Behaviour of Soft Soil Improved with Vertical Drain Accelerated … · 2015-12-10 · Behaviour of...

Behaviour of Soft Soil Improved with Vertical Drain

Accelerated Preloading Incorporating Visco-Plastic

Deformation

A thesis in fulfilment of the requirement for the award of the degree

Doctor of Philosophy

from

University of Technology, Sydney

by

Babak Azari, BSc Eng, MSc Eng

School of Civil and Environmental Engineering, Faculty of Engineering and Information Technology

May 2015

I

CERTIFICATION

I, Babak Azari, declare that this thesis, submitted in fulfilment of the requirements

for the award of Doctor of Philosophy, in the School of Civil and Environmental

Engineering, University of Technology, Sydney, is wholly my own work unless

otherwise referenced or acknowledged. The document has not been submitted for

qualification at any other academic institution.

Babak Azari

May 2015

II

I would like to dedicate my thesis to my beloved parents

III

ABSTRACT

Creep also known as time dependant viscous behaviour of soil is a

significant part of the soft soil settlement, which may cause substantial deformation

in the long-term. Post-construction settlement of soft soils can be significant

throughout the life time of the structure. Consequently, to minimise the post-

construction deformation and improve the bearing capacity and the shear strength of

the soft soil deposits, preloading combined with vertical drains is frequently used as

a ground improvement technique.

Soil disturbance induced by the installation of vertical drains results in

reducing the horizontal soil permeability and the shear strength in the disturbed zone.

Thus, the soil disturbance contributes to the reduced hydraulic conductivity and

overconsolidation ratio (OCR) of the soil in the vicinity of drains, influencing soil

deformation. Based on the available literature, there is a lack of understanding with

respect to the combined effects of the overconsolidation ratio and the hydraulic

conductivity profiles in disturbed zone and the nonlinear visco-plastic behaviour of

soft soils. These combined effects influence the creep parameters and the settlement

rate and accordingly deformation of soft soils improved using vertical drains assisted

preloading.

In this research, the elastic visco-plastic model has been incorporated in the

consolidation equation to investigate the effects of soil disturbance induced by the

installation of vertical drains on the long term performance of soft soil deposits. The

elastic visco-plastic model consists of a nonlinear creep function with a creep strain

limit. The applied elastic visco-plastic model is based on the framework of the

modified Cam-Clay model, capturing the soil creep during the excess pore water

pressure dissipation. Finite difference formulations for fully coupled one dimensional

axisymmetric consolidation have been adopted to model the time dependent

behaviour of the soft soil, combining both vertical and radial drainage. Crank-

Nicholson scheme is applied in formulating the finite difference procedure, since this

scheme uses two steps in partial differentials of pore water pressure over distance,

stabilising the process quicker.

IV

An array of laboratory tests were carried out using Oedometer and small and

large Rowe cells apparatus to verify the developed numerical code for the

axisymmetric solution. The Oedometer tests were conducted to choose the soil

mixtures for disturbed and intact zones. Two sets of small Rowe cell tests were

carried out on selected soil mixes to obtain the elastic visco-plastic model

parameters. A large Rowe cell was used to carry out the vertical drain assisted

consolidation tests by installing a vertical drain in the centre of the cell. To simulate

the disturbed zone for the area surrounding the vertical drain, a different mix with

reduced permeability was used. A compacted sand column covered with flexible

porous geotextile was installed in the centre to simulate the vertical drain. The cell is

fully instrumented and consists of a vertical displacement gauge at the surface level

and nine pore water pressure transducers on the sides and at the base of the cell.

Comparison of laboratory measurements and numerical predictions shows that the

proposed finite difference procedure incorporating the elastic visco-plastic soil

behaviour is appropriate for the consolidation analysis of preloading with vertical

drains.

Two case studies of vertical drains assisted preloading were numerically

simulated to investigate the effects of soil disturbance caused by the installation of

vertical drains. Different variations of the overconsolidation ratio and hydraulic

conductivity in the disturbed zone in combination with time dependant behaviour of

soft soils were considered. Different OCR and initial hydraulic conductivity profiles

in the disturbed and transition zones result in various visco-plastic strain rates and

creep strain limits. Consequently, the induced changes in visco-plastic strain rate and

creep strain limit influence the settlement rate at any given time. Therefore, the

selection of OCR and initial hydraulic conductivity profile in the disturbed zone has

a significant effect on selecting unloading time and therefore the post construction

settlement. It was observed that the creep coefficient and the creep strain limit vary

during loading and unloading and also during excess pore water pressure dissipation.

The creep coefficient and the creep strain limit are functions of the vertical effective

stress and time. The proposed solution can readily be used by practicing engineers

considering layered soil deposits, time dependent loading and unloading, while

incorporating combined effects of soil disturbance and visco-plastic behaviour.

V

ACKNOWLEDGEMENTS

One of the joys of completion is to look over the journey past and remember all the

friends and family who have helped and supported me along this long but fulfilling

road.

First of all, I pay homage to my principal supervisor, Dr. Behzad Fatahi, for all the

support and encouragement he gave me throughout my research. Under his guidance,

I successfully overcame many difficulties and learned a lot.

I would like to say thank you to my co-supervisor, A/Prof. Hadi Khabbaz, for his

valuable suggestions and concise comments on my research. He was abundantly

helpful and offered invaluable assistance, support and guidance.

I gratefully acknowledge the funding received towards my PhD from Australian

Research Council and Menard-Bachy Pty Ltd which made my research possible.

Special thanks to Ali Parsa-Pajouh (former PhD candidate at UTS) for his

collaboration and kind assistance during the experimental phase of the project. Ali

and I conducted the experimental part of this research together to be used in our

theses.

My appreciation is likewise extended to UTS laboratory and workshop staff in

particular Antonio Reyno as well as the former PhD student Thu Minh Le for their

invaluable assistance and contribution in carrying out the laboratory tests.

My gratitude also goes to my friends and fellow students at the University of

Technology, Sydney, particularly, Masoud Ameri, Ali Parsa-pajouh, Pascal

Linossier, Lucia Moretti, Amir Zad, Hamed Rezapour, Reza Afshar Mazandaran and

Hamed Mahdavi for keeping the student life more enjoyable and pleasant.

I would not have contemplated this road if not for my parents who helped me at

every stage of my personal and academic life, and longed to see this achievement

come true. A big thank you to my parents. My Sisters have also been the best of

friends along this journey.

VI

PUBLICATIONS

Azari, B., Fatahi, B., and Khabbaz, H. (2015). “Numerical analysis of vertical drains

accelerated consolidation considering combined soil disturbance and visco-plastic

behaviour.” Geomechanics and Engineering, An International Journal 8(2), pp. 187-

220.

Azari, B., Fatahi, B., and Khabbaz, H. (2014). “Assessment of the Elastic-

Viscoplastic Behavior of Soft Soils Improved with Vertical Drains Capturing

Reduced Shear Strength of a Disturbed Zone.” International Journal of

Geomechanics, in-press (DOI: 10.1061/(ASCE)GM.1943-5622.0000448).

Azari, B., Fatahi, B., Khabbaz, H., and Vincent, P. (2014). Elastic Visco-Plastic

Behaviour of Soft Soils Improved with Preloading and Vertical Drains. GeoHubei

International Conference 2014, Hubei, China, 20-22 July, pp. 17-24 (DOI:

10.1061/9780784478547.003).

Azari, B., Fatahi, B., and Khabbaz, H. (2013). Long-term Viscoplastic Behaviour of

Embankments Built on Improved Soft Soil Using Vertical Drains. In Geo-Congress

2013-Stability and Performance of Slopes and Embankments III, ASCE, San Diego,

California, 3-6 March 2013, pp. 2117-2125.

Azari, B., Fatahi, B., and Khabbaz, H. (2011). Application of Creep Ratio Concept or

Estimating Post-Construction Settlement of Deep Soft Clay Deposits. International

Conference on Advances in Geotechnical Engineering, Perth, Australia, 7-9

November 2011, pp. 127 – 134.

VII

TABLE OF CONTENT

ABSTRACT ........................................................................................................................................ III

ACKNOWLEDGEMENTS ................................................................................................................. V

PUBLICATIONS ............................................................................................................................... VI

TABLE OF CONTENT ................................................................................................................... VII

LIST OF FIGURES ............................................................................................................................. X

LIST OF TABLES .......................................................................................................................... XVI

LIST OF SYMBOLS ...................................................................................................................... XVII

1 INTRODUCTION ....................................................................................................................... 1

1.1 GENERAL ............................................................................................................... 1 1.2 CLAY COMPRESSION............................................................................................... 2

1.2.1 Preloading and vertical drains ......................................................................... 3 1.3 HYPOTHESES TO ESTIMATE SOIL CREEP .................................................................. 5 1.4 OBJECTIVES AND SCOPE OF THE PRESENT STUDY ................................................... 6 1.5 ORGANISATION OF DISSERTATION .......................................................................... 7

2 LITERATURE REVIEW ........................................................................................................... 9

2.1 GENERAL ............................................................................................................... 9 2.2 MECHANISMS OF CREEP DEFORMATION ............................................................... 11

2.2.1 The breakdown of interparticle bonds ........................................................... 11 2.2.2 Jumping bonds ............................................................................................... 12 2.2.3 Sliding between particles ............................................................................... 12 2.2.4 Double porosity ............................................................................................. 12 2.2.5 Structural viscosity ........................................................................................ 13

2.3 HYPOTHESIS A ..................................................................................................... 13 2.3.1 Creep ratio concept ........................................................................................ 14

2.3.1.1 Primary consolidation ......................................................................................... 14 2.3.1.2 Secondary compression ...................................................................................... 16 2.3.1.3 Deformation prediction of clays .......................................................................... 17 2.3.1.4 Reduction of secondary compression using surcharge ....................................... 25

2.4 HYPOTHESIS B ..................................................................................................... 34 2.4.1 Empirical models ........................................................................................... 35

2.4.1.1 Taylor and Merchant’s model ............................................................................. 35 2.4.1.2 Suklje’s model ..................................................................................................... 38

VIII

2.4.1.3 Bjerrum’s model .................................................................................................. 39 2.4.1.4 Garlanger’s model ............................................................................................... 42 2.4.1.5 Kabbaj’s model .................................................................................................... 45 2.4.1.6 Yin’s model .......................................................................................................... 47

2.4.2 Rheological models ....................................................................................... 48 2.4.2.1 Gibson and Lo’s model ........................................................................................ 49 2.4.2.2 Wahls’ model....................................................................................................... 50 2.4.2.3 Barden’s model ................................................................................................... 51 2.4.2.4 Aboshi’s model .................................................................................................... 53 2.4.2.5 Rajot’s model....................................................................................................... 54

2.4.3 General stress-strain-time models ................................................................. 56 2.4.3.1 Overstress theory ................................................................................................ 56 2.4.3.2 Non-stationary flow surface theory .................................................................... 58

2.5 PRELOADING WITH VERTICAL DRAINS ................................................................. 61 2.5.1 Vertical drains assisted preloading ................................................................ 61 2.5.2 Vacuum preloading with membrane .............................................................. 64 2.5.3 Membraneless vacuum preloading ................................................................ 65

2.6 SOIL DISTURBANCE INDUCED WHILE INSTALLING VERTICAL DRAINS ................... 66 2.7 ANALYTICAL FORMULATION FOR VERTICAL DRAIN ASSISTED PRELOADING ........ 68 2.8 NUMERICAL SIMULATION OF VERTICAL DRAIN ASSISTED PRELOADING ............... 79 2.9 SUMMARY ........................................................................................................... 83

3 FINITE DIFFERENCE SOLUTION FOR 2D AXISYMMETRIC CONSOLIDATION

EQUATION CONSIDERING NONLINEAR ELASTIC VISCO-PLASTIC MODEL ...... 86

3.1 GENERAL ............................................................................................................. 86 3.2 NONLINEAR ELASTIC VISCO-PLASTIC BEHAVIOUR OF SOILS ................................. 87 3.3 FINITE DIFFERENCE SOLUTION FOR AXISYMMETRIC CONSOLIDATION EQUATION . 92

3.3.1 Solution to general parabolic differential equations ...................................... 92 3.3.2 Axisymmetric consolidation equations.......................................................... 97

3.4 DEVELOPING A CODE INCORPORATING CREEP MODEL .......................... 102 3.5 CAPABILITIES OF THE DEVELOPED FINITE DIFFERENCE SOLUTION ..................... 104 3.6 SUMMERY .......................................................................................................... 105

4 LABORATORY EXPERIMANTS AND VERIFICATION OF THE NUMERICAL

SOLUTION .............................................................................................................................. 106

4.1 GENERAL ........................................................................................................... 106 4.2 TESTING APPARATUS AND EXPERIMENTAL PROCEDURE ................................... 107

4.2.1 Large Rowe Cell Apparatus ........................................................................ 107 4.2.2 Material Properties ...................................................................................... 112

4.2.2.1 Soil samples ....................................................................................................... 112 4.2.2.2 Consolidation tests on reconstituted samples .................................................. 114

IX

4.2.2.1 Small Rowe cell tests on reconstituted samples ............................................... 119 4.2.3 Preparation of large Rowe cell and initial sample ....................................... 137

4.3 PRE-CONSOLIDATION PROCESS AND PREPARATION OF THE FINAL SAMPLE W ..... 139 4.3.1 Initial drainage and de-airing of the Rowe cell system................................ 142 4.3.2 Vertical drain assisted consolidation test procedure .................................... 144

4.4 RESULTS AND DISCUSSION ................................................................................ 145 4.5 SUMMARY ......................................................................................................... 164

5 EFFECTS OF SOIL DISTURBANCE DUE TO PVD INSTALLATION ON LONG TERM

GROUND BEHAVIOUR ........................................................................................................ 166

5.1 GENERAL ........................................................................................................... 166 5.2 VÄSBY TEST FILL CASE STUDY ................................................................ 167

5.2.1 RESULTS AND DISCUSSION .................................................................. 174 5.3 SKÅ-EDEBY TEST FILL CASE STUDY ....................................................... 184

5.3.1 RESULTS AND DISCUSSION .................................................................. 194 5.4 SUMMARY ......................................................................................................... 204

6 CONCLUSIONS AND RECOMMANDATIONS ................................................................. 207

6.1 SUMMARY .......................................................................................................... 207 6.2 CONCLUSIONS ................................................................................................... 208 6.3 RECOMMENDATIONS FOR FUTURE RESEARCH ................................................... 212

REFERENCES ................................................................................................................................. 214

APPENDIX A ................................................................................................................................... 229

X

LIST OF FIGURES

Figure 1.2. Typical cross section of vertical drain assisted preloading .................................................. 4

Figure 2.1. Effect of sample thickness on the amount of primary consolidation for normally

consolidated clay (after Jamiolkowski et al. 1985) ............................................................. 10

Figure 2.2. Definition of , and (after Mesri et al. 1994) ........................................................ 15

Figure 2.3. Consolidation stages due to one load increment ................................................................ 16

Figure 2.4. Pore water pressure versus time ......................................................................................... 18

Figure 2.5. Dynamic viscosity of water at 1atm as a function of temperature ..................................... 19

Figure 2.6. Coefficient of secondary compression versus consolidation pressure at different

temperatures for remoulded organic Paulding (after Mesri 1973) ...................................... 20

Figure 2.7. Variation of the secondary compression index with the consolidation pressure for

undisturbed Mexico City clay (after Mesri et al. 1975)....................................................... 21

Figure 2.8. Compression index versus consolidation pressure for undisturbed Mexico City clay (after

Mesri et al. 1975) ................................................................................................................ 23

Figure 2.9. (a) Void ratio versus time; (b) void ratio versus effective vertical stress (after Mesri and

Castro 1987) ........................................................................................................................ 24

Figure 2.10. Total and effective surcharge ratios definition (after Feng 1991) .................................... 26

Figure 2.11. values in terms of and , for (after Feng 1991) ....................... 27

Figure 2.12. Elapsed times definition used to estimate post-surcharge secondary compression (after

Feng 1991) .......................................................................................................................... 30

Figure 2.13. Post-surcharge secondary compression index in terms of (after Mesri et al. 1994)

............................................................................................................................................. 31

Figure 2.14. Ideal model of surcharging to reduce secondary compression (after Mesri et al. 1994) .. 32

Figure 2.15. Relationship between void ratio and vertical effective stress throughout the consolidation

process (after Taylor and Merchant 1940) .......................................................................... 38

Figure 2.16. Effect of sustained loading on results of oedometer tests (aging effect) (after Bjerrum

1967) ................................................................................................................................... 41

Figure 2.17. Definition of instant and delayed compression compared with primary consolidation and

secondary compression (after Bjerrum 1967) ..................................................................... 42

Figure 2.18. Creep oedometer tests in Batiscan Clay (after Leroueil et al. 1985) ................................ 48

Figure 2.19. Rheological models proposed by Barden: (a) Barden’s proposed non-linear model, and

(b) model solved by Barden (after Barden 1965) (Note: N and L stand for non-linear and

linear, respectively) ............................................................................................................. 52

Figure 2.20. Effect of drainage path on experimental compression curves (after Aboshi 1973) ......... 54

Figure 2.21. Rajot’s Rheological mechanical model (after Perrone 1998) .......................................... 56

Figure 2.22. Perzyna’s (1963) viscoplastic theory (after Perrone 1998) .............................................. 58

Figure 2.23. Olszak and Perzyna (1966) viscoplastic theory (after Perrone 1998) .............................. 59

Figure 2.24. Schematic diagram of embankment ................................................................................. 62

XI

Figure 2.25. Vertical drain installation patterns; (a) square pattern, (b) triangular pattern .................. 64

Figure 2.26. Cross section of the disturbed zone surrounding a vertical drain, (a) two zones

hypothesis, (b) three zones hypothesis ................................................................................ 68

Figure 3.1. Schematic fitting curves for instant, reference, equivalent and limit time lines ................ 88

Figure 3.2. One dimensional rod of length L ....................................................................................... 92

Figure 3.3. Region and the mesh points (after Kharab and Guenther 2012) ..................................... 93

Figure 3.4. Schematic form of the Forward-difference method (after Kharab and Guenther 2012) .... 94

Figure 3.5. Schematic form of the Crank-Nicolson method (after Kharab and Guenther 2012) .......... 96

Figure 3.6. Schematic 3D-axisymetric consolidation ........................................................................... 97

Figure 3.7. (a) Location of finite difference nodes at any given time; (b) time steps ......................... 101

Figure 3.8. Boundary conditions for (a) soil layer surrounded by two permeable layers (drains) at the

top and bottom; (b) soil layer surrounded by impervious layer at the bottom and highly

permeable layer (drainage blanket) at the top ................................................................... 101

Figure 3.9. Flowchart of the developed MATLAB code ................................................................... 103

Figure 4.1. Large scale Rowe cell apparatus (a) schematic diagram of the cell and (b) locations of the

pore pressure transducers at the base of the cell ................................................................ 108

Figure 4.2. A photographic view of the GDS pressure/volume controller device .............................. 109

Figure 4.3. Infinite volume controller instrument .............................................................................. 109

Figure 4.4. Schematic diagram of Rowe cell set-up ........................................................................... 110

Figure 4.5. Established setup in the laboratory (large Rowe cell) ...................................................... 111

Figure 4.6. Grain size distribution curve for vertical drain sand ........................................................ 113

Figure 4.7. Pre-consolidation process prior to the oedometer test; (a) cylinder contacting reconstituted

sample and (b) samples under pre-consolidation pressure ................................................ 115

Figure 4.8. Preparing the samples for the oedometer test, (a) placing the oedometer ring, (b) cutting

the extra top part, (c) cutting the extra bottom part, and (d) the final sample ................... 116

Figure 4.9. Consolidation test, (a) placing the prepared sample and (b) oedometer apparatus connected

to the data logger ............................................................................................................... 116

Figure 4.10. Variation of permeability against void ratio (sample S1) .............................................. 117



Figure 4.13. Schematic diagram of the small Rowe cell apparatus .................................................... 119

Figure 4.14. Testing procedure, (a) filling the Rowe cell with soil sample and levelling the surface of

sample, (b) placing the porous plate on top of the sample, (c) fixing the top cap, and (d)

Applying a pressure to ensure full saturation .................................................................... 120

Figure 4.15. Established setup in the laboratory (small Rowe cell) ................................................... 121

Figure 4.16. Consolidation test results on reconstituted sample S1 (loading) .................................... 122

Figure 4.17. Excess pore water pressure measurement on reconstituted sample S1 (loading) ........... 123

Figure 4.18. Consolidation test results on reconstituted sample S1 (unloading) ................................ 123

Figure 4.19. Excess pore water pressure measurement on reconstituted sample S1 (unloading) ....... 124

Figure 4.20. Consolidation test results on reconstituted sample S1 (reloading) ................................. 124

XII

Figure 4.21. Excess pore water pressure measurement on reconstituted sample S1 (reloading) ........ 125

Figure 4.22. Consolidation test results on reconstituted sample S3 (loading) .................................... 125

Figure 4.23. Excess pore water pressure measurement on reconstituted sample S3 (loading) ........... 126

Figure 4.24. Consolidation test results on reconstituted sample S3 (unloading) ................................ 126

Figure 4.25. Excess pore water pressure measurement on reconstituted sample S3 (unloading) ....... 127

Figure 4.26. Consolidation test results on reconstituted sample S3 (reloading) ................................. 127

Figure 4.27. Excess pore water pressure measurement on reconstituted sample S3 (reloading) ........ 128

Figure 4.28. Variation of void ratio versus effective vertical stress (Sample S1) .............................. 129

Figure 4.29. Variation of void ratio versus effective vertical stress (Sample S3) .............................. 129

Figure 4.30. Comparison between predicted numerical creep strain and laboratory measurements at

800 kPa (sample S1) .......................................................................................................... 131

Figure 4.31. Comparison between predicted numerical creep strain and laboratory measurements at

800 kPa (sample S3) .......................................................................................................... 132

Figure 4.32. Changes of versus vertical effective stress for reconstituted sample S1 ................ 132

Figure 4.33. Changes of versus vertical effective stress for reconstituted sample S3 ................ 133

