmetric and plane strain methods compared to the constant

discharge capacity assumption.

2.6.4.5 Lee et al. (1997) - Time Factor Analysis

They assume that the time for two systems to achieve a 50% and 90% degree of

consolidation are the same, then the following simple expression is obtained, as shown

below:

"pi B 1 Tr50 Tr90 S

R Th50 '1,90 ftdw

(2.48)

0 ___N

_? 20h 0 O

| 40

60 o _o

a o o

i 80 _

Q 100

0.01

Elasto-plastic analysis

F E M axisymmetnc

FEM plane strain (varied qwp)

i i i i i i i i i i i i i i i

0.1 1

Time factor, T^

Figure 2.18 Comparison of average degree of consolidation (data from Chai et al,

1995)

54


0

-1

a -2 xf +-» OH

_

Q -3

-5

^r^_rrr^r_~J~^^_Axisymmetric

\ s 1 \ \

Plane strain \ ^ (constant qwp) \ \ 'Plane strain

\ \ (varied q^) \ \ \ \

\

Excess pore pressure \ -at periphery, Th=0.27 . (elastic analysis)

1 1 1 1 1 1 1 L... i i

i i I

i! i

!\ i . i . . .

0 2 4 6 8 Excess pore pressure, kPa

Figure 2.19 Comparison of excess pore pressure variation (data from Chai et al., 1995)

2.6.4.6 Indraratna and Redana (1997) - Rigorous Solution for Parallel Drain Wall

Indraratna and Redana (1997) converted the vertical drain system shown in Figure 2.17

into an equivalent parallel drain wall by adjusting the coefficient of soil permeability.

They assumed that the half width of unit cell _5; the half width of drains bw; and the half

width of smear zone bs are the same as their axisymmetric radii R, r>. and rs respectively,

then the average degree of consolidation in plane strain condition given as:

— u U hp = 1 = 1 — exp

r-8Thp^

\ VP (2-49)

where, T. = time factor in plane strain condition , and

Pi a+(p)k-^{ei2lz-z2) k "hP

(2.50)

where, k. and k' are the equivalent horizontal permeability of undisturbed and smear "hp

zone of the model, respectively. These parameters can be related by:

55


kfjp-r-

k, "hp

In fn\

+ ln{s)-0.75 + 7t\2h-z2)^-

(2.51)

The associated geometric parameters a, /3 and the flow term 6 are given by:

2 2b, a = 3 B

2 \

,- +A_ B 3B2

(2.52a)

P = ^s-bJ+^-{3b2w-b

2s)

B 3B-(2.52b)

0 = chp 2kL ( b

hpii B 1 _____

V B (2.52c)

where, qz = 2qw/7tB : the equivalent plane strain discharge capacity.

To verify the above model a finite element analysis was undertaken for both

axisymmetric and equivalent plane strain models. As an example, a unit drain was

considered installed to a depth of 5 m below the surface at 1.2 m spacing. The model

parameters and soil properties were: rw=0.03 m, rm=0.05 m, fa=l x 10" m/s, kh=5 x 10

-10 9 m/s, and the corresponding equivalent plane strain permeability were khp-5.02 x 10

m/s, and /cv=2.97 x IO"9 m/s based on Eqn. (2.65). The water table was assumed to be at

the surface, and rs=5rm (based on experimental result). For the elasto-plastic finite

element analysis, Modified Cam-Clay model (Roscoe and Burland, 1968) was used as

follows: X = 0.2, K = 0.04, M=1.0, ecs = 2 and Poisson's ratio v = 0.25, with a

saturated unit weight of ys =18 kN/m3

56


The results of both axisymmetric and plane strain analysis are plotted in Figure

2.20, where the average degree of radial consolidation Uy, (%) is plotted against the time

factor Th for perfect drain conditions. As illustrated, the proposed plane strain analysis

agreed with the axisymmetric analysis, with the maximum deviation between the two

methods being less than 5%.

_N

rf o • _H

-*_ -3

"o V}

d

o u

O 0>

a. Q

0

20

40

60

80

100

Perfect drain (no smear)

Plane strain Axisymmetric

0.01 0.02 0.05 0.1 0.2 0.5 1 Time Factor, Tj,

10

Figure 2.20 Average degree of consolidation (modified after, Indraratna et al., 2000)

Figures 2.21 and 2.22 illustrate the settlements and excess pore pressure variations

over time for single drains, including smear plus well resistance, where again the

axisymmetric model agreed with the equivalent plane strain model. It is important to

note that the inclusion of well resistance reduces errors.

Based on the above single drain analysis, Figures 2.20 - 2.22 provide sufficient

evidence to prove that the equivalent (converted) plane strain model is an excellent

substitute for the axisymmetric model. In finite element modelling, 2-D plane strain

analysis is. expected to cut down computational time considerably compared to that

taken by a 3-D, axisymmetric model, especially for multi-drain analysis.

57


Plane strain with smear and well resistance

Axisymmetry with smear

and well resistance

100 150 Time (days)

200 250

Figure 2.21 Comparison of the average surface settlement (Indraratna et al., 2000)

100 150

Time (days)

250

Figure 2.22 Comparison of the excess pore pressure (Indraratna et al, 2000)

2.6.5 Simple Method of Modelling (1-D)

A simple approximate method for modelling the effect of PVD is proposed by Chai et

al. (2001). Because PVD increases the mass permeability of subsoil in the vertical

direction it is therefore logical to establish a value of vertical permeability which

approximately represents both the effect of vertical drainage of natural subsoil and

radial permeability towards the PVD. This equivalent vertical permeability {kve) was

58


derived based on equal average degree of consolidation under the ID condition. To

obtain a simple expression for kve< an approximation equation for vertical consolidation

is proposed as follows:

Uv=l-exp(-CdTv) (2.53)

where, Tv =time factor for vertical consolidation; and C_/=constant (3.54), determined

from curve fitting (Figure 2.23). The equivalent vertical permeability kve can be

expressed as:

kve ~ j 2.51% 2u \

fa (2.54) MDeK j

where, /: drain length, De: equivalent diameter of unit cell, and

^ k, , / x 3 it2Vk M = lnl + ^ /„(,)_£ + ____-__ (2.55)

\sj K * 3<1

The effects of smear and well resistance have been incorporated in the derivation of the

equivalent vertical permeability.

2.7 Evaluation of Design Parameters

The primary parameters governing the rate of consolidation are the coefficient of

permeability and coefficient of consolidation. In practice, the vertical coefficient of

consolidation (cv), thereby fa, is estimated from the results of standard oedometer

testing. Hence, the value of the horizontal coefficient of consolidation (ch) is estimated

by assuming an appropriate ratio of cyjcv. The parameters of the smear zone are more

complicated to estimate and all the available theoretical and empirical solutions cannot

provide satisfactory and consistent estimates.

59


0 i,..,.

c __ +->

CO ~3

o c/_ c

o o

20-

40-

60-

80-

100-

TT

\ \ ^ 0^—3.2,

Q=3.54

0.0001

-L/v=l-exp(-Qfv) -Terzaghi's IDsoIution

0,001 0.01 0.1

Time factor, Tv

10

Figure 2.23 Effect of Cdon degree of consolidation (Chai et al. 2001)

2.7.1 Vertical Coefficient of Consolidation and Permeability

The vertical coefficient of consolidation (cv) can be obtained from the standard

oedometer tests. By using Terzaghi's one-dimensional theory cv can be estimated as:

= TvH2/t (2.56)

where, Tv is the time factor for a certain degree of vertical consolidation (e.g. common

useful values are T50=0A91 and r«.=0.848) , t is the corresponding time for Tv, and is

usually obtained by using curve fitting methods such as: Taylor's 4t method (1948) or

Cassagrande's log . method (1936). Hd is the length of the longest drainage path.

60


The coefficient of vertical permeability (fa) of the soil can be indirectly estimated

from the results of oedometer test. The relevant relationship is given by:

K=cvYwav/{l + e0) (2.57)

where, av is the coefficient of compressibility, which is the slope of the e~ <JV curve,

i.e. av = Aej Aav , and eo is the initial void ratio of the sample.

During consolidation the void ratio and permeability may decrease significantly.

To permit changes in soil permeability during consolidation a number of formulas

relating the coefficient of permeability and void ratio have been proposed. For clays, the

most popular empirical formula is suggested by Taylor (1948) as follows:

logk=logk0-{e0-e)/Ck (2.58)

where, fa and e0 are the initial values of permeability and void ratio, respectively, and

Ck is the permeability change index. Tavenas et al. (1983a) found that the Ck could be

related to the initial void ratio e0 of natural clays byQ *0.5e0. Babu et al. (1993)

proposed, based on experiments on remoulded soils, thatQ. * 0.24eL, in which eL is the

void ratio at liquid limit.

Based on experimental result Samarasinghe et al. (1982) suggested a formula that

would be generally applicable to normally consolidated clays:

k = Ce"/(l + e) (2-59)

in which the power n typically in the order of 4-5, and C is a reference permeability

indicating the soil characteristics.

61


2.7.2 Horizontal Coefficient of Consolidation and Permeability

The horizontal coefficient of consolidation (ch) is usually estimated from:

Ch=(kh/kv)cv = rkcv (2.60)

where, r„: permeability anisotropy, is a significant characteristic for most natural soft

clays, approximately given based on the type of soil in which the drains are installed.

Tavenas et al. (1983b) reported that for soil tested in a conventional oedometer the rk

was varies between 0.91 and 1.42 for intact natural clays and from 1.2 to 1.3 for

Matagami varved clay. Leroueil et al. (1990) found that n is generally small (^ 1.15)

for natural clays and r„=3.5-5.5 for varved clay. Through a laboratory study Bergado et

al. (1991) reported that the kp/ks varied between 1.5 and 2 with an average of 1.75, and

more significantly, rk was found to be almost unity within the smear zone. Shogaki et al

(1995) reported that the average values of rk were in the range of 1.36-1.57 for

undisturbed isotropic soil samples taken from Hokkaido to Chugoku region in Japan.

According to the experimental results plotted in Figure 2.11 (Indraratna and

Redana, 1995), the value of n in the smear zone varies between 0.9 and 1.3 with an

average of 1.15. Hansbo (1987) argued that for extensive smearing, the horizontal

permeability coefficient in the smear zone (kh) should approach that of the vertical

permeability coefficient (fa), suggesting that the ratio rk could approach 1. The

experimental results shown in Figure 2.11 (Indraratna and Redana, 1995) seem agree

with Hansbo (1987). For applied consolidation pressures it is observed that the value of

rk varies between 1.4 and 1.9 with an average of 1.63 in the undisturbed zone.

62


2.7.3 Coefficient of Consolidation with Radial Drainage

2.7.3.1 Log U vs t Approach

Aboshi and Monden (1963) presented a curve fitting method using log U and linear t

developed by taking 'log' on both sides of Barron's solution (Eqn. 2.38 or 2.44), and

rearranging with the use of Eqn. (2.39):

V ' I F JD2

where, F is given by Eqn. (2.37) for the perfect drains case (i.e. F=F(n)) and by Eqn.

(2.42) when the smear effect is taken into consideration (i.e.F = v).

From the Eqn. (2.75), the coefficient of radial consolidation chcan be written as:

_ D2Fdln(l-U) (262) Ch~ ~T~ dt

where dln\1~U) [s the slope of the graph of logarithm of the average degree of

dt

consolidation against linear consolidation time using settlement data (Figure 2.24).

Instead of settlement data, pore water pressure data can also be plotted in this method.

Note that the coefficient of consolidation calculated in the case of F=F(n) is the

coefficient of average consolidation with radial drainage, whereas F = v will yield the

coefficient of radial consolidation of undisturbed soil.

63


______ (o. U=-Z-ro

50 100

— (sec/cm2) — dc

250

Figure 2.24 Aboshi and Monden (1963) method for determining Ch

2.7.3.2 Plotting Settlement Data (Asaoka, 1978; Magnan et al., 1980)

Asaoka (1978); Magnan and Deroy (1980) developed a method to find the coefficient of

consolidation (applicable to ID consolidation; consolidation around vertical drains; and

a combination of both) and the maximum settlement using the available settlement data.

The following steps should be carried out to find those parameters:

(i) The observed time-settlement curve plotted to an arithmetic scale is divided

into equal time intervals J/. The settlements, pl,p2, corresponding to times

.;, t2, • • ..are read of and tabulated.

(ii) The settlement values pltp2, are plotted as points (A_/>A) in a

coordinate system with axis /?,_, and/?,. A straight line, /?,_, =/>,- is also

drawn (Figure 2.25).

64


(iii) The plotted points are fitted by a straight line whose corresponding slope is

read as/? and its intercept with the ordinate axis isy_?0. The point of

intersection with the 45° line gives the final consolidation settlement; and the

coefficient of consolidation can be found from the following equations

depending on drainage conditions.

Pi

Pi-r

Figure 2.25 Asaoka (1978) method to determine ch

In the case of ID problem, the vertical coefficient of consolidation is given by:

.,=-___/'___ 12 At

(2.63)

In the case Of radial drainage, the horizontal coefficient of consolidation is given by:

ch=-D2vlnp

8 At (2.64)

65


where, v is expressed in Eqn. 2.42. To obtain the average coefficient of consolidation v

should be replaced by the drain spacing factor F(n), as expressed in Eqn. 2.37, which

gives:

ch = D2F(n)lnj3

8 At (2-65)

For combined radial and vertical drainage, Asaoka's equation was modified by

Magnan and Deroy such that the coefficients of consolidation are related by only one

unique equation as:

8c h , X2 Cv

D2v 4 H'

InB

At (2.66)

It is difficult to decide on values of Cf, and cv respectively but very useful to

estimate the coefficient of consolidation for an isotropic soil. For isotropic soil the Eqn.

(2.80) will become:

ch =

In fi

At f 8 n2^

+

(2.67)

D2v 4H2

It is noted that, in the above equations In J3 can be replaced w i t h — - — , i.e.

In/3

66


2.8 Constitutive Models for Soils

By definition the constitutive model is a mathematical model that describes the physical

behaviour of material perceived mentally. A properly established constitutive model

should therefore be able to simulate at least the prominent physical phenomenon that

has been understood for a given material in a qualitative and quantitative fashion.

2.8.1 Linear Elastic Model

The majority of traditional deformation analysis in geotechnical engineering often

assumes a linear elastic material at small stresses which is probably true for over-

consolidated clay although most soils exhibit plastic behaviour as stresses increase.

2.8.2 Elastic-Perfectly Plastic Model

The behaviour of material can also be modelled as elastic-perfectly plastic where the

first part of the stress-strain curve remains linear-elastic until the material yields. These

models are widely used and various yield criteria can be implemented to define when

the material changes from elastic to plastic.

2.8.3 Critical State Models

Currently, constitutive models based on plasticity theories are popular and widely used

soil engineering problems. A more sophisticated model has been introduced utilising the

critical state concept based on the theory of plasticity in soil mechanics to represent the

behaviour of clay (Schofield and Wroth, 1968). The Cam-Clay model (Roscoe et al,

1963) and Modified Cam-Clay model (Roscoe and Burland, 1968) for yielding soils

67


based on plasticity theory has received wide acceptance due to its simple and elegant

predictions, especially for normally and lightly over consolidated clays. In these models

the shear strength of the soil is related to the void ratio. The main variables in critical

state soil mechanics theory are the effective mean stress p , the deviator stress q, and the

specific volume v (or the void ratio e). These are defined as follows:

p = °l+°2+°3 = Cr]+(J2+CT3 _ u y 3 3

1 t r V / I i\2 / t i\2

o-}-a2\ +\o-2-o-3\ +\o-3-or2 (2.69)

where, cr7, cr2,cr5 are major, intermediate and minor principle stress (effective),

respectively; and u is the pore water pressure.

In critical state theory, the virgin compression, swelling, and recompression lines

are assumed to be straight in (lnp'-V) plots with slope of - X and - K , respectively, as

shown in Figure 2.26. The isotropic virgin compression line or isotropic normal

consolidation line (INCL) is expressed as:

V = N-Aln(pj (2-7°)

2

where, N is the value of specific volume Kwhen p =1 kN/m .

The isotropic swelling and recompression lines are expressed as:

V = VK-Kln\p\ (2-7l)

68


The X and K can be related with compression (Cc) and recompression (Cr) indexes as

follows:

X = CJ 2.303 and K = Cj 2.303 (2.72)

V

Normal Consolidation Line (NCL)

Recompression Line

In(p')

Figure 2.26 Isotropic normal consolidation line plot in critical state theory

The straight line in the q-p'plot is called Critical State Line (CSL) as shown in Figure

2.27. The slope of the critical state line M is expressed as:

q = M p (2.73)

In the V-lnp 'plot the critical state line can be expressed as:

V = r-Xln(p (2.74)

69


Critical State Line (CSL)

Figure 2.27 Position of the critical state line

Combining the C S L Eqn. (2.87) into the M o h r circle plot, the relationship between

drained angle of friction (^ ) and M may be given by:

M = 6 sin 0

3 - sin <f>

(2-75)

The initial specific volume can be estimated at any given depth below the ground level

once/?', q and p'c are known, and the V-lnp'plot is shown in Figure 2.28.

The intersection between the swelling line and the CSL is assumed to be at point

A given by coordinates VA and pA. Point P represents the intersection between the initial

specific volume F and the effective mean normal stressp'. Then the following relation

may be established:

VA=r-Xln\pA (2.76)

where, pA=pcl2 for Modified Cam-Clay and pA=pcl2.718 for Cam-Clay.

70


PA P PC In(p')

Figure 2.28 Position of the initial specific volume

Along the swelling line (K -line) passing through the initial stress state at point P, the

following relation can be applied:

V = VA-kln\p jpA (2.77)

Substituting VA from Eqn (2.90) gives:

V = f - (X - k)ln\ pA - Kin I p (2.78)

Note that r = N-(X-tc)ln2 for Modified Cam-Clay and r = N-(X-K) for Cam-

Clay.

71


2.8.3.1 Cam-Clay Model

The introduction of isotropic hardening plasticity into soil mechanics led to the family

of soil models of strain hardening cap type developed at Cambridge University. The two

of those models are widely used known as Cam Clay and Modified Cam Clay Model.

One of the key assumptions of Cambridge theory is that the flow rule follows the

normality condition, which leads to:

_______. _____ (2 79.

de$ dp

A second key assumption (flow rule), arises from a consideration of the work dissipated

during shear. Thurairajah described the energy dissipation term during plastic

deformation as:

This led Roscoe, Schofield, and Thurairajah to complete the energy balance equation

from a thermodynamic point of view as:

pde$ + qds^ = Mpd£ps (2-81)

This will then lead to the plastic dilatancy ratio as:

dep.

where, 77 = qj p :stress ratio.

72


Using the Eqns. (2.93) and (2.96), Cam-Clay yield locus (Figure 2.29a) can be obtained

as:

q = Mp In^pjpj (2.83)

Substituting Eqn. (2.87) into the above equation p'A can be found asp'A =p'c/2.718

So Eqn. (2.92) will lead to:

V = r + X-k-(X-k)ln(p'c]-Kln(p (2-84)

By eliminating pc between Eqns. (2.97) and (2.98), the Stable State Boundary Surface

(SSBS) equation becomes:

Mp {X-K)

r + X-k-V-Xln\p (2.85a)

Or alternatively (the preferred form):

vA=r+(x-kli-Tj/M) (2.85b)

p( 2.72

P'c P'

(a)

/>;

(b)

P( P'

Figure 2.29 Yield locus of C a m Clay and Modified Cam Clay model

73


2.8.3.2 Modified C a m Clay Model

It was found that the Cam-clay model was deficient in some aspects of modelling the

stress-strain behaviour of soil, namely the shape of the yield locus at increased pc and

the predicted value of Ko (the coefficient of earth pressure at rest). Therefore Modified

Cam-Clay was introduced to address those set backs (Burland, 1965; Roscoe and

Burland, 1968). The obvious difference between Modified Cam-Clay and Cam-Clay

model is the shape of the yield locus, where the yield locus of Modified Cam-Clay is

elliptical as shown in Fig. 2. 29b. The flow rule for Modified Cam Clay is given by:

dsP M2-TJ1

dep. 2rj (2.86)

This came from the energy balance equation:

r t

p de? + qdej! = p {deP)+(Mds?) (2.87)

The Modified C a m Clay yield locus is given by:

q2+M2p2=M2ppc (2.88)

The equation of the SSBS is given by:

Vk =r + (X-K){ln(2)-ln(l + (T]/M)2)} (2.89)

The incremental stress-strain law during yielding is defined as (for M C C ) :

'dev de.

'dev \de.

+ • \dsv \de.

C,} C]2

C21 C22.

dp

,dq (2.90)

74


where, Cu =—7 + k (X-K)

op up M2 + TJ2 (2.90a)

C12 ~ C21 ~ _{X-KY

up

2TJ 2 , „2 A T +7?

(2.90b)

C 22 _ 1 , (*-*) 3G up

( A 2 \

4TJ

M4-q4 (2.90c)

In an undrained condition the plastic shear strain can be expressed as:

de? = 4KA

v

( ,2 >

M4 -TJ4 drj (2.91)

K where, A = 1 :plastic volumetric strain ratio. X

2.9 Salient Aspects of Numerical Modelling

Currently, pore pressures, settlements, lateral displacements, and stresses of the field

site with vertical drains can be accurately analysed using sophisticated finite element

software. Commercial packages such as PLAXIS, ABAQUS, and SAGE-CRISP are

capable of performing fully coupled consolidation analysis. From past experience, finite

element analysis of lateral deformation has been relatively poor compared to settlements

(Indraratna et al., 1994). The recent finite element models applied to vertical drains are

described below.

75


2.9.1 Drain Efficiency by Pore Pressure Dissipation

Indraratna et al. (1994) studied the performance of an embankment stabilised with

vertical drains at Muar clay, Malaysia, using the finite element code CRISP (Britto and

Gunn, 1987). The effectiveness of the prefabricated drains was evaluated according to

the rate of excess-pore pressure dissipation at the soil drain interface.

A plane strain analysis was applied to a single drain and to the whole PVD

scheme. As explained in detail by Ratnayake (1991), the prediction of settlement using

single drain analysis over-predicts the measured settlement, even though the smear

effect was included. In the case of multi-drain analysis underneath the embankment, the

over-prediction of settlement is more significant than single drain analysis and therefore

it was necessary to consider the dissipation of the excess pore pressure at the drain

boundaries at a given time more accurately.

To elaborate on technique the average un-dissipated excess pore pressures could

be estimated by finite element back-analysis of the settlement data at the centreline of

the embankment, as shown in Figure 2.30 where 100% represents zero dissipation when

the drains are fully loaded. Accordingly, at the end of the first stage of consolidation

(ie., 2.5 m of fill after 105 days), the un-dissipated pore pressures decrease from 100%

to 16%. For the second stage of loading the corresponding magnitude decreases from

100% to 18% after 284 days during which the embankment has already attained its

maximum height of 4.74 m. It can be deduced from Figure 2.30 that perfect drain

conditions are approached only after 400 days. Although the general trends between the

finite element results and field data agree during the initial stages, the marked

discrepancy beyond 100 days is too large to be attributed solely to the plane strain

76


assumption. These excess pore-pressures reflect retarded efficiency in the vertical drains

(partial clogging). A better prediction was obtained for settlement, pore pressure, and

lateral deformation when 'non-zero' excess pore pressures at the drain interface were

put into the finite element model simulating 'partially clogged' conditions.

100;

0

1 st stage ^nd, stage Loading loadingf

0 100 200 300 400 500 Time (days)

Figure 2.30 Percentage of undissipated excess pore pressure at drain-soil interfaces

(Indraratna et al., 1994)

2.9.2 Deformation as a Stability Indicator

Indraratna et al. (1997) investigated the effect of ground improvement by preloading

together with geogrid and vertical band drains, and sand compaction piles constructed

on Muar clay in Malaysia. The settlement and lateral displacement of the soft clay

foundation were analysed using plane strain finite element formulation and the findings

were compared to the field measurements.

The analysis employed critical state soil mechanics, and the deformations were

predicted on the basis of the fully coupled (Biot) consolidation model incorporated in

77


the finite element code CRISP (Britto and Gunn, 1987). In the analysis, the soil

underneath the embankment was discretised using linear strain quadrilateral (LSQ)

elements. The vertical drains were modelled as ideal and non-ideal, where the well

resistance factor was ignored in the former. This study shows that an accurate

prediction of lateral displacement depends on correctly assessing the value of the Cam-

clay parameters, the shear resistance at the embankment-foundation interface, and the

nature of assumptions made in the modelling of drains and sand piles. The actual soil

properties are influenced by the working stress range and the assumed stress path of the

sub-soil at a given depth. The normally consolidated parameters associated with the

Cam-clay theories over-estimate lateral displacement and settlements if the applied

stresses are smaller than the pre-consolidation pressure.

The normalized deformation factors for a few trial embankments are compared in

Table 2.6. The ratio of maximum lateral displacement to fill height{fi,), the ratio of

maximum settlement to fill height {j32), and the ratio of maximum lateral displacement

at the toe to the maximum settlement at centre line (a), are used as normalised

parameters. Compared to the unstabilized embankment constructed to failure, the

stabilized foundations are characterized by considerably smaller values for a and Pi,

which elucidates their obvious implications on stability. The normalized settlement (P2)

on its own is not a proper indicator of instability but is still a useful stability indicator

when taken in conjunction with a and Pi. For example the foundation having SCP gives

the lowest values of Pi and P2, clearly suggests the benefits of sand compaction piles

over band drains.

78


Table 2.6 Normalized deformation factors (modified after Indrarataa et al. 1997)

Ground Improvement Scheme a Pi P2

Sand compaction piles for pile/soil stiffness ratio of 5 0.185 0.018 0.097

(/z=9.8 m, including lm sand layer)

Geogrids + vertical band drains in square pattern at 0.141 0.021 0.149

2.0 m spacing (/V=8.7 m )

Vertical band drains in triangular pattern at 1.3 m 0.127 0.035 0.275

spacing (/.=4.75 m )

Embankment rapidly constructed to failure on 0.695 0.089 0.128

untreated foundation (h=5.5 m )

2.10 S u m m a r y

In this Chapter, literature on consolidation analysis, constitutive models for soft soil,

and related merits and demerits have been described. A brief summary of this critical

review section is given below:

1. Vertical drains have been widely used to accelerate primary consolidation of soft

soils. However, it is difficult to predict the settlements and pore pressures

accurately due to the complexity of estimating the correct values of soil

parameters inside and outside the smear zone and therefore appropriate laboratory

techniques must be used to measure them.

2. The soil adjacent to the drain mandrel is disturbed and the fabric of the soil is

distorted during the installation of a vertical drains. Due to this disturbance (smear

effect), the compressibility increases and the permeability decreases in the soil

79


within the smear zone. The combined effect of changes in both the

compressibility and the permeability within the smear zone results in a decrease in

the consolidation rate of the entire soil mass.

3. The soil in the smear zone behaves so differently from the undisturbed soil,

therefore effectiveness of vertical drain systems depend on correct assessment of

the extent of smear zone and its permeability. However, these are often difficult to

quantify and determine from laboratory tests. So far, there is no comprehensive or

standard method for measuring them. In the past, for example, Indraratna and

Redana (1998), Sharma and Xiao (2000) have conducted laboratory tests to

evaluate the smear zone parameters. Indraratna and Redana (1998) proposed that

the estimated smear zone could be as large as 4-5 times the equivalent drain radius

and that the horizontal to vertical permeability ratio is close to unity in the smear

zone. Sharma and Xiao (2000) proposed that the radius of the smear zone is about

four times the equivalent mandrel radius, and the horizontal permeability of the

clay layer in the smear zone is approximately 1.3 times smaller than that in the

undisturbed zone. This laboratory results indicate that the extent of smear zone

and its permeability depends on many factors, such as, type of soil, method of

installation. Therefore it is important to find an analytical solution to evaluate the

smear zone parameters.

4. Existing analytical solutions provide a satisfactory framework for analysing

axisymmetric consolidation (e.g. unit cell). For large construction sites where

many PVDs are installed, 2D plane strain analysis is most convenient given its

computational efficiency. Recently developed axisymmetric to plane strain

conversions provide good agreement with measured data and these simplified

80


plane strain methods are now widely and successfully used in finite element

analysis.

5. Vacuum preloading through PVD and surface membrane systems effectively

promotes radial consolidation while controlling the soil's lateral yield compared

to conventional surcharge embankment loading that can generate large lateral

displacements in very soft clays.

6. Laboratory and field evidence tends to indicate that the capacity of most

commercial PVDs is large enough so it has less influence on the rate of

consolidation of clay.

81

Chapter 3 Theoretical Background

3 THEORETICAL BACKGROUND

3.1 Prediction of Smear Zone Caused by Mandrel Driven Vertical Drains using

Cavity Expansion Analysis

3.1.1 General

Cavity expansion analysis has attracted the attention of many researchers because it has

numerous applications in geomechanics. Geotechnical engineers have benefited from

earlier works, albeit they were primarily concerned with metals (Bishop, Hill and Mott,

1944) because they provide a theoretical groundwork and guidelines for deriving

solutions. In the field of geotechnics some of the areas related to cavity expansion are

pile driving (Carter et al., 1978), tunneling (Atkinson and Potts, 1977), and soil testing

(Ladayi, 1963). In this study, an attempt is made to estimate the extent of the

disturbance zone (smear zone) using the Cylindrical Cavity Expansion theory

incorporating the Modify Cam Clay model (this model has been widely adopted for

describing the elastic-plastic behaviour of soil incorporating the effect of stress

histories).

When a mandrel is driven into the ground it displaces soil equal to its volume.

With small penetrations up to about ten times the radius of the mandrel, some ground

surface heave occurs but at greater depths, soil is predominantly displaced outwards in a

radial direction. This has led to the installation process being modelled as the expansion

of a cylindrical cavity with a final radius equal to the mandrel. In reality, the mandrel is

band shaped hence the equivalent mandrel radius is evaluated by comparing the

perimeter between the assumed circular cross-section and the true rectangular shape.

82


3.1.2 Basic Assumptions and Definition of the Problem

Figure 3.1 shows a cavity with an initial radius ao and an initial internal pressure<J0.

Compressive stresses and strains are taken as positive. The cavity expands to a radius of

a as the infernal pressure increases from a0 to aa while an element initially at a radial

distance ro from the centre of the cavity moves to a new radial position r from the

centre, resulting in a displacement % = r-r0. The soil on the cavity wall will yield

when the pressure is sufficiently large while further increases in pressure will lead to a

plastic zone forming around the cavity. The radial distance of the plastic zone around

the cavity is denoted by rp while the soil beyond this would remain in a state of elastic

equilibrium.

»• r

Elastic zone

Figure 3.1 Expansion of a cavity

Development of analytical framework for analysing cavity expansion is based on

an assumption that the soil obeys Hooke's elasticity law until yielding commences. The

yielding of soil is described by the Modified Cam Clay model.

83


3.1.3 Elastic Analysis

Considering an element at a radial distance r, the equation of equilibrium (applicable in

both elastic and plastic region) for the case of a cylindrical cavity is:

^ + &-£el = 0 (3.1) dr r

where, o~r and cr0 are radial stress and circumferential stress respectively. These

stresses can be written in terms of mean stress (p) and deviator stress (q) as follows:

ar=p + -r=q (3.2a)

o~e = p — j = q (3.2b)

By Hooke's law the radial strain (_?,.) and circumferential strain (se) can be related to

corresponding stresses as follows (in plane strain condition):

= —[v/-v,)°V-i/c r0_ dr 2G

(3.3a) K 1

r 2G

and £r-£e=r^- (3.3b) dr

where, G: shear modulus; and v: Poisson's ratio

Substituting Eqn. (3.3a) into (3.3b), the following equation can be obtained:

84


crr-are =r (7_ v)_^_._v_^_

dr dr (3.4)

The Eqns. (3.1) and (3.4) can be solved for stresses and displacement by using the

boundary conditions, i.e. <jr = arp at r=rp and ar = a0 when r -> oo as:

ar =CT0+{crrp-cr0){rp/rf

cre =o-0-(arp-o-0)(rp/rf

(3.5a)

and £>*-<")M 2G

_p_

r (3.5b)

The Eqn (3.5a) leads to a mean total stress of:

(crr+cre) P = 2 = ~0=A)

(3.5c)

The volume change for an undrained case is zero, i.e.,

du = -/c\dp j p \ = 0 (3.6)

r

This implies that the mean effective stress (p ) is constant in the elastic zone since the

value of /.-(slope of unloading-reloading line in v-lnp space) is non-zero and Eqn.

(3.5c) indicates that the mean total stress p is constant. Consequently, excess pore

pressure would be zero in the elastic zone.

85


3.1.4 Plastic Analysis

3.1.4.1 Stress at Elastic-Plastic Boundary

After initial yielding at the cavity wall a zone of soil extending a radial distance rp will

become plastic as cavity pressure continues to increase. For soil obeying the MCC

model, the yielding criterion is:

rj = M \{P'CIP)-- (3.7)

where, pc : the stress representing the reference size of yield locus.

From the elastic analysis, the mean effective stress at the elastic-plastic boundary is

equal to (p0), so the stress ratio {TJ) at this boundary can be found as:

7z

f \

yp )r=.

o f=M4R~::I Po

(3.8)

where, R is the isotropic overconsolidation ratio defined as the ratio of m a x i m u m stress

on yield locus (pc0 ) and the in situ mean effective stress (p0). There is a small

deviation between the R and the conventional overconsolidation ratio (OCR). The R can

be related with OCR and the slope of critical state line-Af as follows (detailed

derivations in Appendix 1):

R = 3 (45 - 12M + M2) OCR

(6-M) 6 + M + 2{6-M)OCR

3M ".>

6+M

(3.8a)

86


The Eqns. (3.2a) and (3.2b) can be rewritten in terms of elastic-plastic boundary stresses

as:

1 arp-Po+-j=ap (3.9a)

1 C70p=Po-^(lp (3.9b)

3.1.4.2 Strain in Plastic Zone

Following conventional terminology in critical state theory, two strain parameters

known as volumetric strain sv and shear strain^, for an undrained cylindrical cavity

expansion, are defined as:

sv = sr + se = 0 (3.10a)

£s=Jj(£r~£0) (3-10b)

Since an undrained deformation is necessarily isochoric, the conservation of

volume gives the following relationship between r, the current radius of a material

element which was initially at r# and current and initial radii of the cavity a and ao,

respectively.

r2-r02=a2-a20 (3.11)

This leads to the radial speed of the soil element (a>) in terms of the rate of cavity

(da\ expansion — as:

{dt)

87


_ dr _(a\da

dt \r)dt (3.12)

So the radial, circumferential, and shear strain rate can be expressed as follows:

der

dt

dsQ

dt

dco a da

dr r2 dt

CD a da

r r2 dt

(3.13a)

de, 1 i \ 2a da and —s- = —r=[er-ee)-dt J3 43r2 dt

(3.13b)

The later equation can be written in terms of the initial position of the particle r0 as:

de 2a da

dt 43~[a2 +rg -a20)dt (3.14)

Since r0 is fixed for a given particle the Eqn. (3.14) can be integrated to give the finite

logarithmic shear strain as:

e, = — i = l n

4s ' 7 _ ___^1 V

(3.15a)

In undrained condition the shear strain of a M C C model is (Eqn. 2.91):

des =—dq + 4KA

3G v __1__J

M4-rj4 drj (3.15b)

where, v. specific volume; A : plastic volumetric strain ratio=l-K://l; and X: the

slope of normal compression line in v-lnp space.

88


Substituting, Eqns. (3.15a) into (3.15b) and integrating them with the boundary

condition (Eqn. 3.8), a relationship between the stress ratio and radial distance from the

centre of the cavity (r) can be obtained as:

In [a2-alj] 2(l + v) K rr KA , x

1-i ^ =- T7T77—TI -ri-Ul—f(M, rj,R) rl j 3V3(l-2v) v

vM (3.16)

where f(M,rj,R) = — ln \M-Tj)\ + ylR-\

-tan" \^)+tm~]{4R-[) (3.16a)

KM) V ;

3.1.4.3 Effective Stress in the Plastic Zone

In the plastic zone the total volumetric strain consists of two components: the elastic

and plastic volumetric strains. Under the undrained condition, the sum of the elastic and

plastic volumetric strains should be zero, that is:

dp dpc

up upc

(3.17)

Integrating the above equation by considering the condition at the elastic-plastic

boundary, the following equation can be obtained:

r 1

Pc = PcO

1 ,

'V7+; f ' v

PoJ

Rp KPo)

(3.18)

Substituting Eqn. (3.18) into Eqn. (3.7) leads to:

P =Po R

\ + {TJIM)2 (3.19)

89


Eqns. (3.16) and (3.19) are closed form solutions for effective stresses in the plastic

zone. The effective stress can be evaluated from Eqn. (3.19) for a given stress ratio,

therefore the deviator stress and location can be found from Eqn. (3.16).

3.1.4.4 Total Stress in the Plastic Zone

Total stress in the plastic zone can be determined from Eqn. (3.1) but, it is impossible to

integrate to give a closed form solution because Eqns. (3.16) and (3.19) cannot be

integrated directly. Therefore a simple numerical integration is needed to solve this

problem (author used an Excel spreadsheet formulations). The solutions for total

stresses are:

2 Prq *,.-*,.„ \ , \^dr (3-20a)

r * 43 <r

„ _- __?__ + ___ \ldr (3-20b)

**-*" 4!43- }r

rP

- >-*>-h+TsFr (320c)

A series representation of (q/rfr) is determined from Eqns. (3.16) and (3.19). Then the

variation of total stresses with radial distance can be obtained from the above equations

(using a numerical integration technique).

3.1.4.5 Pore Water Pressure in the Plastic Zone

Pore water pressure can be calculated from the following equation:

u = p-p <«»

90


3.1.4.6 Prediction of Smear Zone

The author suggest that the extent of the smear zone as the region in which the pore

water pressure is greater than the initial overburden stress (total) based on a qualitative

assumption that the soil is severely disturbed and thus the anisotropy with respect to its

permeability coefficient is almost entirely destroyed at which u = a^.

3.1.4.7 Solution Procedure

(i) Evaluate the isotropic overconsolidation ratio R;

(ii) Determine the stress ratio TJ at the elastic-plastic boundary using Eqn. (3.8);

(iii) Determine the radius of plastic zone rp, by substituting q = rjpm Eqn. (3.16);

(iv) Tabulate the TJ value between rjp and M (take sufficient number of points);

(v) Corresponding to the tabulated TJ effective mean stress, deviator stress and radius

can be found using Eqns. (3.19) and (3.16) respectively;

(vi) Using the above tabulated values and numerical integration techniques (eg. Excel

spread sheet), pore water pressure variation with radial distance can be estimated;

and

(vii) The extent of smear zone can be determined using the graphical techniques.

3.1.4.8 Illustrated Application

Calculation of smear zone for a typical soil is given in Table 3.1 (the input parameters

are bolded). The normalized pore pressure variation with radial distance is plotted in

Figure 3.2, which shows that the extent of the smear zone is about 2.77205 times the

mandrel radius.

91


Table 3.1 Input M C C parameters and calculation sheet

Prediction of smear zone using cavity expansion analysis

M=

X=

K=

V0=

V=

A=

1

0.966

0.968 0.970

0.972

0.974

0.977

0.979 0.981

0.983

0.986

0.988 0.990

0.992

0.995 0.997

0.999 1.001

1.004

1.006 1.008

1.010

1.013

1.015

1.017

1.019

1.022 1.024

1.026

1.028

1.031

1.033

1.035

1.037

1.040

1.042

1.19

0.50

0.05

2.80

0.25

0.90

P

37.86

37.80 37.73

37.67

37.61 37.55 37.48

37.42

37.36 37.30

37.24 37.17

37.11 37.05

36.99

36.93

36.86

36.80

36.74

36.68

36.62

36.56

36.50

36.43

36.37

36.31 36.25

36.19 36.13

36.07

36.01 35.95

35.89

35.83

35.77

q

36.56 36.58

36.60

36.63

36.65 36.67

36.70 36.72

36.74

36.76

36.79 36.81

36.83

36.85 36.87

36.90 36.92

36.94

36.96 36.98

37.00 37.02

37.04

37.06

37.08

37.10 37.12

37.14

37.15

37.17

37.19 37.21

37.23

37.25

37.26

OCR=

- v o = i

" v O *

R=

K0=

Po=

Po=

r/rm

7.79593 7.73951

7.68376

7.62866 7.57419

7.52035

7.46710 7.41444 7.36235

7.31081 7.25980

7.20932

7.15935 7.10987

7.06086 7.01233

6.96425

6.91661

6.86940 6.82260

6.77620

6.73020

6.68458

6.63933

6.59443

6.54988

6.50567

6.46178

6.41821

6.37495

6.33197

6.28929

6.24687

6.20472

6.16283

1.6

90.00

50.00

1.66

0.64

37.86

77.86

q/(r/rm)

4.689

4.726 4.764

4.801 4.839

4.877 4.914 4.952

4.991

5.029 5.067

5.106 5.144

5.183

5.222 '

5.262 5.301

5.340

5.380 5.420

5.460 5.500

5.541

5.582

5.623 5.664

5.705

5.747

5.789

5.831

5.874

5.916

5.959

6.003

6.047

rp/rm

r/rm Vlr™)

0.000 0.266 0.530

0.794 1.056

1.318 1.578 1.838

2.097 2.355

2.613 2.870

3.126 3.381 3.636

3.891

4.145

4.398

4.651 4.904

5.156 5.408

5.660 5.912

6.163

6.415

6.666

6.917

7.169

7.420 7.672

7.923

8.175

8.427

8.680

PP=

9P=

~rp=

°"eP=

~lp=

rp/rm=

rs/rm=

P

77.860 78.153 78.445 78.735 79.025 79.314

79.601 79.888 80.174

80.459 80.744 81.027

81.310

81.593 81.875

82.156 82.437

82.718 82.998

83.278

83.558 83.837

84.117

84.396

84.675 84.954

85.233

85.513

85.792

86.071

86.351

86.631 86.912

87.193

87.474

37.86

36.56

98.97

56.76

0.966

7.796

2.77205

u

40.000 40.356 40.710 41.063 41.416 41.767 42.117

42.466 42.814 43.162

43.508 43.854

44.199 44.544

44.888

45.231 45.574

45.916 46.258 46.599

46.940

47.281

47.621 47.962

48.302 48.642

48.982

49.322

49.662

50.002

50.343

50.683

51.024

51.365

51.707

u/c-vo

0.44

0.45 0.45 0.46

0.46 0.46 0.47 0.47

0.48 0.48 0.48 0.49

0.49 0.49

0.50

0.50 0.51

0.51 0.51 0.52

0.52

0.53

0.53

0.53 0.54

0.54

0.54

0.55

0.55

0.56

0.56 0.56

0.57

0.57

0.57

92


1.044

1.046

1.049

1.051

1.053

1.055

1.058

1.060

1.062

1.064

1.067

1.069

1.071

1.073

1.076

1.078

1.080

1.082

1.084

1.087

1.089

1.091

1.093

1.096

1.098

1.100

1.102

1.105

1.107

1.109

1.111

1.114

1.116

1.118

1.120

1.123

1.125

1.127

1.129

1.132

1.134

1.136

1.138

1.141

1.143

1.145

1.147

1.150

1.152

1.154

1.156

1.159

35.71

35.65

35.59

35.53

35.47

35.41

35.35

35.29

35.23

35.17

35.11

35.05

34.99

34.93

34.87

34.81

34.76

34.70

34.64

34.58

34.52

34.46

34.41

34.35

34.29

34.23

34.17

34.12

34.06

34.00

33.94

33.89

33.83

33.77

33.71

33.66

33.60

33.54

33.49

33.43

33.37

33.32

33.26

33.20

33.15

33.09

33.03

32.98

32.92

32.87

32.81

32.76

37.28

37.30

37.32

37.33

37.35

37.37

37.38

37.40

37.41

37.43

37.45

37.46

37.48

37.49

37.51

37.52

37.54

37.55

37.57

37.58

37.59

37.61

37.62

37.63

37.65

37.66

37.67

37.69

37.70

37.71

37.72

37.74

37.75

37.76

37.77

37.78

37.80

37.81

37,82

37.83

37.84

37.85

37.86

37.87

37.88

37.89

37.90

37.91

37.92

37.93

37.94

37.95

6.12118

6.07976

6.03857

5.99760

5.95683

5.91626

5.87587

5.83566

5.79562

5.75574

5.71600

5.67640

5.63693

5.59757

5.55832

5.51916

5.48009

5.44109

5.40215

5.36326

5.32441

5.28558

5.24677

5.20795

5.16912

5.13025

5.09134

5.05237

5.01331

4.97416

4.93489

4.89549

4.85593

4.81618

4.77623

4.73605

4.69560

4.65486

4.61380

4.57237

4.53054

4.48827

4.44549

4.40216

4.35822

4.31359

4.26820

4.22196

4.17476

4.12649

4.07699

4.02611

6.091

6.135

6.180

6.225

6.270

6.316

6.362

6.409

6.456

6.503

6.551

6.600

6.648

6.698

6.748

6.799

6.850

6.901

6.954

7.007

7.061

7.115

7.170

7.226

7.283

7.341

7.400

7.459

7.520

7.582

7.644

7.709

7.774

7.840

7.908

7.978

8.049

8.122

8.197

8.273

8.352

8.433

8.517

8.603

8.692

8.784

8.880

8.980

9.083

9.192

9.306

9.426

8.932

9.186

9.439

9.693

9.948

10.203

10.459

10.716

10.974

11.232

11.491

11.752

12.013

12.276

12.540

12.805

13.072

13.340

13.610

13.881

14.154

14.429

14.707

14.986

15.268

15.552

15.839

16.128

16.421

16.717

17.015

17.318

17.624

17.935

18.249

18.568

18.892

19.222

19.557

19.898

20.246

20.601

20.963

|_ 21.334

21.714

22.104

22.505

22.918

23.344

23.785

24.243

24.719

87.756

88.038

88.321

88.604

88.889

89.174

89.460

89.747

90.035

90.325

90.615

90.907

91.200

91.494

91.790

92.088

92.387

92.689

92.992

93.297

93.605

93.914

94.227

94.542

94.859

95.180

95.503

95.830

96.161

96.495

96.833

97.175

97.522

97.873

98.230

98.592

98.959

99.333

99.713

100.101

100.496

100.900

101.312

101.735

102.168

102.612

103.069

103.540

104.027

104.531

105.054

105.599

52.049

52.391

52.734

53.078

53.422

53.767

54.112

54.459

54.807

55.155

55.505

55.856

56.208

56.562

56.917

57.273

57.631

57.991

58.353

58.717

59.083

59.451

59.821

60.194

60.570

60.949

61.330

61.715

62.103

62.495

62.890

63.290

63.694

64.103

64.516

64.935

65.360

65.791

66.228

66.672

67.124

67.584

68.053

68.532

69.021

69.522

70.035

70.562

71.105

71.665

72.243

72.844

0.58

0.58

0.59

0.59

0.59

0.60

0.60

0.61

0.61

0.61

0.62

0.62

0.62

0.63

0.63

0.64

0.64

0.64

0.65

0.65

0.66

0.66

0.66

0.67

0.67

0.68

0.68

0.69

0.69

0.69

0.70

0.70

0.71

0.71

0.72

0.72

0.73

0.73

0.74

0.74

0.75

0.75

0.76

0.76

0.77

0.77

0.78

0.78

0.79

0.80

0.80

0.81

93


1.161

1.163

1.165

1.168

1.170

1.172

1.174

1.177

1.179 1.181

1.183

1.186

1.188

1.188

1.189

1.189 1.190

1.190

1.190 1.190

1.190

32.70 32.64

32.59

32.53

32.48 32.42

32.37

32.31

32.26 32.20

32.15 32.09

32.04

32.03

32.02

32.00

31.99

31.99

31.99 31.99

31.99

37.96 37.97

37.98

37.98

37.99

38.00

38.01 38.02

38.03 38.03

38.04

38.05

38.06

38.06

38.06

38.06

38.06

38.06 38.06

38.06

38.06

3.97364

3.91935

3.86292

3.80399

3.74206

3.67650

3.60643 3.53062 3.44721

3.35323

3.24340 3.10659

2.91051 2.84861

2.77205

2.66759

2.47984

2.44649

2.40538 2.35056

2.26325

9.552 9.687

9.831

9.985 10.153

10.336

10.539 10.768

11.031 11.342

11.729 12.248

13.075 13.360

13.730 14.268

15.349

15.558 15.824 16.193

16.818

25.217

25.740 26.290

26.874

27.498

28.169 28.901

29.708

30.618 31.669

32.936 34.576 37.059 37.877

38.914

40.376 43.156

43.672 44.317

45.194

46.635

106.169 106.767

107.398 108.067

108.782

109.553 110.392

111.320 112.366

113.575 115.034 116.923

119.785 120.729

121.926 123.613 126.823

127.418 128.163 129.176

130.840

73.469

74.123

74.809 75.533 76.304

77.129 78.024 79.007

80.107

81.371 82.884

84.828 87.745

88.701 89.910 91.610

94.831 95.427 96.173 97.188

98.853

0.82

0.82

0.83 0.84

0.85 0.86 0.87

0.88

0.89 0.90 0.92 0.94

0.97 0.99

1.00 1.02 1.05

1.06 1.07 1.08

1.10

0 2 ~ 6 ~ rlrm

Figure 3.2 Normalized pore water pressure variation with radius

94

• . Chapter 3 Theoretical Background

3.2 Analytical Solution for Vertical Drain with Vacuum Preloading

3.2.1 General

The first conventional procedure for radial consolidation by vertical drains with

surcharge loading was proposed by Baron (1948). This was modified later by various

researchers including Yoshikuni and Nakanodo (1974), and Hansbo (1981), to include

the effect of smear and well resistance (reviewed in previous Chapter). Although

consolidation around vertical drains is axisymmetric, most finite element analyses are

based on the plane strain assumption. Therefore the equivalence between plane strain

and axisymmetric analysis needs to be established to use a realistic 2-D finite element

analysis for vertical drains. Hird et al. (1992) introduced an equivalent plane strain

solution which can be conveniently simulated in numerical modelling. Because plane

strain finite element analysis has become popular, Indraratna and Redana (1997) further

modified Hird et al. (1992) solutions to include the effect of both smear and well

resistance.

In the absence of vertical drains Mohamedelhassan and Shang (2002) modelled

the application of vacuum pressure with surcharge load along the surface based on 1-D

consolidation. The mechanism of vacuum-assisted consolidation is comparable but not

the same as conventional surcharge. In earlier studies vacuum preloading was often

simulated with an equivalent surface load or by modifying the surface boundary

condition. Figure 3.3 shows consolidation by the conventional method and vacuum-

assisted preloading. The consolidation rate attributed to vacuum-assisted preloading is

greater than the conventional method because the lateral hydraulic gradient increases.