Figure 4.34. Time dependant stress-vertical strain relationship for reconstituted sample S1 ............ 133

Figure 4.35. Time dependant stress-vertical strain relationship for reconstituted sample S3 ............ 134



Figure 4.38. Comparison between predicted numerical settlements and laboratory measurements for

small Rowe cell (sample S1) ............................................................................................. 136

Figure 4.39. Comparison between predicted numerical settlements and laboratory measurements for

small Rowe cell (sample S3) ............................................................................................. 136

Figure 4.40. Placing of PVC and brass pipes as the reduced permeability zone boundary and the

vertical drain border, (a) top view, (b) side view and (c) a typical cross section of the Rowe

cell ..................................................................................................................................... 138

Figure 4.41. Sample placement, (a) filling the intact area (intact zone) with the prepared soil and (b)

the setup after placing PVC and Brass pipes as the reduced permeability zone boundary and

vertical drain border .......................................................................................................... 139

Figure 4.42. Rig set up, (a) geotextile filters, (b) pre-consolidation loading rings, (c) the first two

loading rings with drainage grooves and holes, (d) placing of the first loading ring and (e)

full loading condition ........................................................................................................ 140

Figure 4.43. Testing procedures, (a) Pouring the vertical drain material and (b) Pulling out the outer

pipe .................................................................................................................................... 141

Figure 4.44. Testing procedures, (a) pulling out the inner pipe and (b) cutting the extra part of the

filter paper ......................................................................................................................... 141

Figure 4.45. Testing procedures, (a) levelling the top surface and (b) placing the geotextile on top

surface ............................................................................................................................... 142

Figure 4.46. Testing procedures, (a) filling the cell with water and (b) placing the cell top .............. 142

XIII

Figure 4.47. Schematic diagram of the de-airing process .................................................................. 143

Figure 4.48. Schematic diagram of the instrumentation plan, (a) the cross section of bottom of the

Rowe cell and (b) plan view of the body of Rowe cell ..................................................... 146

Figure 4.49. Comparison of the excess pore water pressure predictions and laboratory measurement

versus time throughout loading at PWPT B2 .................................................................... 147








versus time throughout unloading and reloading at PWPT B2 ......................................... 150








versus time throughout loading at PWPT A1 .................................................................... 152








versus time at PWPT A1 ................................................................................................... 155





Figure 4.64. Measured excess pore water pressure at transducers located on the bottom of the cell

(loading) ............................................................................................................................ 156

Figure 4.65. Measured excess pore water pressure at transducers located on the bottom of the cell

(unloading and reloading) ................................................................................................. 157

Figure 4.66. Measured excess pore water pressures from transducers located on the sides of the cell

(loading) ............................................................................................................................ 157

XIV

Figure 4.67. Measured excess pore water pressures from transducers located on the sides of the cell


Figure 4.68. Variations of excess pore water pressures with the vertical distance from the bottom of

the impermeable boundary ................................................................................................ 159

Figure 4.69. Variations of excess pore water pressures with the radial distance from the centre of the

drain .................................................................................................................................. 160

Figure 4.70. Predicted creep coefficient ( ) values versus time ...................................................... 161

Figure 4.71. Creep strain limit values predictions versus time........................................................... 161

Figure 4.72. Comparison between predicted numerical settlements and laboratory measurements

(loading) ............................................................................................................................ 163

Figure 4.73. Comparison between predicted numerical settlements and laboratory measurements


Figure 5.1. Väsby test field (after Chang 1981) ................................................................................. 168

Figure 5.2. Soil profile beneath the Väsby test fill ............................................................................. 169

Figure 5.3. Consolidation tests results on Väsby post glacial clay samples for vertical stresses between

5 kPa and 160 kPa ............................................................................................................. 169

Figure 5.4. Time dependant stress-vertical strain relationship for Väsby post glacial clay ............... 170

Figure 5.5. Changes of versus vertical effective stress............................................................... 170

Figure 5.6. Permeability changes versus void ratio ............................................................................ 171

Figure 5.7. Cross section of the disturbed zone surrounding a vertical drain, (a) two zones hypothesis,

(b) three zones hypothesis ................................................................................................. 172

Figure 5.8. Variations of initial permeability profile for Cases A to F............................................... 174

Figure 5.9. Excess pore water pressure values predicted by developed code versus time for Cases A to

F ........................................................................................................................................ 175

Figure 5.10. Variations of excess pore water pressure with time for Case A ..................................... 177

Figure 5.11. Variations of the excess pore water pressure values just before unloading ( )

for Cases A to F ................................................................................................................. 177

Figure 5.12. Predicted creep coefficient ( ) values versus time for Cases A to F ........................... 178

Figure 5.13. Creep strain limit values predicted by the developed code versus time for Cases A to F

........................................................................................................................................... 179

Figure 5.14. Comparison of the settlement predictions for Cases A to F and the field measurements at

the ground surface ............................................................................................................. 180

Figure 5.15. Comparison between the settlement predictions for Cases A to F and the field

measurements at 3.8 m depth ............................................................................................ 180

Figure 5.16. Comparison between post construction settlement prediction for Cases A to F and the

field measurement at the ground surface ........................................................................... 181

Figure 5.17. The required time to achieve 500 mm of settlement for Cases A to F at the ground surface

........................................................................................................................................... 182

Figure 5.18. Variations of permeability profile versus time for Case A ............................................ 182

XV

Figure 5.19. Variations of permeability ratio with time in disturbed zone for Case A ...................... 183

Figure 5.20. Skå-Edeby test field (After Hansbo 1960) ..................................................................... 185

Figure 5.21. Soil profile beneath the Skå-Edeby test fill .................................................................... 185

Figure 5.22. Initial void ratio profile versus depth ............................................................................. 186

Figure 5.23. Preconsolidation pressure profile versus depth .............................................................. 186

Figure 5.24. Consolidation tests results on Skå-Edeby glacial clay samples for vertical stresses

between 21.6 kPa and 338.3 kPa ....................................................................................... 187

Figure 5.25. Time dependant stress-vertical strain relationship for Skå-Edeby glacial clay .............. 189

Figure 5.26. Changes of versus vertical effective stress............................................................. 189

Figure 5.27. Permeability changes versus void ratio .......................................................................... 190

Figure 5.28. Cross section of the disturbed and transition zones surrounding a vertical drain .......... 191

Figure 5.29. Permeability profile in the disturbed and transition zones for all cases ......................... 193

Figure 5.30. Variations of overconsolidation ratio profile for Cases A to E at depth of 2.5m ........... 193

Figure 5.31. Comparison of the developed code excess pore water pressure predictions for cases A to

E and the field measurements at depth of 2.5m ................................................................. 195


E and the field measurements at depth of 5m .................................................................... 195


E and the field measurements at depth of 9m .................................................................... 196

Figure 5.34. Variations of excess pore water pressure with time for Case B ..................................... 197

Figure 5.35. Variations of the excess pore water pressure values at the end of loading ( )

for Cases A to E ................................................................................................................ 198

Figure 5.36. Predicted creep strain limit ( ) values versus radial distance at the end of loading

( ) for Cases A to E ......................................................................................... 199

Figure 5.37. Predicted visco-plastic strain rate ( ) values versus radial distance at the end of loading

( ) for Cases A to E ......................................................................................... 200

Figure 5.38. Predicted visco-plastic strain rate ( ) values versus radial distance at 200 days for Cases

A to E ................................................................................................................................ 201

Figure 5.39. Comparison of the settlement predictions for Cases A to E and the field measurements at

the ground surface ............................................................................................................. 201

Figure 5.40. Comparison of the settlement predictions for Cases A to E and the field measurements at

2.5m depth ......................................................................................................................... 203

Figure 5.41. Comparison between the settlement predictions for Cases A to E and the field

measurements at 5m depth ................................................................................................ 204

XVI

LIST OF TABLES

Table 2.1. Values of for natural soil deposits (modified after Mesri & Godlewski, 1977) ........ 22

Table 2.2. A summary of the Hypothesis B models presented ............................................................. 60

Table 2.3. Conversion relationships suggested for a rectangular drain ................................................ 63

Table 2.4. Proposed analytical solutions for vertical drain assisted preloading ................................... 78

Table 2.5. Summary of numerical studies conducted to simulate PVD assisted .................................. 81

Table 4.1. Properties of the adopted soil samples in this study .......................................................... 112

Table 4.2. Important sizes for vertical drain sand .............................................................................. 113

Table 4.3. Mix design for the reconstituted samples .......................................................................... 114

Table 4.4. Properties of the reconstituted samples ............................................................................. 114

Table 4.5. Permeability of mixtures (Surcharge = 20 kPa) ................................................................ 118

Table 4.6. Properties of the intact zone, the reduced permeability zone, and drain ........................... 118

Table 4.7. Details of loading stages using small Rowe cell (Sample S1)........................................... 121

Table 4.8. Details of loading stages using small Rowe cell (Sample S3)........................................... 122

Table 4.9. The calculated values of and at different vertical effective stress for intact

zone (Sample S1) .............................................................................................................. 130

Table 4.10. The calculated values of and at different vertical effective stress for intact

zone (Sample S3) .............................................................................................................. 131

Table 4.11. Elastic visco-plastic model parameters for soil samples S1 and S3 ................................ 134

Table 4.12. Details of consolidation loading stages ........................................................................... 144

Table 5.1. Adopted soil properties for Väsby post glacial clay .......................................................... 171

Table 5.2. Various available permeability variation equations .......................................................... 173

Table 5.3. Fitting parameters for disturbed zone permeability profile for Cases A-F ........................ 173

Table 5.4. Adopted soil properties for Skå-Edeby glacial clay .......................................................... 188

Table 5.6. Fitting parameters for disturbed zone overconsolidation ratio profile for Cases A to E ... 194

XVII

LIST OF SYMBOLS

experimental constant

the compressibility of the linear spring

constant over load increment for small void ratio changes


the compressibility of the spring


secondary compression index

post-surcharge secondary compression index

post-surcharge secant secondary compression index

the rate of secondary compression

conventional compression index

conventional recompression index (unloading and reloading data)

permeability change index

coefficient of consolidation


initial void ratio

the initial value of void ratio

void ratio at effective stress equal to on reference time line

void ratio for a particular applied effective stress

void ratio when the excess pore water pressure has fully dissipated

visco-plastic settlement rate

layer depth

hydraulic head (static pressure head)

i horizontal node coordinator

j vertical node coordinator

initial permeability

vacuum pressure reduction factor by depth

vacuum pressure reduction factor by radius

average disturbed zone permeability for Case A

average disturbed zone permeability for Case B

average disturbed zone permeability for Case C

average disturbed zone permeability for Case D

average disturbed zone permeability for Case E

average disturbed zone permeability for Case F

coefficients of permeability for horizontal direction for disturbed zone

coefficients of permeability for vertical direction for disturbed zone

XVIII

the effect of both work hardening and strain rate hardening

coefficients of permeability for horizontal direction for intact zone

the horizontal coefficient of permeability of remoulded soil

coefficients of permeability for vertical direction for intact zone

total depth

coefficient of volume compressibility

model parameter

the applied vacuum pressure

The flow in the slice at a distance

the constant loading

disturbed zone radius

effective surcharge ratio

radial coordinate

disturbed zone radius

the equivalent influence radius

partial disturbed zone radius

smear zone radius

vertical drain zone radius

drain spacing

average total settlement

shear strength for disturbed and transition zones

normally consolidated shear strength of soil

horizontal time factor

curve-fitting parameter related to the choice of reference time line

equivalent time

the time that post-surcharge secondary compression reappears after the removal of the

surcharge

the time required for completion of the primary consolidation

the time required for completion of the post-surcharge primary consolidation

time to EOP compression under surcharge

the surcharging time

maximum calculation time

The degree of consolidation

he average degree of consolidation for axisymmetric flow

excess pore water pressure

the average pore water pressure

pore water pressure at any point in the natural soil zone

XIX

pore water pressure at any point in the smear zone

vacuum pressure at any point

the excess pore water pressure within vertical drain

vertical coordinate

Greek symbols

permeability ratio parameter

the instant deformation per unit thickness and unit load

secondary compression rate per unit thickness and unit load



saturated unit weight of soil

unit weight of water

the change in the void ratio during the primary consolidation

radial distance increment

time step

vertical distance increment

he total surcharge pressure

elastic compression

time dependant compression

soil vertical strain

vertical strain at stress level

vertical strain at

reference time line strain

vertical strain at

creep strain limit

strain limit

the visco-plastic strain rate

creep compression strain

the temperature

time-dependent multiplier

material parameter describing the elastic stiffness of the soil

material property describing the elastic-plastic stiffness of the soil

specific volume ( )

average effective stress

′ unit stress

material property

the final effective vertical stress after the removal of surcharge

XX

the maximum vertical effective stress reached immediately before removal of the surcharge

the viscosity of the dashpot

the non-linear viscous resistance of the dashpot

fluidity parameter

viscous nucleus

initial creep coefficient

1

CHAPTER ONE

1 INTRODUCTION

1.1 GENERAL

As a result of increasing and continuous social and infrastructural growth,

appropriate ground for living and development becomes progressively scanty.

Consequently, the waterfront areas alongside lakes, rivers, and marine coasts are

being considered for an alternative land source for developers. The prevalent soils in

these areas are considered as soft soils. Soft soils generally consist of remarkable

amount of water and as a result, the foundations overlying these soils have

significantly low shear strength, high compressibility, and low hydraulic

conductivity. Accordingly, while dealing with soft soils in engineering design and

construction, long term deformation of soft soils is one of the main challenges.

The time dependent behaviour of soft soils, particularly the ground

settlements, is considered as an important issue, which has been studied for many

decades. Nonetheless, predicting the long-term behaviour of soft soils (e.g.

settlement and lateral deformations) under highway or railway embankments is a

challenging task for geotechnical engineers. Creep (i.e. time dependant viscous

behaviour of soil) is an important part of the soft soil settlement, which may result in

significant deformation in the long-term. Creep deformation may be considered as

destruction or adjustment of the soil structure under a constant effective stress.

Although a number of soil improvement techniques are available, the

application of preloading in combination with vertical drains is often used in practice

when enough construction time is available or when the ground improvement budget

is limited. The installation of vertical drains reduces the water drainage path and

speeds up the dissipation of excess pore pressure generated during preloading. The

fact that the flow is predominately in the horizontal direction (except near the top

surface or close to a highly permeable silt/sand seam) further helps the process

2

because, owing to depositional anisotropy, the hydraulic conductivity is generally

greater in horizontal direction than in vertical direction. Vertical drains installation in

the field results in considerable remoulding of the subsoil, particularly in the

immediate vicinity of the mandrel. The disturbed zone will have reduced shear

strength and permeability, adversely affecting the soil consolidation.

This chapter describes how preloading in combination with vertical drains

can speed up the consolidation process. Furthermore, the proposed conceptual

approaches for simulating long term behaviour of soft soils are briefly explained. The

chapter concludes with an outline of aims and content of this thesis.

1.2 CLAY COMPRESSION

Soil comprises of solid particles and voids. The voids in the soil structure

can be filled with air, water or a combination of both. The reduction of void ratio

under vertical loads may take place in three stages (Figure 1.1): (i) immediate

settlement, (ii) primary consolidation and (iii) secondary compression. Immediate

settlement happens instantly after the application of vertical loads with zero volume

change (i.e. the shape change only). In saturated soils (i.e. no air) the increase in the

vertical pressure immediately transferred to water, which is incompressible. Then

water may seep out of the soil, which results in dissipation of the excess pore water

pressure and transformation of the pressure to the soil skeleton (primary

consolidation).

Secondary compression might be determined as the continuation of the

mechanism of volume change following primary consolidation. This mechanism

consists of deformation of the individual particles, and also the relative movements

of particles due to the normal stresses or shear displacements at particle contacts

induced by shear stresses exceeding the bond shear resistance of the contacts (Mesri

1973). Moreover, the settlement under a constant effective stress is generally called

creep or secondary compression (Taylor and Merchant 1940; Bjerrum 1967; Le et al.

2012). However, it should be mentioned that creep deformation should be

differentiated from the settlement under the constant effective stress since creep may

also occur while the excess pore water pressure is dissipating.

3

As stated by Bjerrum (1967) and Taylor (1942), creep compression

increases the resistance of the soil structure against further compression. The creep

compression results in not only excessive settlement of soft soil under an applied

stress, but also influences other soil properties such as the preconsolidation pressure.

In contrast, the time–dependent compression is observed to be primarily influenced

by time, strain rate and stress rate. Consequently, creep compression is an important

contributor to the time-dependant characteristics of soft soils. Moreover, creep

compression changes the pattern of hydraulic conductivity and shear strength of

vertical drain improved ground in short and long term.

Figure 1.1. Typical oedometer test results

1.2.1 Preloading and vertical drains

Preloading is applied prior to building of structures to improve areas with

unsuitable ground conditions. Preloading consists of applying a load, equal to or

greater than the entire load of a planned structure, over the site earlier than

constructing the structure. Preloading, which is normally an earth fill, applies

compression to the underlying soil, which is being partially or fully removed while

the required settlement has taken place. Then, the structure is built, which applies a

load equivalent to or smaller than the preload. Preloading is selected in a way that

Settl

emen

t

Time 0

Initial compression

Primary consolidation

Secondary compression

4

construction restrictions on preloading time and post-construction settlements are

both taken into consideration. Preloading with vertical drains can be used to decrease

the soil settlement time.

Vertical drain assisted preloading improves the shear strength of clay and

reduces the post construction settlements to tolerable levels. There are two classes of

vertical drains: displacement and non-displacement. The non-displacement drains

involve removal of the in situ soil and backfilling with more a permeable material,

usually sand. Holes may be formed by driving, jetting, or auguring with typical

diameters of 200 to 450 mm (Hausmann 1990). Displacement type drains are

prefabricated and are forced into the soil with a hollow mandrel. The mandrel is then

removed leaving the drain in place. Prefabricated vertical drains (PVD) consist of a

core surrounded by a filter sleeve. Figure 1.2 shows a typical cross section of a site

improved with vertical drain assisted preloading.

Figure 1.2. Typical cross section of vertical drain assisted preloading

There are occasions where the use of surcharge loading with vertical drains

is too slow or inappropriate for the site, e.g. when specified construction time may be

very short or there is no access to suitable fill material. In such cases, it is necessary

to use more refined techniques instead of, or in combination with surcharge loading.

One of the ways to hasten water flow in soil is applying a vacuum to the soil surface

and along the vertical drains. During vacuum preloading an external negative load is

Embankment

Sand layer

Clay layer

Vertical drains

5

applied to the soil surface in the form of vacuum (Choa, 1989). Thus, a higher

effective stress is achieved by rapidly decreasing the pore water pressure, while the

total stress remains unchanged.

When vertical drains are installed in the soft ground, the soil surrounding

the drain is disturbed as mandrels or augers/drills are inserted and withdrawn. The

effects associated with this installation disturbance are detrimental to radial

consolidation. Compared to the undisturbed soil, permeability and shear strength in

the smear zone are reduced and compressibility is increased. The extent of

disturbance depends on the mandrel size and the soil type (Eriksson et al., 2000; Lo,

1998).

1.3 HYPOTHESES TO ESTIMATE SOIL CREEP

Numerous approaches have been presented to simulate the time dependant

behaviour of soft soils. Generally, two broad concepts are suggested by researchers,

namely, Hypotheses A and B. Hypothesis A states that void ratio at the end of

primary consolidation is unique for thin and thick samples although creep occurs

during primary consolidation (Ladd et al. 1977; Mesri and Choi 1985). However,

Hypothesis B states that the primary consolidation consists of creep and the end of

primary consolidation void ratio cannot be unique. In Hypothesis A, soil settlement

is divided into two parts, the primary consolidation followed by the secondary

compression. Nonetheless, in Hypothesis B, soil settlement is estimated using a

constitutive model simulating soil creep and consolidation settlement simultaneously.

In literature, models based on Hypothesis B are classified in three categories namely

empirical models, rheological models and general stress-strain-time models

(Liingaard et al. 2004; Karim and Gnanendran 2014).

Empirical models are presented by closed form solutions or differential

equations (Karim and Gnanendran 2014). Empirical models are generally based on

fitting experimental results from creep, stress relaxation, and constant rate of strain

tests. Rheological mechanical models contain different arrangements of springs,

dashpots and sliders to represent soil behaviour (i.e. elastic, viscous, or plastic

behaviour). Non-linear behaviour of the elements represents the structural or fabric

variations that happen at the particle level. General constitutive laws define not only

6

viscous effects but also the rate dependant behaviour of soil under any possible

loading conditions.

1.4 OBJECTIVES AND SCOPE OF THE PRESENT STUDY

The main objective of this study is to develop a numerical code for the

analysis of long term behaviour of soft soils involving vertical drains and preloading

considering smear zone and creep effects. Various methods are presented in literature

to study the effects of time dependant behaviour of soft soils or capturing the

reduction of hydraulic permeability caused by vertical drains installation. However,

there is a lack of consideration with respect to the combined effects of the hydraulic

conductivity or the shear strength profile in the disturbed zone and the visco-plastic

behaviour (creep) of soil on the settlement rate and consequently the deformation of

the soft soil improved using preloading and vertical drains.

To achieve the research objectives, the following stages are included in this

research: (i) a numerical solution adopting an elastic visco-plastic model with

nonlinear creep function in combination with the 2D axisymmetric consolidation

equations was developed, (ii) laboratory testing using oedometer and small and large

Rowe cells apparatus to verify the developed numerical code by comparing measured

and predicted settlement and excess pore water pressure values at different height

and radiuses from the vertical drain was conducted, and (iii) two field case studies to

investigate the long term behaviour of soft soils considering the disturbance caused

by installation of vertical drains (i.e. permeability and shear strength reduction in the

vicinity of vertical drains) were simulated.

It should be mentioned that the proposed solution can readily be applied to

layered soil deposits incorporating time dependent loading and unloading, while

considering combined effects of soil disturbance effects and visco-plastic behaviour.