The application of vacuum with PVDs requires modification of existing theories. In this

95


study, a comprehensive analytical solution for vacuum preloading in conjunction with

vertical drains using Hansbo's (1981) and Indraratna and Redana's (1997) approach is

introduced under axisymmetric and equivalent plane strain conditions. The effect of

various factors such as drain spacing, well resistance and smear effect are also

examined.

-UL

_«; Cl)

"D to 01

__ n </> <n a) L_

00

100

0

-100

>

>

I.

Ap (preloading pressure)

t ^

Time

TO a. __: CO CO 0

co CO CD L_

55

100

-100

Ap (preloading pressure)

p0 (Vacuum pressure) ~^me

CD

a. _«:

CO IO CD L_

a o CL CO CO

0 u X LU

00 Q_

CO CO 0

CD

> o _ _ TO o _ >

100

-100

Maximum excess pore pressure

100

-100

Time

Time

(a)

ro -L

3 CO CO 05

O Q. CO CO 0 O X LU

TO 0.

__: CO

co c_ 0 > _ 0 TO o __ 0 >

100

-100

100

-100

Maximum excess pore pressure

Time

Time

(b)

Figure 3.3 Consolidation process (a) conventional loading (b) vacuum preloading

96


In the laboratory (using a large-scale consolidometer) the author measured the

vacuum pressure at several points along the drain which indicated that the vacuum

pressure is immediately developed within the drain (trapezoidal distribution) in addition

to the uniformly applied surface suction. In reality, the rate of vacuum development

within the drain may depend on the length and type of PVD (core and filter properties)

even though some field studies suggest it develops rapidly even if the PVD are long (Bo

et al., 2003). Given these factors the assumption of an immediate vacuum development

(rather than gradual) is used as boundary condition. Since the drain spacing is relatively

small, it is realistic to assume constant vacuum distribution across the soil and that

decreases linearly along the drain, i.e., the vacuum varies along the drain length (/) from

uvac to cuvac, where c is the vacuum propagation factor.

3.2.2 Modelling of Axisymmetric Solution with Applied Vacuum Pressure

Figure 3.4 shows the schematic illustration of a soil cylinder with a central vertical drain

where _v=the radius of the drain, rs= the radius of smear zone, R =the radius of soil

cylinder and /= the length of the drain installed into soft ground. The coefficient of

permeability in the vertical and horizontal directions is fa and fa, respectively, and kh is

the coefficient permeability in the smear zone. The axisymmetric analysis described by

Hansbo (1981) is extended to include the vacuum pressure as follows. Considering

Darcy's linear law, the radial velocities of water in the undisturbed zone (v_) and smear

zone (vr) are given by Eqns. (3.22) and (3.23), respectively, as follows:

kh (du\ vr =•

r \drj (3.22)

97


du

r w dr V J

(3.23)

where, YW is m e unit weight of water and u and u are excess pore water pressure in

undisturbed zone and smear zone at radius r.

Drain Smear zone

Figure 3.4 Schematic of soil cylinder with vertical drain (adapted from Hansbo, 1979)

It is postulated that the flow of pore water through the boundary of the cylinder

with radius r is equal to the change in volume of the hollow cylinder with outer radius R

and inner radius r, such that:

2xvr=4R2-r2)—

r v 'dt

(3.24)

98


where, e is vertical strain in the (z) direction. Substituting Eqn. (3.22) into Eqn. (3.24),

and subsequent rearranging gives the following equation for pore pressure gradient in

the undisturbed soil (rs <r<R):

du Yy

dr 2fa

(R2

r de_

dt (3.25)

Similarly, in the smeared zone(rw < r < rs), the corresponding pore pressure gradient is

given by:

du _ Yv

dr

fR2 ^

2fa J

de_

dt (3.26)

Considering the horizontal cross-sectional slice of thickness dz of a circular cylindrical

drain with radius rw, the total change in flow from the entrance face to the exit face of

the slice is given by:

2u (*2\ dQi =

7tr k w

r w ay

Kdz dzdt for r < r„ (3.27)

where, fa is coefficient of permeability of drain (representing well resistance).

The horizontal inflow of water into the slice from the surrounding area is given by:

dQ2 Jwk-

f '\ du

r V. dr V J

dzdt for r = r„ (3.28)

For ensuring continuity of flow, the following equation must be satisfied:

dQ}+dQ2=0 (3.29)

99


It is assumed that at the boundary of the drain (r=rw) there is no sudden drop in pore

pressure so u = u . Substituting Eqns. (3.27) and (3.28) in Eqn. (3.29) and rearranging

with the above boundary condition yields:

f 7 >\ d2u + __________

rw KW

f i\

du

dr = 0 (3.30)

Combining the Eqns. (3.26) and (3.30), leads to:

f 7 > d2u

dz2 = _z^(^_7)

kw dt (3.31)

r=ru,

where, n=R/rw: spacing ratio.

Integration of Eqn. (3.31) in the z direction is carried out subject to the following

boundary conditions:

,/ ./.,„. ai _: ... a n d ^ - - ^ ( l - c ) a t _ - / 'vac dz I

(3.31a)

This leads to excess pore pressure at r=rw as:

r~rw ""W

Y^dji A_, dt

{n2-l)[t-2

+ u vac /-C-c)f (3.32)

Integrating Eqn. (3.26) in the r direction using the above boundary condition (Eqn.

(3.32)), the following expression for u can be derived:

100


U =-7w Se

2khdt

^~Uvac 1

g> In r -?''r- + k> [n> -rw 2 kw

-(i-c)z-*• _

-l\2lz-z2)

(3.33)

Integrating Eqn. (3.25) in the r direction with the boundary condition as ur=r = ur=r J 's

the following expression can be derived forw :

u = Yw de

2kh dt

Rlln r r2-r2 +k/

2 k

2 _ 2\

R2lns.rjL-JjL

V J

+ ^(n2-l\2lz-z2)

+ u vac /-(7-c)i (3.34)

where, s=rs/rw: smear ratio.

Let u be average excess pore water pressure for the whole unit cell at a given time .,

then:

un:\R2 -r2)l= l]2ffru drdz+lfarudrdz

0 r... 0 TV

(3.35)

Substituting Eqns. (3.33) and (3.34) for u and u into Eqn. (3.35) and rearranging, the

following expression for u can be obtained:

„=Z-î + a n+c^

2ku dt 'vac

(3.36)

V * J

where,

101


n p = -j nL-\

m^_km(s)_l s fa 4

+ n2-\ An2

+ -fa 1 f J

v*m S-4-s2+\ ' An2

2fanl1( \_ 2

v n J 3?V

(3.36a)

Or in a simplified form:

P . n kh , / \ 3 ln — + -~ln\s)— s fa 4

+ IfaTlV

2<lw (3.36b)

Eqn. (3.36) may be combined with the time-dependent compressibility governed by the

following well-known consolidation expression:

de _ du kh du

dt v dt chYw dt (3.37)

where, mv is the coefficient of volume compressibility and ch is the horizontal

coefficient of consolidation.

Substituting Eqn. (3.37) into Eqn. (3.36) and integrating subject to the boundary

condition that u = usur at f=0 gives the following expression:

U Uvac + f - \ , Uvac

usur usur \ usur J

8T^ exp

K P J (3.38)

where, usur is the applied surcharge pressure and uvac = uvac (l + c)j2 is the average

applied vacuum pressure.

This yields an average degree of consolidation Uh of the whole layer as:

102


— ut-u Usur-u U h = —•—=—= — - — = — = 1-exp

8T^

Ui — U f Usur ~^vac \ P ) (3.39)

where, ut and u / are the initial and final excess pore water pressure respectively, and,

(a) // = s h 4

+ 2khKV

2<]w (with both smear and well resistance)

/ 3

(b) p = ln—+-7- /«(_.) — (smear effect only) s fa 4

(c) // = Inn -0.75 + 2/Cft.rr

3^w

(well resistance only)

(d) fi = Inn-0.75 (perfectdrain)

In the above expressions, qw is the drain discharge capacity at the unit hydraulic

gradient.

In the absence of vacuum pressure (i.e. surcharge pressure only), the Eqn. (3.39) is the

same as the Eqn. (2.49), which was developed by Hansbo (1981) and is rewritten below:

8T,^

u \ p (3.40)

In the absence of surcharge pressure (i.e. vacuum pressure only, usur-Q), the Eqn. (3.39)

leads to:

— Uj-U u 1

Uh = - — = — = = — = 1-exp Ui -Uf Uvac

P J

(3.41)

103


Effect of vacuum pressure distribution on average excess pore water pressure

The following parameters are used in this analysis: w-10, s=2, fa/fa=3, uvac= -50 kPa,

usur=25 kPa and three different values of c (1, 0.5, 0). The excess pore water pressure

distribution (Figure 3.5) shows that with the increase of vacuum propagation factor (c),

the rate of pore water pressure dissipation is accelerated. Note that the average excess

pore pressure equals to the applied surcharge pressure at time t=0 and to the applied

average vacuum pressure at the end of consolidation.

20 --3

__

_ 0 -_

-

& -20 t/2

§3 g CD

fc -40 > <

-60 0.001

-

Vacuum pressure on

Both va

-

—

i i i i i i i 1 1

^*s_* v "•

\\ s

/Sss_. v

ly / \

cuum+surcharge

i — 1

c=0.5

c=0

i i . i . i 1 1 1

"^\ Surcharge only

^ N \ — — — _

\ \ — .-\ \ \ \ \

\ A \ ~ __

\ \ v

\ \ v ^ \ \ >. v

' • i i i i i i 1 i

0.01 0.1 Time factor, Th

10

Figure 3.5 Average excess pore water pressure distribution with different vacuum

pressure distribution

104


Influence of drain spacing on average excess pore water pressure

The following parameters are used in this analysis: 5=2, fa/fa=3, uvac= -50 kPa, c=1.0,

usur=25 kPa and three different values of n (5, 10, 20). The excess pore water pressure

distribution for different drain spacing ratio is shown in Figure 3.6. This clearly shows

that with the increase of drain spacing, the rate of pore water pressure dissipation is

retarded, because when the drain spacing is large the influence of drain is less.

40

e_

o -

5a 2 0 -< -

CM CM

_ — _ CS

s-

o a 3 "20 _ u X a o % § -40 <

-60

0.001

spacing ration n=5

spacing ration n=l0

spacing ration w=20

i i i 1 1 1 | —

0.01

1 1—II M i l )

0.1 Time factor, Th

i — i — i i i i i n - i — i — i i i 1 1 1

10

Figure 3.6 Average excess pore water pressure distribution with different drain spacing

105


Effect of smear zone parameters on average excess pore water pressure

Figure 3.7 presents the variation of excess pore pressure distribution with different

smear zone parameters. For this comparison the following parameters are used: «=20,

uVac= -50 kPa, c=1.0, usur=25 kPa and three combinations of smear zone parameters

(5=2, fa/fa=3); (5=2, falfa=5) and (5=4, falfa=5). The distribution for a perfect drain is

also plotted in the same figure for comparison and as expected, the rate of pore pressure

dissipation decreases as the smear zone parameters increase.

0.001 0.01

1 1 — [ M M

0.1 Time factor, T^

Figure 3.7 Average excess pore water pressure distribution with different smear zone

parameters

106


3.2.2.1 Excess Pore Water Pressure

By combining Eqns. (3.33), (3.36) and (3.38), the excess pore water pressure (at any

point within the smear zone) with time can be found as:

u = _\usur-uVgc)kh ( 8Ti

R2p •yexp

K P J

2 2

R2ln^-r

w

-*L+*L(n2-lhz-z2)

2 fa„X A ' (3.42)

+ u vac i-d-«)f

By combining Eqns. (3.34), (3.36) and (3.38), the excess pore water pressure (at any

point outside the smear zone) with time can be found as:

Wsur -~uvac)

R2p exp

8TU

V P

.2 ,.2 R2lnL-Cz!L+hL\B

2ins--> 2 2^ r, -r,

2

+ *L(n2-ll21z-z2)

+ u. i-if-c^

(3.43)

3.2.2.2 Hydraulic Gradient

Differentiating Eqn. (3.42) in the r direction and rearranging, the hydraulic gradient (i )

with time at any point within the smear zone can be found as:

.' _\usur~uvac)kh

rwRp fa exp

8Th R r

{ P )L r R] (3.44)

Differentiating Eqn. (3.43) in the r direction and rearranging, the hydraulic gradient (/)

with time at any point outside the smear zone can be found as:

• _,("«--«»*) eJ_ EL rwRp P )

R r_

r R (3.45)

107


3.2.3 Modelling of Plane Strain Solution with Applied Vacuum Pressure

dqx _ —

m ' • ' : .

: ••) .

— * • 1

X

dqx

K

ui utn

B

\dz

Unit cell: width = 2B

Figure 3.8 Plane strain unit cell

Plane strain analysis described by Indraratna and Redana (1998) may be extended to

include the application of vacuum pressure as follows. Considering Darcy's linear law

the horizontal velocity of water in the undisturbed zone (vx) and smear zone (vx) is

given by Eqns. (3.46) and (3.47), respectively, as follows:

Vx =

khP (du)

dx r (3.46)

w \vxj

vv chp du

r w dx V J

(3.47)

108


where, khp and khp are the coefficient of horizontal permeability in a plane strain

condition in the undisturbed and smear zone respectively; YW *S me un^ weight of

water; u and u are excess pore water pressure in the undisturbed and smear zone at

distance x ; and x is the prescribed direction of flow coordinate.

Consider a horizontal slice of thickness dz of the unit cell (Figure 3.8). For the

plane strain model it is postulated that the flow in the slice at a distance x from the

centreline of the drain is equal to the change in volume within a block of soil of width

(B-x), such that:

vx=^(B-x) (3.48) dt

where, e is the strain in the vertical (z) direction. Substituting Eqn. (3.46) into Eqn.

(3.48) and rearranging gives the following equation for the pore pressure gradient in the

undisturbed zone (fry < x < B):

?l=hL^(B-x) (3-49) dx khp dt

Similarly, in the smeared zone (bw < x < bs) the corresponding pore pressure gradient

is given by:

^=.£L *£(__„) (3.50) fix h-' dt

hp

For vertical flow in the z direction of the drain, the change of flow from the entrance

face to the exit face of the slice is given by:

109


dqz =• d u

r> ydz2 ,

dzdt for x < b „ (3.51)

The horizontal inflow of water into the slice from the surrounding area is given by:

dqs ^hp

f <\ du

r w dr V J

dzdt for x = b Ml (3.52)

To ensure flow continuity the following equation needs to be satisfied:

dqz + 2dqx = 0 (3.53)

It is assumed that at the drain boundary (x=bw) there is no sudden drop in pore pressure

so u =u . Substituting Eqns. (3.51) and (3.52) into Eqn. (3.53) and rearranging with the

above boundary condition yields:

f 7 '> _______

i dz2 j

2k ' ( „ <\

+ hp

Qz

du

dx , = 0 (3.54)

Combining Eqns. (3.50) and (3.54) leads to:

d2u

Kdz2 _____ £(__..)

qz dt

(3.55)

Integrating Eqn. (3.31) in the z direction, subject to the following boundary conditions

(assume the same vacuum pressure gradient):

u =uvac at z = 0 ; a n d ^ - = - ^ ( l - c ) a t z = / dz- I

(3.55a)

This leads to excess pore pressure dXx=bw as:

110


U 2(B-bw)Ywde

qz dt V 2) + «,. M^)f (3.56)

Integrating Eqn. (3.50) in the x direction using the above boundary condition (Eqn.

(3.56)), the following expression can be derived for u

u = rw d*

2khp dt

x(2B-x)-bw(2B-bw) + 2{B K)hp [2lz-z2)

(3.57)

+ u vac M/-)f

integrating Eqn. (3.49) in the x direction with the boundary condition as ux=bs =ux=bs

the following expression can be derived for u :

u = rw d£

2khp dt

c(2B-x) + 2{B-bw)khp

az

(21Z-Z2)

^{bs-bw\2B-bs-bw)-bs(2B-bs) khP

+ u vac l-{l-c)-\ (3.58)

Let u be the average excess pore water pressure for the whole unit cell at a given time .,

then:

lb, f IB u(B-bw)l= J Jw dxdz+ ^judxdz

OK

(3.59)

Substituting Eqns. (3.57) and (3.58) for u and u into Eqn. (3.59) and rearranging, the

following expression can be obtained for u :

- 7WB Pp de u=- -+u 2khp dt vac

'1 + c}

{ 2 j (3.60)

111


where,

£ a+p-^+e

hP (3.60a)

In the above equation the geometric parameters a ,p and 0 are given by:

a = 2 (n- sf

3 n2(n-l)

pjAlÛn-s-^+s + l) 3 n (n-1)

and (3.60b)

0 = 4k hp 3Bq. M

I n)

V

B R b r 2 ( 1^ where, n = — = — , s = — = — , and note that a + p = - 1 —

h r h r 3 \ n J

Eqn. (3.60) may now be combined with the time-dependent compressibility governed

by the following well-known consolidation expression:

de _ du __ khp du

dt v dt chpYw dt (3.61)

where, chp is the horizontal coefficient of consolidation under plane strain condition.

Substituting Eqn. (3.61) into Eqn. (3.60), and then integrating subject to the boundary

condition that u = usur, at .=0 gives the following expression:

•+ 1-usur usur U

exp 8T, hp

sur J

(3.62)

112


This yields the average degree of consolidation Uhp of the whole layer as:

UhP = = U j U U r.j.y. U

sur = 1 - exp Uj Uf U sur —Uvac pp,

(3.63)

In the absence of vacuum pressure (i.e. surcharge pressure only), the Eqn. (3.63) yields

the same equation developed by Indraratna and Redana (1997):

u U hp = 1— = 1-exp u pp,

(3.64)

In the absence of surcharge pressure (i.e. vacuum pressure only, usur=0), the Eqn. (3.63)

leads to:

UhP = Ui —u u = = — = 1 - exp Ui -Uf Uvac

(3.65)


By combining Eqns. (3.57), (3.60) and (3.62), the excess pore water pressure with time

at any point within the smear zone can be found as:

u \usur~Uvac)

B2M,

___£_ f

chp exp

' _8T^

PP.

x(2B-x)-bw{2B-bw)

+ **-**)** fas)

+ u vac i-(i-cy

(3.66)

By combining Eqns. (3.58), (3.60) and (3.62), the excess pore water pressure with time

at any point outside the smear zone can be found as:

113


u = \Usur~Uvac)

B Pl exp

PP ;

x(2B-x) + 2{B-K)h'{2lz-z2)

+ • (bs-bw\2B-bs-bw)-bs{2B-bs) chp

(3.67)

+u vac i-d-cY-,


Differentiating Eqn. (3.66) in the x direction and rearranging, the hydraulic gradient (i )


/ 2\usur-uvac)khp

YwBPP k exp

hp

8ThP

Pp . B

(3.68)

Differentiating Eqn. (3.67) in the x direction and rearranging, the hydraulic gradient (/')

with time at any point outside the smear zone can be found by:

2\Usur — Uvac )

YwBP exp

w"r-p

8Thp^

v Pp .

1-B

(3.69)

3.2.4 Comparison of Axisymmetric vs Plane Strain Conditions

For a perfect drain (neglecting both smear and well resistance) the average excess pore

pressure variation in axisymmetric (Eqn. 3.38) and plane strain (Eqn. 3.62) unit cell can

be rewritten as follows:

114


For axisymmetric,

U = U vac + \Usur - U vac jexp 8T^

V P J (3.70)

For plane strain,

U=Uvac + \Usur - U vac )exp ' SThp^

p (3.71)

P J

2 where, p = Inn-0.7 5 and Pp = — nj

Assuming the same value for the axisymmetric and plane strain case, say «=10, fa~fap,

usur = 25, and uvac = -50kPa, the average excess pore pressure distribution is plotted in

Figure 3.9. It can be seen that the pore pressure dissipation rate is higher in the plane

strain cell. This is attributed to using the same parameters and therefore it is important

to use the proper matching procedure when conducting plane strain analysis instead of a

true axisymmetric situation.

03

_ _ CO CO

cu _ cu c_

O DH CO 02

u o X u CD 00 c_

__ u > <

40

20

-20

•40

-Equal to applied surcharge

\

\

Axisymmetry

Plane strain

Equal to applied average vacuum pressure

-60 0.0001

1 1 , i i i i ' 1 1 1 1 1 • i i ' i i

0.001 0.01 0.1

Time factor, Th ot Tfo

10

Figure 3.9 Comparison of average excess pore water pressure distribution

115


3.2.5 Matching Approach and Theoretical Considerations

In practice, the clay foundation will usually have a large number of vertical drains

beneath an embankment. In such cases, finite element modelling using the plane strain

model is the common approach where it is pertinent to convert the vertical drains

system into an equivalent drain wall. The equivalent plane strain theory can convert a

row of individual drains to a continuous drain wall based on geometry and permeability

transformations. Figure 3.10 shows the conversion of an axisymmetric vertical drain

into an equivalent drain wall. This can be achieved in several ways (Hird et al. 1992,

Indraratna and Redana, 1997), for example:

(i) Geometric matching - the drain spacing is matched while the same

permeability coefficient is maintained;

(ii) Permeability matching -coefficient of permeability is matched while keeping

the same drain spacing; and

(iii) Combination of (i) and (ii), with the plane strain permeability calculated for a

convenient drain spacing.

This equivalence makes the computational efficiency much greater by reducing

convergence time and the required computer memory, while still giving the correct

time-settlement response. Various researchers have described these advantages for field

studies where a large number of drains are used and for which a true 3D analysis may

be cumbersome and impractical (e.g. Hird et al., 1992, Chai et al., 1995, Indraratna et

al., 1997).

In the method proposed here, the vertical drain system is converted into

equivalent parallel drain walls by adjusting the coefficient of soil permeability while

keeping the half width of unit cell B, the half width of drains bw, and the half width of

116


smear zone bs of plane strain cell are the same as their axisymmetric radii R, rw and

respectively.

Drain

Smear zone

du

3z 0

\

». «

» « • «

• *

• «

• « • «

a) Axisymmetric b) Plane Strain

Figure 3.10 Conversion of an axisymmetric unit cell into plane strain condition (adapted

from, Hird et al., 1992; Indraratna and Redana, 1997)

At each time step and at a given stress level the average degree of consolidation for both

axisymmetric (Uh) and equivalent plane strain (Uhp) conditions are made equal, hence:

Uh=Uhp (3.72)

By substituting Eqns. (3.41) and (3.63) into the above equation, the following equation

can be obtained:

•117


*hp _ hp Pp Th h P

(3.73)

By substituting Eqns. (3.36b) and (3.60a) into the above equation, the following

expression for the equivalent plane strain permeability is obtained:

^hp ~

a+p-^+e khp

, n kh , / \ 3 2kh7rV ln- + -fln[s)— + — s —

5 fa 4 3qw

(3.74)

By ignoring the both smear and well resistance in Eqn. (3.74), the equivalent plane

strain permeability in the undisturbed zone can readily be determined as:

hp -^h

2

3

N V nj

n)-0.75 (3.74a)

By ignoring the well resistance in Eqn. (3.74), the influence of the smear effect can be

isolated and the equivalent plane strain permeability in the smear zone written as:

khp -' Pk. hp

lhp In +

f \

\kh) ln(s)-0.75

(3.74b)

-a

Well resistance is derived independently and yields an equivalent plane strain discharge

capacity of drains, as proposed earlier by Hird et al. (1992):

*<=*«» (3.74c)

118


In this study Eqns. (3.74a)-(3.74c) are incorporated into the numerical analysis

(employing PLAXIS and ABAQUS) to compare laboratory data and study selected case

histories, as discussed in the following Chapters.

For convenience of practical use the equivalent permeability in terms of spacing

ratio (n), smear ratio (s) and the permeability ratio of axisymmetric cell \khjkh are

graphically illustrated in Figures 3.11 and 3.12.

40 60 Spacing ratio, n

100

Figure 3.11 Ratio between coefficients of permeability of undisturbed zone of

equivalent plane strain cell to those of axisymmetric cell as a function of n

The ratio of the coefficient of permeability of the undisturbed zone of equivalent

plane strain cell to that of the axisymmetric cell (khp/kh) is shown in Figure 3.11 as a

function of n. When the spacing ratio (n) increases the value of (khp/kh) sharply

decreases up to a value of 0.3, and then decreases at a diminishing rate.

119


0.5-

0.4-

_g-0.3H •J*

0.2-

0.1-

n | i | r 0 20 40 60 80 100

Spacing ratio, n

S m e a r ratio, s=3

V„'*=3

- i | i | i | i | r -

0 20 40 60 80 100 Spacing ratio, n

Smear ratio, .v=5

v„;,3

Wh=4

1 1 1 1 | r

0 20 40 60 80 100 Spacing ratio, n

' r ° i — • — i ' i • r 0 20 40 60 80 100

Spacing ratio, n

Figure 3.12 Ratio between smear zone permeability to undisturbed zone permeability of

equivalent plane strain cell as a function of n, s and \kh/kh

The ratio of coefficients of permeability of the smear zone to the undisturbed zone

of the equivalent plane strain cell is shown in Figure 3.12 as a function of n, s, and

kh/kh . When the spacing ratio (n) increases the value of khp/khp increases and attains

a maximum (w 0.45) at about «=10, and then decrease at a diminishing rate. Since, the

effect of smear will play a major role at small n, the khp/khp value increases up to a

certain extent and then decreases due to the insignificant effect of smear (at large n).

120


3.3 Plane Strain Consolidation Equation for a Single Drain under Non-Darcian

Flow

3.3.1 General

Many researchers (e.g. Hansbo, 1960; Miller & Low, 1963; Olsen, 1985; Dubin &

Moulin, 1986) pointed out there was a deviation from Darcy's law at small hydraulic

gradients. Based on the laboratory investigations on Ska-Edeby clay, Hansbo (1960)

proposed that the Darcy's flow law v = ki (v = seepage velocity, k = coefficient of

permeability and i = hydraulic gradient) should be replaced with a non-Darcian flow as

defined by the exponential flow correlation v =Ki" (Figure 3.13).

h H i (n-l)/n) .1

Hydraulic gradient, /'

Figure 3.13 Exponential correlation (modified after Hansbo, 2001)

Permeability tests on clay samples by Hansbo (1960) indicated that for an

exponent (ri) of 1.5 the threshold hydraulic gradient (/0) values of 1 to 4 correspond to

the limiting hydraulic gradient (/,) values of 3 to 12. Permeability tests carried out by

121


Dubin and Moulin (1986) on high-plasticity, moderately organic clay showed similar

correlations to those by Hansbo (1960). According to their tests, the i{ value was in the

range of 8 to 35. In the following section a plane strain consolidation solution is

described under non-Darcian flow for a single drain incorporating smear effect only

(well resistance is neglected).

3.3.2 Proposed Plane Strain Solution

The flow velocity (vx) under non-Darcian low (exponential flow correlation) can be

written as:

vx = hP mv Yw i" (3J5)

where, subscript p stands for plane strain, and coefficient of plane strain consolidation

hP = KhPl

mvYw . KhP is the coefficient of horizontal permeability in the undisturbed

zone.

On the assumption that the flow in the slice at a distance x from the centreline of the

drain is equal to the change in volume within a block of soil of width (B-x), such that:

v,=-(»-«Kf (376)

Inserting Eqn. (3.76) into Eqn. (3.75) gives:

. _ 1 du

Yw dx

B du^

hPYw d* J

n f ]_

X ]n

V.

\-±\" (3-77)

122


By rearranging Eqn. (3.77) the pore pressure gradient in the undisturbed soil can be

found by (bs <x<B):

du

dx

n-1

= v n i w

-\ B du

^ hp dt j y B

n f Y\ 1 - - (3.78)

Similarly, in the smear zone(bw <x<bs), the corresponding pore pressure gradient is

given by:

du — = y "

- / w OX

B du f „ „-xzf N 1 - -

Bj

i

v hP dt; (3.79)

The last term in the above expression can be expanded in the following binomial series:

V,

B ) n

c X {n-\) B 2\nd

(w-l)(2w-l)

3\n \B) (3.80)

(»-l)(2w-l)(3w-l)

4!» KBj

Substituting Eqn. (3.80) into Eqn. (3.79) and integrating term by term, and inserting the

boundary conditions u =0at x=bw, the excess pore water pressure within the smear

zone u at a given time t becomes:

n-$

u =B7w" _______ 1„ dt

\ SP J L _ j

< x^ n,— V Bj

-8, n,-__L B for (bw<x<bs) (3.81)

where,

123


8. {n,y) = y

1- 1(V 2!

{n-l)(y]2 (n-l)(2n-l)(y ,3 1

\n) 3! I « 4! n

(n-$(2n-\){?>n-\)( y^.

5! \nj

(3.81a)

In the above equation the variabley represents — or — and the function gp{n,y) for B B

..=1.2, 1.3 and 1.5 is graphically illustrated in Figure 3.14. This indicates that the

function gp(n,y) increases from zero to a maximum value of 0.6 for increasing values of

n andy.

0.2 0.4 0.6 Ratio, v

0.8

Figure 3.14 The variation of function gp (n, y) with y for selected n

Substituting Eqn. (3.80) into Eqn. (3.78) and integrating term by term, and

incorporating the boundary conditions u'x=bs =ux=bs, the following expression can be

124


derived for excess pore water pressure u in the undisturbed zone (bs < x < B) at a given

time t:

n-1

u = B7wn _______

hP dt

g, r x^

V B j g> ( V K B j

+

1

Khp

yKsp j gi

bs) ( b \ Si n,-

\ u J B

(3.82)

Let u be the average excess pore water pressure at a given time /, then:

bs . B

u(B-bw)= ji. dx+ \udx

bw bs

(3.83)

Substituting Eqns. (3.81) and (3.82) into Eqn. (3.79) and integrating, the average excess

pore water pressure u at a given time t becomes (assuming that bw and bs are negligible

compared with B, i.e.(B-bw)& B and(_5 -_>s)« B):

u = Yy

n-\ C —\— n+\

' 1 cO" ^ hP dt j

B " P. (3.84)

where,

(~ \

P» = K hp f U \

w fv n,^ -f {KsP J L ' V B j

J P

( b >

\ B) + fp\n> B

(3.84a)

The following convergent series can be used in the above equation.

125


fP(n»y)=

I . 1 (w-l) (w-l)(2w-l) (w-l)(2»-l)(3w-l)

2! 3!« 4!T72 5!T2~ &.n

•y-(»4) 2 (2/i-l) 3 J/i-lX3»-l) 2!TZ

• ^

3!T2^ y

4\n: •y (3.84b)

(7.-IX272-IX477-I) 5 (7.-1X272-1X377-1X577-1) 6 4 y .-. * -^

5!T7 6!T7"

The variable y represents — or —— and the function fp(n,y) for n=\.2, 1.3 is _5 B

graphically illustrated in Figure 3.15. The function fp(n,y) increases with the exponent n,

while it decreases wheny increases and converges to zero at about y ~ 0.36.

5 0.2

a o '4—

o

a 3

0.04 1 I I I 1 L

0.1 0.2 Ratio, y

.7=1.2 1 /i=1.3 I — __ «=1.5

J I I L

0.3

Figure 3.15 The variation of function/^ (n, y) with y for selected n

Since n > 1, the Eqn. (3.84) can be integrated using the boundary conditions u = uo at

t=0, to give:

126


4a ( 2 Yw D

Ahp iu)n~ W 1 . (3.85)

where,

aP = tt

4(77-1) (3.85a)

Introducing the average degree of consolidation,

u Uhp=l-^-WO

(3.86)

The time t required for a given degree of consolidation U hp found by:

t = AaP Yw' B

X

n-\ Dn+\ w_ - \n-l hP (wo)" k-v»Y

(3.87)

Rearranging the above equation, the degree of consolidation U hP at a given time . is:

Uhp=\- 1 + -hP

l

ap(2Bf

f ~ \"~]

up l-n

(3.88)


Substituting Eqn. (3.84) into Eqn. (3.81) and rearranging, the normalized excess pore

u water pressure =— with time, at any point within the smear zone can be found as: wo

127


W

wo

;

Khp

\^Ksp J "(l-Uhp) (3.89)

Similarly, substituting Eqn. (3.84) into Eqn. (3.82) and rearranging, the normalized

w excess pore water pressure = - with time, at any point outside the smear zone is given wo

by:

w

uo

_\t-Uhp)

Pi

Si ( x^

\ Bj -Si

r b n,-B

Khp + I \KSP J

g,

V » )

f b \

V B j Sx n,-B

(3.90)


>

Differentiating Eqn. (3.89) in the x direction and rearranging, the hydraulic gradient (/ )


. =

Khn

Y w yKsP j

i

n(l-Uhp)uQ

BPP 1 - *

B

(3.91)

Differentiating Eqn. (3.90) in the x direction and rearranging, the hydraulic gradient (/)

with time at any point outside the smear zone is given by:

i = (l-Uhp)ui

Y w B0. 1 - *

B

(3.92)

128


Influence of exponent n on consolidation process

In order to study the influence of exponent on the consolidation process, a plane strain

analysis has been executed with the following parameters using different values of n

(1.00-1.50 in 0.05 interval). _5=0.7m, _>==0.105m, &w=0.035m, khp =Khp =0.008 m/yr,

KP =KsP =0.002 m/yr, chp=Xhp =0.4 m

2/yr and two points (x=0.07 m, within smear

zone and x=0.35 m, outside the smear zone) are considered for comparing excess pore

pressure and the hydraulic gradient. The estimated average degree of consolidation,

excess pore pressure and hydraulic gradient after a consolidation period of 0.5 years are

plotted in Figures 3.16-3.18.

1.0 l.i 1.2 1.3

Exponent, n 1.4

Figure 3.16 Variation of average degree of consolidation with exponent n

Figure 3.16 shows that the average degree of consolidation increases from about

5 2 % to 6 8 % when 77 varies from 1.00 to 1.50, i.e., when 77 increases from 1 to 1.5, the

129


degree of consolidation is increased by approximately 3 0 % . From a practical

perspective this is a significant deviation from conventional consolidation theory,

emphasising the influence of the exponent 77.

0.6

o 3

_

t/a

CD

<D

_-» - s (U

_

o P. __ _n _

o X <D

0.4

0.3 -

at a point 0.35m from the centre (outside the smear zone)

at a point 0.07m from the centre (inside the smear zone)

-a N ___; 0.2 c_

O

1.0 1.1 1.2 1.3 1.5

Exponent,«

Figure 3.17 Variation of normalized excess pore pressure with exponent n

Figures 3.17 and 3.18 show that variation of normalized excess pore pressure and

hydraulic gradient at selected points (within and outside the smear zone) with exponent

n. When 77 increases from 1 to 1.5, excess pore pressure within the smear zone and

outside the smear zone is decreased by 24% and 35%, respectively and the

corresponding hydraulic gradients decrease by 52% and 75%. This means that not only

is the consolidation process in the undisturbed zone influenced more than in the smear

zone by the value of 77, but also the hydraulic gradient is more sensitive than the excess

pore pressure to variations of n. This is because the permeability of the undisturbed

zone is considerably higher compared to the reduced permeability in the smear zone.

130


o J3 4.0

£ _ T3 3.0 cd (-i

bO _

1 _3 -_ (D N

ed

O

z

1.0

at a point 0.35m from the centre (outside the smear zone)

at a point 0.07m from the centre (inside the smear zone)

o.o _L _l_

1.0 1.1 1.2 1.3 Exponent, n

1.4 1.5

Figure 3.18 Variation of normalized hydraulic gradient with exponent n

3.3.3 Matching with Axisymmetric Consolidation

At each time step and at a given stress level, the average degree of consolidation for

both axisymmetric (Oh) and equivalent plane strain (Uhp) conditions are made equal,

hence:

Ur = Uhp (3.93)

By substituting Eqns. (2.56) and (3.88) into the above equation, the following equation

can be obtained (assume the dimensions of both unit cells are equal):

lhp (a^

2 n+l Ka J

(3.94)

Substituting a _ and a , which are given in Eqns. (3.85a) and (2.56) respectively, in the

above equation, the converted permeability can be related by:

131


khp=2k} 'n-lBvy

2n2 P (3.94a)

where, /?_ and /? are given in Eqns. (3.84a) and (2.56a), respectively.

Ignoring the smear effect in Eqn. (3.94), the equivalent plane strain permeability in the

undisturbed zone can readily be obtained as:

f (

K hp _ hp

Kh X = 2

ft n, o B

2f

V D J ( r \ n,-R

(3.94b)

V A J J

where, fp

1960):

npL is given in Eqn.(3.84b) and / f r >

K K J is given below (Hansbo,

fP ' r "\ n

n-1

n-\ (n-lf 3n - 1 n(3n - l\5n -1) V.n1 (5n - lpn -1)

n-1 3n-l fv \

2n \R) +

1 1 77-1 • +

2n 3n-l 2n2{3n-l)_

fr V

\RJ

+ 77-1

+ (îf n-\

n{3n-l\5n-l) 4n3(5n-l) 2n2(3n-l)

fr V 5n-l

(3.94c)

KRJ + ..

For convenience of practical use the equivalent parameters in terms of spacing ratio

(Blbw), smear ratio (bs/bw) and permeability ratio of axisymmetric cell for different 77 are

graphically illustrated in Figures 3.19-3.22. Once the field permeability values and

geometric parameters are known (or assumed) then the equivalent plane strain

parameters can be readily obtained from such plots.

132


i 111111 1 i i i m i j — i i i 11 i n , — i i 11'.in

10 100 1000 10000 Blbw {=Dldw)

1 1 n u i — i i 1 1 1 I I I I — i i i i m . | — i i i i n n

2 10 100 1000 10000 Blbw {=D!dJ

0.5-

T 10 100 1000

Blbw {=Dldw)

10000

"\

_" s '3 J3 in

_ a

J3 n.

I "3 > 'B o-

w

"I /,. \ 0.3

0.2-

0.1-

Perfect drain

bs=2bw

bs=lbw bs=Abw

,Kh/Ks=4

/Ks=3

nTTTTj I I I I llll| 1 I I I 111 (| 1 I I Mill

10 100 1000 10000 Blbw (=Dldw)

Figure 3.19 Equivalent plane strain ap value as a function of Blbw, bjbw and KH/KS of

axisymmetric cell for different 77 values

As shown in Figure 3.19, when the B/bw ratio increases, the value of ap

(equivalent plane strain) increases and attains a maximum. For KhJKs >2, a distinct

peak of a is followed by a post-peak reduction. Because the effect of smear plays a

major role at small B/bw, the a _ increases up to a certain extent and then decreases due

to the insignificant effect of smear (at large B/bw) and converges to a value

corresponding to that of a perfect drain (i.e. negligible smear). The equivalent plane

strain parameter p is plotted in Figure 3.20, which also shows a similar trend to a _.

133


10 100 1000 B/bw (=D/dw)

10000

n=1.2

^ . = 4

Kh/Ks=3

i i m i l 1—i 11 IIIII 1—i i i u n i 1 i i i m i

2 10 100 1000 10000 B/bw(=D/dw)

a.0. __ a '3 b

0.6-

0.4-

o-w 0

2-

n=1.3

Kh/Ks=4

Kh/Ks=3

Perfect drain

- _»,=46_, i , 1 1 H I — i r i u u i j — i 1 1 1 i u i | — i i ci

2 10 100 1000 10000 Blbw (=_ /_/w)

'! Illlll| [ l . l l l l i | 1 I I (1111| I I I M i l l

10 100 1000 10000 Blbw (=£>/_/J

Figure 3.20 Equivalent plane strain pp value as a function of Blbw, bJK and KH/KS of

axisymmetric for different n values

The ratio of the coefficient of consolidation (or permeability of undisturbed zone)

of the equivalent plane strain cell to axisymmetric cell (Xhp/X = Khp/fch) is shown in

Figure 3.21, as a function of B/bw. When the Blbw ratio increases the value of

Xh IX oxKhpJKh increases and attains a maximum, and then decreases at a diminishing

rate.

134


~ — i — i i i u | 1 1 — i — i i i M | 1 1 — i — i i i i i

100 1000 10000 B/bw (=D/dw)

Figure 3.21 Ratio between coefficients of consolidation of undisturbed zone of

equivalent plane strain cell to those of axisymmetric cell as a function of Blbw

The ratio of undisturbed zone permeability to smear zone permeability of the

equivalent plane strain cell (KhpJKsp ) is shown in Figure 3.22. This shows that the

value of Kh IK gradually increases with an increase in both Blbw and KH/KS . It is

noted that the ratio KhpJKsp increases at a faster rate at large Blbw ratios when the smear

effect becomes less. These converted parameters can then be incorporated into the finite

element analysis.

135


25-n=l._

BIbw{=Dldw) 100 200

i i i i 1 1 1 1 — i — i i i i 1 1 1

4 10 100 200 B/bw (=D/dw)

i i i i 1 1 1 1 i i i i i 1 1

4 10 100 200 Blbw (=D/dw)

i i i i 1 1 1 1 — i — i i i i 1 1

4 10 100 200 B/bw(=D/dJ

Figure 3.22 Ratio between undisturbed zone permeability to smear zone permeability of

equivalent plane strain cell as a function of B/bw, bjbw and Kh/Ks of

axisymmetric cell for different 77 values

3.4 Summary

Analytical formulations related to consolidation analysis have been explained in this

Chapter. A brief summary of this formulation is given below:

1. An analytical solution to estimate the extent of smear zone has been formulated

based on Cylindrical Cavity Expansion analysis incorporating the Modified Cam

Clay theory.

136


2. The existing axisymmetric (Hansbo, 1981) and plane strain (Indraratna and Redana,

1997) theories of a unit cell were modified to include the vacuum pressure

application. It was assumed that the applied vacuum pressure is constant along the

top surface and propagates immediately along the length of drain in a trapezoidal

pattern.

3. New plane strain lateral consolidation equations neglecting the well resistance of

vertical drains are formulated and can be applied for both Darcian and non-Darcian

flow. The new parameters are graphically illustrated for practical use. Finally, plane

strain matching procedure for the proposed solution has been explained.

137

— Chapter 4 Laboratory Testing and Analysis of Results

4 LABORATORY TESTING AND ANALYSIS OF RESULTS

4.1 General

Bergado et al. (1991); Indraratna and Redana, (1995); Xiao (2000) conducted laboratory

tests to study the behaviour of vertical drains installed in soft clay using a specially

designed large-scale consolidation apparatus. Bergado et al. (1991) used a transparent

PVC cylinder (455 mm x 920 mm x 10 mm wall thickness) with a steel base plate but

the excess pore water pressure measurements were considered unrealistic with this

setup. The cylinder was filled with soft remoulded Bangkok clay and a PVD (Ali drain -

4 mm x 60 mm) was installed using the 6 mm x 80 mm mandrel, and then the

settlement behaviour was monitored under a surcharge pressure of 47.8 kPa. The

permeability coefficients were calculated from conventional oedometer tests, carried out

for horizontal and vertical specimens taken at several locations.

Indraratna and Redana (1995) used a large-scale consolidometer (450 mm x 950

mm) to investigate the effect of smear due to the installation of prefabricated vertical

drains. They examined the reduction of soil permeability to assess the extent of the

smear zone around vertical drain installed by a mandrel. Xiao (2000) also conducted a

series of large-scale tests to study the behaviour around vertical drains installed in soft

clay using remoulded kaolin clay.

In this research, a fully instrumented large-scale consolidometer was used to

measure the pore pressure development during mandrel installation to assess the extent

of the smear zone around a vertical drain. In addition, the extent of the smear zone was

estimated using the permeability and water content measurements by taking samples

138

Chapter 4 Laboratory Testing and Analysis of Results

(horizontally and vertically) at different locations. A number of approaches were used to

predict the smear zone using the permeability and water content measurements, such as:

(i) From the variation of horizontal permeability;

(ii) Using the permeability anisotropy (horizontal to vertical permeability ratio, M:.)

- Indraratna and Redana (1998) approach;

(iii) Using the normalized lateral permeability (kh/khu), i.e., the coefficient of lateral

permeability (kh) over the maximum undisturbed zone permeability (/./,„);

(iv) From the variation of water content; and

(v) Using the normalized water content reduction, i.e., (wmax-w)/wmax, where wmax=

maximum water content and w= water content at any point.

In the Indraratna and Redana (1998) approach, the vertical permeability was shown to

be relatively unaffected by mandrel installation. Therefore, in the current analysis the

normalized lateral permeability (kh/khu) and normalized water content reduction were

taken to be more realistic than the ky/kv ratio in deteraiining the effect of smear. Also, a

correlation between the permeability reduction (i.e. difference between the undisturbed

and smear zone values) and water content reduction was proposed. The proposed

empirical equation is very useful in practice, because by measuring the water content,

the horizontal permeability may be estimated. Details of the apparatus and test

procedures are explained in the following section.

139

— . — . Chapter 4 Laboratory Testing and Analysis of Results

4.2 Experimental Set-up and Testing Procedure

4.2.1 Apparatus

The large-scale radial drainage consolidometer (Figure 4.1) consists of two cylindrical

(stainless steel) half sections, each of which has a flange running the length of the

cylinder so they can be bolted together. The cell is 650 mm internal diameter x 1040

mm height x 8 mm thick, and it has a 1.5 mm thick Teflon sleeve fitted around the

internal cell boundary to reduce friction and is then placed onto a steel base. A LVDT

(Linear Variable Differential Transformer) transducer is placed on top of the piston to

monitor surface settlement, and strain gauge type pore pressure transducers (PPT's) are

also installed to measure the pore water pressures at various depths. In addition, an array

of strain gauge type pore pressure transducers are installed radially at a depth of 0.5 m

from the top surface (the plan view is shown in Figure 4.2) to monitor pore pressure

development during mandrel installation.

The transducers used in this laboratory studies are based on the use of strain gauge

technology and manufactured by Durham Geo Enterprises. Design of these transducers

incorporated a ceramic pressure element in a stainless steel enclosure and bleed valve to

eliminate air traps (see Fig. 4.3). Model E-120 pore pressures are used in this study has

a measurement range of 0-30 psi (0-207 kPa) and readability to 0.1 psi (0.69 kPa). The

transducers were calibrated using a Budenburg dead weight testing machine. A simple

computer programme was written using the calibration data to convert the transducer

output in voltage to an appropriate pore pressure and settlement units (kPa and mm).

140


4.2.2 Testing Procedure

Because this cell is very large (the volume is about 0.34m3), it is almost impossible to

obtain undisturbed samples of this size. Therefore, reconstituted alluvial clay from

Moruya, NSW was used to make large samples. The geotechnical properties (Table 4.1)

of the selected soils were determined from Atterberg limits and specific gravity tests

and were classified using the Casagranade Plasticity Chart or Unified Soil Classification

System as high plasticity (CH) clay.

Figure 4.1 Large-scale radial drainage consolidometer

141


Table 4.1 Engineering properties of selected sample

L L PL Clay particles Silt particles Specific Unit weight Moisture

(%) (%) (<2pm) (<10pm) Gravity (kN/m3) content (%)

70 30 45-55 45-60 2.66 17.5 48

LL: Liquid Limit; PL: Plastic Limit

(b) (c)

Figure 4.2 Location of pore pressure transducers and cored samples

4.2.2.1 Preparation of Reconstituted Clay

• The plastic limit, liquid limit, and natural water content of the clay sample were

determined for soil classification.

• The clay was thoroughly mixed with water so the water content was equal to or

greater than the liquid limit.

142


• The clay slurry was kept in a closed container for several days to ensure full

saturation.

• To ensure that the clay was fully saturated, a couple of small cylindrical specimens

(38mm x 76mm) were cored and tested in the triaxial equipment. Skempton's B

parameter of 0.99 or more was determined.

4.2.2.2 Preparation of the Apparatus

• Calibrated the pore pressure transducers using a Bundenburg dead weight testing

machine. Load cell and LVDT were also calibrated, and a simple computer

program was written using the calibrated data to convert the data logger output (in

voltage) to an appropriate unit (pore pressure-kPa, load-kN and settlement-mm).

• Each part of the apparatus was cleaned and the pressure chamber lubricated to

prevent internal friction.

• The O-ring and elastic seal were placed into the groove in the bottom plate to

prevent air leakages. An elastic seal should be applied along the flanges of the two

half cylinders and left for couple of days to dry.

• Placed a round plastic sheet with a hole at the centre and a round geotextile sheet

onto the bottom plate to prevent clay sticking to it and the clay particles from

eroding. Subsequently, a 1.5 mm thickness Teflon sheet was placed around the

inner periphery of the cell to minimize the friction between the side wall of the

cylinder and the clay.

143


4.2.2.3 Preparation of the Vertical Drain

• A 1.5 m long prefabricated vertical band drain (100 mm x 3 mm, equivalent radius

is 33mm using Eqn. 2.3) was taken and kept in water to make it saturate.

• Inserted the PVD into the slot in the specially designed rectangular mandrel (Figure

4.3), which was slightly larger than (125 mm x 25 mm, equivalent radius is 48mm

using Eqn. 2.3) the PVD. The end of the drain was attached to a shoe to ensure the

drain remained in the proper position when the mandrel is withdrawn after

insertion.

Figure 4.3 Mandrel, Guider, PPT and prepared sample

4.2.2.4 Testing Procedure

The following testing procedure is recommended.

• Fill the cell, with the prepared reconstituted clay in 150 mm layers by compacting

or vibrating to expel the air trap in the clay slurry before adding the next layer, to a

total height of 950 mm.

• Install the pore pressure transducers at selected locations.

144

. Chapter 4 Laboratory Testing and Analysis of Results

• Place a round plastic sheet followed by the top plate, and subsequently place the

load cell onto the top of plate to measure the applied load from the pressure

chamber.

• In this study, four tests were conducted with different initial consolidation pressure

(i.e. 20, 30, 40, and 50 kPa). The initial consolidation pressure was applied and left

for two weeks (or until about 90% of the consolidation was obtained). It is noted

that all four samples have same characteristics.

• The pressure chamber, load cell, top plate and plastic sheet were removed at the end

of the preconsolidation phase and then the PVD was installed, at an average

penetrating rate of 0.5 m/min using a specially designed guider, during which the

pore pressure was recorded. The mandrel was withdrawn after it touches the bottom

of the cell.

• The top plate, load cell, pressure chamber and settlement transducer (LVDT) were

re-installed and the surcharge load was applied in stages in increments of 50, 100,

and 200 kPa to promote radial consolidation. It is noted that the time duration

between two stages is 30 days.