Furthermore, well resistance and discharge capacity and shear creep are not

considered in this research which can be easily implemented into the developed

numerical solution in future studies.

7

1.5 ORGANISATION OF DISSERTATION

In Chapter 1, a brief introduction is presented where the aim and scope of

the present research are highlighted. Chapter 2 presents a comprehensive survey of

the literature associated with the present work. A review of the mechanisms of creep

is presented followed by detailed explanation of the constitutive models proposed to

simulate the behaviour of soft soils. Hypotheses A and B as two broad concepts,

proposed by researchers to estimate the time dependant deformation of soft soils, are

presented (Ladd et al. 1977). The creep ratio concept, one of the main settlement

calculation approaches supported by Hypothesis A, is discussed. Furthermore, the

constitutive models based on Hypothesis B presented in literature are divided into

three categories namely empirical models, rheological models, and general stress-

strain-time models and explained. Moreover, application of preloading and vertical

drain and also the disturbance associated with the installation of vertical drains are

described in details. Finally, the analytical and numerical methods presented in

literature to investigate the time dependant settlement of soil dealing with

disturbance due to vertical drains installation are explained.

Chapter 3 presents the finite difference solution for 2D axisymmetric

consolidation equation considering nonlinear elastic visco-plastic model. The applied

elastic visco-plastic model is based on the framework of modified Cam-Clay which

comprises the nonlinear variations of the creep coefficient with the effective stress

and time and creep strain limit. Finite difference formulations for fully coupled one

dimensional axisymmetric consolidation are adopted. In formulating the finite

difference procedure, the Crank-Nicholson scheme has been used. In this method,

two steps have been used in partial differentials of pore water pressure over distance

to stabilise the process quicker. The finite difference solution of elastic visco-plastic

model is coded in MATLAB to model the time dependent behaviour of the soft soils.

In Chapter 4, the procedure for a laboratory consolidation test is described

and the numerical code developed in Chapter 3 is verified using the experimental

results. A large Rowe cell with a diameter of 250 mm and a height of 200 mm was

used to carry out the PVD assisted consolidation tests. The cell was fully

instrumented and comprised of a vertical displacement gauge at the surface level and

nine pore water pressure transducers on the sides and at the base of the cell. The time

8

dependent vertical displacements of the sample were captured using GDSLab

software, with an LVDT (Linear Variable Differential Transformer). A series of

pressure lines connected to the enterprise level pressure/volume controllers, that are

filled with de-aired water, are used to apply pressure to the cell (cell pressure) and to

the jacket (back pressure). To solve the problem of manually filling or emptying the

controllers, two parallel pressure/volume controllers (primary and secondary)

connected to an infinite volume controller (IVC) device were used for each pressure

line. A sand vertical drain has been installed in the centre of the sample and a zone

with reduced permeability adjacent to the vertical drain has been considered. Two

reconstituted clay samples were prepared by mixing Q38 kaolinite, ActiveBond23

bentonite, and fine sand for the reduced permeability zone soil and the intact zone

soil. To carry out the test, five loading (25 kPa to 400 kPa), one unloading (50 kPa),

and three reloading (100 kPa, 200 kPa, and 400 kPa) stages were applied to conduct

the PVD assisted consolidation tests.

In Chapter 5, two case studies are numerically simulated to investigate the

effect of time dependant settlement in combination with the disturbance due to

installation of vertical drains (i.e. permeability and shear strength reduction). Skå-

edeby test fill case study is used to study long term behaviour of soil considering

different shear strength profiles in the disturbed zone. Furthermore, to study the

effect of permeability profile in the disturbed zone in combination with creep, Väsby

test fill case study is adopted. It should be mentioned that in the simulations of case

studies, the variations of initial vertical effective stress and void ratio with depth and

the variations of permeability and overconsolidation ratio with depth and time are

considered.

Chapter 6 presents the conclusions that can be drawn from the current

research and provides recommendations for future work, followed by the list of

references.

9

CHAPTER TWO

2 LITERATURE REVIEW

2.1 GENERAL

Following Terzaghi’s (1923) outstanding one dimensional consolidation

theory concerning the rate of excess pore water pressure dissipation, laboratory

results and field measurements showed that settlement continues even after the

excess pore water pressure has completely dissipated. To distinguish between the

two components of settlement, the term “primary consolidation” is used to denote the

time dependant process resulting from the change in volume due to the expulsion of

water from the voids, and shifting loads from the pore water to the soil particles,

while settlement under a constant effective stress is generally called creep or

secondary compression (Taylor and Merchant 1940, Bjerrum 1967, Le et al. 2012).

Creep deformation should be differentiated from settlement under constant effective

stress because creep may also occur while the excess pore water pressure is

dissipating. Although research on the long-term deformation of soils has become

significant and has been developed for many decades, there is still no unified method

of defining the mechanism of creep deformation, with the resulting that various

schools of thought and different methods of estimating soil deformation have arisen.

Furthermore, different ways of estimating the time dependant deformation

of soft soils have also been proposed, although researchers generally suggest two

broad concepts: (i) although creep occurs during dissipation of excess pore water

pressure (the primary consolidation process), the void ratio when the excess pore

water pressure had fully dissipated at the end of primary consolidation ( ) was

unique for thin and thick samples, so all the subsequent calculations were based on

this assumption (Hypothesis A) (e.g. Ladd 1973, Ladd et al. 1977, Mesri 2001; Mesri

and Rokhsar 1974; Mesri and Feng 1991; Mesri et al. 1994), and (ii) because primary

consolidation consists of creep deformation, which increases over time, the value of

for thin and thick soil samples cannot be unique and the equations should

10

embrace the results of this assumption (Hypothesis B) (e.g. Suklje 1957, Barden

1965, 1969, Bjerrum 1967, Yin and Graham 1989, 1990, and 1996). In Hypothesis

A, soil settlement is divided into primary consolidation (during the dissipation of

excess pore water pressure) followed by secondary compression (where the

remaining excess pore water pressure is insignificant). However, in Hypothesis B the

soil settlement was estimated using a constitutive model to simulate creep

deformation and excess pore water pressure dissipation simultaneously and

continuously; as a result, the longer the duration of primary consolidation (Thicker

sample), the more significant the difference between Hypothesis A and B

(Figure 2.1).

Figure 2.1. General effect of sample thickness on the amount of primary consolidation for normally consolidated clay (after Jamiolkowski et al. 1985)

Because the post-construction settlement of soils such as clays, silts, and

peats during the life time of the structure may be significant, preloading combined

with vertical drains is commonly used to minimise settlement and improve the

bearing capacity and shear strength of soil deposits. Preloading, which is commonly

an earth fill, involves applying a load that is equal to or greater than the entire load of

a planned structure over the site before construction begins, and which partially or

fully removed while the required settlement takes place.

Axi

al st

rain

Log time (t)

Thin sample Thick sample

Hypotheses A & B

Hypothesis A

Hypothesis B

11

In this chapter the mechanisms of creep deformation and a summary of

previous studies on long term settlement of soils based on Hypotheses A and B have

been reviewed. The references consist of field measurements and laboratory tests

associated with secondary compression and theoretical studies for modelling the time

dependant behaviour of soil. A critical review of the preloading assisted

consolidation method was carried out, and the analytical and numerical methods

proposed in literature to simulate the behaviour of soft soils are discussed in detail.

2.2 MECHANISMS OF CREEP DEFORMATION

Unlike other materials such as metal or glass, soil is considered to be a

complex structure due to its heterogeneous composition. From a macroscopic

perspective, it is assumed that an element of a clayey soil contains particles of clay,

coarse grains, and water where free water can flow as a result of the hydraulic

gradient. As Taylor and Merchant (1940) and Terzaghi (1941) explained, the

compression process containing primary consolidation and secondary compression

(or creep) is defined according to the transfer of stress and the adjustment of soil

structure. Free water flows out of the soil under the effect of applied stress which in

turn rearranges the soil structure, and may lead to an increase in the solid-to-solid

contacts in the soil. This section attempts to explain the mechanism of creep for

clayey soils according to numerous relevant studies.

2.2.1 The breakdown of interparticle bonds

The flow of free water takes time because it is a function of soil

permeability; consequently, primary consolidation is a time dependent process.

Primary consolidation increases contact between soil particles and also decreases the

voids between them. The effective stresses increase throughout primary

consolidation as the total stress shifts from pore water to inter-particle bonds, but this

increase in effective stress can break or destroy the contacts between particles. This

breakdown in the bonds between soil particles may lead to further rearrangement and

therefore additional settlement or compression, which is called creep. There are

numerous sources for the breakdown of interparticle bonds; it can be the relative

12

movements of particles with respect to each other caused by shear displacement or

variations in the particle spacing due to a change in the net inter-particle forces

(Mesri, 1973).

2.2.2 Jumping bonds

Murayama and Shibata (1961), Christensen and Wu (1964) and Mitchell

(1964) explained creep mechanism based on the theory of rate process where creep

deformation stems from the movement of atoms and molecules under constant

effective stress to a new equilibrium position. Since virtual energy barriers resist the

movement of atoms and molecules (named flow unit), enough energy is needed to

conquer the barriers. As Kuhn and Mitchell (1993) explained, soil creep is the

displacement of oxygen atoms, which are seen as flow units, within the contact

surface between clay mineral particles, where the flow units do not remain static,

they vibrate with a certain frequency. As Mitchell et al. (1968) mentioned, since

creep is defined as a rate process, the activation energy depends on the deviatoric

stress and the elapsed time of creep.

2.2.3 Sliding between particles

Kuhn and Mitchell (1993) proposed that creep deformation is the result of a

sliding movement between the particles. This proposed concept can be compared to

the deformation of interparticle bonds mentioned earlier. By means of viscous

friction, the sliding movement is due to the tangential component of the contact

forces between soil particles, and the deformation is defined as the relationship

between the sliding velocity, the sliding force, and the friction ratio between the

tangential force and normal force.

2.2.4 Double porosity

The theory of double porosity is defined as an assumption of the double

structure levels of soils comprising micro-pore and macro-pore. Creep is the transfer

of pore water from the microstructure to macrostructure. This creep theory was

13

originally proposed by De Jong and Verruijt (1965) and has been pursued by many

researchers, including Berry and Poskitt (1972), Zeevaart (1986), Navarro and

Alonso (2001), Mitchell and Soga (2005), and Wang and Xu (2006). It may be

assumed that under the applied stresses, the water inside the micropores of the clay

clusters will be expelled to the larger pores (or macropores). This flow of water may

cause deformation inside the clay clusters such as a reduction in spacing between the

clay minerals or a relative movement between the particles inside the clusters.

2.2.5 Structural viscosity

There is a school of thought that believes that structural viscosity has a

significant impact on soil creep. This theory is supported by Terzaghi (1941), Barden

(1969), Bjerrum (1967), Garlanger (1972), Christie and Tonks (1985), Graham and

Yin (2001) and others. Viscosity is generally defined as the resistance of a fluid to

flow or deformation under an applied stress. Fredlund and Rahardjo (1993) defined

viscosity as the frictional drag of one plate of fluid sliding over another platelet, for

example, clayey soils show viscosity in their structure at the contacts between the

soil particles, where the viscosity imposed by the absorbed water layer surrounding

the particles and may induce a plastic resistance against any relative movement

between them.

2.3 HYPOTHESIS A

In Hypothesis A, even though creep occurs during primary consolidation

(i.e. throughout the dissipation of excess pore pressures), the void ratio at the end of

primary consolidation, regardless of the thickness of the sample, is considered to be

constant. Figure 2.1 shows that Hypothesis A predicts that the thickness of the

sample, and consequently its drainage height ( ) and the time required for pore

pressure dissipation, has no effect on the location of the end of primary (EOP)

compression curve ( ).

One of the main approaches to calculating settlement supported by

Hypothesis A is the concept of creep ratio which is discussed in this section. In the

creep ratio concept, soil settlement is divided into primary consolidation (during the

14

dissipation of excess pore water pressure) followed by secondary compression (while

the remaining excess pore water pressure left is insignificant and soil settles under

almost constant vertical effective stress).

2.3.1 Creep ratio concept

2.3.1.1 Primary consolidation

As soon as the vertical effective stress begins to increase, the volume of the

soil structure begins to change. According to Mesri and Rokhsar (1974), primary

consolidation is the change in volume related to the hydrodynamic transfer of pore

pressure to the effective stress. Mesri and Feng (1991) indicated that primary

consolidation ends when the excess pore water pressure approaches a minor value,

e.g. 1kPa. According to Mesri and Godlewski (1977), the duration of primary

consolidation is related to drainage boundary conditions.

As presented by Mesri and Choi (1985), the principle equation for

estimating one-dimensional primary consolidation is as follows:

( 2.1)

where is the change in the void ratio during primary consolidation,

is the void ratio, is the elapsed time, is the vertical effective stress, stands for

the time required for primary consolidation to end, is the compressibility of

the soil at time , as a result of an increase in the effective vertical stress, and

is the compressibility of the soil structure at a vertical effective stress equal to

, due to the passage of time.

According to many researchers, although primary consolidation takes much

longer in the field than in laboratory tests and primary consolidation also consists of

creep, analyses of laboratory and field observations support the independency of the

end-of-primary consolidation void ratio ( from the duration of primary

consolidation ( (Hanrahan 1954, Lee and Brawner 1963, Adams 1965, Berre

1969, Barden 1969, Berre and Iversen 1972, Aboshi 1973, Mesri 1973, Mesri and

Choi 1985, Mesri and Feng 1991, Mesri 2001). Mesri and Feng (1991) stated that it

15

is possible to have a correlation between and and an interrelationship

between and that makes independent of . According to

Mesri (1973), the most important factor which contributes to the difference between

the coefficients of secondary compression in the field and laboratory, is the departure

from the K0-condition in the field.

Figure 2.2. Definition of , and (after Mesri et al. 1994)

Mesri et al. (1994) presented the secant compression index ( ) shown in

Figure 2.2 for the primary consolidation settlement calculation. A recompression

line from Point ( , ) with slope delineates Point P at the preconsolidation

pressure ( ). The secant compression index ( ) is defined as the slope of the lines

connecting Point P to various points on the compression curve. The value of the

secant compression index ( ), related to , can replace for the primary

consolidation settlement prediction (see Equation ( 2.4)).

Void

ratio

(e)

Vertical effective stress (logσ'v)

p

P0e

0,v

rCSlope

cCSlopev

cCSlope

16

Void

ratio

(e)

Time (t)

2.3.1.2 Secondary compression

Mesri (1973) noted that secondary compression can be determined as a

continuation of the mechanism of volume change following primary consolidation.

This mechanism consists of deformation of the individual particles and the relative

movements of particles due to normal stresses or shear displacements at contacts

induced by the shear stresses that exceed the bond shear resistance.

Figure 2.3. Consolidation stages due to one load increment

For a given time, the duration and the magnitude of secondary compression

settlement depends on the time needed to complete primary consolidation. As a result

Mesri and Choi (1985) proposed Equation ( 2.2) to estimate the secondary

compression.

( 2.2)

where is the change in the void ratio due to the secondary

compression, is the void ratio, is the elapsed time, is the effective stress, and

stands for the time required for the completion of primary consolidation.

Figure 2.3 shows the consolidation and compression stages due to one load

increment. As mentioned earlier, and as depicted in Figure 2.4, primary consolidation

Primary consolidation stage

Secondary compression stage

17

ends when the excess pore water pressure is insignificant (i.e. 1kPa according to

Mesri 2001), whereas secondary compression commences there.

2.3.1.3 Deformation prediction of clays

2.3.1.3.1 Compression formulation

Mesri and Rokhsar (1974) proposed the following equation by combining

Equations ( 2.1) and ( 2.2) to obtain the total soil compression.

( 2.3)

As stated before, calculates primary

consolidation and calculates secondary compression, which is endless.

In Equation ( 2.3), represents the phase of consolidation through which effective

vertical stress increases and is a function of the permeability of the soil and

drainage boundary conditions. Beyond the effective stress is constant and thus

there is no contribution from primary consolidation in this part. It should be

mentioned that and are not constant soil properties. As Mesri (2001)

stated, and change during the primary consolidation and secondary

compression stages, and values of throughout primary consolidation and

secondary compression are not quite the same. As Mesri (2001) stated, creep (time

dependant settlement) acts during the primary consolidation period, though, not as a

separate phenomenon because and are interconnected and both depend

on and . Since evaluating and are not readily possible during the

primary consolidation stage, Equation ( 2.3) is hardly ever used to calculate total

compression.

According to Mesri’s (2001) experiments, the secondary compression

behaviour of any soil can be calculated by considering the creep ratio, where

represents the slope of and stands for the slope of curves as

follows:

18

Pore

wat

er p

ress

ure

(u)

Time (t)

( 2.4)

where is the total vertical strain during compression, represents the

void ratio at the beginning of loading, is time that the secondary compression

should be calculated for, represents the period in which vertical effective stress

changes, which is the time required for the completion of primary consolidation.

Figure 2.4. Pore water pressure versus time

It is generally accepted that preconsolidation (overconsolidation) can reduce

secondary compression, and the degree of decrease will be a function of the degree

of overconsolidation. However, as Mesri (1973) pointed out, when stresses are less

than the preconsolidation pressure, the secondary compression coefficient increases

with increasing recompression stress and reaches its maximum at a stress level

higher than the preconsolidation pressure. Mesri et al. (1997) also indicated that as

soon as a soil is subjected to secondary compression, it builds up a preconsolidation

pressure ( ), and after reloading, soil shows a recompression response instead of

compression. Mesri (1973) also mentioned that the rate of secondary compression is

influenced by the duration to which soils were subjected to previous loads.

T

19

Figure 2.5. Dynamic viscosity of water at 1atm as a function of temperature

The other factor that can influence secondary compression is temperature.

Mesri (1973) explained that the effect of temperature on the rate of secondary

compression has been given more importance than it merits because that the

temperature increases the speed of primary consolidation by decreasing the viscosity

of free water, as shown in Figure 2.5. Mesri (1973) carried out consolidation tests on

normally consolidated organic Paulding clay at four different temperatures where

four identical tests were carried out at each temperature. The differences between the

average values at different temperatures were much smaller than the amount of

scatter in from the average value at each temperature. Figure 2.6 shows the results

of these four tests. In one test, the temperature was increased from 25˚C to 50˚C

during secondary compression and was held at 50˚ C for a number of days. There

was an instant compression, but after a couple of days the coefficient of secondary

compression at 50˚ C was equal to the coefficient at 25˚ C.

0.00E+00

4.00E-04

8.00E-04

1.20E-03

1.60E-03

0 20 40 60 80 100

Dyn

amic

Vis

cosi

ty (P

a.s)

Temperature (˚C)

20

Figure 2.6. Coefficient of secondary compression versus consolidation pressure at

different temperatures for remoulded organic Paulding (after Mesri 1973)

2.3.1.3.2 and caculations and dependency

The secondary compression index ( ) is the slope of the secant line which

connects the end of primary ( ) to each point of time on the curve

(Figure 2.9). Mesri and Godlewski (1977) mentioned that according to the existing

evidence, the measured values of for a particular target final effective stress ( )

are not a function of the load increment ratio, although experimental and

interpretational issues do arise when very small load increment ratios are applied.

When the effective stress is constant, the volume changes and excess pore water

pressures related to the tendency of the soil structure to change is more significant

compared to where effective stress increases. This makes it difficult to differentiate

between the primary and secondary stages and define the secondary compression

index ( ) (Mesri and Godlewski 1977). When different values of are obtained

from different values of load increment, then the corresponding values of would

also be different so a comparison of the relations obtained using different

load increment ratios is required. Although such data are rare, obviously depends

on the final effective stress.

Mesri et al. (1975) denoted that the secondary compression index ( )

increases as the consolidation pressure increases and at a certain level of stress it

reaches a maximum, and then it decreases as the consolidation pressure increases.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

10 100 1000 10000

Cα,

perc

ent

Consolidation pressure (kPa)

25˚ C 35˚ C 45˚ C 50˚ C

21

Figure 2.7 shows the relationship between the secondary compression index ( ) and

the consolidation pressure for undisturbed clay at Mexico City.

Mesri and Godlewski (1977, 1979) stated that the effective stresses are

stated in terms of the critical pressure taken from the curve obtained from

the results taken at the end of primary consolidation, and the term critical pressure

refers to the vertical compressive pressure at which considerable structural changes

occur in a natural soil when it is loaded. In the curve, the critical pressure

is associated to the point with the sharpest slope. The source of critical pressure, as

well as preconsolidation, could be the sustained secondary compression and

thixotropic hardening, as well as the chemical changes. Mitchell (1960) described

the thixotropic hardening mechanism as time dependent changes in the

arrangement of particles, adsorbed water structure, and allocation of ions.

Experimental evidence shows that for compressive pressures less than the critical

pressure, the secondary compression index ( ) increases with time, while

decreases with time for pressure values larger than the critical pressure (see

Figure 2.7).