• The horizontal and vertical undisturbed specimens (total of 32 specimens per test)

were taken (size of 38mm x 76mm) at the end of the test from several locations to

find the permeability coefficients and the water content using the standard (one-

dimensional) oedometer test

145


4.3 Presentation of Results

4.3.1 Evaluation of Compressibility Indices

The available finite element programs are generally based on the elastic-plastic Mohr-

Coulomb model or soft soil model (Cam-Clay model), so it is essential that the soil

parameters be determined. Figure 4.4 shows the relationship between the void ratio and

applied consolidation pressure, while Table 4.2 shows the evaluated soil parameters. It

is noted that the C a m clay parameters and the angle of shear resistance were estimated

from triaxial tests (conducted by Redana, 2000).

1.9

1.2

1.1

Cc=0.34 =>>.=0.15

Cr=0.14 = > K = 0 . 0 6

' I ' I l _ L

10 20 50 100 200

Consolidation Pressure (kPa)

500 1000

Figure 4.4 Variation of void ratio with consolidation pressure

146


Table 4.2 Modified C a m Clay Parameters

*

26

X

0.15

K

0.06

M

1.02

N

2.57

r

2.51

u0

2.8

V

0.25

4.3.2 Pore Pressure Variation during Mandrel Installation

Pore pressure transducers, T1-T5 having a radial distance from the centre of 95, 105,

120, 140 and 190 mm, respectively, were installed 0.5 m below the surface (Figure

4.2a) to measure variation of pore water pressure during installation. Figure 4.5 shows

these variations over time elapsed for each initial surcharge pressure, during mandrel

installation and withdrawal. The pore pressure response shows an increase in magnitude

until the maximum value, which occurs when the mandrel tip just passes the depth at

which the PPT is located. The pore pressure continuous to drop as the mandrel is driven

deeper. Subsequently the pore pressure drops rapidly and then converges to a small

residual value when the mandrel is withdrawn. This also shows that the developed pore

pressure decreases in the radial direction (pore water pressure is higher at transducer TI

which is close to the mandrel and less at T5 which further away from the centre of

drain).

147


40-

30-cd

u c/a 00 _ s_ O H

_

o OH

2 0 -

10-

0-

Mandrel pass the PPT's

Before mandrel installation

Mandrel withdrawal

TI T2

T3

T4

T5

After mandrel withdrawal

0 -| i r -100 200

Time Elapsed (sec)

(a) Initial Pressure = 20 kPa

300

100 200 Time Elapsed (sec)

(b) Initial Pressure = 30 kPa

300

148


100 200 Time Elapsed (sec)

(c) Initial Pressure = 40 kPa

300

0 100 200 Time Elapsed (sec)

(d) Initial Pressure = 50 kPa

Figure 4.5 Pore pressure variation during mandrel installation for different initial

surcharge pressure (preconsolidation pressure)

149


In order to determine the smear zone, the normalized pore pressure (i.e., pore

pressure / surcharge pressure) was plotted against the normalized radial distance (i.e.,

radial distance / equivalent mandrel radius). In Figure 4.6, the maximum normalized

pore water pressure at the location of each PPT is compared with the predicted,

normalized pore water pressure based on the proposed (described in the previous

Chapter) cavity expansion theory (CET) using the modified C a m clay parameters given

in Table 4.2. The predicted pore pressure ratios are very close to those measured, and

the predicted extent of smear zone is 2.401, 2.449, 2.588 and 2.623 times the equivalent

mandrel radius, corresponding to the initial surcharge pressure of 20, 30, 40 and 50 kPa,

respectively. This suggests that the extent of smear zone might be dependent on the

depth of overburden.

rlrm

(a) Initial surcharge = 20 kPa

150


•Normalized pore water

pressure - Measured

Normalized pore water pressure

(calculated using C E T )

Calculated extent of smear zone, rs/rm = 2.4491

rlr m

(b) Initial surcharge = 3 0 kPa

1.2 Normalized pore water

pressure - Measured

Nonnalized pore water pressure

(calculated using C E T )

Calculated extent of

smear zone, rslrm = 2.5876

rlr. m

(c) Initial surcharge = 4 0 kPa

151


0 1 2 3 4 5 6 ~ rlrm

(d) Initial surcharge =50 kPa

Figure 4.6 Normalized pore water pressure variation with distance

4.3.3 Permeability Tests

At the end of the large-scale consolidation test, horizontal and vertical specimens were

cored at 0.5 m below the surface to measure the coefficient of permeability and the

extent of smear zone. Variations of the horizontal and vertical coefficient of

permeability (fa and /cv), as well as the permeability ratios (/c„/frv) and the normalized

permeability (fa/khu) for different mean applied consolidation pressures are plotted in

Figure 4.7.

152


1 *• o

i

o

4=

•a _ DH

c o N •c

o

4-

2 - Q

0-

Smear zone

Applied pressure -

0 rlr

m (a) Variation of horizontal permeability

o b

c_

u

u OH

1. o >

(b) Variation of vertical permeability

153


«4_

c_

_

0.

U

— OJ CL,

2.000

1.800-

1.600-

1.400-

1.200-

1.000

1.000-

J 0.900H _^

3 0.800H

•o 0.700H 4) __ "c_

| 0.600H

0.500-

c_

0

(c) Variation of permeability ratio

Highly disturbed zone »*-

Marginal disturbance

Insignificant disturbance

r/r, m

(d) Variation of normalized permeability

Figure 4.7 Variation of (a) horizontal permeability, (b) vertical permeability, (c)

permeability ratio, and (d) the normalized permeability, with radial distance

154


Figure 4.7a shows that the horizontal permeability gradually increases up to a

certain extent and then remains relatively constant whereas the Figure 4.7b shows that

the variation in vertical permeability with radial distance is less. This indicates that the

mandrel installation have more affects on the horizontal soil parameters than those of

vertical. The permeability ratio between horizontal and vertical is illustrated in Figure

4.7c, which shows a similar trend to the variation of horizontal permeability in Figure

4.7a. The variation of normalized permeability is plotted in Figure 4.7d. This clearly

demonstrates that close to the drain boundary (highly disturbed zone), the value of

normalized permeability ratio increases rapidly with radial distance, whereas further

away the change in normalized permeability becomes insignificance. It is of interest to

note that the irrespective of the applied pressure, all curves are confined within a narrow

band, clearly defining the extent of the smear zone. From this data, one may conclude

that the smear zone is at least 2.5 times the equivalent mandrel radius (rm), and the

normalized lateral permeability ratio within the smear zone varies from 1.0943 and

1.6437 (an average of 1.3429). Moreover, very close to the drain, the k\/kv ratio is

expected to approach unity.

4.3.4 Variations of Water Content

Various researchers (e.g. Taylor, 1948; Samarasinghe et al. 1982; Tavenas et al. 1983;

Babu et al. 1993) have discussed the effect of the void ratio on the coefficient of

permeability and concluded that it is dependent upon the void ratio and the water

content of soils. Therefore, it is logical to argue that installation of mandrel not only

affects the horizontal coefficient of permeability but also the water content. In this

research, an attempt is made to examine the extent of the smear zone from the variation

155

• , Chapter 4 Laboratory Testing and Analysis of Results

of water content and the normalized water content reduction, i.e., (wmax-w)/wmax. For

this purpose, samples were collected at five different vertical levels and eight different

radial locations.

The variation of water content and the normalized water content reduction with

radial distance at a depth of 0.5 m, for different applied pressure is illustrated in Figure

4.8. The variation of water content (Figure 4.8a) shows a similar trend to the variation

of fa as shown previously in Figure 4.7a. The normalized water content reduction is

shown in Figure 4.8b, which shows that all plots are confined within a narrow band,

again demonstrating the effect of smear.

The variation of the water content with depth and radial distance is shown in

Figure 4.9 for an applied pressure of 200 kPa. As expected, the water content decreases

towards the drain, and also the water content being greater towards the bottom of cell at

all radial points. Based on these curves the extent of smear zone can be estimated to be

at least 2.5 times the equivalent mandrel radius. This agrees well with the estimated

extent of smear zone based on fa/fa ratio (Figure 4.7c), fa/fan ratio (Figure 4.7d). These

results are also in good agreement with the CET predicted smear zone extent as

previously indicated in Figure 4.6. It confirmed that the extent of smear zone could be

quantified using the proposed cavity expansion theory.

156


_ s_ o o OJ

•*—

0.04-

0.02-

Figure 4.8 Variation of (a) water content and (b) normalized water content reduction,

with radial distance at a depth of 0.5 m

157


70-

68-

66-

64-

62-

cd

Smear zone wmax = ws = 6 9 %

Location of the smaple from bottom (mm)

• 0 (bottom)

• 200

400

600

800

T 4

rlr, m

Figure 4.9 Variation of water content with depth and radial distance for an applied

pressure of 200 kPa

Correlation between permeability and water content within smear zone

A correlation between the permeability reduction (i.e. difference between the

undisturbed and smear zone values) and water content reduction is shown in Figure

4.10. This relationship is almost perfectly linear (R2 > 0.99), and the following empirical

expression can be proposed:

Ak = C 'Aw] wo,

(4.1)

158


where, fa: permeability of undisturbed zone; A/c: permeability reduction; w0: water

content of undisturbed zone; Aw. reduction in water content (Aw = wo-w). The

empirical coefficients C and n are 8.32 and 1.1, respectively, for the current test results.

The above equation can be very useful in practice, because by measuring the water

content, the horizontal permeability may be estimated if the initial water content and

initial horizontal permeability are known.

-0.4-

-0.6-

^

o

-0.8-

-1.0-

•1.2-

-1.4-

-2.2

Linear fitted line (R2=0.993)

25kPa

50kPa

100 kPa

200 kPa

-2.0 •1.8 -1.6

Log (AW/WQ)

-1.4 -1.2

Figure 4.10 Correlation between the reduction of permeability and the water content

within the smear zone

159


4.3.5 Comparison of Surface Settlement

The surface settlement was monitored using the LVDT transducer placed on top of the

plate. The measured and predicted surface settlements are compared in Fig. 4.11a, and

found to agree (only the result of Test 1 is considered because the surface settlement

was almost the same in all the tests). The numerical predictions were carried out using

the finite element code PLAXIS incorporating the modified Cam- clay parameters given

in Table 4.2. Hence, the length of the drain is small (about lm), the well resistance is

neglected in the analysis and the extent of smear is taken as 5 times the equivalent drain

radius («150 mm). The finite element mesh of the large-scale cell consisting of 15-

node triangular elements with 12-point Gauss integration is shown in Fig. 4.1 lb.

_

S -40 _ t_ cz_

-60

-80

Time (days)

20 30 40

Surcharge pressure increase

• Measured

— Predicted

50 60

(a) (b)

Figure 4.11 (a) Comparison of surface settlement, and (b) FE mesh used in Plaxis

160


4.3.6 Comparison of Excess Pore Water Pressures during Consolidation

The measured and predicted excess pore water pressures (neglecting well resistance) of

Test 1 are compared in Fig. 4.12. The excess pore water pressure measured at

transducer T4 (185 mm from the centre and 0.5 m below the surface) was in good

agreement with the numerical predictions. While there was a very small deviation

between the predicted and measured excess pore pressure at transducer TI (95 mm from

the centre and 0.5 m below the surface) during Stage 1 loading, a good agreement was

found during Stage 2 loading. Transducer TI shows an accelerated dissipation of excess

pore water pressure compared to transducer T4, because, TI is closer to the boundary of

the drain.

-

c_

___ _ 80-3 BO

u __ 2 _« £ « O P-i

„ 40-§3

-

n i u T

i

I m i i

>%

. __ I • \

\ - ^ . x?_ **_. N» '•l-l

• Measured EPP at TI

• Measured EPP at T4 O -i • I i - . , 1 T7T>T_ ..1 T1 rerdicted hrr at 1J

_. . jêrdicteci i__rr at 14

1 \ 1 \

\ •• \ V • v

^ *_.

^5%_ ^ * _ * \?»*_

1 1 1 ' "T 20 40

Time (days) 60

Figure 4.12 Comparison of excess pore water pressure

161


4.4 Summary

The effect of smear zone due to the installation of prefabricated vertical drains was

investigated in the laboratory, and compared with the analytical and numerical results.

From the laboratory measurements, the extent of smear zone was found to be 2.5 times

the equivalent mandrel radius. The calculated extent of smear zone was about 2.515

(average) times the mandrel radius using the cavity expansion theory proposed in

Chapter 3, which is very close to the value evaluated in the laboratory. The

experimental results also show the horizontal permeability in the smear zone varying

from 1.0943 to 1.6437, with an average of 1.3429 times smaller than the undisturbed

zone. The above findings are comparable with previous research reported in the

literature (Table 2.4). For example, Hansbo (1979), Indraratna and Redana (1998) Chai

and Miura (1999) proposed that the extent of smear zone is about 1.5 ~ 3, 2 ~ 3, and 2

~ 3 times the mandrel radius, respectively, which is close to the extent of smear zone

found in this studies ( about 2.5 times the mandrel radius).

The measured excess pore water pressure and surface settlement were also

compared with predicted values using the PLAXIS software and found to be in good

agreement. The experimental results shown in this Chapter confirm that predicting the

extent of the smear zone is feasible using the proposed solution.

An expression between the change in water content and change in horizontal

permeability within the smear zone is found and this empirical expression is very useful

in practical sense for estimating lateral smear zone permeability by measuring the water

content. It is also noted that initial water content and initial horizontal permeability

should be known.

162

Chapter 5 Case Study: Sunshine Motorway. Australia

5 CASE STUDY 1: SUNSHINE MOTORWAY (QLD, AUSTRALIA)

5.1 General

The Sunshine Coast is one of Australia's fastest growing regions but this continued

economic and population growth has increased the pressure on the region's main traffic

corridor, the Sunshine Motorway. Site investigation at the proposed development route

revealed that the subsoil consists of very soft, highly compressible, saturated marine

clays of high sensitivity, which presented difficulties developing the new alignment. In

order to study the foundation response upon loading, and evaluate the effectiveness of

various ground improvement techniques on these marine clays, a fully instrumented

trial embankment was constructed in 1992 at Area 2A of the proposed Sunshine

Motorway (Figure 5.1), located in the Maroochy Shire, Queensland, Australia. This trial

embankment was monitored by the Queensland Department of Main Roads (QDMR),

Brisbane, Australia.

The subsoil conditions are relatively uniform throughout the site, consisting of

sensitive silty clay about 10-1 lm thick, overlying a layer of sand approximately 6m

thick. A thin soft clay layer underlies the sand and extends to a depth of 18m. Another

sand layer is encountered below 18m. The water content, Atterberg limits, bulk unit

weight, undrained shear strength and the compression index ratio of the top sensitive

silty clay with depth are given in Figure 5.2. The compression index ratio varies from

0.15 to 0.50, and the recompression ratio was found to be about 10 times smaller than

the compression index ratio (QDMR Report, No.R1765, June 1991). The over

consolidation ratio (OCR) of the soils varies from 1.0 to 1.6 (lightly overconsolidated

soil).

163

Chapter 5 Case Studv: Sunshine Motorway. Australia

Maroochydore

Study

Area

Figure 5.1 M a p of Australia showing the location of the study area

u_ ~ Q

0

-

3

4

5

6

7

8 _

9

IO

WATER

CONTENT

% o o o o g o W . I t - •_ _

GRADING % SMALLER THAN 2Jp r, O O O

.t t/m'

BULK PARTKL-DENSITY (BP01

LI PI %

UNDRAINED SHEAR STRENGTH IkPo)

_, o in o m o tn___________oo I _J___5__

_ *% UT

o o o

(mVyr)

ORGANIC

CONTENT

<l 17'

-.A,.

1.66

L5Z

_3_* 1,40

1.34

I.3S

2.63

_.60~

2.55

1.7 2,5 2.3 2.6

2.4

1.9

S

27 28-35

40 55 46

' i !

0-9

0-4

5.3

10.7

/

1 /

1.60

1.54

.64

2.60

2.55

2.62

17

31

33

25

3l

ii® 0 3 4-6

^-LIQUID LIMIT

OLD ALIGNMENT o - VANE SHEAR ® - UL) TRIAXIAL - 50mm Thin wolled sarnct.

NEW ALIGNMENT A OU TRIAXIAL - DH.I9 100mm P:s<_r< samples 0 DHI9 \VANE SHEAR +• DH20J

Figure 5.2 Profile of the Geotechnical characteristics (Sunshine Motorway Stage 2

Interim Report, 1992)

164


The base area of the trial embankment was approximately 9 0 m x 4 0 m and

incorporated 3 separate sections (Figure 5.3), identified as Sections A, B, and C,

respectively. Sections A and B (each 35m in long) were the primary sections of the trial

embankment and they represented the zones of vertical drains (at lm intervals) and 'no

drains', respectively. Section C (approximately 20m in long) was an intermediate case

with vertical drains at 2m apart. The vertical drains (Nylex Flodrain) in Sections A and

C were installed in a triangular pattern.

Tensar SS2 geogrids were laid longitudinally on the natural ground surface,

followed by a lightweight needle-punched geotextile (Bidim U12). The SS2 was

considered to be a light reinforcement whilst the U12 acting as a separator. A working

platform 0.65m thick (500mm of 7mm screenings drainage layer plus 150mm of

selected fill) was placed on top for construction traffic access.

Prefabricated vertical drains were installed from the working platform to a depth

of 1 lm under the Sections A and C, followed by a geogrid reinforcement placed in the

lateral direction (Tensar ER200). Then the embankment was constructed in stages using

a loosely compacted granular material (yt,«20 kN/m3) up to a height of 2.85m.

1 i n • • • • *-

: ^ H A .'-Sr±'+ B -•-."<: -i-.-

_ _ _ _ _ _ _ 1.

Figure 5.3 Plan view of Trial Embankment

165


w-* .

isUii/

Inclinometer (IA2/IB4/IC5)

7m Working Platform (0.65m) w

Readout Box w 5 m *\X I \ w ^-f

|_tent of ER200 15m from centre

PPA13/PPB3; /PPC43

SCC5 I SCA1/SCB3

PVA4.' PVC39 t P V B 2 2

Reinforcement (ER 200) Sensitive Silty Clay

Inclinometer (IA1/IB3) Sand Soft Clay I2m

10.5m

5.5m

, Sand

Figure 5.4 Typical cross-section of embankment with selected instrumentation points

Two berms, 5m in width on the instrumented side and 8m wide on the other side

as shown in Figure 5.4 were constructed to increase the stability of the embankment.

Half of the cross-section was intensively instrumented to capture the foundation

response upon loading. The instrumentation consisted of settlement gauges (SC),

piezometers (pneumatic-PP, vibrating wire-PV and standpipe), inclinometers (I),

horizontal profile gauges, sondex settlement systems, strain gauges and earth pressure

cells. Typical cross-section of embankment with selected instrumentation points is

shown in Figure 5.4.

The deformation and pore water pressure responses below each section of the

embankment were predicted using a plane strain finite element analysis and then

compared with the available field data.

5.2 Finite Element Analysis

The multi-drain plane strain analysis was carried out using the finite element code

PLAXIS (Version 8). In the analysis, the soil layers were divided into many elements.

The element types were selected to deal with the relevant geometrical and material non-

linearities. The soft soil model based on Modified Cam-clay theory was used to

166


replicate the clay layers, whereas the Mohr-Coulomb model was employed to analyse

the sand layers. The soil parameters used in the finite element analysis are given in

Tables 5.1 and 5.2 on the basis of information given by QDMR.

Table 5.1 Modified Cam-clay parameters used in the finite element analysis

Depth (m) Soil type M X K v eQ J- 3 A v e r a S e Pc

(kN/W) (kPa)

0.0-2.5 Silty clay 1.20 0.270 0.027 0.30 1.85 16.4 20

2.5-5.0 Soft silty clay 1.20 0.480 0.048 0.30 3.10 13.7 31

5.0-10.5 Silty clay 1.18 0.260 0.026 0.3 1.75 15.9 66

10.5-16.0 Sand [ See Table 5.2]

16.0-18.0 Soft clay 1.08 0.240 0.024 0.3 1.70 16.1 120

Table 5.2 Mohr-Coulomb parameters of the sand layer

ys(kN/m3) v c'(kPa) /(degrees) E(MPa)

15.0 0.3 13.5 35 7.5

5.2.1 Element Types used in PLAXIS

The element types used in the finite element code PLAXIS are shown in Figure 5.5. A

choice can be made between 15-node triangular elements and 6-node triangular

elements. The 15-node element has 15 nodes and 12 Gaussian integration stress point,

167


whereas the 6-node element has 6 nodes and 3 Gaussian integration points. While the

powerful 15-node element provides an accurate calculation of stresses and failure loads,

the 6-node elements permit more rapid computation. In addition to the triangular

elements, which are generally used to model the soil, geogrid elements and interface

elements may be used to model the soil-structure interaction.

(a) 15-node element (b) 6-node element

Figure 5.5 Types of element used in P L A X I S (Version 8)

• nodes

x stress point

soil element

x ^—s s interface element f—*• - • — • — 1 >

(a) 10-node interface element (b) 6-node interface element

Figure 5.6 Distribution of nodes and stress points in interface elements and their

connection to soil elements

Figure 5.6 illustrates how interface elements are connected to soil elements.

When using 15-node soil elements the corresponding interface elements are defined by

five pairs of nodes whereas for 6-node soil elements the corresponding interface

168

Chapter 5 Case Study. Sunshine Motorway. Australia

elements have three. In the figure the interface elements are shown to have a finite

thickness but in reality they have zero thickness. Each interface is assigned a 'virtual

thickness' which is an imaginary dimension used to define its material properties which

are related to the adjacent soil properties. Also, interfaces may be used in a

consolidation analysis to block the flow perpendicular to the interface to simulate an

impermeable screen.

5.2.2 Types of Material Models used in PLAXIS

The mechanical behaviour of soils may be modelled with various degrees of accuracy.

For example, Hook's law of linear, isotropic elasticity may be conceived as the simplest

available stress-strain relationship. As it involves only two input parameters (Young's

modulus and Poisson's ratio), it is generally too 'crude' to capture essential soil

behaviour. To replicate the non-linear stress-strain behaviour a variety of constitutive

laws are available, and even though most advanced models require a relatively large

number of parameters the Mohr-Coulomb and the Soft-Soil models are often considered

to be sufficient to model the soil behaviour in practice.

5.2.2.1 Mohr-Coulomb Model

The elastic-plastic Mohr-Coulomb model involves five input parameters, i.e. Young's

modulus and Poisson's ratio for soil elasticity; friction angle and cohesion for soil

plasticity; and the dilation angle. This model is often recommended for

overconsolidated soils (e.g. surface crust).

169

— —Chapter 5 Case Studv: Sunshine Motorway. Australia

5.2.2.2 Soft Soil Model

This is a Modified Cam-Clay type model especially suitable for primary compression of

normally consolidated soils. The Soft-Soil model requires the following material

* X constants: X - modified compression index = ; K - modified swelling index

l + e0

=- ; M- slope of critical state line; c - cohesion; 4> - friction angle; y - dilatancy l +eQ

angle; v - Poisson's ratio; and Konc- coefficient of lateral stress in normal

consolidation.

5.2.3 Variation of Extent of Smear Zone

The extent of the smear zone with depth was predicted using the proposed cavity

expansion theory (CET), as explained in Chapter 3 (Section 3.1), incorporating the

modified Cam clay parameters given in Table 5.1. The predicted normalised pore water

pressure (_/av0) variation with radial distance for each soil layer is shown in Figure 5.7.

The CET predicted extent of the smear zone (i.e., the distance from the centreline at

which w/avo^l) is illustrated in Figure 5.8, which shows that the smear zone decreases

from 6.6rw to 4.9rw («230-170mm) when the depth increases from 0 to 10.5m. In this

case the equivalent radius of the drain rw was 35 mm.

170


1.0-

0.9-

o

s n R—I 0.8-

0.7-

0.6-

Depth (m)

0.0-2.5

2.5-5.0

5.0-10.5

Range of smear zone

0 n I r

4 i r n i r i 1 r T i r

8 12 16 20 rlr.

w

Figure 5.7 Normalised pore water pressure variation with radial distance

Figure 5.8 Variation of extent of smear zone with depth

5.2.4 Plane Strain Permeability

The finite element analyses were executed using the equivalent plane strain conversion

described in Chapter 3 (Section 3.2.5). The equivalent plane strain permeabilities are

given in Table 5.3. It is noted that the axisymmetric (in situ) permeabilities are identical

to the equivalent permeabilities of Section B.

171

— Chapter 5 Case Study: Sunshine Motorway. Australia

Table 5.3 Equivalent Plane Strain permeabilities of embankment sections

Depth

(m)

0.0-2.5

2.5-5.0

5.0-10.5

16.0-18.0

Sand

Section B (10"10 m/s)

kh=khp ks=khp

48.89 18.45

28.09 10.60

21.07 7.95

13.12 4.95

86.79 32.75

Section A (10"10 m/s)

K

14.76

8.48

6.36

13.12

86.79

**

4.89

2.75

2.04

4.95

32.75

Section C(l(

K

11.66

6.70

5.03

13.12

86.79

f10 m/s)

K

3.41

1.85

1.34

4.95

32.75

5.2.5 Numerical Calculations and Comparison with Field Observations

The finite element mesh, which contained 15-node triangular elements, is shown in

Figure 5.9. The full width of embankment had to be modelled because the loading was

not symmetrical. The prefabricated vertical drains were modelled with zero thickness

drain elements. For these elements, the excess pore pressure along the boundary is zero,

i.e., well resistance is neglected. The smear zone was modelled with the same soil

properties as the adjacent soil except for coefficient of permeability. The reinforcement

in the embankment was modelled with geogrid elements and these were combined with

interface elements to replicate the interaction with the surrounding soil. The locations of

instruments were conveniently placed in the mesh in such a manner that the measuring

points coincided with the mesh nodes.