Figure 2.7. Variation of the secondary compression index with the consolidation

pressure for undisturbed Mexico City clay (after Mesri et al. 1975)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

10 100 1000 10000

Seco

ndar

y co

mpr

essi

on in

dex

(Cα)


Critical pressure

22

Table 2.1. Values of for natural soil deposits (modified after Mesri & Godlewski, 1977)

Type of soil Reference

Whangamarino clay 0.03-0.04 Newland and Allely (1960)

Norfolk organic silt 0.05 Barber (1961)

Calcareous organic silt 0.035-0.06 Wahls (1962)

Amorphous and fibrous peat 0.035-0.083 Lea and Brawner (1963)

Canadian muskeg 0.09-0.10 Adams (1965)

Leda clay 0.03-0.055 Walker and Raymond (1968)

Leda clay 0.04-0.06 Walker and Raymond (1969)

Peat 0.075-0.085 Weber (1969)

Post-glacial organic clay 0.05-0.07 Chang (1969)

Soft blue clay 0.026 Crawford and Sutherland (1971)

Organic clays 0.04-0.06 Ladd (1971)

Sensitive clay, Portland 0.025-0.055 Ladd (1971)

Peat 0.05-0.08 Samson and La Rochelle (1972)

San Francisco Bay mud 0.04-0.06 Su and Prysock (1972)

New Liskeardvarved clay 0.03-0.06 Quigley and Ogunbadejo (1972)

Silty clay C 0.032 Samson and Garneau (1973)

Nearshore clays and silts 0.055-0.075 Brown and Rashid (1975)

Fibrous peat 0.06-0.085 Berry and Vickers (1975)

Mexico City clay 0.03-0.035 Mesri, et al. (1975)

Hudson River silt 0.03-0.06 Mesri, Personal files

Leda clay 0.025-0.04 Mesri and Godlewski (1977)

New Haven organic clay silt 0.04-0.075 Mesri and Godlewski (1977)

Organic Silt 0.043 Iyer (1989)

Batiscan clay 0.03 Mesri et al. (1999)

Singapore Marine clay 0.032 Mesri et al. (1999)

St. Espirit clay 0.038 Mesri et al. (1999)

La Grande clay 0.052 Mesri et al. (1999)

23

Mesri and Godlewski (1979) stated that the definition and calculation of

critical pressure has three aspects: (I) using incremental loading or a continuous

loading test to attain an relationship; (II) the curve is related to

a specific time (incremental loading test) or strain rate (constant rate of strain test)

used, and (III) for a specified curve, the practical construction method is

used to define the critical pressure.

Mesri (2001) concluded that the relationship between and time ( ) is

directly correlated to which is changing with the consolidation pressures ( ).

generally decreases, remains constant, or increases with time, in the range of

consolidation pressure where decreases, remains constant, or increases with ,

respectively. Furthermore, as Ladd (1973) pointed out, in normally consolidated

soils, remains almost constant or decreases slightly in soils with a constant .

Figure 2.8. Compression index versus consolidation pressure for undisturbed Mexico

City clay (after Mesri et al. 1975)

As stated earlier, is the slope of curve passing through each

stress point. The compression index ( ) increases as the consolidation pressure

increases, and at a certain level of stress it reaches its maximum and then decreases

as the consolidation pressure increases. Figure 2.8 shows the variation of with

0

2

4

6

8

10

12

1 10 100 1000 10000

Com

pres

sion

inde

x (C

c)


Critical pressure

24

consolidation pressure for undisturbed clay from Mexico City, as reported by Mesri

et al. (1975).

Figure 2.9. (a) Void ratio versus time; (b) void ratio versus effective vertical stress (after Mesri and Castro 1987)

Mesri and Castro (1987) explained that three or four pairs of and

values are usually enough to estimate the for one type of soil. The secondary

compression index ( ) at each compression pressure is taken from the slope of

the secant line connecting the point representing the end of primary consolidation to

a given point on the curve. The value of at the same consolidation

pressure is obtained from the slope of the curves (Mesri and Castro, 1987).

As a result, only one log time cycle of secondary compression is enough to calculate

the that corresponds to the from the curves. Some values of for

natural soil deposits are shown in Table 2.1. In all the discussions associated with the

concept of creep ratio , stands for the slope of the curve in both the

recompression and compression ranges, while the term makes a distinction

g ( ) ( )

e

tlog vlog

e)log( veEOP

cCSlope

CSlope

1,pt

2,pt

3,pt

1,C2,C

3,C

1,cC

2,cC

3,cC

1,v 2,v 3,v

)(a )(b

1,v

2,v

3,v

25

between the recompression index , and the compression index (Mesri and

Castro, 1987). As Table 2.1 shows, the values of change between 0.025 and 0.1

for different types of soils.

Figure 2.9(a) depicts the void ratio changes versus time ( ) curves,

where the curves show changes of in relation to time. Figure 2.9(b) depicts the

changes in the void ratio versus effective vertical stress ( ) curves. The

curves were drawn for different specified times normalised against the

time corresponding to the end of primary consolidation ( ). The compression index

( ) can be readily obtained by measuring the slope of the void ratio versus vertical

effective stress ( ) curve corresponding to the final vertical effective stress

( ) that settlement should be calculated for. Furthermore, can be calculated by

measuring the slope of the void ratio versus time ( ) curve corresponding to

the time the settlement should be calculated. As a graphical demonstration, three sets

of and are depicted in Figure 2.9.

( 2.5)

and ( 2.6)

such that ( 2.7)

Although the ratio varies with the type of soil, it was assumed constant

for each soil and can be estimated by carrying out three or four consolidation tests,

where is measured during the test using a pore water pressure transducer. By

measuring these parameters the secondary compression can be calculated. The

benefit of using this method is that long secondary compression tests are not required

to be conducted in the laboratory.

2.3.1.4 Reduction of secondary compression using surcharge

As explained earlier, preloading indicates the load application that is equal

to or greater than the total proposed load over the site prior to constructing the

structure. Preload is used to compress the underlying soil and can be partially or fully

removed after the desired settlement has occurred. Surcharging is a type of

preloading where the applied load is greater than the structural load to reduce the

26

preloading time or to minimise post-construction compression. Surcharging is

generally carried out to reduce post-construction settlement, to improve shear

strength, and to reduce the rate of secondary compression in the field. For example, a

surcharge can be used where secondary compression turns out to be significant

during the useful life of a structure and vertical drains are used to accelerate primary

consolidation.

Figure 2.10. Total and effective surcharge ratios definition (after Feng 1991)

Generally, the higher the surcharge load, the less surcharging time is needed

to achieve a desired settlement, but the bearing capacity of the soil places an

important restriction on the maximum surcharge load that can be applied. The

surcharging time is the duration that surcharge fill will be in place before being

removed. As Mesri and Feng (1991) pointed out, the surcharge effort is stated in

terms of total surcharge ratio ( ) and surcharging time ratio ( ).

( 2.8)

where , is the final effective vertical stress after

removing the surcharge, is the total surcharge pressure, is the surcharging time, and is time to end of primary (EOP) compression under surcharge.

Com

pres

sion

, Reb

ound

Vertical effective stress (logσ'v)

27

Mesri and Feng (1991), on the other hand, stated that the surcharging effort

can be expressed using the effective surcharge ratio , as shown in Equation ( 2.9).

( 2.9)

where is the maximum vertical effective stress reached immediately

before removing the surcharge.

Figure 2.11. values in terms of and , for (after Feng 1991)

According to Equations ( 2.8) and ( 2.9), while the surcharging time ratio is

equal to 1, then , but the effective surcharge ratio ( ) is greater than the

total surcharging ratio ( ) when the surcharging time ratio ( ) is greater than one

and vice versa. Mesri and Feng (1991) mentioned that unless very large values of

total surcharging ratio are utilized, surcharging may be ineffective at any time and

depth in soil where and where . Moreover, surcharging may not be

economical when corresponds to because the most practical values

of can be selected in a way that . When , the

effective surcharge ratio is not a constant within the soil profile and an average

effective surcharge ratio ( ) is used in post-surcharge secondary compression

analysis. Moreover, Feng (1991) stated that for soil elements where the vertical

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.6

For surcharging efforts

located below this line,

for

28

effective stress was smaller than the final effective stress when the surcharge was

removed, the soil elements will continue to compress as a result of primary

consolidation and secondary compression. In this case, surcharging has no effect on

the secondary compression characteristics of soil. The definitions of total and

effective surcharge ratios are shown in Figure 2.10.

Feng (1991) proposed that to calculate for a layer of soil, Terzaghi’s

theory of consolidation for a single layer can be used to develop a relationship

between , and (Figure 2.11). For given loading conditions (i.e. and

), by definition , and . In Figure 2.11, curve

AB signifies that less surcharging effort was needed to reduce secondary

compression. For surcharge cases below curve AB, not only will some primary

consolidation remain after the surcharge has been removed, secondary compression

will occur in its full rate (i.e. ) under final effective stress. Moreover, by

analysing the excess pore water pressure isochrones to ensure zero post-surcharge

primary consolidation for any given value of , the minimum value of must be

0.33. As a result, the area between the line and the curve AB in

Figure 2.11 is a transition zone for surcharging resulting in some post-surcharge

primary consolidation as well as reduced post-surcharge secondary compression.

While , not only no post-surcharge primary consolidation remain, there

will also be a significant reduction in the secondary compression.

Mesri and Feng (1991) stated that removing the surcharge leads to a

rebound consisting of the primary rebound up to , a secondary rebound which

levels off at , followed by a post-surcharge secondary compression. and were

measured from the time the surcharge load ( ) was removed. Mesri et al. (1994)

pointed out that the time needed for a primary rebound depends on the rebound

characteristics of the soil and the permeability and drainage boundary conditions of

the soil. Consequently, the duration of primary rebound was computed from a time-

rate of rebound analysis, while the the secondary rebound duration was calculated

from the empirical correlation between and .

29

Feng (1991) stated that when soil is over consolidated due to mechanical

unloading, primary consolidation is substituted by primary rebound followed by

secondary rebound. However, secondary compression still exists but its magnitude

has probably decreased due to the decreasing vertical effective stress. Consequently,

while the effective surcharge ratio is small, primary and secondary rebounds are

small while secondary compression is predominant, and post-surcharge secondary

compression reappears shortly after the surcharge is removed. By increasing the

effective surcharge ratio, not only does secondary rebound increase, secondary

compression decreases further, whereas increasing the effective surcharge ratio

means that the reappearance time of post-surcharge secondary compression is

prolonged. As a result of the above, a large increase in will appear with an

increase in . This means that increasing the effective surcharge ratio will postpone

the reappearance of secondary compression or increase the secondary rebound time.

Feng (1991) indicated that the time to end primary rebound ( ) is

proportional to the magnitude of the effective surcharge ratio ( ). It should be

noted that by definition , where OCR is the overconsolidation ratio.

Consequently, it is expected that the reduced time to reach the end of primary

rebound ( ) increases with an increase in the magnitude of the effective surcharge

ratio. However, it is expected that the rate of increase at the end of primary rebound

( ) will be insignificant for values between 0.2 and 1.0, whereas the rebound

index ( ) and permeability of the soil will change slightly. However, the

reappearance time for the post-surcharge secondary compression ( ) depends mostly

on the magnitude of .

As Choi (1982) stated, where the surcharge load is removed at the end of

primary consolidation ( ), may be approximately estimated using

Terzaghi’s theory of one dimensional consolidation by an appropriate selection of the

coefficient of swelling ( ). On the other hand, where compression

reappears very quickly after the surcharge is removed, and the rebound is small, in

fact, part of the soil near the drainage boundaries is rebounding while the remainder

is compressing. The rebound rate and compression rate are estimated by and ,

respectively, and the magnitude of each mode of deformation is estimated by the

coefficients of compressibility and . A theory is needed to estimate the time-

30

rate of settlement after the surcharge is removed, so the model must consider changes

in and with an increase in OCR during the rebound.

Figure 2.12. Elapsed times definition used to estimate post-surcharge secondary compression (after Feng 1991)

Mesri et al. (1994) proposed Equation ( 2.10) for calculating post-surcharge

secondary compression.

( 2.10)

where is the time that post-surcharge secondary compression reappears

after the surcharge is removed, and is a post-surcharge secant secondary

compression index from to any time t ( ). Mesri et al. (1994) mentioned

that the post-surcharge secondary compression can be explained and predicted by the

law of compressibility, so post-surcharge secondary settlement can be estimated

using Equation ( 2.11).

( 2.11)

where corresponds to . For any soil, the and values at on the

compression curve, in combination with , can be used to calculate from

Figure 2.13.

Com

pres

sion

, Reb

ound

31

Mesri et al. (1994) stated that it is generally expected that is small

initially and then gradually increases, becomes constant, or decreases with time. The

common shape of any recompression or compression EOP curve proposes

that should decrease when the compression time is very long (time approaching

infinity). The post-surcharge secondary compression may be expected to begin with

a less than 1.0 and then will automatically enlarge until the recompression

curve merges with the compression curve. The succeeding variations of with time

are related to the variations of with .

Figure 2.13. Post-surcharge secondary compression index in terms of (after Mesri et al. 1994)

Since is not constant with time, for a practical settlement analysis, a

secant is described from the time that post-surcharge secondary compression

begins to any time where post-surcharge secondary compression is to be calculated.

The definition of elapsed time used to estimate the post-surcharge secondary

compression is shown in Figure 2.12. As Feng (1991) denoted, investigations show

that both and are functions of effective surcharge ratio ( ) or in other words,

both and may be estimated when is known. Ladd (1973) reported that the

0

0.1

0.2

0.3

0.4

0.5

1 10 100 1000 10000

Soft Clays and Silts

32

post-surcharge secondary compression data showed that decreased from 1 to 0.1

as the effective surcharge ratio increased from 0.07 to 0.5.

Figure 2.14. Ideal model of surcharging to reduce secondary compression (after Mesri et al. 1994)

Feng (1991) explained that Figure 2.14 is an ideal model for the reduction

of secondary compression after the surcharge is removed. The solid curve represents

the relationship of without surcharging, whereas, the dashed curve represents

the relationship of for the soil upon reloading while the soil is loaded to the

Loading without surcharge

Reloading after previously loaded to

33

surcharging pressure ( ) and then unloaded to the final structure pressure ( . As

a result, at , the value of is reduced to by surcharging. It should be

mentioned that the key assumption in Figure 2.14 is that the primary consolidation is

achieved under the surcharge pressure before the surcharge is removed.

Choi (1982) pointed out that surcharging does not always lead to a reduction

in the secondary compression index ( ) because if a soil is loaded slightly more

than its preconsolidation pressure and then unloaded, upon further reloading can

go beyond within a certain pressure range. Choi (1982) indicated that, while

natural soft clays were considered, can exceed unity, but when soil is loaded

beyond its preconsolidation pressure, the main structural change commences and

and increases quickly. When soil is unloaded from to it experiences

secondary compression at at a rate related to when the main structural changes

began, but if a surcharge is not applied, the main structural change of the particles

has not yet begun at . Consequently, under and without surcharge may

be smaller than the value after the surcharge.

Mesri (2001) proposed Equations ( 2.12) and ( 2.13) as the empirical

correlations between and for inorganic clays and peats, respectively. It should

be mentioned that the values of for peats are much smaller than those for soft

clay and silt deposits. According to Mesri (2001), it would appear that the

fundamental tendency for rebound to the tendency for compression is less in peats

than in clays, indeed a significant part of the pore water in fibrous peat fabric is held

as free water outside and within the particles so the free pore water squeezed out by

loading has no physico-chemical tendency to return to the peat fabric after an

unloading process.

( 2.12)

( 2.13)

Mesri (2001) indicated that while is small, is also small and thus post-

surcharge secondary compression appears soon after removing the surcharge, while

increases rapidly with time, but when is large, is also large and thus the

34

post-surcharge secondary compression appears long after removing the surcharge,

while increases slowly with time.

Based on a large number of surcharge tests (5 to 6 values of in the range

of 0.2 to 1.0) carried out by Mesri and Ajlouni (1997) on six undisturbed specimens

of soft clay and silt deposits, a typical value of was determined, in fact,

is not a constant and increases with the reduction in , and since approaches

zero, the secondary compression rate after unloading is the same as the continuation

of the secondary compression rate.

Hypothesis A has a huge experimental base and has been used reliably for

many years by practising engineers. Since most of the creep ratio concept equations

and parameters’ relations are empirical, the correct way of using them needs careful

study. Although the creep ratio concept is practical, it has limitations such as a lack

of excess pore water pressure simulation and difficulties with numerical calculations

when the nonlinear variation of permeability and the void ratio influence the

dissipation rate of excess pore water pressure. Furthermore, this lack of excess pore

water pressure simulation creates difficulties in predicting the end of primary

consolidation when removing the preloading. Furthermore, Mesri et al. (1975)

reported that the compression index used in the creep ratio concept as a function of

the vertical effective stress and time during consolidation, makes it difficult to

determine the compression index.

2.4 HYPOTHESIS B

Hypothesis B assumes that since the viscous behaviour of soft soils occurs

during primary consolidation, the void ratio at the end of primary consolidation

( ) cannot be equal for thick and thin layers of soil. A number of constitutive

models that support Hypothesis B are presented in the literature, and for the sake of

classification they can be divided into three categories: empirical models, rheological

models, and general stress-strain-time models. These models are explained in the

next section.

35

2.4.1 Empirical models

Empirical models are generally presented by closed form solutions or

differential equations because they are mostly based on fitting experimental results

from creep, stress relaxation, and constant rate of strain tests. Furthermore, to the

extent that the boundary conditions comply with laboratory experiments, the

empirical models may provide practical solutions to engineering problems. Empirical

constitutive models are usually attained by directly fitting the experimental test data

with mathematical functions. The empirical models reviewed in detail were selected

for at least one of two reasons: (i) they present major steps for understanding the

physics of secondary compression; (ii) they raise major unresolved issues for

consolidation analysis.

2.4.1.1 Taylor and Merchant’s model

Taylor and Merchant (1940) suggested a revision to Terzaghi’s (1923)

consolidation theory to include secondary compression effects, in fact their

consolidation theory was the first to contain a time dependant effective stress model,

while their method of deriving the consolidation equation has subsequently been

used by many researchers.

The same flow equation as in Terzaghi’s consolidation theory is used in

Talyor and Merchant (1940). The following equation defines the change in volume

of a soil element by dissipating the excess pore water pressure:

( 2.14)

where is the void ratio, is the coefficient of permeability, is the unit weight

of water, and stands for excess pore water pressure.

The following assumptions were adopted to obtain the above equation:

- The soil is homogeneous and saturated,

- The soil skeleton and the flow of water are compressed in a vertical direction,

- The pore water and solid grains are incompressible,

- Darcy’s law describes the flow of water, and the permeability is constant,

36

- Only small changes in the void ratio were considered because the deformations

are small,

- The change in the void ratio is proportional to the change in the applied vertical

effective stress (i.e. ), and

- is constant over a load increment for small changes in the void ratio.

Taylor and Merchant’s (1940) consolidation theory differs from Terzaghi’s

theory in the last two assumptions; Taylor and Merchant (1940) assumed that the

compression of an element of the soil skeleton is a function of effective stress and

time. Consequently, the void ratio of soil can be defined by Equation ( 2.15).

( 2.15)

From Equation ( 2.15), the rate of change in the void ratio for a soil element

can be explained by Equation ( 2.16).

( 2.16)

Referring to Equation ( 2.16), part of the decrease in the void ratio stems

from instantaneous consolidation (the first term), while the other part is the result of

secondary compression under a constant vertical effective stress (the second term).

Taylor and Merchant (1940) assumed that for any value of vertical effective stress

there is a unique final value of deformation corresponding to a linear relationship

between the void ratio and vertical effective stress. In Figure 2.15, the line related to

the final value of the void ratio under current loading is defined by segment AB with

the slope of .

Taylor and Merchant (1940) also hypothesized that instantaneous

compression was proportional to the vertical effective stress. Segment AG with the

slope of in Figure 2.15 defines this component. As a result, the instantaneous

compression can be calculated by Equation ( 2.17).

( 2.17)

Taylor and Merchant (1940) suggested that hydrodynamic retardation

results in the ( , ) point on Figure 2.15 following an ACB curve as the effective

stress increases from to with time, but at some time throughout the

consolidation process, the total deformation would correspond to the change in the

37

void ratio from Point A to Point C. The contribution of instantaneous compression

corresponds to the difference in the void ratio between Point A and Point E, and the

contribution of delayed compression corresponds to the difference in the void ratio

between Point E and Point C.

Lastly, Taylor and Merchant (1942) assumed that the rate of secondary

compression at Point C may be proportional to the residual compression needed to

reach the value of final deformation under the current vertical effective stress (i.e. the

distance between Points C and D). As a result, the rate of secondary compression can

be calculated by the following equation:

( 2.18)

where is a constant.

Substituting Equations ( 2.17) (instantaneous compression rate) and ( 2.18)

(secondary compression rate) in Equation ( 2.16) results in the model equation for a

soil element, as follows:

( 2.19)

The void ratio and vertical effective stress in a layer of soil differ with depth

( ), and actually, this expression for relates to the partial derivative in the flow

equation (Equation ( 2.14)). Moreover, for a constant total load, the variation of

vertical effective stress relates to the dissipation of excess pore pressure (

). Thus Taylor and Merchant’s (1940) equation can be presented as:

( 2.20)

where and it is assumed to be constant.

An experimental evaluation of the precision of Taylor and Merchant’s

(1940) theory of consolidation necessitates considering the variation of the void ratio

with the variation of vertical effective stress throughout the period of pore pressure

dissipation. Taylor (1942) carried out oedometer tests with excess pore pressure

measurement at the base of the specimen and found that the results were

encouraging.

38

Figure 2.15. Relationship between void ratio and vertical effective stress throughout the consolidation process (after Taylor and Merchant 1940)

2.4.1.2 Suklje’s model

The model presented by Suklje (1957) is based on a system of void ratio-

effective stress lines at constant strain rate, named isotaches. The void ratio and

effective stress used in this theory are the average values over the thickness of a

sample or a soil layer. To calculate the average effective stress, it is assumed that

pore pressure isochrones are parabolic. Throughout secondary compression, the

maximum excess pore pressure was calculated by Buisman’s (1936) equation for

settlement. Suklje (1657) assumed that the same expression for pore pressure is valid

during primary consolidation when the coefficient of secondary compression is

replaced by instantaneous slope of the compression curve. If we consider these

assumptions, then the compression curves for several load increments on a sample

can be transformed into isotaches. For the settlement of a clay layer, the slope of the

compression curve is iterated in two steps: (i) for a given slope, calculate the average

strain rate and maximum pore pressure (and therefore the average effective stress),

and (ii) the current void ratio must coincide with the value on the corresponding

isotache at the current effective stress.

Void

ratio

Vertical effective stress

Voi

A

B

D C

E

G D

1

1

39

Suklje (1957) stated that since isotaches are derived from experimental

compression curves for several load increments, the variation of permeability

throughout consolidation is considered implicitly. Suklje (1957) also presented a

graphical construction of the consolidation curves for layers whose thicknesses were

several orders of magnitude larger than the thickness of the sample. The

consolidation curves gradually merged into the same linear relationship,

and were compared to solutions based on Terzaghi’s theory of consolidation. It

should be mentioned that the experimental excess pore pressures did not agree with

the computed values based on the slope of the experimental consolidation curves.