172

Chapter 5 Case Study. Sunshine Motorway. Australia

Figure 5.9 Finite element mesh used to analyse Section B

5.2.5.1 Displacement Boundary

Only 20m depth of the foundation was considered due to the existence of the sand layer

(below the overlying soft clay layer), which was dense enough to neglect any associated

deformations. To minimize the boundary effect, lateral boundaries were placed 150m

(7.5 times the vertical dimension) from the centreline of the embankment and horizontal

movements along this boundary were fixed.

5.2.5.2 Drainage and Loading Boundary

Both the top (open boundary) and bottom (sand layer) surfaces of the subsoil foundation

were assumed to be free draining and water table coincided with the ground surface.

Embankment loading (Figure 5.10) was simulated by applying incremental vertical

loads to the upper boundary.

173


3.0-

2.5-

& 2.(H •4—»

_3

'53 X ~ 1.5-

1.0-

0.5-

Section A

Section B

Section C

•

0 20 40 Time (days)

Figure 5.10 Construction History of the Trial Embankment

60

5.2.5.3 Comparison of Centreline Settlement

The predicted and observed surface settlements under each section of embankment are

compared in Figure 5.11. The settlement gauges under Sections A, B, and C, namely,

SCA1, SCB3, (both under the centreline) and SCC5 (lm to the left of centreline) were

selected for the purpose of comparing the field data with the numerical results. Figure

5.11 shows that the predicted values are in good agreement with field data, and as

expected, the settlement rate increases as the drain spacing decreases. The settlement

responses of Sections A and C are comparable whereas the centreline settlement under

Section B (no drain) is only about 60% of the settlement of the other two sections. This

suggests that the installation of vertical drains significantly reduces the consolidation

time (from years to days) and thereby reducing the subsequent post-construction

settlement. Generally, the settlement rate is sensitive to drain spacing but in this study

174


the differences under Section A and C are small and therefore any benefits derived from

installing the vertical drains at lm spacing as compared to 2m spacing are minimal.

However, this could be negated by the smear effect caused by the vertical drain

installation.

0-*

200-

400-

_

£

? 60(H _/.

800-

1000-

FieldData-SCAl

Field Data-SCB3

Field Data-SCC5

FEM-SCA1

FEM-SCB3

FEM-SC5

0 20 40 60 Time (day)

80

Figure 5.11 Centreline settlement of trial embankment sections

The settlement contours at the end of construction and after 100 days for Sections

B and C are illustrated in Figures 5.12 and 5.13, respectively. These plots indicate that

the settlements under Sections B and C can be negligible below a depth of 18m;

therefore, the assumption of fixed supports at a depth of 20m is justified. Furthermore, a

small heave is seen to occur at the toe of Section B but the amount of heave has become

less due to the vertical drain installation (Section C).

175

Chapter 5 Case Study: Sunshine Motorway, Austral la

-290 -260 -230 -200 -170 -140 -110 -80

(a) Settlement (mm)-Section B

_490 -440

-50 -20

-390 -340 -290 -240 -190

(b) Settlement (mm)-Scction C

140 -90 -40

Figure 5.12 Settlement contours at the end of construction

10

10

-470 -422

-720 -646

-374 -326 -278 -230 -182 -134

(a) Settlement (mm)-Section B

-86 -38

i

-572 -498 -424 -350 -276 -202

(b) Settlement (mm)-Scction C

-128 -54

Figure 5.13 Settlement contours at the end of 100 days

176

20


Even though the settlement contour patterns under both Sections B and C are

similar, the contour lines intensified within the vertical drain zone, i.e., the post-

construction settlements were concentrated within the improved zone.

5.2.5.4 Comparison of Lateral Displacements

Lateral deformation measured by the inclinometers installed at the middle of the main

batter and toe of the berm are compared with the numerical predictions in Figures 5.14

and 5.15. Inclinometers IA1 and IB3 were installed in the centre of the main batter of

Sections A and B, whereas inclinometers IA2, IB4, and IC5 were installed at the toe of

the berm of Sections A, B, and C, respectively.

0-

4-

_ 8 -B

c_

Q 12-

—

16—

••

_

"# °

[i

20-f

o o^ V • • D \

/ •

ym y __-

y • ^ ^ ^

. 1

\ n o "*--,

• Field Data-IAl (56days)

o Field Data-IA 1 (1 OOdays)

• Field Data-IB3 (56days)

n Field Data-IB3 (1 OOdays) FT7A4 T A 1 l^fÂ-iirc\

I'Liivi-iAi (poaaysj TTCTv/l T A 1 (\ C\C\A<*iic\

— r.civi-i_A.i tiouaaysj FEM-IB3 (56days)

FEM-IB3 (1 OOdays) 1 1 '

0 100 200 Lateral displacement ( m m )

300

Figure 5.14 Lateral displacement profile at the middle of the main batter

177


•B

u _

Q

a. _

O Q O D

DO aqj

Field Data-1A2 (56days)

Field Data-IA2(1 OOdays)

Field Data-1B4 (56days)

Field Data-IB4(1 OOdays)

Field Data-IC5 (56days)

Field Data-IC5(l OOdays)

FEM-IA2 (56days)

FEM-IA2 (1 OOdays)

FEM-IB4 (56days)

FEM-IB4(1 OOdays)

FEM-IC5 (56days)

FEM-IC5 (1 OOdays)

40 80 120 Lateral displacement (mm)

Figure 5.15 Lateral displacement profile at the toe of the berm

160

Figure 5.14 illustrates the predicted and observed lateral displacement profiles at

inclinometer locations IA1 and IB3 after 56 days (at the end of construction) and at 100

days. The results indicate that the predicted lateral displacement below a depth of 3m

represents an acceptable match with the field data, whereas a noticeable discrepancy is

found near the ground surface due to surface crust. It is also interesting to note that the

installation of vertical drains increased the lateral displacement at the middle of the

main batter by an average of about 52%.

178


The predicted and observed lateral displacement profiles after 56 and 100 days at

the toe of the embankment sections are plotted in Figure 5.15. Once again an acceptable

agreement is found below a depth of 3m and as expected, the vertical drains

significantly curtailed lateral deformation. Vertical drains installed at lm intervals

(Section A) reduced lateral displacement by about 46% but only by 22% at 2m spacing

(Section C).

Lateral displacement contours at the end of construction and after 100 days for

Section B and C are shown in Figures 5.16 and 5.17, respectively. These figures clearly

show that lateral displacements are negligible beyond a distance 50m and 35m from the

centreline of the embankment Sections B and C, respectively. Therefore, the assumption

of a fixed lateral boundary at a distance of 150m is warranted. As expected, Figure 5.16

shows maximum lateral displacement occurs at a depth of approximately 5m («1.7 x fill

height) beneath the berm and this location moves towards the toe of the main batter as

consolidation progresses (Figure 5.17).

5.2.5.5 Comparison of Excess Pore Pressure Variation

The predicted and observed variations of excess pore pressure at selected points beneath

the embankment centreline and the middle of the berm are shown in Figures 5.18 and

5.19, respectively. The selected pneumatic piezometers PPA13, PPB31 and PPC43

were installed at a depth of 3.85m beneath the middle of the berm of embankment

Sections A, B, and C, respectively. In contrast, the selected vibrating wire piezometers

PVA4, PVB22, and PVC39 were installed close to the centreline of the embankment at

a depth of 5.5m, 6.6m, and 6.75m under the embankment Sections A, B and C,

respectively.

179

Chapter 5 Case Study: Sunshine Motorway, Australia

-90 -72 -54 -36 -18 0 18 36 54 72 90

(a) Lateral displacement (mm)-Section B

n -70 -56

T _42 -28 -14 0 14 28 42

(b) Lateral displacement (mm)-Scction C

56 70

Figure 5.16 Lateral displacement contours at the end of construction

-

-200 -160 •120 -80 -40 0 40 80 120

(a) Lateral displacement (mm)-Scction B

160 200

170 -136 -102 -68 -34 0 34 68 102 136 170

(b) Lateral displacement (mm)-Scction C

Figure 5.17 Lateral displacement contours at the end of 100 days

180


Figure 5.18 shows that all sections generated significant excess pore pressure

due to embankment loading and the predictions are in acceptable agreement with the

field data. As expected the induced excess pore pressure under Section B (no drain) is

significantly higher than the other sections while the excess pore pressure developed

under Sections A and C are comparable. Although the drain spacing of Section A (lm)

is smaller than Section C (2m), differences in the rate of dissipation is low because

piezometer PPC43 is further away from the boundary of the influence zone whereas the

piezometer PPA13 is closer to boundary.

30-

.3

^ 20-\ =3 i/i Ui

<u u p. *_ o & Ui Ui

_ o X W

Fielddata-PPA13

Field data-PPB31

Field data-PPC43

FEM-PPA13

FEM-PPB31

FEM-PPC43 U

\ \

\ \

10-

0 20 40 60 Time (day)

80 100

Figure 5.18 Excess pore pressure variation with time beneath the middle of the berm

181

.Chapter 5 Case Studv: Sunshine Motorway. Australia

60-

03

_>

Ui Ui

0. — _ c_

0. _H

O OH CW 1/3 (D

o

40-

20-

Field data-PVA4

Field data-PVB22

Field data-PVC39

F E M - P V A 4

FEM-PVB22

FEM-PVC39

40 60 Time (day)

80 100

Figure 5.19 Variation of excess pore pressure below the embankment centreline

The predicted and measured excess pore pressure variations at the embankment

centreline for each section are plotted in Figure 5.20. Again an acceptable match

between predictions and field data are found. Surprisingly, the generated excess pore

pressure under Section B, which has no drain, was lower than the other sections. This

could be the result of less amount of fill placement over the section B compared to the

other sections (since the centreline settlement of Section B < Section C < Section A, the

amount of fill required to make the same reduced level at the embankment centreline at

the end of construction would be in the same order). The excess pore pressure

developed under Sections A and C (vertical drains are installed at lm and 2m spacing,

respectively) are comparable. It is also important to note that the rate of pore pressure

182

— Chapter 5 Case Studv: Sunshine Motorway. Australia

dissipation at piezometer locations PPA13 and PPC43 (beneath the berm, Figure 5.18)

is higher than that of piezometer locations PVA4 and PVC39 (beneath the embankment

centreline, Figure 5.19). This is because the piezometers PVA4 and PVC39 were placed

close to the drain influence zone boundary where the flow was constrained.

5.3 Summary

The performance of a trial embankment consisting of three different sections

constructed with and without prefabricated vertical drains was analysed using 2D multi-

drain (plane strain) finite element analysis. The effect of smear associated with the

installation of PVD was considered but the effect of well resistance was neglected since

the discharge capacity of prefabricated vertical drains was large enough. The extent of

the smear zone was evaluated using the cavity expansion theory explained in Chapter 3

(Section 3.1). Predictions incorporating the proposed solutions, namely, the plane strain

matching and cavity expansion theory were made using the finite element code

PLAXIS. It shows that the inclusion of exact extent of the smear zone and the

application of the plane strain matching, improves the accuracy of the numerical

predictions.

The predicted centreline settlement, excess pore water pressure beneath the berm

and the centreline of the embankment section, and the lateral movements at the toe of

the embankment and middle of the main batter of each sections were compared with the

available observed data. Good agreement between the predicted and measured

settlements was found, whereas an acceptable matching was found between the

predicted and measured excess pore pressures. Even though lateral displacements were

harder to match an acceptable comparison was found below a depth of 3m.

183


The settlement response of the embankment sections indicated that the installation

of vertical drains significantly decreased the consolidation time, whereas the benefits

derived from installing vertical drains at lm spacing as compared to 2m spacing were

insignificant. Closer spacing invariably contributes to increased smear (the total extent

of smear zone under Section A was about 18.9m but only 9.7m under Section C).

The predicted and measured lateral movements showed that the installation of

vertical drains increased lateral movement under the middle of the main batter while

decreasing it at the toe of the embankment. These results also verified that maximum

lateral movement occurred at a depth of about 1.7 times the fill height. The pore

pressure dissipation rate under Sections A and C was comparable which proved that the

effect of smear was greater under Section A.

184

Chapter 6 Case Studv: Second Bangkok International Airport. Thailand

6 CASE STUDY 2: SECOND BANGKOK INTERNATIONAL AIRPORT

(SUVARNABHUMI, THAILAND)

6.1 General

Construction of the Second Bangkok International Airport (SBIA) has been planned

since the 1960s to accommodate the rapid growth of air traffic at the Bangkok

International Airport at Don Muang (Thailand). The SBIA site is located at Nong Ngu

Hao (in Samutprakan Province), about 30 km east of capital city Bangkok (Figure 6.1).

This site is situated on a swampy land in a flat marine deltaic deposit with an average

elevation of less than one meter above mean sea level (MSL). In the past, most of this

area was covered by fishponds or agricultural land usage with several canals.

Site investigations at the SBIA showed that the subsoil conditions are relatively

uniform throughout the site, consisting of a weathered crust formed by cyclic wetting

process together with natural cementation with total thickness of 1 to 2m, overlying soft

Bangkok clay layer extending to about 8-1 lm below the surface, followed by medium

stiff to stiff clay to a depth of 20m or more. Groundwater level is at about 0.5m depth

from the surface. The major concern for the airport construction was the presence of the

8 to 11m thick soft Bangkok Clay which often has a natural water content more than

100 percent with low shear strength. The general properties and compressibility

parameters of the in-situ subsoil are summarized in Figures 6.2 and 6.3, respectively.

The compression index ratio varies from 0.18 to 0.53 and the recompression ratio was

found to be about 10 times smaller than the compression index ratio. The over

consolidation ratio (OCR) of the soils varies from 1.0 to 2.5.

185


Bjingkok International Airport

SBIA SITE HOENGSAO

LAEN CHABANG EASTERN SEABOARD DEVELOPMENT

Figure 6.1 Location of SBIA site (after M o h and Lin, 1997)

Figure 6.2 General soil properties at SBIA site (modified after Sangamala, 1997)

186


shear strength Initial void ratio compression index Over consolidation (eo) Cc/(1+e0) ratio (OCR)

10 15 20 25 30 1.5 2 2.5 3 3.5 0.2 0.3 0.4 0.5 0.6 1.2 1.6 2 2.4 2.8

(kN/m2)

u

2

4

Depth (m)

CO

CO

10

12

14

• |

•

•

•

•

•

•

•

•

•

•

•

•

<

, | ' | ' ' ' I ' •

•

•

•

•

•

•

•

•

•

•

•

•

• | |

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

Figure 6.3 Compressibility parameters at SBIA site (adopted from Sangmala, 1997)

Due to the underlying high compressibility and low strength soft Bangkok clay,

ground improvement was needed to overcome potential problems related to excessive

and differential settlements. Several well-instrumented test embankments were

constructed using Prefabricated Vertical Drains (PVDs) with preloading. In this Thesis,

the behaviour of embankments (namely, TV1 and TV2) improved by prefabricated

vertical drain system with vacuum and surcharge preloading are analysed. The vertical

drains (Mebra MD-7007, 100mm x 3mm) were installed beneath TV1 and TV2 in a

triangular pattern at lm spacing to a depth of 15m and 12m, respectively. The total base

area of each embankment is 40m x 40m and the cross sections of embankments with

key instrumentation that includes surface settlement plates, extensometers,

inclinometers, and piezometers are shown in Figure 6.4. In embankment TV1, a hyper

net drainage system was used while perforated pipes were utilized in embankment TV2.

187

Chapter 6 Case Study: Second Bangkok International Airport. Thailand

10m »w ______

Geomembrane Liner (LLDPE) Qe o n et ^

4 5m >k

L B M S 4 Vacuum Pump

Bentonite

Inclinometer

Legend D Surface settlement plate 0 Extensometer • Electrical piezometer * Stand-pipe piezometer

13m. S3 _____

Vertical drains

Geotextile

S2 Fi_l\ SI ,2.2m 10.3m

- S

l 1 — — I 1

@3m (a) TVI (PVD's are installed at lm interval upto a depth of 15m)

*L Perforated pipe Geomembrane Liner (LLDPE)

I.In

0.3fL'M 0.5ir

-H 10m ' 10m

Vacuum Pump

Inclinometer

(b) T V 2 (PVD's are installed at lm interval upto a depth of 12m)

Figure 6.4 Cross section of embankments with key instrumentation at SBIA (modified

after Indraratna et al., 2005)

The drainage blanket, which serves as a working platform, was constructed with

sand (unit weight of 18 kN/m3) to a thickness of 0.3m for TVI and 0.8m for TV2. Both

embankments have a water and air-tight linear low-density polyethylene (LLDPE)

geomembrane liner on top of the drainage system. The borders of the geomembrane

liner were completely sealed off from the atmosphere by placing the liner borders at the

bottom of a trench, which was filled with a 0.30 m thick layer of sand-bentonite mix.

188


The water collection system in each embankment was connected to a vacuum pump

having a capability of supplying a continuous vacuum pressure of 70 kPa. Both

embankments were constructed in stages up to a maximum height of 2.5m, the loading

history of which is shown in Figure 6.5.

_E_

'<3 X •4—"

c __

I

3.0-

2.5-

2 . 0 -

1.5-

1.0-

0.5-

TV1 TV2

No PVD

o.o-

Vacumm + Surcharge

Vacumm only*—|—• (unit weight of fill is 18 kN/m3) i—i—r—i—i—i—i—|—i—i—i—|—i—i—i—|—' i ' | ' r~

0 20 40 60 80 100 Time (Days)

120 140 160

Figure 6.5 Construction loading history

6.2 Finite Element Analysis

A plane strain multi-drain analysis was carried out using the finite element code

HKS/ABAQUS (Version 6.3), incorporating the Modified Cam-clay theory. The Cam-

clay parameters for each soil layer are given in Table 6.1, which includes the slope of

the critical state line (M), gradients of compression (X) and swelling (K) line on

v-lnp space, Poisson's ratio (v), the initial void ratio (co), the saturated unit weight of

t

soil (Ys), and the preconsolidation pressure (pc).

189


Table 6.1 Modified Cam-clay parameters of SBIA site (Indraratna et al., 2005)

Depth (m) M X K V e( 0 ys Average pc

(kPa) kN/m3

0.0-2.0 1.19 0.3 0.03 0.30 1.8 16 58

2.0-8.5 1.19 0.7 0.08 0.30 2.8 15 45

8.5-10.5 1.19 0.5 0.05 0.25 2.4 15 70

10.5-13.0 1.19 0.3 0.03 0.25 1.8 16 80

13.0-15.0 1.07 1.2 0.10 0.25 1.2 18 90

6.2.1 Element Types used in A B A Q U S

The types of elements used in the finite element code ABAQUS are illustrated in Figure

6.6. The basic element type is a 4-node bilinear displacement and pore pressure element

(CPE4P) consisting of 4 displacement and pore pressure nodes at the corners. A higher

order element consists of a 20-node tri-quadratic displacement and tri-linear pore

pressure nodes with reduced integration (C3D20RP), which contain 20 displacement

nodes and 8 pore pressure nodes. The common element type used in the case history

analysis in this Chapter is the CPE8RP, which contains 8 displacement nodes and 4 pore

pressure nodes. In finite element analysis the pore pressure shape function is usually

one order less than the displacement shape function. The pore pressure shape function is

linear in most of the elements shown in Figure 6.6 while the displacement shape

functions are either quadratic or cubic

190


<¥> O — < S >

6 o

CPE6P CPE4P

® O—<£) CPE8RP

Q

O

G>

-©-^

O

-e CINPE5R

C3D20RP

-i ASI3

(interface element)

pore pressure node O displacement node

Figure 6.6 Types of elements used in A B A Q U S (Hibbirt et al., (2004)

Interface elements are most appropriate to simulate soil-drain interaction. Since

the thickness of PVD is relatively thin compared to its spacing, the interface element is

envisaged as the soil element having properties similar to the adjacent soil, except for

permeability. A 3-node interface element (ASI3) is shown in Figure 6.6 where there are

2 pore pressure nodes at the ends.

6.2.2 Variation of Extent of Smear Zone

The extent of the smear zone with depth was predicted using the cavity expansion

theory as explained in Chapter 3 (Section 3.1), incorporating the Modified Cam clay

parameters given in Table 6.1. The predicted normalized pore water pressure (i.e., UIGV0)

variation with radial distance for each soil layers is shown in Figure 6.7. As expected, it

clearly shows that the induced pore pressure during mandrel installation is very high

close to the mandrel boundary but gradually decreases with radial distance. For better

191


clarity, the predicted extent of the smear zone with depth is shown in Figure 6.8, which

indicates that the smear zone decreases form 6.30rw to 3.10rw when the depth varies

from 0 to 15m.

Depth (m)

0.0-2.0

2.0-8.5

8.5-10.5

- - - 10.5-13.0

— 13.0-15.0

Figure 6.7 Normalized pore water pressure variation with radial distance

6.2.3 Plane Strain Permeability

The multi-drain plane strain numerical analysis was executed using two models:

Model 1- Assuming a constant extent of smear zone (i.e. rs=150mm, taken as 6 times

the equivalent drain diameter of rw=25mm, Indraratna et al. 2005).

Model 2- Varying extent of smear zone (as illustrated in Figure 6.8).

192


In the analysis described here, axisymmetric to equivalent plane strain conversion was

executed using the permeability matching method proposed in Chapter 3 (Section

3.2.5). The axisymmetric and equivalent plane strain permeabilities (calculated using

the Eqns. 3.74a and 3.74b) are given for both models in Table 6.2.

ex _

u—

__.

3—

_

6-J

_

9-

12-

i ^

(

"c.

Q

i

)

Extent of Smear zone

Predic Expan

Assun (Indra

ted based on Cavity sion Theory

led Value ratna et al., 2005)

1 1 ' 1 2 3

1 1 •

4 5 i

.

i

7

's^w

Figure 6.8 Variation of extent of smear zone with depth

Table 6.2 Axisymmetric and Plane Strain permeabilities for both embankments

Depth

(m)

0.0-2.0

2.0-8.5

8.5-10.5

10.5-13.0

13.0-15.0

K

10"9m/s

30.1

12.7

6.0

2.6

0.6

10"9m/s

15.1

6.4

3.0

1.3

0.3

K

10"9m/s

8.064

3.403

1.607

0.697

0.161

K (io-

Model 1

3.474

1.474

0.690

0.299

0.069

9 m/s)

Model 2

3.482

1.465

0.679

0.288

0.066

193


6.2.4 Numerical Predictions and Comparison with Field Data

The finite element mesh, which contained 8-node bi-quadratic displacement and bilinear

pore pressure elements, is shown in Figure 6.9. Due to the symmetrical nature of the

embankment, it was sufficient to model half width of embankment. For the area with

PVDs and smear zone, a finer mesh was employed so that each unit cell represented a

single drain with the smear zone on either side of the drain. The instrumentation points

were placed in the mesh in such a manner that the measuring points coincided with the

mesh nodes. Only the top surface of the clay was assumed to be free draining because

the presence of the stiff clay layer at the bottom was considered to be impervious. The

embankment loading was simulated by applying incremental vertical loads to the upper

boundary whereas the vacuum pressure was modelled by applying negative pressure

along the surface and drain. The value of vacuum pressure was assumed to be constant

over the soil surface but varying linearly to zero along the length of drain.

Figure 6.9 Finite element mesh used in the analyses (Model 2)

194

Chapter 6 Case Study: Second Bangkok International Airnort. Thailand

160—r

c_

_ _ (fl u -_ OH

_ O

E2

Applied vacuum pressure

Measured surface vacumm pressure D t«

Assumed surface vacuum

pressure in FEM analysis 12m

n 1 r 80 Time (Days)

(a) TVI

160

u •~

P^ V u o o H

80 Time (Days)

(b)TV2

Figure 6.10 Measured total pore pressure and simulated vacuum pressure at surface

195


Figure 6.10 illustrates the measured pore pressure at various depths of the

embankment from electrical piezometers installed 0.5m away from the centreline. A

discrepancy between the measured and applied vacuum pressure is noted. The suction

head in the field could not be maintained because of possible air leaks. Therefore, in the

numerical analysis, the magnitude of applied vacuum pressure at the surface with time

was adjusted based on the field measurements, which is also plotted in Figure 6.10.

The results of the multi-drain plane strain analysis (neglecting well resistance)

based on both models, together with the available field data, are shown in Figures 6.11-

6.13. The predicted and measured settlements at various depths are illustrated in Figure

6.11. These plots show that the predictions based on Model 2 are much closer to the

field data than the predictions based on Model 1, i.e., the predictions based on the

changing smear zone with depth (estimated using cavity expansion analysis) are more

accurate than the results based on a constant extent of the smear zone. Even though the

total fill height and drain spacing are the same, and the length of drain under

embankment TVI is high, the rate of settlement under the embankment TV2 is greater

than TVI. In fact, the settlements under embankment TVI are about 22.52% smaller

than those of TV2. These differences could be attributed to the presence of geotextile in

embankment TVI, as well as the different loading rate, and the variations in vacuum

pressure.

Comparisons between the predicted and measured excess pore water pressure 3m

below ground level for both embankments are shown in Figure 6.12. Once again, the

predicted excess pore pressures based on Model 2 agree well with the field data,

implying the correct assessment of the extent of the smear zone. It is important to note

that after about 100 days (end of embankment construction), the field measurements

196

Chapter 6 Case Study: Second Bangkok International Airport Thailand

indicated an increase in pore water pressure, which further suggests that the constant

suction could not be sustained during the entire construction period.

80 Time (Days)

B O

•*->

B B 33 00

-40-

-80-(b) TV2

Symbols: Field data

Solid lines: Model 1 Dot lines: Model 2

•120-

Depth (m)

0 40 80 Time (Days)

120 160

Figure 6.11 Settlement variation with depth for embankments (a) TVI and (b) TV2

197

Chapter 6 Case Study: Second Bangkok International Airport Thailand

30-

Sa _

Ui Ui <D

u DH

_ S-l

O PH <z> CO

_ O X

W -20-

(a) TVI

0

0.

S-i

_ _

_ 1-

O OH

<D CJ

!><

w

Field Data

Model 1

Model 2

40 80 Time (Days)

120 160

Field Data

Model 1

Model 2

80 Time (Days)

160

Figure 6.12 Variation of excess pore water pressure at 3m depth below ground level,

0.5m away from the centreline for embankments (a) TVI and (b) TV2

At the end of monitoring period (150 day) the lateral deformation measured by

the inclinometer installed at the toe of the embankments together with the predicted

results are shown in Figure 6.13. Unlike settlement, the observed lateral displacements

could not be matched well, but an acceptable agreement could be found below the

middle of the very soft clay layer using Model 2. But the field observations closer to the

198

Chapter 6 Case Study: Second Bangkok International Airport, Thailand

ground surface do not support the significant 'inward' lateral movements as indicated

by the numerical predictions. These results confirm the difficulties associated with

accurate modelling of the surficial compacted crust.

(a) TVI

• Field Data

Model 1 Model 2

i 1 ' | r | r | 50 100 150 200 250

Lateral Displacement (mm)

• Field Data

Model 1

-- Model 2

40 80 Lateral Displacement (mm)

160

Figure 6.13 Lateral displacement profiles (after 150days) through the toe of the

embankments (a) TVI and (b) TV2

199


Previous studies on embankments constucted on soft clay have shown that the

accurate prediction of lateral movement is a difficult task compared to vertical

displacement (Tavenas et al., 1979; Indraratna et al., 1997). The errors made in the

prediction of lateral movements can be numerous and are attributed to soil anisotropy

and the assumption of 2D plane strain. The embankment corner effects are not properly

modelled in 2D plane strain. The behaviour of the stiff crust just below the ground

surface cannot be modelled using conventional Cam-Clay properties but requires the

accurate assessment of its highly over-consolidated (compacted) parameters as

discussed by Indraratna et al., (1994). Given the difficulties of modelling lateral

displacements in plane strain, Model 2 still provides an acceptable comparison.

6.3 Summary

The performance of two test embankments stabilised with vertical drains subjected to

vacuum preloading was investigated using a plane strain (multi-drain) finite element

analysis. The effect of smear associated with PVD installation in conjunction with the

applied surcharge load and vacuum was considered. The predictions were made based

on two different models, namely, Model 1- assuming a constant extent of smear zone,

and Model 2- varying the extent of the smear zone.

The predicted centreline settlement at different depths, excess pore water pressure

and lateral movement of the soil were compared with the available field data. Good

agreement between the predicted and measured settlements was found but the pore

pressures and lateral displacements were more difficult to match. Also, the settlement

analysis showed that the drainage efficiency of the embankment using perforated pipe

(TV2) is better compared to embankment using hypernet (TVI) drainage system. It was

200

— Chapter 6 Case Study: Second Bangkok International Airport. Thailand

demonstrated that the predictions based on Model 2 with the inclusion of time and depth

dependent vacuum pressure distribution improves accuracy.

From the field studies, Choa (1989) also confirmed that the propagation of

vacuum pressure decreases substantially with depth due to various practical limitations,

improper sealing, and the nature of soil conditions (e.g. presence of fissures and macro-

pores). Hence, in this study, the assumption of diminishing vacuum pressure along the

drain length is warranted in the finite element analysis.

Even though the conventional preloading (surcharge) method is more economical

than the vacuum assisted soft clay improvement, the use of a sufficient vacuum pressure

with a properly sealed surface membrane will significantly accelerate pre-construction

settlement, thereby compensating for the initial capital costs by enhanced speed of

construction and minimising post-construction settlement. It is also important to note

that the application of vacuum pressure substantially decreases lateral displacement,

thereby minimizing the risk of shear failure.

201

— Chapter 7 Numerical Modelling and Design Implications

7 NUMERICAL MODELLING OF VERTICAL DRAINS AND DESIGN

IMPLICATIONS

7.1 General

Construction of structures on subsiding or derelict ground is becoming a major task in

today's civil engineering work. Due to increasing loads as well as the need to construct

in areas of soft and compressible soils necessitate the improvement of the soil

properties. The choice of the appropriate ground improvement technique has to be made

depending on the type of soil, the application of loads (loading rate) and the time

available for the improvement process. Evaluating the design and performance of each

structure and foundation is accomplished by analytical, semi-analytical (empirical), or

numerical methods. Design charts or certain design methods based on well known

governing equations are available for embankments stabilized with prefabricated

vertical drains. In general the embankments are constructed on variable ground with

multi-drain system, therefore, numerical computation has become popular among

design engineers.

For construction sites with a large number of PVDs, two-dimensional (2D) plane

strain conversion is the most convenient approach with regards to computational

efficiency. It is far less time consuming than a three dimensional (3D) multi-drain

analysis where each drain has its own axisymmetric (3-D) zone which substantially

affects mesh complexity and the corresponding convergence. In the analysis described

in this Chapter, axisymmetric to equivalent plane strain conversion was executed using

the method described in Section 3.2.5 (Indraratna et al., 2005).

202

Chapter 7 Numerical Modelling and Design Implications

Numerical analysis in geotechnical engineering must fulfil certain basic

requirements, i.e., evaluation of initial stress, embankment slope, construction rate, the

spacing of vertical drains and the possibility to simulate the sequential construction

stages. In multi-stage construction a rest period is allowed after each stage of loading so

that excess pore water pressure in the foundation can dissipate. Such dissipation is

accompanied by consolidation and a gain in the soil strength. This increased shear

strength enables the embankment to be raised to a greater height during the next stage of

construction. In this Chapter, selected numerical studies have been carried out to

investigate the effect of the embankment slope, construction rate, drain spacing, extent

and parameters of the smear zone, stage loading, and the influence of surface crust on

the potential failure of a soft clay foundation.

7.2 Design of E m b a n k m e n t Constructed on Soft Clay without Vertical Drains

In this section the effect of embankment slope and rate of embankment loading are

studied using the finite element code PLAXIS. The subsoil profile is assumed to consist

of 4 sub-layers. The Modified Cam-clay parameters and the soil properties are shown in

Table 7.1. The finite element mesh (15-node triangular element), which is used for this

analysis, is shown in the Figure 7.1 where only half the embankment (half base width is

20 m) is considered by symmetry. A foundation depth of 15 m was considered

adequate for the purpose of analysis because of the existence of a stiff clay layer

beneath this depth. The lateral boundary of the finite element is defined 100 m away

from the centreline of the embankment and the ground water table is assumed to be at

the ground surface.

203


The embankment fill is modelled as a fully drained elastic-perfectly plastic

material with Mohr-Coulomb criteria. The fill is defined as having a bulk density of

/'bulk =20 kN/m3; elastic parameters are £=105 kN/m2 and v = 0.3; and the strength

parameters are ^ =30 and c = 10 kN/m .

Table 7.1 Soil parameters used in finite element analysis

y Permeability (m/day)

Depth e0 X* K M v

(kN/m3) fa fa

0-2 1.8 0.107 0.011 1.2 16.0 0.30 0.0025 0.00125

2-8 2.8 0.211 0.021 1.0 14.5 0.30 0.0010 0.00050

8-10 2.4 0.147 0.015 1.2 15.0 0.25 0.0005 0.00025

10-15 1.8 0.107 0.011 1.2 16.0 0.25 0.0003 0.00015

^^\

\

H - -T t

/

. .-T H H

/

f ff ff ft !

7\

\ / \ yi\\

1 y

B-n

/ \

yi \ \

- \ I ft ft tt ft ft tt tt tt tt

\

- \ t «• 1

z

__.

\

\

~7\/\ / l__--r-_A_/

• ' - \ -

A

-

—U

Hj

Jl — 4

t T T T T T T T T T T T T r T

Figure 7.1 Finite element mesh (consists of 15-node elements) used in this analysis

204


7.2.1 Effect of Embankment Slope on Foundation Failure

The effect of embankment slope on foundation failure has been studied through plane

strain finite element analysis using the mesh shown in Figure 7.1. Three different slopes

(1:1, 2:1 and 3:1) are considered in the analysis and the influence of embankment

loading is simulated by a continuous loading of 0.1m per week until failure. Failure is

identified when the solution fails to converge and the displacement increases

continuously without further addition of load. It was noticed that excess pore pressure

under the embankment would not provide a clear indication of failure.

7.2.1.1 Surface Settlements and Displacement

The predicted centreline settlement and heave at the toe with embankment height and

the displacement contours when the fill height is 1.5 m are illustrated in Figures 7.2,

7.3, and 7.4, respectively. Figures 7.2 and 7.3 show that settlement increases at a low

rate as the embankment height increases (up to about 1.75 m), after which there was a

significant and sudden change in the settlement rate is observed that was close to

failure. This clearly indicates that as expected a decrease in the embankment slope

would contribute to a greater height at failure and in this analysis the embankment

height at failure increases from 1.85 m to 2.25 m (increased by 21.6%), when the slope

decreases from 1:1 to 3:1.

The displacement contours are shown in Figure 7.4 when the embankment height is 1.5

m it is close to the failure embankment height. This indicates that when the

embankment is about to reach failure height, maximum displacement takes place near

the embankment toe. Once again, it proves that a decrease in embankment slope would

205


contribute to a greater embankment height at failure, as well as decreasing

displacement. In this analysis when the slope decreases from 1:1 to 3:1 the maximum

displacement decreases from 784.50 mm to 609.72 mm (a reduction of 22.3%).

0-

-200-

3 -400-

g 3 -600H ty_

-800-

-1000

0.0

Slope =1:1

Slope = 2:1

Slope = 3:1

0.5 1.0 1.5 Embankment Height (m)

2.0 2.5

Figure 7.2 Surface settlement at the embankment centre with fill height

1000-

800-

600-

8 400H X

200-

Slope=l:l

Slope = 2:1

Slope = 3:1

0.5 1.0 Embankment Height (m)

2.0

Figure 7.3 Surface heave at embankment toe with fill height

206


0

(a) Slope =1:1 (Maximum displacement = 784.50 mm)

0 80 160 240 320 400 480 560 640 720 800 Total displacement (mm)

(b) Slope =2:1 (Maximum displacement = 691.92 mm)


(c) Slope =3:1 (Maximum displacement = 609.72 mm)

61 122 183 244 305 366 427 488 549 610

Total displacement (mm)

Figure 7.4 Displacement contours for different slopes

207


7.2.1.2 Lateral Displacement

The lateral displacement contours for different embankment slope are shown in Figure

7.5 when the fill height is 1.5 m. This illuminates that the effect of embankment slopes

are more influential on lateral displacement than settlement. When the embankment

slope decreases from 1:1 to 3:1 the maximum lateral displacement decreases from

628.23 mm to 331.42 mm (a reduction of 47.3%), while settlement is only reduced by

22.3%.

(a) Slope =1:1 (Maximum lateral displacement = 628.23 m m )

0 64 128 192 256 320 384 448 512 576 640 Lateral displacement (mm)

(b) Slope =2:1 (Maximum lateral displacement = 432.27 mm)

• _ _ _ _ _ — . ^ ^ — n i i i


(c) Slope =3:1 (Maximum lateral displacement = 331.42 mm)


Figure 7.5 Lateral displacement contours for different slopes

208



Figure 7.6 shows the excess pore water pressure contours when the fill height is 1.5m,

which indicates there is no significant different in excess pore pressure with changes in

the embankment slope. As a result the excess pore pressure measurement cannot be

used as a clear indication of failure height in comparison to settlement or lateral

displacements.

(a) Slope =1:1 (Maximum excess pore water pressure = 28.42 kPa)

0 10 15 20 25 Excess pore water pressure (kPa)

(b) Slope =2:1 (Maximum excess pore water pressure = 28.31 kPa)

0 10 15 20 Excess pore water pressure (kPa)

(c) Slope =3:1 (Maximum excess pore water pressure = 28.18 kPa)

0 5 10 15 20 25 Excess pore water pressure (kPa)

Figure 7.6 Excess pore pressure contours for different slopes

209

30

Chapter 7 Numerical Modelling and Design Tmpli^tinn.

7.2.2 Effect of Loading Rate on Foundation Failure

Loading rates on foundation failure has been studied using the finite element mesh

shown in Figure 7.1. Embankment construction is simulated with three different loading

rates (0.1, 0.2 and 0.25 meters per week), with an embankment side slope of 3:1.

Surface settlement at the embankment centreline and toe are depicted in Figures 7.7 and

7.8, respectively, which shows how failure height of embankment is influenced more by

the loading rate than the embankment slope. As expected, the slower construction rate

permits a greater embankment height at failure because this gradual rate allows more

dissipation of pore water pressure. When the loading rate increases from 0.1 to 0.25

meters per week the failure height of embankment decreases from 2.25 to 1.3 m.

0-

6

I &_

-200-

-400-

-600-

-800-

•1000-

0.0

0.1 m/week

0.2 m/week

0.25 m/week

0.5 1.0 1.5 Embankment Height (m)

2.0 2.5

Figure 7.7 Surface settlement at embankment centreline with fill height

210

. Chapter 7 Numerical Modelling and Design Implications

1000—i

800-

1 600-

200-

I i - | i 1 1 j

0.0 0.5 1.0 1.5 2.0 Embankment Height (m)

Figure 7.8 Displacement (heave) at embankment toe with fill height

Figure 7.9 shows the contour plot of total displacement, lateral displacement, and

excess pore water pressure for a loading rate of 0.2 meters per week when the fill height

is 1.5 m. When the construction rate is 0.2 meters per week the maximum total and

lateral displacements are 1020 and 984.84 mm (Figure 7.9a and 7.9b), whereas these

values are reduced to 609.72 and 331.42 mm (Figure 7.4c and 7.5c) when the loading

rate is reduced to 0.1 meters per week. In other words, when the construction rate

increases from 0.1 to 0.2 meters per week the total and lateral displacements increase by

67.29% and 197.16%, respectively. This result also confirms that, not only the rate of

construction influences the failure height of embankment, but also the failure could be

monitored on the basis of lateral displacement which can be measured by an

inclinometer.

When the embankment height is 1.5 m the maximum excess pore water pressure

for the construction loading of 0.1 and 0. 2 meters per week are 28.18 and 35.15 kPa,

211


in a respectively (Figure 7.6c and 7.9c). As expected, the small construction rate results i

smaller excess pore water pressure because the slower rate of construction allows the

soft clay to dissipate the pore pressure more effectively.

(a) Total displacement contours (Maximum displacement = 1020 m m )


(b) Lateral displacement contours (Maximum lateral displacement = 984.84 mm)


(c) Excess pore pressure contours (Maximum value = 35.15 kPa)

7.5 15 22.5 30 Excess pore water pressure (kPa)

37.5

Figure 7.9 Contour plots for a loading rate of 0.2 m/weck (the height and the slope of

the embankment are 1.5m and 3:1, respectively)

212


7.3 Design of Embankment Constructed on Soft Clay with Vertical Drains

As discussed in the previous section embankments can only be built up to 1.5m to 3m

high on very soft clays without any improvement to subsoil layers. However, higher

embankments are often needed and their rapid construction is often required given the

usual stringent deadlines. To achieve these goals, special construction measures such as

light-weight embankment fill, the provision of geogrid reinforcement at the foundation,

suitable ground improvement techniques, and staged construction may be considered. It

is often advantageous to install prefabricated vertical drains in the soft clay foundation

to decrease the length of drainage path and thereby speed up consolidation, as described

in details in Chapter 2.

The effect of vertical drains on embankment stability has been investigated via the

finite element code PLAXIS incorporating the 2-D plane strain solution proposed

earlier. An embankment slope of 3:1 is simulated at a construction rate of 0.2 meters per

week on a soft clay foundation improved with prefabricated vertical drains installed in a

square pattern. Since the dimensions of the prefabricated vertical drains (PVDs)

commonly in use today are quite small and the discharge capacities are big enough to

neglect well resistance, the PVDs are modelled using zero thickness drain elements to

avoid an unacceptable aspect ratio. In this section, five different analyses have been

carried out to examine:

i. Effect of drain spacing (assuming perfect drain condition)

ii. Effect of the extent of smear zone (assuming constant kh / kh ratio)

iii. Effect of smear zone permeability for a given smear zone area

213

— . Chapter 7 Numerical Modelling and Design Implications

iv. Effect of stage loading, and

v. Effect of surface crust

7.3.1 Effect of Drain Spacing

The effect of drain spacing on embankment stability is studied by assuming two

different drain spacing. The converted equivalent plane strain permeability (/chp) of the

undisturbed zone is given in Table 7.2 as a function of drain spacing. The predicted

displacements are compared with the 'no drain condition'. Surface settlement at the

embankment centreline and the displacement at the embankment toe are shown in

Figures 7.10 and 7.11, respectively. Figure 7.10 shows how the installation of drains

significantly increases the settlement rate while displacement at the toe (Figure 7.11),

represents the effect of spacing on the potential failure height. Figure 7.11 demonstrates

that the failure height increases from 1.3 m (no drain) to more than 5 m when vertical

drains are installed at 1 m intervals.

Table 7.2 Equivalent permeability

Depth

0-2

2-8

8-10

10-15

214

faP (m/day)

5=1 m S=2 m

0.000670 0.000539

0.000268 0.000216

0.000134 0.000106

0.000080 0.000065

Chapter 7 Numerical Modelling and Design Impl'

§ OJ

CO

-200-

-400-

-600-

-800-

•1000-

0.0

No drains

Drains at 2.0 m spacing


- | i | i | i | i

0.5 1.0 1.5 2.0 Embankment Height (m)

Figure 7.10 Surface settlement at embankment centreline for different drain spacing

1000

800-

600-

_

i_L 400-

200-No drains



r 2 3

Embankment Height (m)

Figure 7.11 Displacement at embankment toe for different drain spacing

215


The lateral displacement contours for different drain spacing are plotted in Figure

7.12 when the embankment height is 1.5m. The maximum lateral displacement for 'no

drain' condition and for drains installed at 2 m and 1 m intervals, are 984.84, 417.94

and 270.15 mm, respectively. This result proves that the installation of vertical drains

curtail the lateral displacement because the PVD's decrease the lateral yield due to a

rapid dissipation of pore water pressure.

(a) Drain at 2 m intervals (Maximum lateral displacement = 417.94 m m )


(b) Drain at lm intervals (Maximum lateral displacement = 270.15 mm)


Figure 7.12 Lateral displacement contours for different drain spacing

7.3.2 Effect of the Extent of Smear Zone

In the field, the vertical drains are installed using a steel mandrel. This process causes

significant remoulding of the subsoil, especially in the immediate vicinity of the

216

Chapter 7 Numerical Modelling a nd Desitm [mplir--,..-.-,.

mandrel. The resulting smear zone will have a reduced lateral permeability, which

affects the pore pressure dissipation rate. In this section the effect of the extent of the

smear zone is studied by assuming a constant permeability ratio (kjk'h =2), and two

different areas of smear zone (rs/rw= 4 and 6). Tlie converted plane strain permeabilities

are given in Table 7.3 as a function of the extent of the smear zone for a drain spacing

of 2m. The predicted settlement and displacements are compared with those of 'perfect

drain' condition. Figures 7.13 and 7.14 illustrate surface settlement at the embankment

centreline and displacement at the toe, respectively.

Figure 7.13 shows that the degree of consolidation is affected by the extent of the

smear zone (i.e., the degree of consolidation decreases as the extent of the smear zone

increases). Displacement at the embankment toe for two different extent of the smear

zone is plotted in Figure 7.14. This indicates that the displacement increases with the

extent of the smear zone, thereby decreasing in the failure height of embankment. For

example, when the embankment height is 2m the displacement increases from 541.41 to

778.97 mm (an increased of about 44%), while the extent of smear zone ratio (rs/rw)

varies from 1 to 6.


Depth

0-2

2-8

8-10

10-15

fav (m/day)

0.000539

0.000216

0.000106

0.000065

khp

rs/rw=4

0.000168

0.000067

0.000034

0.000020

(m/day)

rs/rw=6

0.000192

0.000077

0.000038

0.000023

217


-0

v 0 B «_•

IB V_

-1.

0-,

4-

8-

2-

f. u

^ ^ 5 r > -\_-»

T. r i •

Perfect drain With smear (rs=4rw)

With smear (rs=6rw)

1 , | i |

'V

\ ^ --V

1 ' 0.0 0.5 1.0 1.5

Fill Height (m) 2.0 2.5

Figure 7.13 Surface settlement at embankment centreline for two different smear zones

1.6-

1.2-

8 CD O

'E, Ui

0.8-

0.4-

0-| T

0.0

Perfect drain

With smear (rs=4rw)

With smear (rs=6rw)

0.5 1.0 1.5 Fill Height (m)

2.5

Figure 7.14 Displacement at embankment toe for two different smear zone

218


7.3.3 Effect of Smear Zone Permeability

In the previous section the effect of the extent of smear zone was studied and it was

concluded that the extent of the smear zone would play a major role in the prediction of

embankment behaviour. Here, an attempt is made to study the role of smear zone

permeability on embankment behaviour by assuming a constant area of smear zone

(rs/rw is taken as 6). The converted plane strain permeabilities are shown in Table 7.4 as

a function of the axisymmetric permeability ratios.

Figures 7.15 and 7.16 present the predicted surface settlement at the centreline

and displacement at the embankment toe for four different permeability ratios where as

expected a higher settlement is shown by the lower permeability ratio (Figure 7.15).

Also, Figure 7.15 shows there is no sudden change in the settlement rate when the

permeability ratio is small (2 or 3), but when it's high (5-10), the settlement rate

suddenly changed when the embankment height was about 2.5m high. This is a clear

indication that failure height is influenced by the permeability of the smear zone.


Depth

0-2

2-8

8-10

10-15

fap (m/day) -

0.000539

0.000216

0.000106

0.000065

hP

kjkh=3

0.000117

0.000047

0.000023

0.000014

(m/day), when

kjkh=5

0.000066

0.000016

0.000013

0.000008

kjkh=\0

0.000031

0.000012

0.000006

0.000004

219


0.0-r

1 2 Fill height (m)

Figure 7.15 Surface settlement at embankment centreline for different permeability

ratios

0 0.5 1 1.5 Fill height (m)

Figure 7.16 Displacement at embankment toe for different permeability ratios

220


Figure 7.16 shows the displacements at embankment toe for different permeability

ratios. It is observed from Figure 7.16 that irrespective of permeability ratios, the rate of

displacement is almost the same at small embankment heights (less than 1 m). This

analysis reaffirms that the performance of an embankment is not only influenced by the

extent of the smear zone, but also by the accurate evaluation of the smear zone

permeability.

7.3.4 Effect of Stage Loading

In those situations discussed above the embankment is likely to fail at smaller heights

because the strength of the foundation soil is not sufficient to withstand higher loads. In

this situation potential failure can be avoided by halting construction and allowing a

sufficiently period of time to dissipate the pore pressure. The corresponding increase in

shear strength contributes to an increased height of embankment during the next loading

stage.

To evaluate how staged construction affects failure height of embankment a plane

strain analysis was conducted for a perfect drain (the spacing is assumed to be 2m) with

a loading condition as shown in Figure 7.17. The predicted surface settlement at

embankment centreline with time and fill height is plotted in Figure 7.18. This clearly

shows that the settlement rate is almost the same for all loading stages but in the case of

continuous loading the displacement rate suddenly increased when the fill height

reached the critical fill height of about 2.5 m.

221

.Chapter 7 Numerical Modelling and Design Implications

5-1

4-

1 " _3 _-* 2 ^ E

1-

0-

Loading rate = 0.2 m/week

" Stage loading

Continuous loading

0 100 Time (days)

200 H 300

Figure 7.17 Construction loading history

0.0-1

40 80 120

Time (day)

(a)

160 0.0 1.0 2.0

Fill Height (m)

(b)

3.0

Figure 7.18 Surface settlement at embankment centreline with (a) time, (b) fill height

When the embankment height is 1.6 m the total and lateral displacements

illustrated in Figures 7.19 and 7.20, respectively. This shows that a 25 days resting

period after lm loading decreases the total and lateral displacements by about 15% and

43%o, respectively. This is an indication that the resting period is beneficial to increases

the maximum (failure) height of embankment.

222


(a) Displacement contours-for continuous loading (Maximum value = 909.08 mm)


(b) Displacement contours-for stage loading (Maximum value = 769.20 mm)


Figure 7.19 Displacement contours when the fill height is 1.6 m

(a) Lateral displacement contours-continuous loading (Max. value = 778.81 mm)


(b) Lateral displacement contours-stage loading (Maximum value = 443.66 mm)

0 45 90 135 180 225 270 315 360 405 450

Lateral displacement (mm)

Figure 7.20 Lateral displacement contours when the fill height is 1.6 m

223


7.3.5 Effect of Surface Crust

To demonstrate the effect of the surface crust the above example has been re-analysed

by assuming that the top 2 m of the surface as weathered compacted clay. Because the

surface crust has a high overconsolidation ratio it can be modelled using the Mohr-

Coulomb model with the following elastic properties: Young's modulus, __.=25 MN/m

and Poisson's ratio, v = 0.3 .