2.4.1.3 Bjerrum’s model

The effects of time on the compressibility of clay in terms of secondary

compression were first formulated by Buisman (1936), who presented the results of

long oedometer tests on clay and peat along with the settlement records of a road

embankment and a levee. According to laboratory tests, Buisman (1936) found that

settlements and the logarithm of time were linearly correlated after about one day for

the clay and about one minute for the peat. Moreover, settlements of a road

embankment and a levee throughout the period of observation (about two years)

linearly increased with the logarithm of time. Based on an assumption that settlement

is proportional to the load, Buisman (1936) suggested Equation ( 2.21) to estimate the

soil settlement for values of time corresponding to the log-linear part of the

settlement diagram.

( 2.21)

where is the soil vertical strain, is the load increment, is time in min,

represents the instant deformation per unit thickness and unit load, and stands

for secondary compression rate per unit thickness and unit load.

Buisman (1936) assumed the superposition of time dependant settlement for

staged loading where secondary compression equals the sum of secondary

compressions of previous loadings, while each loading is considered individually.

Furthermore, Buisman (1936) pointed out that, as defined in Terzaghi’s (1923)

consolidation theory, typical compressibility consists of the direct load and time

dependant effects. The conclusions of the proposed method are as follows:

40

- Based on experiments, there is a linear relationship between the logarithm of

time and soil deformation except for that time immediately after loading,

- The slope of the deformation diagram ( ) is proportionally related to the

applied load, and

- The superposition of loads results in a superposition of the deformation

diagrams.

Following Buisman (1936), Taylor (1942) reported that there is a family of

curves called “time lines” for the one-dimensional compression of clay. The

logarithm of effective stress versus the void ratio diagram was used to define the

time lines, with each line corresponding to a given duration of loading. At the end of

primary (0.1 day), 1, and 10 days, the time lines can be gained in the laboratory. By

assuming that settlements indefinitely increase linearly with the logarithm of time

throughout secondary consolidation, the 1 year and 10 year time lines can be

obtained by extrapolation.

Bjerrum (1967) denoted that a layer of clay deforms when a deposit is

constructed and then settlement continues as a function of time under a constant

vertical effective stress. Thus, the compressibility of soil cannot be defined by a

single curve in a logarithm of vertical effective stress versus a void ratio.

Bjerrum’s (1967) theory of time dependant compressibility stands on two

baselines: (i) a system of parallel time lines or curves can be defined in a logarithm

of effective vertical stress versus void ratio figure. A sample of the time lines system

is depicted in Figure 2.16, where each line represents the void ratio equilibrium for

various values of vertical effective stress at a given time of a sustained loading; (ii) a

unique relationship between vertical effective stress, void ratio, and time is defined

by the time lines system.

Bjerrum (1967) stated there is an equivalent time of load suspension and a

certain rate of delayed consolidation for any specific value of vertical stress and void

ratio, which is independent of the way clay have reached these values. This means

the delayed component of the compression of a soil at any given void ratio and

vertical effective stress can be identified based on the initial void ratio, vertical

effective stress, and the fixed system of lines (or the age of the clay deposit).

41

Figure 2.16. Effect of sustained loading on results of oedometer tests (aging effect) (after Bjerrum 1967)

Consequently, soil settlement is divided into two parts: an instant

compression that occurs concurrently with an increase in the vertical effective stress,

and a delayed compression. Instant and delayed compressions are alternatives to

primary consolidation and secondary compression which divides soil settlement into

two components that cover the before and after dissipation of excess pore water

pressure. Figure 2.17 shows a comparison between the definition of instant and

delayed compressions, as well as primary consolidation and secondary compression.

Bjerrum (1967) supported his theory by: (i) comparing the overburden

pressure of a clay deposit in Drammen with the preconsolidation pressure profile; (ii)

the preconsolidation pressure increased due to sustained loading in the laboratory;

(iii) the deformation records and analyses of six buildings in Drammen. Nearly 40

standard oedometer tests were carried out on deposits of Drammen clay to obtain a

profile of critical pressure, which is the pressure at yielding in standard oedometer

tests, versus the maximum load applied in the past. Based on geological studies, the

clay was normally consolidated and also subjected to the existing overburden

pressure for almost 3000 years. The critical pressure profile showed that delayed

compression produced a 60% increase in the critical pressure compared to the

present overburden pressure profile.

1.1

1.2

1.3

1.410 100

Void

ratio

Vertical stress (kPa)

28 days delayed consolidation 1 hour

5 hours 1 day

4 days 28 days

Instant compression

42

Figure 2.17. Definition of instant and delayed compression compared with primary consolidation and secondary compression (after Bjerrum 1967)

Bjerrum (1967) stated that the increase in the critical pressure due to

delayed compression is called the ageing effect. Figure 2.16 shows how an

oedometer sample was loaded in steps over a period of 24 hours each. A loading of

130 kPa, which was sustained for 28 days, increased the critical pressure by 23% to

160 kPa. Various combinations of step loading always results in the same time lines

system, thus this time lines system is unique.

2.4.1.4 Garlanger’s model

Garlanger (1972) was the first to propose a numerical model to consider the

effects of time on yielding. His model was developed based on Bjerrum (1967) time

effe

ctiv

e st

ress

Time

Com

pres

sion

Instant

Primary

Delayed

Secondary Period of pore water pressure dissipation No excess pore water pressure

43

lines. Garlanger (1972) defined the rate of variation in the void ratio of a soil element

throughout consolidation by Equation ( 2.22).

( 2.22)

By referring to the time lines (Figure 2.16), Garlanger (1972) expressed the

change in the void ratio due to a loading from to by Equation ( 2.23).

( 2.23)

where is a reference time line that Garlanger (1972) assigned to the

instant compression line in the theory of time lines, while is time measured from

the moment of the load application. It was assumed that the final value of the vertical

effective stress ( ) was larger than the critical pressure ( ) which is located on the

instant compression line.

As suggested by Hansen (1969), Garlanger (1972) decided to use a linear

relationship between the logarithm of void ratio, the logarithm of effective stress, and

the logarithm of time.

( 2.24)

with , , and replacing , , and , respectively. As Garlanger (1972)

stated, the first two terms on the right side of Equation ( 2.24) represent instant

compression and the last term represents secondary compression. Furthermore, for

any point ( ) on the left of the instant compression line (refer to Figure 2.16),

the instant compression rate would be as follows:

( 2.25)

While on the instant compression line, the instant compression rate would

be:

( 2.26)

To obtain the rate of delayed compression, Equation ( 2.24) can be

differentiated with respect to time. Equation ( 2.27) shows the rate of delayed

compression:

( 2.27)

where,

44

( 2.28)

It should be mentioned that Equation ( 2.27) defines the delayed

compression only throughout the secondary compression, so to derive an equation to

capture the rate of delayed compression at any time during consolidation (i.e. while

vertical effective stress is increasing), Garlanger (1972) made supplementary

assumptions.

Based on time line theory, each combination of overburden pressure and

void ratio relates to an equivalent time of sustained loading and a certain rate of

delayed compression. At a given point ( ), the equivalent time of sustained

loading relates to the duration of creep under the final vertical effective stress needed

to reduce the void ratio from an instant line to the current value of the void ratio.

Substituting for in Equation ( 2.24) and solving it for results in an equation for

the equivalent time of sustained loading that can be related to any combination of

void ratio and vertical effective stress.

The rate of delayed compression throughout secondary compression is a

function of sustained loading time. This relationship was derived by differentiating

Equation ( 2.24) with respect to time. Garlanger (1972) proposed Equation ( 2.29) by

substituting instant and delayed compression rate equations into Equation ( 2.22).

( 2.29)

The numerical solution proposed by Garlanger (1972) for a load increment

applied to a clay layer consisted of three equations: (i) The flow equation of

Terzaghi’s consolidation theory; (ii) Equation ( 2.22) for the rate of change in the

void ratio in a soil element (Equations ( 2.25) and ( 2.26) were used to obtain the rate

of instant compression); and (iii) Equation ( 2.27) was used to calculate the rate of

delayed compression. Dimensionless terms were presented in these equations which

were substituted by their finite difference calculations and considered concurrently to

estimate changes in the void ratio and the excess pore water pressure with time

during consolidation.

45

A numerical solution was used to simulate oedometer tests on samples of

various thicknesses and to calculate the settlement of three buildings. As Garlanger

(1972) explained, the model parameters for laboratory tests, particularly for

parameters and , were obtained by fitting the compression curves under a

constant vertical effective stress. Garlanger (1972) indicated that the rate of delayed

compression attained in the laboratory over a short period of time may overestimate

the actual rate of delayed compression in the field. As a result, it is suggested to

assume year and to deduce the creep rate for computation from the amount

of delayed compression in the field and the age of the clay.

A unique feature of Garlanger’s (1972) model may be explained as a

hardening of the instant component of compression resulting from either instant or

delayed compression. It should be mentioned that this feature is consistent with the

theory of time lines. The definition of the rate of delayed compression applied by

Garlanger (1972) was partly based on the theory of time lines, and it resulted in

excessive values of the rate of delayed compression throughout the duration of pore

water pressure dissipation. Garlanger (1972) stated that the creep rate calculated

from Equation ( 2.27) could be larger than the total rate of change in the void ratio

allowed by the drainage which causes the excess pore water pressure to remain

constant over the time step. This point may result in some concerns about the

performance of the model.

2.4.1.5 Kabbaj’s model

Bjerrum’s (1967) theory of time lines was used by Kabbaj (1985) as the

physical basis of the model. Kabbaj’s (1985) approach consisted of the influence of

the strain rate on the value of preconsoldiation pressure. Kabbaj (1985) assumed that

total compression is a combination of instant (elastic) compression and delayed (time

dependant plastic) compression, so the total strain rate was proposed as:

( 2.30)

where is elastic compression and is time dependant compression. The

rate of instant elastic compression is given by:

( 2.31)

where is the initial void ratio, and is the model parameter.

46

Kabbaj (1985) used the description of clay behaviour proposed by Leroueil

et al. (1985) to derive a model of time dependant plastic compression. Based on

Leroueil et al. (1985) proposal, the following relationships can describe the

behaviour of clay:

- A relationship between preconsolidation pressure and strain rate, and

- A relationship between strain and normalised effective stress. Normalised

effective stress is assumed to be the value of effective stress divided by the

preconsolidation pressure (the normalised stress strain curves related to constant

values of strain rate).

Kabbaj (1985) mentioned that the proposed description of clay behaviour by

Leroueil et al. (1985) is the result of observations in the normally consolidated range

which comprise mostly plastic strain. Consequently, Kabbaj (1985) proposed the

observations as the time dependant plastic behaviour. A linear relationship between

the logarithm of preconsolidation pressure and the logarithm of plastic strain rate,

and also a piece-wise linear relationship between the logarithm of effective stress and

the plastic strain were assumed by Kabbaj (1985).

( 2.32)

( 2.33)

where , , , and are experimental constants.

To obtain the plastic strain rate, the preconsolidation pressure in Equation

( 2.32) was eliminated using Equation ( 2.33).

( 2.34)

Substituting the expressions of the elastic strain rate and the rate of time

dependant plastic strain in Equation ( 2.30) leads to the model equation by Kabbaj

(1985).

( 2.35)

47

2.4.1.6 Yin’s model

An elastic visco-plastic model was originally proposed by Yin and Graham

(1989) based on Bjerrum’s equivalent time line concept which can calculate the

behaviour of soil measured in the field and the laboratory. According to Yin and

Graham (1989), the strain at any given vertical effective stress can be calculated by

the following equation:

( 2.36)

where is the strain at the reference point, is a material parameter

describing the elastic stiffness of the soil where is the specific volume

, and is the initial void ratio, and are material properties, is defined by

the slope of creep strain plotted versus , and is the equivalent time.

According to Yin and Graham (1989), equivalent time can be defined at the end of

standard oedometer test. Based on Equation ( 2.36), the equivalent time can be

estimated by Equation ( 2.37).

( 2.37)

While the incremental strain rate can be written as follows:

( 2.38)

where is a material parameter describing the elastic stiffness of the soil.

The elastic visco-plastic model can be obtained by substituting Equation ( 2.37) into

Equation ( 2.38).

( 2.39)

Considering Equations ( 2.40) and ( 2.41), Equation ( 2.39) can be rewritten

as Equation ( 2.42).

( 2.40)

( 2.41)

( 2.42)

48

Figure 2.18. Creep oedometer tests in Batiscan Clay (after Leroueil et al. 1985)

The logarithmic creep function will provide an infinitive creep settlement as

the creep time approaches infinity. The oedometer tests revealed that (such as

Figure 2.18) the relationship between strains (void ratio) and logarithmic of time was

not a straight line, so Yin (1999) proposed a new creep function to describe the creep

behaviour of soils. The proposed nonlinear creep function is expressed as follows:

( 2.43)

where,

( 2.44)

where is the creep strain limit, and is the initial value of at

2.4.2 Rheological models

Rheological models consist of arrangements of springs, dashpots, and

sliders to represent soil behaviour (i.e. elastic, viscous, or plastic behaviour).

Rheological models were developed by Gibson and Lo (1961), Wahls (1962), Barden

(1965), Aboshi (1973), and Rajot (1992) to study particular aspects of time-

dependent soil behaviour (i.e. pore pressure dissipation and settlement rates

throughout the early stages of the consolidation process and secondary compression).

67kPa 78kPa

90kPa 98kPa

109kPa

121kPa 133kPa 139kPa 151kPa

0

5

10

15

20

25

0.1 1 10 100 1000 10000 100000 1000000

Verti

cal s

train

(%)

Time (min)

49

These models cannot capture the effects of preconsolidation pressure, which means

that using these models requires calculating various unfamiliar parameters whose

values cannot be determined directly from standard laboratory tests. Consequently,

rheological models are not widely used.

2.4.2.1 Gibson and Lo’s model

For soils with substantial secondary compression, a lot of deformation

occurs after the excess pore water pressure has dissipated. A modification of

Terzaghi’s (1923) consolidation theory was proposed by Gibson and Lo (1961) to

calculate secondary compression, and it may be considered as the first conceptual

approach to estimate secondary compression. Gibson and Lo (1961) considered the

same assumption as Terzaghi’s (1923) theory to attain the flow equation (Equation

( 2.45)):

( 2.45)

Gibson and Lo (1961) assumed that a linear spring with a Kelvin element

could be used to model the compressibility of the soil skeleton such that when

vertical effective stress increases, the linear spring generates an instantaneous

compression that represents primary compressibility. The response of the Kelvin

element, retarded by the viscosity of the dashpot, relates to the secondary

compression. The compression of the linear spring ( ) for an increase of vertical

effective stress is as follows:

( 2.46)

where is the compressibility of the linear spring at a given time .

This increase in the vertical effective stress is also tolerated by the Kelvin

element where the linear spring and linear dashpot each carry part of the load, and

the dashpot carries the load related to the strain rate. As a result, compression of the

Kelvin element ( ) due to a loading is as follows:

( 2.47)

where is the compressibility of the spring, and is the viscosity of the dashpot.

Equation ( 2.47) can be integrated formally. The total compression of the

model at time when it is subjected to an effective stress equal to is as follows:

50

( 2.48)

where is time variation between 0 and . Derivation of Equation ( 2.48)

with respect to time gives the strain rate in a soil element.

( 2.49)

For a load increment applied at a time , the change in the vertical

effective stress is a function of excess pore water pressure ( ).

Combining Equations ( 2.45) and ( 2.49) leads to Gibson and Lo’s (1961)

consolidation equation, as follows:

( 2.50)

The Laplace transform method was used by Gibson and Lo (1961) to solve

the equation. Gibson and Lo (1961) verified their theory by simulating long term

oedometer tests and found that a comparison to the experimental data was

satisfactory.

2.4.2.2 Wahls’ model

Wahls (1962) carried out several series of incremental oedometer tests (each

one with a given value of load increment ratio) on remoulded and undisturbed

samples of calcareous organic silt from southwest Chicago. Calcareous organic silt

was selected due to its remarkable secondary compression. To determine the

secondary compression characteristics, each load increment was applied for

sufficient time, with a minimum duration of two days. As a result, after a period of

time, soil deformation becomes linearly related to the logarithm of time. Thus,

secondary compression may be defined by the rate of change in the void ratio per log

cycle of time ( ). Wahls (1962) pointed out that the rate of secondary

compression increased during the primary consolidation and eventually reached the

rate during secondary compression.

Based on experimental results, Wahls (1962) drew the following

conclusions about the secondary compression rate:

51

- The rate of secondary compression ( ) is independent of the load increment

and the load increment ratio, but it depends on the void ratio and as a result on

the effective stress.

- represents the maximum rate at which secondary compression takes place

throughout the load increment

- The time required for the rate of secondary compression to reach is

correlated with the time required for the completion of primary consolidation.

Walhs (1962) suggested a method to calculate the magnitude of primary

consolidation for deformation curves without an inflection point, based on fitting the

settlement curve with a theoretical model. The analytical solution of Terzaghi’s

consolidation equation was used as a reference to model primary consolidation by an

infinite series of Kelvin elements, while an infinite series of nonlinear dashpots was

used to model secondary compression.

2.4.2.3 Barden’s model

Barden (1965) stated there are three experimental facts that do not agree

with Terzaghi’s (1923) consolidation theory: (i) the rate of mid-plane pore pressure

dissipation in the early stages of the oedometer tests was more than predicted

according to the value of the consolidation coefficient subtracted from the

compression curves, and (ii) creep also occurs during primary consolidation, and (iii)

the total settlement was affected by the loading conditions. Consequently, Barden

(1965) proposed a system with a non-linear spring and dashpot (Figure 2.19a).

Barden (1965) simplified his model by assuming a linear spring, and by adopting

Kelvin’s element (Figure 2.19b).

The load increment ( ) is carried by the linear spring, the dashpot, and the

excess pore pressure.

( 2.51)

where is the initial value of void ratio, is the void ratio, is the

compressibility of the linear spring, is the non-linear viscous resistance of the

dashpot, and is the excess pore water pressure at a particular time. According to

52

the power law of Ostwald, the non-linear viscous resistance of the dashpot can be

defined as follows:

( 2.52)

where and are model parameters.

Since the load is transferred from the dashpot to the linear spring, the void

ratio will reach a final value , which is:

( 2.53)

Substituting Equation ( 2.53) in Equation ( 2.51) and applying the viscous

resistance of the dashpot results in Barden’s (1965) model equation for a load

increment.

( 2.54)

(a) (b)

Figure 2.19. Rheological models proposed by Barden: (a) Barden’s proposed non-linear model, and (b) model solved by Barden (after Barden 1965) (Note: N and L

stand for non-linear and linear, respectively)

It should be noted that the soil skeleton yielding and non-linear stress-strain

behaviour of soils are not considered in Barden’s (1965) model because the final

value of compression must be defined, which means that the secondary compression

is in a non-linear relationship with the logarithm of time (Equation ( 2.53)).

L N N N

53

2.4.2.4 Aboshi’s model

Aboshi (1973) presented settlement curves attained for single load

increments in oedometer tests on samples of different sizes, to investigate the

similarity between field and laboratory deformations of soft clay. Five samples, 2cm,

4.8cm, 20cm, 40cm, and 100cm thick and with a diameter/thickness ratio of 3 were

used. The 2cm, 4.8cm, and 20cm thick samples were tested in the laboratory and the

remaining samples were tested in the field. A trench, 15m long by 10m wide by 1.5m

deep was cut in a sand layer and filled with marine clay slurry to carry out the tests

on site. The liquid and plastic limits of the clay were 100.2% and 58.2%,

respectively. The solid matter consisted of 5% sand, 68% silts, and 27% clay. A thin

layer of sand was used to consolidate the slurry for six years. At the end of

consolidation, the average undrained shear strength of slurry was 15kPa with the

natural water content of 80%.

The first loading step was 20kPa, which was then increased from 20kPa to

80kPa at the end of primary consolidation to minimise the effects of sustained

loading on compressibility. Figure 2.20 depicts the variations of the vertical stain

versus time for all the samples.

The following observations were made by Aboshi (1973) based on the test

results:

- The consolidation coefficient ( ) estimated from the consolidation test results

increased as the sample increased in thickness.

- The amount of deformation at the end of primary consolidation had a direct

relationship with the thickness of the soil sample. The increasing deformation in

the thicker sample occurred because the secondary compression and primary

consolidation were not independent.

- Independent of sample thickness, the creep strain rate decreased with time to a

minimum constant value throughout secondary compression.

54

Figure 2.20. Effect of drainage path on experimental compression curves (after Aboshi 1973)

During secondary compression, the deformation curves became essentially

parallel, but unlike Suklje’s (1957) isotache theory, they did not merge to a single

line. As Aboshi (1973) denoted, the amount of deformation at the end of primary

consolidation depends on the effective stress loading in time, while the rate of creep

deformation is affected by the loading history.

2.4.2.5 Rajot’s model

The Rajot (1992) elastic visco-plastic model of clay compressibility can

simulate the observed phenomena of time lines, secondary compression, and stress

relaxation. By referring to Figure 2.21, Rajot (1992) applied a mechanism involving

two springs, a dashpot, and a slider to formulate the time dependant constitutive

relationships. The instant component of compression is elasto-plastic with an elastic

spring and a rigid plastic slider. The deformation(s) of the spring relate to

recoverable changes in volume ( ), while deformation(s) of the slider relate to

instant non-recoverable changes in volume ( ) that take place when the vertical

effective stress exceeds the yield effective stress. The non-recoverable creep

component of compression ( ) can be defined by the extended Kelvin element, as

0

2

4

6

8

10

12

0.1 1 10 100 1000 10000 100000 1000000Ve

rtica

l stra

in (%

)

Time (min)

Drainage path = 1cmDrainage path = 2.4cmDrainage path = 10cmDrainage path = 20cmDrainage path = 50cm

End of primary

consolidation

55

shown on the right side of Figure 2.21. Deformation of this part of the model relates

to time dependant non-recoverable changes in volume (i.e. creep). The model

generates simultaneous plastic creep deformation and instant compression when an

extended Kelvin element is placed in series with the instant spring and slider. Since

this occurs with all elastic visco-plastic models, the soil skeleton is always yielding

as a result of any increase in loading.

Rajot (1992) extended Bjerrum’s concept of time lines to include instant

plastic compression of the soil skeleton, whereas the existing vertical effective stress

and void ratio are not on Bjerrum’s instant time line, and creep compression is not

throughout recompression loading. The new model proposed by Rajot (1992)

consists of the following assumptions:

- A strain decomposition that includes instant elastic, instant plastic, and creep

deformations,

- Instant plastic compression takes place while the existing vertical effective stress

matches the yield stress,

- Variations of the yield stress values are a function of the quantity of the plastic

strain (instant or creep) and the creep strain rate,

- Bjerrum’s time lines express a set of yield stress loci related to the creep strain

rate,

- The time lines are equally spaced with respect to the logarithm of time.

In this proposed model, and according to the above mentioned assumptions,

the yield stress and creep are related (see Figure 2.21), and even though the slider is

included in the instant compression component of the model, its properties are also a

function of the amount of creep and the creep rate obtained from the creep

compression component of the model. The preconsolidation stress defined from a

laboratory oedometer test is a yield stress related to the creep rate that takes place

throughout the test. For other loading conditions that generate creep rates which are

not equal to laboratory creep rates, the yield stress is not equal to the laboratory

preconsoldiation stress.

56

Figure 2.21. Rajot’s Rheological mechanical model (after Perrone 1998)

2.4.3 General stress-strain-time models

General constitutive laws define not only viscous effects but also the rate

dependant behaviour of soils under any possible loading conditions. Since an elastic

visco-plastic approach combines the rate dependant elastic and time-dependant

plastic behaviour of soft soils, special attention was given to the elastic visco-plastic

approach in this section. As with elasto-plastic theory, a physical interpretation of the

soil response was applied for elastic visco-plastic approach such that the principal

features of experimentally observed soil behaviours (e.g. dilatacy and soil hardening)

were incorporated. For instance, variations in the state parameters such as the void

ratio, strain rate, and effective stress were defined by the hardening or softening

functions used in visco-plastic formulations. According to Liingaard et al. (2004), the

elastic visco-plastic models can be divided into three classes: (i) overstress theory,

(ii) Non-stationary flow surface theory, and (iii) others.

2.4.3.1 Overstress theory

Satake (1989) reported that the concept of overstress theory was introduced

and developed by Ludwick (1922), Prandtl (1928), Hohenemser and Prager (1932),

Sokolovsky (1948), and Malvern (1951). The elastic visco-plastic approach is based

Instant compression Creep compression

Elastic Plastic Non-linear spring

Tangent stiffness,

Non-linear dashpot

viscosity,

Slider yield stress,

Linear spring

tangent stiffness,

57

on Perzyna’s (1963) visco-plastic over-stress theory and was then used in one

dimensional visco-plastic models for multi-dimensional stress space. Based on

Perzyna’s (1963) elastic visco-plastic theory, only when the stress state reaches the

yield surface, can visco-plastic strains happen, whereas below the yield surface (in

the elastic zone), they are insignificant (Figure 2.22). The visco-plastic strain rate is a

function of the over-stress (the amount that the effective stress surpasses the current

static yield stress). The effects of ageing were not considered in this model, so the

yield surface does not change with time when the visco-plastic strains are held

constant, and when the over-stress is zero, the visco-plastic strains rate is zero.

Perzyna (1963) assumed that the difference between the dynamic loading

function (Equation ( 2.55)) and static yield function (Equation ( 2.56)) is defined as

the excess stress function (Equation ( 2.57)).

( 2.55)

( 2.56)

( 2.57)

where is the stress tensor, is the temperature, stands for the work

hardening parameter, and captures the effect of both work hardening and strain

rate hardening and is a functional of excess stress. The visco-plastic strain rate is

assumed to obey the following non-associated flow rule:

( 2.58)

where is the fluidity parameter and is the viscous nucleus.

58

Figure 2.22. Perzyna’s (1963) visco-plastic theory (after Perrone 1998)

2.4.3.2 Non-stationary flow surface theory

Matsui and Abe (1985) reported that a non-stationary flow surface theory

has been proposed and developed by Naghdi and Murch (1963) and Olszak and

Perzyna (1966, 1970), so the following description is based on Olszak and Perzyna

(1966, 1970). To include the concept of a yield surface that changes in time

according to creep behaviour, Olszak and Perzyna (1966) modified Perzyna’s (1963)

visco-plastic theory such that, although the yield surface does not distinguish

between viscous and non-viscous behaviour, it still represents a specific visco-plastic

strain rate (Figure 2.23). This modified theory consists of a time dependant yield

surface. By considering the associated flow rule, the visco-plastic potential function

is as follows:

( 2.59)

where is the effective stress component in the direction, is visco-

plastic strain, and is a scalar parameter for the time dependant behaviour of

material. Selecting parameter and stating its physical nature differentiates the

various visco-plastic models because in 1-D consolidation, is defined as either

based on a total or delayed strain rate, or on the duration of loading. Using the

associated flow rule:

Mean effective stress ( )

Dev

iato

r stre

ss (

)

Critical state line

Elastic zone ( )

due to over-stress

Yield surface corresponding

to current and

yield

59

( 2.60)

where is the time-dependent multiplier, when , elastic compression

happens and when , visco-plastic flow happens. Consequently, a visco-plastic

flow equation can be obtained as follows:

( 2.61)

Figure 2.23. Olszak and Perzyna (1966) visco-plastic theory (after Perrone 1998)

The time-dependent multiplier ( ) and multi-dimensional visco-plastic

strain can be obtained by substituting Equation ( 2.60) in Equation ( 2.61). Oka

(1981), Matsue and Abe (1985), and Sekiguchi (1984) proposed multi-dimensional

formulations of time-dependent soil behaviour based on visco-plastic flow theory.

Olszak and Perzyna’s (1966) theory was extended by Niemunis and Krieg (1996) to

develop a one dimensional computer program that would allow viscous plastic strain

(creep) to happen inside and outside the reference yield surface.

As mentioned earlier, the methods proposed for Hypothesis B can be

divided into three classes: (i) empirical models, (ii) rheological model, and (iii)

general stress-strain-time models. A summary of these models is presented in the

following table.

Dev

iato

r stre

ss (

)

Mean effective stress ( ) yield

Yield surface corresponding

to current and

Time dependant yield surfaces

Critical state line

60

Table 2.2. A summary of the Hypothesis B models presented

Category Reference Comment about creep

Empirical

model

Taylor and

Merchant

(1940)

Assuming a final settlement under any given loading

and the rate of secondary compression is proportional

to the residual compression to reach the value of final

compression.

Suklje (1957)

Graphical construction of consolidation curves for

layers several orders of magnitude thicker than sample

thickness (isotache theory).

Bjerrum

(1967)

Linear relationship between logarithm of time and soil

deformation.

Garlanger

(1971)

The rate of secondary compression is a function of

sustained loading time.

Kabbaj

(1985)

A linear relationship between the logarithm of the

preconsolidation pressure and the logarithm of the

plastic strain rate, and also a piece-wise linear

relationship between the logarithm of effective stress

and the plastic strain were assumed.

Yin and

Graham

(1989)

Creep is a nonlinear function of time with a finite

creep strain limit.

Rheological

model

Gibson and

Lo (1961)

Secondary compression was modelled by a linear

dashpot.

Wahls (1962) An infinite series of nonlinear dashpots was used to

model secondary compression.

Barden

(1965)

A nonlinear secondary compression in relation to

logarithm of time which leads to an end for total

settlement.

Aboshi

(1973)

Creep rate decreased with time to a minimum constant

value throughout the secondary compression.

Rajot (1992) Creep was modelled based on a combination of a

nonlinear spring and dashpot.

Overstress Perzyna Ageing effects not considered in this model, so the

61

theory (1963) yield surface does not change with time when the

visco-plastic strains are held constant. Moreover, when

the over-stress is zero, the visco-plastic strains rate is

zero.

Non-

stationary

flow theory

Olszak and

Perzyna

(1966)

Visco-plastic flow equation is proposed based on time

dependant yield surfaces.

2.5 PRELOADING WITH VERTICAL DRAINS

Preloading can be considered as an economic and successful ground

improvement technique for stabilising deposits of soft soil. The aim of constructing

a surcharge embankment is to provide the initial vertical stress produced as a result

of the compressive forces the soil will experience after the structure has been

constructed. Therefore, loading the soil with an adequate level of vertical stress prior

to construction is crucial to achieve the desired level of settlement. The void spaces

will be replaced by soil grains to enable the soil to carry the load from the

foundations. If the temporary load exceeds the final load (loading when the structure

is in use), the amount of excess is referred to as a surcharge load. Preloading by

adopting a staged surcharge embankment is one of the most successful and cost

effective techniques for improving the shear strength of low-lying areas because it

loads the ground to with a larger part of the ultimate settlement that it is expected to

carry after construction (Richart 1957, Indraratna and Redana 2000, Indraratna et al.

2005a). Since traditional preloading is time consuming, it is commonly used in

combination with other methods such as vertical drains and vacuum preloading.

2.5.1 Vertical drains assisted preloading

The application of vertical drains was introduced as a ground improvement

method to reduce the duration of consolidation by reducing the seepage path of

excess pore water pressure trapped deep inside the ground. Without vertical drains,

the travel distance of pore water is taken as being equal to either full or half the

thickness of the strata undergoing consolidation, depending on the boundary

62

conditions. The addition of vertical drains means that the drainage path would be

correlated to the vertical drain spacing.

Figure 2.24. Schematic diagram of embankment

A site is prepared for vertical drains by removing vegetation and surface

debris and grading the ground if necessary. Sometimes the initial step is challenging,

particularly for very soft soils, because the construction equipment can get bogged,

which results in severe rutting at the site. Minimising the disturbance to any

weathered surface crust that may generate some strength to the soil and help prevent

lateral spreading under embankment loading is beneficial. Vertical drains are

commonly installed from a sand blanket that is used to create a sound working

platform and let water egress from the drains. Horizontal drains may be used on the

surface to facilitate the drainage of the sand blanket. A typical instrumented vertical

drain scheme is shown in Figure 2.24.

In most vertical drains solutions, it is assumed that pore water pressure

flows into a drain with a circular cross section, however, Wang and Jiao (2004)

presented an analytical solution that did not assume a circular drain and modelled a

polygonal influence area that drained to a similarly shaped but smaller sized

polygon. Rectangular cross sections must be converted to a corresponding circular

one if band shaped drains are to be analysed with solutions developed for cylindrical

drains. The following conversion relationships are suggested for a rectangular drain

with a width and a thickness :

Vertical drains

Sand blanket Embankment

H r

H: Vertical drains height

r: Vertical drains spacing

: Vertical drains radius

: Disturbed zone radius

: Undisturbed zone radius

Vertical drain ( )

63

Table 2.3. Conversion relationships suggested for a rectangular drain

Suggested equation Reference Equation number

Hansbo (1981) ( 2.62)

Atkinson and Eldred (1981) ( 2.63)

Fellenius and Castonguay (1985) ( 2.64)

Long and Corvo (1994) ( 2.65)

Equations ( 2.62) and ( 2.64) are based on the perimeter and area

equivalence, respectively, whereas Long and Corvo (1994) used an electrical analogy

to determine an equivalent diameter. A rectangular drain was painted on electrically

conducting paper with silver paint and the resulting flow net is found with an

analogue field plotter. The size of the equivalent circular drain cross section that best

matches the flow net is defined by Equation ( 2.65), while equation ( 2.63) was

established to justify the throttle that takes place close to the drain.

However, there is no definitive answer as to which of these equations is the

best; Rixner et al. (1986) endorsed Equation ( 2.63) based on finite element studies,

Long and Corvo (1994) believed that Equation ( 2.63) was better than Equation

( 2.62), but Equation ( 2.65) was the most accurate. It can be noted that there is almost

no variance in the consolidation rates that were estimated using any of the above

equations (see Indraratna and Redana 2000, Welker et al. 2000).

Vertical drains are usually installed in square or triangular patterns

(Figure 2.25), and the influence zone is recognised as the area covered by pore water

flowing to a single drain. To change the square or hexagonal influence zones to

circular zones for use in numerical solutions, a circle with an equal area must be

calculated. The equivalent influence radius ( ) for triangular and square spacing

arrangements is a function of the drain spacing ( ) as follows:

64

( 2.66)

( 2.67)

Although the square pattern of drains may be easier to lay out and manage

while being installed in the field, a triangular pattern is often used because it provides a

more uniform consolidation between drains.

(a) (b)

Figure 2.25. Vertical drain installation patterns; (a) square pattern, (b) triangular pattern

2.5.2 Vacuum preloading with membrane

There are situations where the application of a surcharge loading alone is

very slow or unsuitable for the site; the specified construction times might be too

short, the essential load may result in an embankment with a dangerous height, the

space for constructing an embankment might be inadequate or there is no access to

appropriate fill material. These cases require more sophisticated techniques either in

place of, or combined with the surcharge loading. The Vacuum pressure technique

has several advantages over embankment loading, such as: (i) less fill material needs

to be used, (ii) the construction period is generally shorter, (iii) there is no need for

heavy machinery for heavy preloading, and (iv) it is an environmentally friendly

ground improvement method.

In vacuum preloading using the membrane method, vacuum consolidation

consists of a system of vertical drains and a drainage sand blanket on the surface

which in turn is sealed from the atmosphere by an impervious membrane on top.

65

Horizontal drains are installed in the drainage layer and then connected to a vacuum

pump. The ends of the membranes are placed at the bottom of a peripheral trench

filled with bentonite to maintain air tightness. Q negative pressure is generated in the

drainage layer by the vacuum pump which leads to an increase in the effective

stresses in the soil, and which in turn accelerates the consolidation process (Qian et

al. 2003).

New materials have been developed for horizontal drain pipes to advance

the vacuum preloading process, but with combined fill and vacuum preloading,

drainage panels can be used instead of pipes to confirm that the drainage channels

will still function correctly under a high surcharge pressure. These drainage panels

actually provide better channels for distributing vacuum pressure and water

discharging, and some even have slots for a direct connection with PVDs, which also

increases the efficiency of the system.

2.5.3 Membraneless vacuum preloading

When the total area must be sub-divided into a number of sections to

facilitate installation of the membrane, vacuum preloading can only be carried out in

one section at a time, but this may not be efficient when vacuum preloading is used

to reclaim land over a large area. To overcome this problem a common method is to

connect the vacuum channel directly to each individual drain using a tubing system

so that the channel from the top of the PVD to the vacuum line is sealed and

therefore a sand blanket and membranes are not needed. This system was used to

construct the new Bangkok International Airport (Seah, 2006). Nevertheless, because

this system does not provide an airtight condition for the entire area, its efficiency

can be low (Seah, 2006), so in reality, this method only works when the layer of soil

to be improved is predominantly clay with a very low permeability.

Another way of excluding the membrane is the so-called low level vacuum

preloading method (Yan and Cao, 2005) where, by using clay slurry as fill for land

reclamation, the vacuum pipes can be installed at the seabed or a few metres below

the ground surface, and the clay slurry fill can be placed on top of the vacuum pipes.

Since clay has a low permeability, the fill material provides a good sealing cap, thus

membranes will not be required. Nonetheless, this method has some disadvantages;

66

when the top layer is in direct sunlight, it dries and develops tension cracks (Chu et

al., 2008), so unless a drainage blanket is used at the level where the drainage pipes

are installed, or where individual drains are directly connected to the vacuum pipes,

the vacuum pressure may not be distributed properly. Furthermore, installing

drainage pipes or panels underwater is very difficult, but this method does not need

inner dikes for a sub-division and therefore it cuts down the project costs

considerably (Chu et al., 2008).

2.6 SOIL DISTURBANCE INDUCED WHILE INSTALLING VERTICAL DRAINS

Installing vertical drains disturbs the soil around the drain to a certain extent

and also reduces the horizontal permeability in this region. The extent to which the

hydraulic conductivity of the soil changes in the disturbed zone versus the distance

from the vertical drain has not been identified with certainty, and so far there is no

comprehensive or standard method for measuring these characteristics. According to

field and laboratory observations (e.g. Bergado et al. 1991, Madhav et al. 1993,

Indrarantna and Redana 1998, Hird and Moseley 2000), the hydraulic conductivity of

soil varies with the radial distance away from the vertical drain. Although some

efforts have been made to simulate a gradual variation of the hydraulic conductivity

with radius (Madlav et al. 1993, Chai et al. 1997, Hawlader et al. 2002), quantifying

the effects of disturbance has never been a straight forward task (Hansbo 1997). So

to characterise the disturbed zone, two major parameters, including the permeability

( ) and the extent ( ) of the disturbed zone were proposed. Bergado et al. (1991)

stated that the procedure for installing vertical drains, the specifications of the

mandrel, and the type of soil, are the key factors influencing the disturbed zone

characteristics. According to Barron (1948), inserting and withdrawing cased holes,

which are back filled, would distort and remould the soil near the vertical drain, so in

response researchers have suggested two broad concepts to determine the

characteristics of soil surrounding the drain; (i) a two zone hypothesis that consists of

the intact zone surrounding the disturbed zone adjacent to the vertical drain, and (ii)

the three zones hypothesis, consisting of the undisturbed zone surrounding the

transition zone, and the smear zone near the vertical drain.

67

Based on the available literature (e.g. Barron 1948, Onoue et al. 1991,

Madhav et al. 1993, Walker and Indraratna 2006, Basu et al. 2006 and 2010, and

Rujikiatkamjorn and Indraratna 2009), proposed various patterns for the initial

hydraulic conductivity of the soil in the disturbed region are presented in Figure 2.26.

According to most researchers (e.g. Barron 1948; Holtz and Holm 1973; Hansbo

1981; Jamiolkowski et al. 1983; Chai and Miura 1999), the soil inside the disturbed

zone is entirely remoulded, causing an initial constant hydraulic conductivity which

is smaller than the undisturbed horizontal hydraulic conductivity (Figure 2.26a, Case

A). Indeed, Rujikiatkamjorn and Indraratna (2009) stated that the initial hydraulic

conductivity of the disturbed zone may have a linear variation with the radial

distance (Figure 2.26a, Case B), while a parabolic distribution of the permeability in

the disturbed region was proposed by Walker and Indraratna (2006) (Figure 2.26a,

Case C). Madhav et al. (1993) proposed a constant hydraulic conductivity which is

smaller than the undisturbed hydraulic conductivity in the smear zone and a linear

hydraulic conductivity variation in the transition zone (Figure 2.26b, Case D). Onoue

et al. (1991) proposed a bilinear variation for hydraulic conductivity by assuming

that the permeability is changing linearly in the smear and transition zones

(Figure 2.26b, Case E). Furthermore, Basu et al. (2006) assumed that the horizontal

hydraulic conductivity is constant in the smear zone and proposed a bi-linear

variation of permeability in the transition zone (Figure 2.26b, Case F).

Moreover, a mechanical installation of vertical drains inevitably disturbs the

surrounding soil due to induced shear strains. Baligh (1985), and Whittle and

Aubeny (1993) stated that the induced shear strain is a function of the radial distance

and diameter of the vertical drain. Shear strains caused by the installation of vertical

drains help reduce the shear strength and as a result, the over consolidation ratio of

the soil. The extent to which the over consolidation ratio of soil changes in the

disturbed zone versus the distance from the vertical drain has not been identified with

certainty, and so far there is no comprehensive or standard method for measuring

these characteristics.

68

(a) (b)

Figure 2.26. Cross section of the disturbed zone surrounding a vertical drain, (a) two zones hypothesis, (b) three zones hypothesis

2.7 ANALYTICAL FORMULATION FOR VERTICAL DRAIN ASSISTED PRELOADING

Researchers have used different analytical developments to simulate the

behaviour of soft soils. Tang and Onitsuka (2001) proposed a solution using double-

layered ground consolidation with vertical drains, by considering the well resistance

and the smear action. The basic assumption here is that every layer of ground with

vertical drains satisfies the assumption of the consolidation of single-layered ground

Intact Zone Intact Zone

Smear Zone Transition Zone

Smear Zone

Vertical Drain

Vertical Drain

R R

1

1

1

1

1 1

Case A

Case B

Case C Case F

Case E

Case D

69

with vertical drains under a quasi-equal strain condition, apart from the conjunction

plane with two layers. The solution proposed by Tang and Onitsuka is: (i) to adopt

and modify Barron's equal-strain hypothesis such that the modified part is where the

analysis for the vertical consolidation of soil applies to the average value of the pore

water pressure at the same depth, instead of the pore water pressure at any point, as

suggested by Barron, (ii) the horizontal coefficient of the permeability of the smear

zone is less than the natural soil, the coefficient of volume compressibility and the

vertical coefficient of permeability of the smear zone are the same as natural soil, and

(iii) the total inflow of pore water through the boundary of the vertical drains is equal

to the vertical flow of pore water within the vertical drains. The radial water flow in

the vertical drains is omitted. By considering the boundary conditions at the top and

bottom surfaces of the system, general solutions for excess pore water pressure

within a vertical drain ( ) and the average pore water pressure of soil ( ) can be

expressed as follows:

( 2.68)

( 2.69)

where

( 2.70)

( 2.71)

( 2.72)

( 2.73)

( 2.74)

in which , , , , , , , and are the solution parameters that

can be found in Tang and Onitsuka (2001).

The average degree of consolidation for each layer is:

( 2.75)

( 2.76)

70

The overall average degree of consolidation defined by the pore pressure is:

( 2.77)

The overall average degree of consolidation defined by the total settlement

is:

( 2.78)

The results of this study were compared with the finite difference method

and finite element method results provided by Amirebrahimi et al. (1993) and Onoue

(1988), respectively. The key shortcoming of this method was omitting the viscous

behaviour (creep) of soils.