The variation of centreline displacement with embankment height is plotted in

Figure 7.21, which shows that the surface crust plays the role of increasing the potential

embankment height by decreasing settlement. For example, when the embankment

height is 2m the centreline settlement is decreased from 86.01 cm to 46.74 cm (a

reduction of about 46%) due to crust.

0-i

-20-

1 T o

r -40-§ M -60-<D

00 -80-

-100—

~^\\

\

\. \with crust

\ without crust\

i I i | i | i | <

B O

!_ <U> fi <D

a <u oo

-20--

-40-

-60-

-80-

-100—

^\""*--. \^ !-.. with crust \ * \

^*s^

^ v ;

\^ 1 ^v Ns N. \^ \.

without crust *

%-

\ 1 1 • '

0 40 80 120 160 200 Time (days)

0

(a)

1 2 Fill height (m)

(b)

Figure 7.21 Effect of surface crust

Based on the analyses illustrated in this Chapter, it is interesting to note that

contrary to expectation, there is a discrepancy in the critical locations. Figure 7.19, for

example, shows the occurrence of maximum settlement beneath the side slope of the

embankment rather than at the centreline, and Figure 7.20 illustrates the occurrence of

224


maximum lateral displacement at a section through the middle of the side slope rather

than at the toe. The increment of vertical displacement, lateral displacement, and shear

strain are plotted in Figure 7.22 when the height is increased from 1.8 to 2.0 m.

(a) Vertical incremental displacements (maximum= -47.74 m m )

_^^^H

-48 -34 -21 -7 6

Vertical displacement increment (mm) 20

(b) Horizontal incremental displacements (maximum= 35.28 m m )

7.2 14.4 21.6 28.8 Horizontal displacement increment (mm)

36.0

0.0

(c) Shear strain increments (maximum= 1.33%)

^ ^ ^ 1 _1 ^^^^

0.28 0.56 0.84 Shear strain increment (%)

1.12 .40

Figure 7.22 Increment contours when the embankment height increases from 1.8 to 2.0

m (a) vertical displacement, (b) lateral displacement, and (c) shear strain

225

— — — • Chapter 7 Numerical Modelling and Design Implications

Figures 7.22a and 7.22b also indicate a discrepancy in the critical locations and as

expected, the lateral displacement and shear strain contours show a very similar pattern

(Figure 7.22b and 7.22c). The above variations could be the result of shear-induced

displacements (Zhang, 1999). In the following section an attempt is made to explain the

critical location using induced stresses.

7.3.6 Identification of the Critical Location

Generally, it is observed that the maximum settlement and maximum lateral

displacements occur at, or close to the embankment centerline and at the toe of the

embankment, respectively. This is true for most soft soil foundations under highway

embankments (narrow) but this may differ for soil foundations beneath wide

embankments such as airport terminals. The settlement profiles (Figure 7.23), beneath

Muar test embankments in Malaysia, showed that the location of maximum settlement

occurred slightly away from the centerline and gradually moved towards the centerline

as the fill height increased (Indraratna et al., 1992). That is, the locations where the

maximum settlements occurred were 11.0m, 10.5m, 9.0m, and 5.5m corresponding to

the embankment height of 2m, 3m, 4m and 5m. This discrepancy could be the result of

shear-induced displacements (Zhang, 1999).

Induced Normal and Shear Stress Distributions

The stress distributions beneath an embankment are closely related to the settlement

pattern. To facilitate setting up a 'stress criterion' for classifying the foundation

deformation patterns the embankment shown in Figure 7.24 is considered.

226


a.

<D

>

o c_

o

'-5 >

Ground Level

Measured surface settlement

• Fill height,h=2 m



x Fill height,h=5 m

10 20 30 Distnace form centreline (m)

40

Figure 7.23 Measured settlement of Muar test embankment, Malaysia (after Indraratna

et al., 1992)

• x

Figure 7.24 Typical embankment

227


The induced vertical stress (Arjv) and shear stress (Axxz) due to embankment loading at

point A (x, _.) can be derived based on the theory of elasticity, as follows (modified after

Gray, 1936):

Aav = [a(ax + a2 + Pi +p2)+b(al +a2)+ x(a} -a2)] na

(7.1)

A ynz( .

Ka (7.2)

where, y: unit weight of embankment fill; h: height of embankment; and the relevant

angles are:

c.j = tan -l az z1 +{a + b + xfb + x)_

(7.3a)

a2 = tan -l az

z2 +(a + b-x\b-x) (7.3b)

Px = tan * bz

z +x (b + x) (7.3c)

and, Pi = tan -l bz

z -x (b-x\

(7.4d)

Figure 7.25 illustrates the variation of the increment of vertical total stress and

shear stress with horizontal distance at a depth of 6 m for a number of embankments.

This clearly indicates that shear stress is negligible in the central area but is more

significant than the vertical stress under the side slope of the embankment, i.e., shear-

induced displacement will be greater beneath the side slope.

228


>

< S-c

o

&

<D

IH

O

a <u O _> 13

cs

a=2,b=18,h=2, z=6

-- - - a=4, b=16, h=4, z=6

a=6,b=14,h=3,z=6

a=8,b=12,h=4,z=6

N X

< 20-

— a=2,b=18,h=2,z=6

a=4,b=16,h=4,z=6

— a=6, b=14, h=3, z=6

- a=8,b=12,h=4,z=6

/ \

\

1 I "T

10 15 Distance from centreline (m)

Figure 7.25 Variation of induced vertical stress and shear stress

Identification of Location of Maximum Shear Stress

Figure 7.26 shows the variation of shear stress with horizontal distance for two

embankments at different depths. These plots indicate that the largest shear stress in the

foundation induced by embankment loads occurs beneath the middle of the side slopes.

At these locations the distribution of shear stress with normalized depth are shown in

229


Figure 7.27 for eight embankments, which certifies that the maximum shear stress

occurs at a normalized depth of around 1.4-2.0, with an average of 1.7.

0 4 8 Distance from centreline, x (m)

Figure 7.26 Variation of shear stress with horizontal direction at different depths

230


Max. Shear stress (Average, z/h=1.7)

w=20m|

Critical section- 1 Solid lines for 1:1 slope Dot lines for 2:1 slope

10 20 Shear stress, xxz (kPa)

30

Figure 7.27 Shear stress distribution with normalized depth under embankments

Since shear-induced deformation is closely related to the shear stress distribution

(Zhang, 1999) and clarified based on Figures 7.25-27, it is reasonable to argue that the

critical locations under embankments occur at a vertical section through the middle of

the side slope, and a horizontal section at a depth of 1.7 times the fill height. This is in

good agreement with the numerical predictions (for example, see Figures 7.5, 7.9b, and

7.12). Variation of incremental lateral displacement at different cross section is shown

in Figure 7.28 when the embankment height increases from 1.8 to 2.0 m. This also

shows that the maximum lateral displacement occurred in the middle of side slope, 17m

from the centreline.

231


o-

Q

3-

6-

9-

12-

15-

Lateral displacement increment (mm)

10 20 30

^ ' •—-t: ' 40

:-$>

14 m from centreline


17 m from centreline (middle of side slope)


20 m from centreline (embankment toe)

Figure 7.28 Variation of lateral displacement at different cross section

The distance from the embankment centreline to the point where maximum settlement

occurs (L\) can be defined as (Zhang, 1999):

*-rHl (7.4)

where, L is the distance from the embankment centreline within which the settlement

can be considered as 1-D (i.e., the region in which the shear stress is negligible).

232


7.4 Summary

The stability of a typical embankment constructed on soft clay was analysed using a

number of plane strain finite element models. The factors influencing the failure of

embankment were investigated using the following analyses:

> Effect of embankment slope on failure height;

> Rate of construction on failure height;

> Influence of vertical drain spacing on failure height;

> The extent and permeability of smear zone on failure height;

> The use of multi-stage loading; and

> Effect of surface crust.

It was found that a steeper slope and greater construction rates reduce the failure

height. For very soft clays the critical height is about 2 m unless the foundation is

stabilized by vertical drains or other means but higher embankments are often needed

and moreover, their rapid construction is pertinent given the usual stringent deadlines.

In these instances use of vertical drains in soft clay is an economical and effective way

to increase failure height by accelerating consolidation. It is also found that failure

height is more sensitive on extent of smear zone and its permeability. Therefore, it is

important to evaluate those parameters in an accurate way such as presented in this

Thesis.

The analysis with multi-staged construction shows those intermediate rest periods

allow embankments to be raised to considerable heights by allowing pore pressure

dissipation prior to subsequent loading. Numerical predictions with the surface crust

233


show that it increases the embankment height by decreasing settlement. Finally, the

critical locations beneath the embankment have been explained using induced shear

stress. This analysis shows that the critical locations under the embankments occur at a

vertical section through the middle of the side slope, and at horizontal section at a depth

of about 1.7 times the fill height, for the range of properties considered in this study.

234

— Chapter 8 Conclusions and Recommendations

8 CONCLUSIONS AND RECOMMENDATIONS

8.1 General

Although significant progress has been made in the past through analytical and

numerical modelling (Hird et al., 1992; Chai et al., 1995; Indraratna and Redana, 1997

among others), the available literature indicates that there has always been a

discrepancy between the observed and predicted behaviour of embankments stabilized

with vertical drains. This discrepancy is usually attributed to numerous factors such as

the uncertainty of soil properties, estimating the effect of smear, the inability of existing

theories to model consolidation, and improper conversion of the axisymmetric condition

to the plane strain (2-D) analysis of vertical drains. Some of these uncertainties and

difficulties are overcome through this doctoral research by studying the effect of smear

in more detail as well as the proper conversion of the axisymmetric to plane strain with

greater rigour.

The accuracy of the settlement estimation relies on the correct assessment of the

preconsolidation pressure of the clay, as described in detail by Casagrande (1936).

Generally, the values of preconsolidation pressure show considerable scatter when

plotted with depth. In multi-stage construction, it is important to note that the first stage

of loading might not exceed the natural preconsolidation pressure of the foundation soil.

Therefore, appropriate values for compression index (Cr and Cc or K and X) should be

selected which may represent a state of over-consolidation. The overall conclusions are

presented below in addition to the concluding remarks made earlier at the end of

individual chapters.

235

Chapter 8 Conclusions and Recommendations

8.2 Specific Observations

In this research a new method for evaluating the extent of the smear zone using cavity

expansion theory (CET) incorporating the modified C a m clay ( M C C ) theory is

proposed and verified via large-scale consolidometer tests. In addition, an equivalent 2-

D plane strain analysis of embankments stabilized with vertical drains, subjected to

vacuum pressure has been carried out incorporating the smear effects. Well resistance

has been ignored in this analysis because previous studies have shown that the effect of

well resistance could be ignored for most modern prefabricated vertical drains which do

not deform (form kinks) during installation.

The proposed solutions were incorporated into the finite element code PLAXIS

and A B A Q U S to study the behaviour of a few selected embankments stabilized with

vertical drains. The predictions based on the proposed model were compared with the

available field data and an acceptable agreement was found. Finally, the proposed

model was used to predict the failure height of embankments under various conditions.

Specific conclusions, which can be drawn based on this study, are summarised below in

details.

8.2.1 Mathematical Formulations and Modifications to the Existing Theories

1. Previous studies indicated that the effect of smear is a significant and unavoidable

problem caused by mandrel driven prefabricated vertical drains (PVD). The size and

permeability of the smear zone is often difficult to quantify and determine from

laboratory tests and thus far, there is no comprehensive or standard method for

measuring them. In this research, an analytical solution based on C E T was

proposed to estimate the extent of the smear zone and the predicted results were

236

— • — Chapter 8 Conclusions and Recommendations

verified via large-scale consolidometer testing. The extent of smear zone evaluated

from the large-scale laboratory studies (described in Section 8.22) is close to the

value that calculated using the proposed CET solution. This results show that the

CET solution could be used to estimate the extent of the smear zone.

2. Existing theories based on the Darcian linear flow law of a unit cell (Hansbo, 1981,

Indraratna and Redana, 1997) were modified to include vacuum pressure

application. This combined vacuum and surcharge preloading technique follows the

principle of superposition, using the same equation for analysing the degree of

consolidation. However, the combination of vacuum and surcharge pressure further

accelerates the rate of pore pressure dissipation due to an increased hydraulic

gradient generated towards the PVD.

3. New plane strain lateral consolidation equations based on exponential flow were

formulated which are valid for both Darcian and non-Darcian flow. A matching

procedure for the proposed solution was explained, and the new parameters

introduced in this study were graphically presented for the convenience of practical

application.

8.2.2 Laboratory Program

1. Laboratory tests utilizing a large-scale consolidometer revealed that the installation

of vertical drains, significantly increased pore water pressure (during mandrel

installation) in the immediate vicinity of the mandrel, and reduced the horizontal

permeability and water content surrounding the drain. This induced pore pressure

was compared with the predicted value and a good agreement was found. The extent

of the smear zone based on the induced pore water pressure was about 2.4-2.6 times

237

— Chapter 8 Conclusions and Recommendations

the equivalent radius of the mandrel. This is comparable with previous research

reported in the literature. For example, Hansbo (1979), mdraratna and Redana

(1998) Chai and Miura (1999) proposed that the extent of smear zone is about 1.5 -

3, 2 ~ 3, and 2-3 times the mandrel radius, respectively, which is close to the

extent of smear zone found in this studies.

2. Permeability measurements indicated that the horizontal permeability (fa)

significantly decreases within the smear zone, whereas the variation in vertical

permeability (fa) with radial distance is generally negligible. The extent of the smear

zone was estimated based on the permeability anisotropy (fa/fa ratio) and

normalised permeability (fa/fau ratio). These results verified that smear zone was

about 5 times the equivalent drain radius while horizontal permeability (in the smear

zone) was smaller by 1.0943 to 1.6437 (an average of 1.3429) times that of the

undisturbed zone. Moreover, the fa/fa ratio converged towards unity close to the

drain where the soil suffers from maximum disturbance (smear).

3. Variation of water content demonstrated a similar trend to the variations of

horizontal permeability. The normalized water content reduction, (wmax-w)/wmax was

found to be confined within a narrow band for all surcharge pressures, and based on

this result, the smear zone was estimated to be around 5 times the equivalent drain

radius. An empirical expression between the reduction of water content and the

change in horizontal permeability within the smear zone was found. This empirical

expression is useful in a practical sense for estimating the lateral smear zone

permeability by measuring the water content.

4. The excess pore water pressure and the surface settlement of the soil observed in the

large-scale consolidometer were compared with the predicted values. Given the

238

. Chapter 8 Conclusions and Recommendations

good agreement, one can conclude that the proposed analytical solutions can be used

confidently to study the behaviour of soft clay foundations improved with

prefabricated vertical drains.

8.2.3 Case History Analysis

1. The competency of the CET solutions was certified by an acceptable agreement

between the available field data and numerical predictions of the few full-scale

embankments reported in this study. The effect of smear associated with PVD

installation was considered, while the effect of well resistance was neglected for

reasons explained earlier. Based on the proposed CET solution, the extent of the

smear zone was evaluated to be about 4-6 times the equivalent radius of the drain,

and also the smear zone size was found to vary with depth. There is no doubt that

the accurate prediction of the extent of the smear zone is also dependent upon the

correct assessment of the critical state soil parameters.

2. Multi-drain analysis of vertical drains installed at the Sunshine Motorway presented

a very good match between the predicted and measured settlements, whereas an

acceptable agreement was found between the predicted and measured excess pore

pressures. Even though lateral displacements were harder to match, still a reasonable

comparison was found below a depth of 3m. In addition, the following conclusions

can be drawn from this case study:

(i) A closer spacing of drains invariably contributes towards an increased

smear effect because the total area of the smear zone is higher under a closer

239


spacing system (note that the area of the smear zone is proportional to

number of drains).

(ii) Vertical drains increase the lateral movement under the middle of the main

batter but decrease it at the toe.

(m) Maximum lateral movement occurred at a depth approximately 1.7 times the

fill height.

(iv) Near the surface, the lateral displacements were harder to match due to the

existence of the compacted crust, which has a significant benefit on the

stability of the embankment. Further accuracy of the writer's numerical

predictions may be achieved by using improved procedures for accurately

modelling of the properties of the over-consolidated (compacted) crust that

does not obey the modified Cam-clay theory or the simplified Mohr-

Coulomb theory.

3. Selected embankments stabilised with vertical drains subjected to vacuum

preloading were investigated using a plane strain finite element analysis. Predictions

were made based on two different models: (1) assuming a constant extent of smear,

and (2) varying the extent of the smear zone, with depth. It was demonstrated that

the predicted values based on Model 2 are in good agreement with the field data. In

addition, the following conclusions can be drawn from this case history analysis:

(i) The extent of the smear zone variation with the inclusion of time and depth

dependent vacuum pressure distribution significantly improves the accuracy

of predictions.

240

— — Chapter 8 Conclusions and Recommendations

(ii) The drainage efficiency of the perforated pipe is better than the hypernet.

(iii) The application of vacuum generates 'inward' lateral displacement rather

than 'outward', and substantially decreased the lateral displacement, thereby

minimizing the risk of shear failure.

8.2.4 Application of Finite Element Modelling for General Design

1. The height of an embankment, which can be raised on soft clay without any stability

problems, depends on the embankment slope, construction rate, vertical drain

spacing, smear zone parameters, subsoil properties, and the method of construction

(continuous loading or multi-stage loading).

2. A steeper slope and faster construction rate would reduce failure height, but the

critical height of embankment can be increased by installing prefabricated vertical

drains (PVD). Introduction of PVD for subsurface drainage can provide increased

stiffness of the soft clay and curtail lateral displacement substantially, thereby

minimizing the risk of shear failure. The drain spacing and propagation of smear

effect will influence the critical height of embankment. Even though the ultimate

settlement of the soil is not changed by the pattern of PVD, their spacing and length

can still influence the construction rate and final height of embankment.

3. The drain installation method should be carefully implemented to reduce the smear

effect as much as possible. The foundation response upon loading is more sensitive

to nominal changes in the permeability of the smear zone than its size.

241


4. Multi-staged construction showed that intermediate rest periods permit

embankments to be raised to greater final heights by allowing pore pressure

dissipation prior to subsequent loading.

5. The presence of surface crust (over-consolidated) beneath an embankment can resist

lateral displacement, thereby significantly increasing the failure height of the

embankment. The presence of a surface crust has a similar influence as placing a

geogrid reinforcement underneath the embankment.

6. Stress distribution beneath the embankment showed that the role of shear stress is

negligible in the central area but more significant under the side slope of the

embankment. As a result of this shear-induced displacement, the location of

maximum settlement is found to occur slightly away from the centerline, but

gradually moved towards the centerline as the fill height was increased. A similar

trend was also observed beneath the Muar test embankments in Malaysia (Indraratna

etal., 1992).

7. The critical locations beneath an embankment have also been explained on the basis

of induced shear stress. This analysis demonstrates that the critical locations under

the embankments occur in a vertical section through the middle of the side slope,

and at a horizontal section at a depth of about 1.7 times the fill height.

8.3 Suggestions for Future Research

Based on the current study, further analytical, numerical, and experimental studies

associated with embankments stabilised with vertical drains are recommended. Such

future work should focus more on the following aspects:

242

. Chapter 8 Conclusions and Recommendations

1. Available finite element programmes based on coupled consolidation theory (Biot

theory) can be modified to incorporate non-Darcian flow, or a new finite element

code may be written so that it can be used for both Darcian and non-Darcian flow.

2. Anisotropic soil properties have a significant effect on embankment behaviour

(Potts and Zdravkovic, 2000). To predict the embankment responses more

accurately, the use of complex constitutive models such as MIT-E3 (Whittle, 1991)

and Structured Cam Clay model (Liu et.al. 2002) incorporating the anisotropy

observed in the laboratory and field are recommended.

3. The drain-soil interface is often unsaturated, and therefore, the soil constitutive

models employed in this current study should be further extended to become more

realistic for unsaturated soils. For this purpose, a separate subroutine incorporating

moisture characteristic curves may be developed to directly link the laboratory data

and then used at the drain-soil interface.

4. It is important to evaluate the coefficient of permeability of the saturated clay as

accurately as possible. Often, the coefficient of permeability obtained via

conventional oedometer tests is not accurate, in contrast to the field permeability

data (when available). Therefore, if the field permeability values are not available, it

is recommended that a more realistic permeability coefficient should be obtained

through back analysis.

5. Controlled stress path testing of soft foundation soils is also preferred in order to

obtain the most accurate soil properties and to rationally quantify the role of the

applied stresses in relation to the pre-consolidation pressure of the foundation soils.

243


6. The numerical model should also incorporate the role of both axial and lateral

stiffness of vertical drains where warranted. Especially, in plane strain models

where a continuous vertical drain 'wall' is assumed, the increased stiffiiess of this

drain 'wall' may become excessively higher than the soft clay (of same wall

thickness), hence the need for modelling the correct stiffness should be considered.

7. In order to examine the accuracy of soft clay foundation responses, it is

recommended to carry out a parametric study on the assumed plane strain model.

Various parameters such as construction rate, embankment slope, and drain spacing

etc. should be taken into account, and finally, a numerical scheme to estimate

maximum fill height in relation to the PVD patterns may be developed.

8. Comprehensive design charts for a combined vertical and radial drainage system

with vacuum and surcharge pressure application can be developed based on non-

Darcian flow equations incorporating the smear effect.

9. The author used an Excel spreadsheet formulation to evaluate the extent of the

smear zone and equivalent plane strain permeabilities, however, it is advisable that a

separate subroutine is written, which can be directly linked to a selected finite

element code, such as PLAXIS or ABAQUS.

10. The accurate prediction of foundation response requires careful examination of soil

properties and the selection of an appropriate soil model, especially for the topmost

over consolidated (compacted) crust. The available soil models, such as the Mohr-

Coulomb and modified Cam-clay model are not appropriate for modelling the

behaviour of a weathered and compacted crust due to the significant differences in

244


soil fabric and anisotropy. Therefore, it is recommended that the available soil

models may be modified to replicate the effect of the crust.

11. In many classical theories (e.g. Barron, 1948; Hansbo, 1981), the influence of the

smear zone is considered with an idealized two-zone model, i.e., an undisturbed

zone with natural permeability and a smear zone with reduced permeability. In

addition, they assumed that the permeability within the smear zone is constant,

although reduced. However, laboratory measurements indicate that the coefficient of

permeability varies with radial distance within the smear zone. Therefore, it is

recommended that the available axisymmetric and plane strain solutions are

modified to incorporate this permeability variation within the smear zone. A linear

variation of fa/fa ratio may be assumed in the simplest case.

245

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261

Appendix 1 R-OCR Relationship

APPENDIX 1: RELATIONSHIP BETWEEN ISOTROPIC AND

CONVENTIONAL OVERCONSOLIDATION RATIO

To properly account for the effect of initial stress condition, one needs to differentiate

between the overconsolidation ratio defined in terms of either the vertical effective

stress or the mean effective stress. The isotropic overconsolidation ratio, R, is defined as

the ratio of maximum stress on yield locus pc0 and the in situ mean effective stress p0.

That is,

R=^f- (Al) Po

The conventional overconsolidation ratio, O C R , is defined in the one-dimensional

condition as the ratio of the vertical preconsolidation stress crvmax and the in situ

effective vertical stress cv0. I.e.,

0CR=^vnmX_ (A2) CTvO

The in situ mean effective stress is usually related with the in situ effective vertical

stress as:

p0=^{l + 2K0)or'v0 (A3)

where, K0 is coefficient of earth pressure at rest, can be approximated by (Mayne and

Kulhawy, 1982)

262

^ O ^ O n c O C R 8 ^ (A4)

I

where, : effective friction angle; Konc : the value of K0 for a normally consolidated

soil and can be estimated from the well known expression (Jaky, 1944):

Ônc =l-sin^ (A5)

When <7v0 reacheso-vmax, the maximum past mean effective stress p'm becomes

1, . , Pm -^V1 + 2A-0nc)_rvmax (A6)

And the corresponding deviator stress qm is

am = (l-^0nc)°"vmax = sin^ <* vmax (A7)

Substituting Eqns. (A6) and (A7) into the M C C equation, the p'c0 can be found as

PcO = 9(l-Konc)

2 + M2{l + 2Koncf

3M2{\ + 2Konc) vmax (A8)

where, M: slope of critical state line, can be related with effective friction angle as:

., 6sind . / 3 M M = or sin^ =

3-sin^ 6 + M (A9)

Substituting Eqns. (A3) and (A8) into Eqn. (Al) and rearranging with Eqns. (A2), (A4),

and (A9), the isotropic overconsolidation ratio can be found as:

263

Appendix 1 R-OCR Relationship

R = .(45-12M + M JOCR

(—1 (6-A/1 6 + M + 2{6-M)OCR^6+M)

(A10)

Figure Al shows the relationship between the isotropic overconsolidation ratio and the

conventional overconsolidation ratio for different M (slope of critical state line). It

shows that assuming R to be equal to OCR may produce errors up to a 20%, and there is

a need to differentiate R from OCR.

<<

_.

a o

"_J — i "3 00

C

o 0

.3 >

o o .—. OH

o o Ui

M 1.4-

1.0 1.4 1.8 2.2 2.6 Conventional overconsolidation ratio (OCR)

3.0

Figure Al Variation of the isotropic and conventional overconsolidation ratio with slope

of critical state line

264

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