Indraratna et al. (2005) presented an analytical model of vertical drains

combined with vacuum preloading in axisymmetric conditions by considering the

vacuum pressure along the length of the drain. Indraratna et al. (2005) stated that

when a vacuum is applied in the field through PVDs, the suction head may decrease

with depth, as well as laterally, thus reducing its efficiency. To study the effect of

loss of vacuum, a trapezoidal vacuum pressure distribution was assumed. In the

vertical direction (along the drain boundary), the vacuum pressure varied from

to , whereas it varied from to across the soil.

According to Indraratna et al. (2005), the excess pore pressure variation inside and

outside the smear zone can be derived as follows:

( 2.79)

( 2.80)

where and are the variations in excess pore pressure in the smear and

intact zones, respectively, is the applied vacuum pressure, is vacuum pressure

reduction factor by depth, is the depth, is the total depth, is the vacuum

pressure reduction factor by radius, is the radius, is the drain radius, is the

smear zone radius, is the total radius, and and are the smear and intact zones

71

permeability, respectively. By integrating Equation ( 2.79) in a radial direction with a

boundary condition of , the excess pore pressure within the smear

zone is given by:

( 2.81)

By integrating Equation ( 2.80) in a radial direction with the boundary

condition of the excess pore pressure outside the smear zone can be

calculated by:

( 2.82)

Consequently, the average excess pore pressure at a given time is:

( 2.83)

where

( 2.84)

( 2.85)

and , .

The average degree of consolidation can be obtained as follows:

( 2.86)

The finite element analysis using ABAQUS software was used to support

the exact solutions based on unit cell theory, with the result that the details of a

proper corresponding method by transforming the permeability and vacuum pressure

between the axisymmetric condition, is described through analytical and numerical

schemes. The extent and distribution of vacuum pressure on the consolidation of soft

clay were inspected with average excess pore pressure, consolidation settlement, and

time analyses. Moreover, viscous creep was omitted by Indraratna et al. (2005) in the

present method.

Basu et al. (2006) developed closed-form solutions for the rate of

consolidation for four assumptive hydraulic conductivity profiles (i.e. Cases B, D, E,

and F in Figure 2.26) in the disturbed zone using methodology that was similar to

72

Hansbo (1981). In this methodology, a number of drains are installed into the

ground, with each one having an influence zone that performs identically (for

homogeneous deposits), but where water inside one influence zone does not flow

into another; this influence zone is called a ‘unit cell’. One such unit cell with a

circular cross section was considered in the analytical simulation where the effect of

the flow of water in a vertical direction within the unit cell is negligible (Leo 2004).

As a consequence, the interface between the drain and the unit cell is the only

pervious boundary of the unit cell, and it results in a radially convergent horizontal

flow of water into the drain. Flow patterns are identical along any horizontal plane

because they assume a homogeneous deposit with no horizontal strain in the soil

cylinder. To solve this problem, only one such horizontal plane with axisymmetric

flow should be considered. It was assumed that the flow of water would follow

Darcy’s law and the vertical strain within the unit cell is spatially uniform. This

represents the case of ‘equal strain’ consolidation (Richart 1959). By considering

to be the average excess pore pressure throughout the unit cell, the average excess

pore water pressure in Case D, for instance, can be obtained by the following

equation:

( 2.87)

where , , , and represent unit the cell, the disturbed zone, the smear

zone, and the drain radii, respectively, , , are the excess pore water

pressures in the smear zone, transition zone, and unit zone respectively. By

rearranging Equation ( 2.87) we obtain:

( 2.88)

where

( 2.89)

and where , , , and .

73

Equation ( 2.89) is difficult to use in routine design, nonetheless a number of

terms on the right hand side make an insignificant contribution to the value of , but

if we omit these terms, Equation ( 2.89) can be simplified to:

( 2.90)

Since the ratio is close to unity for a typical unit cell and drain

diameters used in practice, it was not included in Equation ( 2.90). Assuming that all

the excess pore pressure due to preloading is generated instantaneously:

( 2.91)

where is the average effective stress in the unit cell due to preloading at the end of

consolidation, is the average excess pore pressure at the time of load application,

and is the coefficient of volume compressibility.

By substituting Equation ( 2.88) into Equation ( 2.91), the following linear

differential equation can be obtained:

( 2.92)

Solving Equation ( 2.92) using the initial condition and the change in

average excess pore pressure with time can be obtained as follows:

( 2.93)

The degree of consolidation at a particular time (or time factor) is the

ratio of excess pore pressure dissipated to the excess pore pressure induced at that

time.

( 2.94)

Substituting Equation ( 2.93) in Equation ( 2.94) gives the following

expression for the degree of consolidation:

( 2.95)

The results of these analyses showed that different variations of the

hydraulic conductivity profiles in the disturbed zone caused various rates of

consolidation, although in the method presented by Basu et al. (2006) the time

dependant behaviour of soft soils was not included.

74

Walker and Indraratna (2006) presented an analytical solution for nonlinear

radial consolidation under equal-strain conditions that included soil disturbance and

ignored well resistance. Walker and Indraratna (2006) considered the following

aspects of non-linearity: (i) non-Darcian flow; (ii) a logarithm-linear void-ratio-stress

relationship; and (iii) a logarithm-linear void-ratio-permeability relationship. As

Hansbo (1981) expressed, the average degree of consolidation for axisymmetric flow

( ) on a horizontal plane at a depth and at time is:

( 2.96)

where the value of for smear effect, assuming no well resistance, is

given by:

( 2.97)

In the preceding, and , is the external radius of unit cell,

and are the radius of the vertical drain and smear zone, respectively, is the

horizontal coefficient of permeability, and is the horizontal coefficient of

permeability in the smear zone, which was assumed to be constant throughout the

smear zone in the Hansbo’s (1981) theory. Hansbo (1981) used a constant value of

, so in the proposed model, is a parabolic function of as follows:

( 2.98)

where , , , and . By considering as

the depth averaged vertical strain rate, the pore pressure gradient in the undisturbed

zone can be obtained as follows:

( 2.99)

Similarly, the pore pressure gradient in the smear zone is:

( 2.100)

The resulting expressions for pore water pressure on either side of the smear

zone boundary, and by substituting Equation ( 2.98) in Equation ( 2.99) and ( 2.100)

are:

( 2.101)

75

( 2.102)

where

( 2.103)

( 2.104)

( 2.105)

If is the average excess pore pressure in the soil cylinder at a depth , then

( 2.106)

Substituting Equations ( 2.101) and ( 2.102) into Equation ( 2.105) and

subsequent solution gives:

( 2.107)

where

( 2.108)

In the preceding, , , is the external radius of unit cell, is the

initial length of the drainage path, and are the radius of the vertical drain and

smear zone, respectively, and is the discharge capacity of the drain. Equation

( 2.107) may now be combined with Terzaghi’s constitutive equation for one-

dimensional compression.

( 2.109)

where is the coefficient of volume compressibility ( in the smear and

undisturbed zone are assumed to be equal) and is the average effective stress.

Combining Equations ( 2.108) and ( 2.109) with the initial condition gives:

( 2.110)

where is the horizontal time factor. The average degree of consolidation

in a radial direction and at a particular depth with well resistance, as in Equation

( 2.96), is now given by:

76

( 2.111)

The behaviour of over consolidated and normally consolidated soils can be

captured by the analytical solution to non-linear radial consolidation theory. Walker

and Indraratna (2006) mentioned that compared to the cases with constant material

properties, consolidation might be faster or slower for non-linear material properties,

with the difference being a function of the compressibility/permeability ratios, the

preconsolidation pressure and the stress increase. It is worth noting that the method

proposed by Walker and Indraratna (2006) did not include the viscous behaviour of

soft soils.

Indraratna et al. (2008) proposed a new technique to model consolidation by

vertical drains below a circular loaded area where the system of vertical drains in the

field was transformed by a series of equivalent and concentric cylindrical drain walls.

Soil that has consolidated around an individual vertical drain can easily be analysed

as a single unit cell, but to analyse a multi-drain system under an axisymmetric

condition, the equivalent soil parameters that give the same time-settlement response

in the field must be determined. In such a transformation, each drain element should

behave as part of the concentric cylindrical drain wall with a perimeter that is

increasing with the radial distance from the centreline. The main assumptions made

in this analysis were: (i) an equal strain assumption (small strain) and Darcy’s law

are valid, (ii) only vertical strains are allowed, (iii) the soil is fully saturated, and the

permeability of the soil is assumed to be constant during consolidation, (iv) well

resistance is omitted because the discharge capacity of the drain is sufficient and thus

the pore pressure at the interface is assumed to be zero, (v) each set of vertical drains

located the same radial distance from the line of axisymmetric is modelled as a

continuous cylindrical drain wall with a radius ( ) where is the spacing of

the drains and is the number of that set ( ). For ( ), each cylindrical

drain wall lies in the middle of a revolving prism of soil which has a thickness of ,

and (vi) it is assumed that the cylindrical drain wall has a negligible thickness.

Indraratna et al. (2008) assumed the radial flow rate to be equal to the rate

of volume change of the soil mass in a vertical direction, so:

( 2.112)

77

The excess pore pressure gradient can be derived by rearranging Equation

( 2.112) as follows:

( 2.113)

Integrating Equation ( 2.113) in the radial direction with the boundary

condition ( ), the distribution of excess pore pressure in Zone

( ) can be expressed by

( 2.114)

Similarly, the excess pore pressure in Zone ( )) is

determined by:

( 2.115)

The subscripts and were denoted for Zones and , respectively, and a

subscript ‘ring’ was denoted for circular loading. The average excess pore pressure is

given by:

( 2.116)

where

( 2.117)

( 2.118)

and for drains installed in a square pattern and an

equilateral triangular pattern, respectively (Holtz et al. 1991). It is interesting to note

that the value of converges to for all values of , and combining

Equations ( 2.116) to ( 2.118) with the well-known compressibility relationship

( ) yields:

( 2.119)

Rearranging the above Equation ( 2.119) and then integrating it by applying

the initial boundary condition at gives:

( 2.120)

78

Table 2.4. Proposed analytical solutions for vertical drain assisted preloading

Reference Suggested theoretical solution for radial consolidation

Tang and

Onitsuka

(2001)

Indraratna

et al.

(2005)

Basu et al.

(2006)

Walker

and

Indraratna

(2006)

Indraratna

et al.

(2008)

Note: is the average degree of consolidation; , , , , , , , and are the solution parameters which can be found in Tang and Onitsuka (2001); ;

; is vacuum pressure reduction factor by depth; is vacuum pressure reduction factor by radius; is depth, is total depth; ; ; and

; is the model parameter related to vertical drains pattern; is the number of the set ( ); is the spacing of the drains.

79

( 2.121)

Indraratna et al. (2008) used the proposed model to study the consolidation

process by vertical drains at Area II of the Skå-Edeby circular test embankment. In

should be mentioned that the proposed method omitted the viscous behaviour of soft

soils. Table 2.4 summarised the proposed solutions for analytical formulations of

vertical drain assisted preloading.

2.8 NUMERICAL SIMULATION OF VERTICAL DRAIN ASSISTED PRELOADING

Geotechnical engineers have used various methods of numerical analysis in

preloading projects to rectify the limitations of using analytical approaches to

simulate complex vertical drains and to estimate the behaviour of soils. To conduct a

numerical analysis, various programs such as CRISP, PLAXIS, ABAQUS, and

FLAC as well as developed codes have been used by researchers, and a number of

numerical investigations already conducted are tabulated in Table 2.5. Indraratna and

Redana (2000) applied the finite element code CRISP92 (Britto and Gunn 1987) to

simulate ground improved by vertical drains under the load of an embankment by

implementing the equivalent coefficient of permeability for the undisturbed soil and

the disturbed zone. The Modified Cam-Clay model as the soil constitutive model was

used to carry out the numerical analysis, while incremental vertical loads were

applied to the upper boundary to model the embankment surcharge. The smear zone

was presumed to be three to four times the radius of the mandrel while permeability

within the smear zone was assumed to decrease linearly. Based on a numerical

simulation carried out by Indraratna and Redana (2000), including the smearing

effects can increase the accuracy of the estimated settlements.

Zhu and Yin (2000) proposed a finite element procedure to analyse the

consolidation of layered soils with a vertical drain by using general one dimensional

(1-D) constitutive models. A Newton-Cotes-type integration formula was used to

formulate the finite element procedure to elude the asymmetry of the stiffness matrix

for a Newton (Modified Newton) iteration scheme. This procedure was then used to

analyse the consolidation of numerous and usual problems by applying linear and

nonlinear soil models. The results from this simplified method were compared with

80

results from fully coupled consolidation analysis by applying the recognised finite

element package. One of the shortcomings of Zhu and Yin’s (2000) method was

assuming that the creep part of equations was equal to zero.

Nash and Ryde (2001) proposed a one-dimensional finite difference

consolidation analysis to simulate the vertical and radial drainage of a multilayer soil

profile in the zone of influence of a vertical drain. The model was illustrated with a

back-analysis of field data from construction of the approach motorways to the new

Second Severn Crossing in the UK with temporary surcharge over estuarine

alluvium. For simplicity, it was assumed by Nash and Ryde (2001) that the creep

coefficient is constant at any vertical effective stress.

Arulrajah et al. (2005) used the PLAXIS 2D V.8 program to simulate the

unit cell and full scale embankment with a finite element model. Based on Arulrajah

et al. (2005), a good agreement was attained between the numerical results and the

field measurements. Settlement estimations attained from the axisymmetric unit cell

and a full scale analysis of vertical drains were found to agree with each other, and

with the actual field measurements. Moreover, to simplify the analysis, the

constitutive model used to simulate the soil considered the creep coefficient to be

constant with time and effective stress.

Rujikiatkamjorn and Indraratna (2006) introduced a numerical model based

on a three-dimensional finite element model to analyse the behaviour of soft soil

improved with prefabricated vertical drains. In the numerical simulation, the cross

section of these prefabricated vertical drains was considered to be rectangular, so the

equivalent drain diameter was back calculated and then compared to the results of

previous studies based on consolidation behaviour. The study confirmed that the

equivalent drain diameter was comparable to that attained from the numerical results.

The predictions from the 3D numerical model and the laboratory measurements from

large-scale consolidation testing agreed with each other. Including the effect of the

disturbed zone, the actual variation of horizontal permeability along the radial

direction, and the void ratio permeability relationship helped in making the

estimation of settlement more accurate, although the estimations made using

conventional analysis (Hansbo 1979) slightly underestimated the laboratory data, and

the creep coefficient was considered to be a function of the square root of time.

81

Basu et al. (2010) used two dimensional finite element analyses to study

how prefabricated vertical drains affected the rate of consolidation of the soil. In this

analysis, the highly disturbed smear and intact zones were separated by a transition

zone lying between them. It was assumed that the permeability in the transition zone

varied linearly from a low value in the smear zone to the original in situ value in the

undisturbed zone. Basu et al. (2010) also carried out a parametric study to study the

effects of the degree of soil disturbance, the size of the smear and the transition

zones, the spacing of the prefabricated vertical drains, and the shape and size of the

mandrel. A comparison with the experimental results showed that a consideration of

the transition zone was enough to make an accurate approximation of the degree of

consolidation. It should be noted that in Basu et al. (2010) study, the time dependant

behaviour of soil was not included.

Table 2.5. Summary of numerical studies conducted to simulate PVD assisted preloading

Reference Objective of the Numerical Study Simulated Case Study

Applied Numerical Program

Inclusion of creep

Comment about creep

Mesri et al. (1994)

To investigate the settlement of embankments on soft clays

improvement using vertical drains - ILLICON Yes

Adopting Hypothesis

A (the creep

coefficient considered

to be constant

with effective stress and

time)

Zhu and Yin (2000)

Proposed a finite element procedure to analyse consolidation of layered soils with vertical drain using general one-dimensional (1-

D) constitutive models

Berthierville Test

Embankment in Quebec, Canada

- No -

Indraratna and Redana

(2000)

To evaluate the performance of soft clay foundation beneath

embankments stabilised with vertical drains by considering

smear effects

Muar clay embankment

Malaysia

CRISP92 2D No -

Nash and Ryde

(2001)

To simulate vertical and radial drainage of a multilayer soil profile

in the zone of influence of a vertical drain

motorways to the new Second

Severn Crossing in the UK

- Yes

Adopting Hypothesis

B (the creep


82

to be constant

with time)

Indraratna et al.

(2004a)

To investigate the effect of unsaturation at a drain boundary on the behaviour of a single PVD subjected to vacuum preloading applying 2D plane-strain model

Large-scale consolidometer

test

ABAQUS 2D No -

Indraratna et al.

(2005a)

To study the performance of a full-scale test embankment

constructed on PVD improved soft clay incorporating vacuum

preloading

Second Bangkok

international airport

ABAQUS 2D No -

Indraratna et al.

(2005b)

To model a single vertical drain incorporating the effects of

vacuum preloading and compare the results with proposed

analytical solutions

- ABAQUS 2D No -

Arulrajah et al.

(2005)

To investigate the deformation behaviour of marine clay under

reclamation fills conducting full-scale analysis of PVDs and compare the results with the

axisymmetric approach

Singapore marine clay at

Changi

PLAXIS 2D Yes

Adopting Hypothesis

B (the creep


to be constant with time

and effective stress)

Rujikiatkamjorn and Indraratna

(2006)

To analyse the behaviour of soft soil improved by prefabricated

vertical drains

Large-scale consolidometer

test - No -

Indraratna and

Rujikiatkamjorn (2007)

To analyse the behaviour of soft soil improved by prefabricated

vertical drains

Sunshine Motorway, Queensland

PLAXIS 2D Yes

Adopting Hypothesis

B (the creep


to be constant with time

and effective stress)

Indraratna et al.

(2008)

To evaluate the consolidation of soft soil beneath a circular

embankment improved with PVD

Skå-Edeby Sweden

ABAQUS 2D No -

Tran and Mitachi (2008)

To evaluate the efficiency of a proposed method to convert an

axisymmetric unit cell to an equivalent plane-strain unit cell

- CRISP 2D No -

83

under embankment loading combined with vacuum

preloading

Rujikiatkamjorn et al.

(2009)

To conduct 2D and 3D numerical modelling of combined surcharge

and vacuum preloading with vertical drains

storage yard at Tianjin Port,

China

ABAQUS 2D & 3D No -

Yildiz and Karstunen

(2009)

To study the performance of the matching procedures proposed by Hird et al. (1991 & 1995) when

complex elasto-plastic models are used in the plane-strain analyses

of vertical drains

Haarajoki embankmentFin

land

PLAXIS 3D No -

Yildiz (2009)

To evaluate the accuracy of three different matching methods for conversion of axisymmetric to

plane-strain conditions by comparing results of 2D and 3D

analyses

Haarajoki embankmentFin

land

PLAXIS 3D No -

Lin and Chang (2009)

To investigate the drainage of a PVD unit cell and full-scale PVD

improved ground with a 3D numerical simulation

Second Bangkok

international airport

FLAC 3D No -

Basu et al. (2010)

To study the effect of soil disturbance induced by

installation of prefabricated vertical drains on the rate of

consolidation by two dimensional finite element analyses

large-scale consolidometer

experiment - No -

2.9 SUMMARY

Due to its heterogeneous compositions, soil has a complex structure, unlike

other materials such as metal or glass. Taylor and Merchant (1940) and Terzaghi

(1941) stated that the process of compression consists of primary consolidation and

secondary compression (or creep), and it is defined based on the transfer of stress and

adjustment of the soil structure. As free water flows out of the soil due to the applied

stress, the structure is rearranged such that there may be an increase in the solid-to-

solid contacts in the soil. Secondary compression (creep) may be due to: (i) the

breakdown of inter-particle bonds, (ii) the bonds jumping, (iii) sliding between the

particles, (iv) double porosity, and (v) structural viscosity. An attempt was made in

this chapter to explain the mechanism of creep for clayey soils.

Various approaches have been suggested to simulate the time dependant

deformation of soft soils, but researchers have suggested the following two broad

84

concepts: (i) Hypothesis A; and (ii) Hypothesis B. In Hypothesis A, regardless of the

fact that creep occurs during primary consolidation, or the thickness of the sample,

the void ratio at the end of primary consolidation is considered constant. However,

Hypothesis B assumes that since creep occurs during primary consolidation and

secondary compression, the void ratio at the end of primary consolidation cannot be

constant for samples with different thickness. This chapter then, provides an

explanation of how these two Hypotheses (i.e. Hypothesis A and B) can be used to

simulate the behaviour of soft soils. The concept of a creep ratio as one of the main

methods supporting Hypothesis A was explained in detail, while a number of

constitutive models that support Hypothesis B proposed in the literature were

discussed. To classify the proposed models, they were divided into three categories;

empirical models, rheological models, and general stress-strain-time models.

As elaborated in this Chapter, the system of vertical drains assisted

preloading has been broadly used as a ground improvement technique for soft soil to

accelerate consolidation and improve the strength of the soil, including its bearing

capacity and shear strength. However, vertical drains disturb the soil, reduce the

permeability and shear strength of the smear zone, and retard the rate of

consolidation quite significantly.

Since different methods are presented in literature to capture the time

dependant behaviour of soft soils or consider the reduction of hydraulic permeability

induced by vertical drains, the combined effects of hydraulic conductivity or shear

strength profile in the disturbed zone and how the visco-plastic behaviour of the soil

influences the creep parameters, rate of settlement, and consequent deformation of

soft soils improved with vertical drains has not been considered.

Researchers have proposed various analytical developments to simulate the

behaviour of soft soils improved with vertical drains assisted preloading, and while

these analytical models considered the reduction in hydraulic conductivity induced

by installation of vertical drains, the resulting reduction in shear strength and time

dependant behaviour of soft soils (creep) were not included in these models.

Moreover, geotechnical engineers have used numerous methods of numerical

analysis to overcome limitations in the analytical approaches to simulate projects

with complex vertical drains. Indeed, most of these numerical models either did not

consider the time dependant behaviour of soft soils (creep) or considered the creep

85

coefficient to be constant with time and effective stress, and the reduction in shear

strength due to vertical drains were not included in the proposed numerical analysis.

In this research, finite difference formulations for fully coupled one

dimensional axisymmetric consolidation are used to model the time dependent

behaviour of soft soil combining vertical and radial drainage, as well as variations of

the settlement and excess pore water pressures with time. The elastic visco-plastic

model developed by Yin (1999) is incorporated in the consolidation equation, while

the selected elastic visco-plastic model can simulate consolidation and creep in a

single constant analysis that consists of a nonlinear creep function, as a function of

effective stress and time, and a creep strain limit. Different possible variations of

horizontal permeability and shear strength in the disturbed zone, and nonlinear

variations of permeability with changes in the void ratio combined with soil creep are

also considered. The effects of different hydraulic conductivity and shear strength

profiles on settlement, the rate of excess pore water pressure dissipation, the creep

strain rate, and the creep strain limit are also investigated and discussed.

86

CHAPTER THREE

3 FINITE DIFFERENCE SOLUTION FOR 2D AXISYMMETRIC CONSOLIDATION EQUATION CONSIDERING NONLINEAR ELASTIC VISCO-

PLASTIC MODEL

3.1 GENERAL

Post-construction deformation of soft soils including clays, silt deposits,

organic soils and peat deposits, may be extreme during the life time of the structure.

Therefore, in order to minimise the post-construction settlement and improve the

bearing capacity and the shear strength of the soft soil deposits, preloading combined

with vertical drains is commonly used. Preloading comprises of applying a load,

equal to or greater than the entire load of a planned structure, over the site earlier

than constructing the structure. Preloading, which is commonly an earth fill, applies

compression to the underlying soil, which is being partially or fully removed, while

the required settlement has taken place. Installation of vertical drains causes soil

disturbance in the vicinity of the drain. The disturbed zone around the vertical drain

possesses reduced strength and hydraulic conductivity in the horizontal direction,

remarkably affecting the excess pore water pressure dissipation rate as well as the

creep rate (Holtz and Holm 1973; Massarsch 1976; Bozozuk et al. 1977; Baligh

1985; Whittle and Aubeny 1993; Lo and Mesri 1994; Bergado et al. 1991; Sharma

and Xiao 2000; Basu and Prezzi 2007; Walker and Indraratna 2006).

Elastic visco-plastic model divides the soil deformation into two parts: (i) an

instant compression as a result of a reduction in the void ratio; and (ii) a delayed

compression representative of the volume decrease under the unchanged effective

stress. Elastic visco-plastic model originally proposed by Yin and Graham (1989) is

based on Bjerrum’s (1967) equivalent time line concept, which is capable of

calculating soil behaviour measured in the field and laboratory. Different numerical

solutions (e.g. finite difference and finite element methods) have been proposed by

87

researchers to solve the elastic visco-plastic models. In this chapter, the elastic visco-

plastic model, considering a nonlinear creep function and creep strain limit is

incorporated in the consolidation equation. Finite difference formulations for fully

coupled one dimensional axisymmetric consolidation are adopted to investigate the

long term behaviour of soft soils considering the disturbance induced by installation

of vertical drains. The finite difference solution of elastic visco-plastic model is

developed using MATLB software to model the time dependent behaviour of the soft

soils.

3.2 NONLINEAR ELASTIC VISCO-PLASTIC BEHAVIOUR OF SOILS

Bjerrum (1967) represented a relationship between the applied stresses,

compression and time, by a system of lines, while each line being a representative of

a unique relationship between stress, strain and time. Time line system concept of

Bjerrum (1967) was adopted by Yin (1990) assuming that time lines are lines with

the same values of “equivalent times” ( ). However, as explained by Yin (1999),

unlike Bjerrum (1967) concept, each time line can be correlated to a unique creep

strain rate. Furthermore, time lines are not necessarily equal to the loading duration.

The time line concept comprises an instant time line, a reference time line, a limit

time line, and a set of equivalent time lines. Figure 3.1 depicts schematic fitting

curves for instant, reference, equivalent, and limit time lines.

As described by Yin and Graham (1989), the instant time line defines the

elastic-plastic settlement and is correlated to the normal consolidation line.

Nevertheless, instant compression is expressed by Yin (1990) as time independent

elastic compression. As explained by Graham and Yin (2001), the normal

consolidation or overconsolidation lines are often generated by the standard

oedometer tests, which may include creep deformation (while the excess pore water

pressure is being dissipated) since the consecutive loadings in the tests are generally

kept constant for 24 hours. As a result, the true instant line should be obtained by

reloading/unloading tests. The instant time line fitting equation can be defined as

follows:

( 3.1)

88

where is a unit stress, is the vertical strain at stress level , is

the vertical strain at , and is a material parameter describing the elastic

stiffness of the soil, in which is the specific volume , and is the

initial void ratio.

Figure 3.1. Schematic fitting curves for instant, reference, equivalent and limit time lines

Yin and Graham (1989) proposed their original elastic visco-plastic model

assuming the creep strain rate is constant. However, according to the field

measurements and laboratory test results (e.g. Yin 1999, Mesri 2001, and Yin et al.

2002) the relationship between the strain (or void ratio) and logarithm of time is not

linear. Consequently, in order to simulate the behaviour of soft soils more accurately,

Yin (1999) proposed a modified creep function incorporating a nonlinear creep strain

rate and the creep strain limit. The author believes that there is an absolute minimum

value for the volume of solid in a soil element while approximately no real void

exists within that soil element. As expressed by Mitchell (1956), regardless of the

pressure or initial orientation, void ratio for a particular soil can reach to a minimum

value. It means that the soil structure cannot deform forever. Thus, the deformation

of the soil under a particular applied pressure must cease after a finite period that can

be counted in years or decades, meaning that there is a finite strain. The compression

Instant time line ( line)

Logarithm of vertical effective stress

Reference time line ( line) (

Ver

tical

Stra

inV

Lines of Equivalent time

Limit time line (Creep strain limit)

(

89

may end under the final effective stress that the ultimate equilibrium inside the soil

structure is reached or when almost no void exist inside the clay mass. Obviously,

creep strain limit measurement is not an easy task, since it is not feasible to carry out

the tests for a very long duration approaching infinity. Hence, it can be assumed that

the limit strain can be reached when the volume of voids within the soil approaches

zero under the applied stress at the infinity time. Yin et al. (2002) proposed that the

creep strain limit may be estimated based on the initial void ratio as follows:

( 3.2)

where is the creep strain limit and is the initial soil void ratio.

However, the author believes that Equation ( 3.2) is an overestimation of the

creep strain limit since the term consists of the conventional consolidation

volume change due to hydrodynamic excess pore water pressure dissipation as well

as the creep. As a result, the soil void ratio at a certain effective stress on the

reference time line should be used to define the creep strain limit for a particular

applied effective stress as follows:

( 3.3)

where

( 3.4)

where is the void ratio at effective stress equal to on the reference

time line, is the soil vertical strain, is a material property, and is the void

ratio at a particular applied effective stress.

As stated by Yin and Graham (1989), the equivalent time line to some

extent is comparable to the equivalent pressure, defined in the critical state soil

mechanics. The equivalent time ( ) is referred to the time that clay needs to wait

after instantaneous loading along the reference line ( line) to get to the required

future state point condition. The time needed for soil to creep from the reference time

line to a state point under the same effective stress is defined by Yin (1999) as the

equivalent time ( ). As explained by Yin et al. (2002), an equivalent time ( ) can

be defined as a function of the state point ( ), as follows:

90

( 3.5)

where , , , and are the model parameters, related to the choice of the

reference time line, the initial creep strain rate, the elastic-plastic stiffness of the soil,

and the vertical strain at , respectively.

Equivalent times above and below the reference time line are negative and

positive, respectively, as shown in Figure 3.1. In the overconsolidated range, the

equivalent time is a function of the consolidation ratio, however, in the normally

consolidation range and multistage loading tests (i.e. constant load increment), the

equivalent time is equal to the duration of the load increment (Yin and Graham

1994). According to Yin (2006), the vertical creep compression strain ( ) can be

calculated by the following equation:

( 3.6)

where

( 3.7)

where is the creep strain rate, and is the initial value of at .

Yin (1990) defined the reference time line as the line with the equivalent

time equal to zero ( ). The reference line can be used to calculate the equivalent

time and the creep strains (Yin 1990). As explained by Yin and Graham (1994), in

time independent soils, while viscosity of the soil is equal to zero, the reference time

line is the elastic-plastic line. The fitting function for the reference time line strain

can be presented as:

( 3.8)

As suggested by Yin and Graham (1994), a unique limit time line in the

space of ( ) exists for both viscous and non-viscous soils. They expressed the

limit time line as the line in which and the creep strain rate approaches zero.

Behaviour of the soil beyond the limit time line is time independent. It is believed

that the creep straining will finally terminate after very long time (approaching

91

infinity) when the soil particles occupy a fixed volume. The fitting function for the

limit time line is as follows:

( 3.9)

where is the strain limit, and is the initial strain on the reference

time line.

According to Yin (2006), considering and combining Equations ( 3.1) to

( 3.9), the elastic visco-plastic model to predict the time dependant behaviour of soft

soils can be obtained using Equations ( 3.10) and ( 3.11):

( 3.10)

where

( 3.11)

Equations (3.10a) and (3.10b) represent the soil behaviour transition from

heavily over-consolidated to lightly over-consolidated and then the normally

consolidated situation. Equation (3.10a) denotes the situation where the soil is

heavily over-consolidated and as a result, the viscous creep component is

insignificant, while Equation (3.10b) denotes the situation where the soil is lightly

over-consolidated or normally consolidated and consequently, the behaviour is a

function of the effective stress and time (Figure 3.1). It should be noted that the

above mentioned equations are in 1-D platform and can be used for one dimensional

consolidation. Further explanation regarding equations in 3-D can be found in Yin et

al. (2002) and Yin (2006).

92

3.3 FINITE DIFFERENCE SOLUTION FOR AXISYMMETRIC CONSOLIDATION EQUATION

3.3.1 Solution to general parabolic differential equations

As presented by Kharab and Guenther (2012), Equation ( 3.12) defines the

one-dimensional partial differential equations of parabolic type for unsteady-state

flow which is derived for heat flow but applies equally to diffusion of material, flow

of fluids, flow of electricity in cables, etc. Equation ( 3.12) is subjected to the

boundary conditions and initial conditions as presented in Equations ( 3.13) and

( 3.14), respectively.

( 3.12)

( 3.13)

( 3.14)

where for example in heat transfer problem, , is thermal

conductivity, is the heat capacity, is the density, represents the

temperature at any time along a thin, long rod of length in which heat is flowing

as depicted in Figure 3.2. It is assumed that the rod is of homogeneous material and

has a cross sectional area that is constant throughout the length of the rod. The rod

is laterally insulated along its entire length.

Figure 3.2. One dimensional rod of length L

General explicit finite difference solution

Finite difference is one of the approaches to approximate the solution to

Equations ( 3.12) to ( 3.14). A network of grid points is first established throughout

Area=

L

93

the rectangular region to approximate the solution for the problem as shown in

Figure 3.3.

Figure 3.3. Region and the mesh points (after Kharab and Guenther 2012)

( 3.15)

Region is partitioned by dividing the interval into equal

subintervals, each of the length and the interval into equal

subintervals, each of length . The corresponding points of the intervals

and are denoted by , for and , for ,

respectively. The points are called the mesh or grid points and can be

calculated as follows:

( 3.16)

The approximate solution of at the mesh point is explained

by and the true solution is explained by . As presented by Kharab and

Guenther (2012), Equation ( 3.17) explains the central difference formula for

approximating

( 3.17)

94

And the forward difference formula for approximating is as

follows:

( 3.18)

By substituting Equations ( 3.17) and ( 3.18) into Equation ( 3.12) and

neglecting error terms and Equation ( 3.19) can be obtained.

( 3.19)

If we set and solve for in Equation ( 3.19), the

explicit difference formula can be obtained as follows:

( 3.20)

For and known as the “Forward Difference”

or “Classical explicit method”. Equation ( 3.20) is schematically shown in Figure 3.6.

Figure 3.4. Schematic form of the Forward-difference method (after Kharab and Guenther 2012)

Kharab and Guenther (2012) defined the solution at every point on

the time level in terms of the solution values at the points , ,

and of the previous time level. Such a method is named an explicit method.

It can be shown that the Forward-Difference method has an accuracy of the order

. The values of the initial condition , for ,

are applied in Equation ( 3.20) to calculate the values of , for . The

boundary conditions, , imply that , for

95

. Once the approximations , are known, the values

of , ,…, can be calculated in a similar manner.

General implicit finite difference solution

In the explicit method explained earlier, the approximate solution

depends on the values , , of at the previous level. Moreover, the

condition places an undesirable restriction on the time step that can be

applied. The implicit method overcomes the stability condition by being

unconditionally stable. As explained by Kharab and Guenther (2012), the finite

difference equation of this method is attained by replacing in Equation

( 3.12) with the average of the centred difference at the time steps and and

with the forward difference.

( 3.21)

By setting as before, Equation ( 3.21) can be written as

follows:

( 3.22)

where for . This method is named “Crank-Nicolson”

method. Figure 3.5 shows the schematic form of Equation ( 3.22). The solution value

at any point on the time level is dependent on the solution values

at the neighbouring points on the same level and three points on the time level.

Since values at the time level are calculated implicitly, the method is called

an implicit method. It can be shown that the Crank-Nicolson method has an accuracy

of the order and is unconditionally stable.

Kharab and Guenther (2012) explained that the classical implicit method

can be obtained by replacing the time derivative by a forward difference and the

space derivative by a centred difference at the forward time step ( ) in Equation

( 3.12) as follows:

( 3.23)

where .

The matrix form of the Crank-Nicolson method is as Equation ( 3.24).

96

( 3.24)

where

( 3.25)

( 3.26)

( 3.27)

The tridiagonal matrix is positive define and strictly diagonally dominant.

It should be mentioned, explained implicit and explicit methods can easily

transferred to 2D situation.

As previously stated, in order to ensure stability and convergence in the

explicit method, the ratio must less than 0.5. However, the implicit

Crank-Nicolson method has no such limitation. Consequently, in this research the

implicit Crank-Nicolson method has chosen to solve the equations.

Figure 3.5. Schematic form of the Crank-Nicolson method (after Kharab and Guenther 2012)

97

3.3.2 Axisymmetric consolidation equations

Barron (1948) proposed the governing equation for estimation of one

dimensional axisymmetric consolidation deformation of a saturated soil considering

both vertical and horizontal drainage as shown in Equation ( 3.28).

( 3.28)

where is the vertical strain, and are the coefficients of permeability

in horizontal and vertical directions, respectively, is the unit weight of water, is

the excess pore water pressure at time , and and are the radial and vertical

coordinates, respectively (Figure 3.6).

The assumptions to obtain this equation are: (i) the soil is fully saturated, (ii)

water and soil particles are incompressible, (iii) Darcy’s law is valid and (iv) strains

are small. Obviously, when the soil consists of horizontal layers with thickness (or

the length of vertical drains) of much lesser than the dimensions of the preloading

area, or for the points located at the centre of the embankment, the average strain or

deformation of the soil can be calculated using 1D (vertical) assumption reasonably

accurate.

Figure 3.6. Schematic 3D-axisymetric consolidation

Equations ( 3.10) and ( 3.28) and the effective stress concept of Terzaghi can

be combined to predict the time dependent behaviour of the soil inside and outside of

the disturbed zone (Equations ( 3.29) and ( 3.30)). Furthermore, defining equations

Vertical drain boundary

Disturbed and transition zoneDrainage surface

Cell

heig

ht

r

z

Cell diameter

98

based on the effective stresses in combination with the consolidation theory

(Equation ( 3.10)) facilitates embracing the effects of excess pore water pressure on

the settlement rate, while considering elastic visco-plastic behaviour of soils. It can

be noted that the term captures both time dependant loading and unloading

processes (i.e. staged construction).

For the disturbed zone:

( 3.29)

For the intact zone:

( 3.30)

where is the total vertical stress, and are the horizontal and

vertical coefficients of permeability in the disturbed zone, respectively, and and

are the horizontal and vertical coefficients of permeability in the intact zone,

respectively. Referring to previous studies (e.g. Hansbo 1987; Bergado et al. 1991;

Indraratna and Redana 1998), horizontal and vertical permeability coefficients in the

disturbed zone can be assumed to be equal (i.e. )

Equations ( 3.29) and ( 3.30) are nonlinear partial differential equations

simulating the consolidation process considering combined vertical and horizontal

drainage conditions. To simulate the consolidation of layered soil profiles as a result

of vertical and radial flow, Equations ( 3.29) and ( 3.30) have been expressed in finite

difference form. The adopted numerical formulation in this research is similar to that

99

in the analysis of heat flow (Equation ( 3.12)). The soil is divided into a series of

horizontal and vertical layers and annuli. The state of the soil in each cell is defined

by the conditions at a central node while the cell boundaries lie halfway between

adjacent nodes. Equations ( 3.31) - ( 3.38) are obtained after applying the general

implicit finite difference solution (Crank-Nicholson finite difference scheme) to

Equations ( 3.29) and ( 3.30).

For the disturbed zone when is below the limit time line:

( 3.31)

For the disturbed zone when is above the limit time line:

( 3.32)

For the intact zone when is below the limit time line:

( 3.33)

For the intact zone when is above the limit time line:

( 3.34)

100

where

( 3.35)

( 3.36)

( 3.37)

( 3.38)

where the subscripts i and j represent the horizontal and vertical node

coordinates as i = 1, 2, 3,…, m and j = 1, 2, 3,…, n, respectively. and , are the

mesh size in vertical and horizontal directions and is the time step, as shown in

Figure 3.7.

Permeability changes are taken into consideration as a function of void ratio

changes. In other words, the slope of the straight line in space

(permeability change index, ) is considered to calculate the permeability

coefficient at each time step based on changes in the void ratio, as presented in

Equation ( 3.39).

( 3.39)

To model the elastic visco-plastic behaviour of the soft soil, according to

Yin and Zhu (1999), the initial values of vertical strain in soil profile can be

calculated applying the following equation:

( 3.40)

where represents the initial vertical strain and is

overconsolidation ratio of soil.

101

Figure 3.7. (a) Location of finite difference nodes at any given time; (b) time steps

(a) (b)

Figure 3.8. Boundary conditions for (a) soil layer surrounded by two permeable layers (drains) at the top and bottom; (b) soil layer surrounded by impervious layer at

the bottom and highly permeable layer (drainage blanket) at the top

Furthermore, two different boundary conditions, which are the most

frequent situations, are taken into consideration. Figure 3.8a represents the situation

when soft soil layer is sandwiched between two permeable layers at the top and

bottom. Figure 3.8b represents the situation when soft soil layer is surrounded by

impervious layer at the bottom and highly permeable layer (drainage blanket) at the

top. The following boundary conditions (Equations ( 3.41) and ( 3.42)) can be applied

for two-way drainage (Figure 3.8a) and one-way drainage (Figure 3.8b) conditions,

respectively.

Vertical drain

Drainage surface

z

r

Drainage surface

Vertical drain

Drainage surface

z

r

Impervious surface

Impervious surface

Impervious surface

(a) (b)

102

One-way drainage system:

( 3.41)

Two-way drainage system:

( 3.42)

3.4 DEVELOPING A CODE INCORPORATING CREEP MODEL

In this study, a MATLAB code has been developed to calculate the excess

pore water pressure and the settlement of soil at any given depth and time. Figure 3.9

presents a detailed flowchart of the developed code. As illustrated in the flowchart of

Figure 3.9, the approach starts with collecting the input data, including the soil

properties, soil disturbance parameters, boundary conditions, and loading history

(Equations ( 3.41) and ( 3.42)) (Stage 1). The first stage is completed by defining the

initial vertical strain applying Equation ( 3.40) followed by calculating the effective

vertical stress and the void ratio. Afterward, Equations ( 3.36), ( 3.38), and ( 3.39) are

applied to calculate , , and the permeability, respectively. Calculating the

limit time line ( ) and the creep strain limit ( (Equations ( 3.3) and ( 3.9)) is

the next step to be conducted. For using Equation ( 3.10), the actual vertical strain for

each point should be compared by strain limit (Figure 3.1 and Equation ( 3.9)). If

actual vertical strain at any point is smaller than the strain limit, Equations ( 3.32) and

( 3.34) are applied for calculation, while, Equations ( 3.31) and ( 3.33) are applied if

actual vertical strain is bigger than the strain limit at any point. A system of

equations involving tridiagonal square matrix is formed to obtain excess pore water

pressures (i.e. , , , , and for

and ). Excess pore water pressures values are

used to calculate vertical strain applying Equations ( 3.10) and ( 3.11) (Stage 2). On

Stage 3, settlement and excess pore water pressure values are reported for each time

step.

103

Figure 3.9. Flowchart of the developed MATLAB code

Start

Calculate limit time line ( ) & creep strain limit ( (Equations 3.3 and 3.9)

Calculate (Equation 3.36) Calculate (Equation 3.38)

Calculate void ratio (e) Calculate effective vertical stress ( )

Calculate permeability (Equation 3.39)

If (Figure 3.1 and Equation 3.9)

Yes

Calculate (Equation 3.35)

No Calculate (Equations 3.31 to 3.38)

Calculate vertical strain ( ) (Equations 3.10 and 3.11)

If ( is the maximum

calculation time)

No

Yes

Calculate Settlement (Equation 3.44)

End

Read soil parameters ( , OCR, , , , , , and )

Read boundary conditions )

(Equations 3.41 or 3.42)

Print the results (Settlement, Excess pore water pressures, etc)

Calculate initial vertical strains ( ) (Equation 3.40)

Inpu

t (St

age

1)

Num

eric

al a

naly

sis

(Sta

ge 2

) R

epor

ting

the

resu

lts (S

tage

3)

104

The average time dependant settlement in the influence area of a drain

( ) at the top of the soil layer ( ) for a specific soil thickness ( ) is

given by:

( 3.43)

Applying finite difference method to Equation ( 3.43) provides the following

equation to estimate settlement at time .

( 3.44)

where the subscripts i and j represent the horizontal and vertical node

coordinates. The mentioned steps are applied in a loop until the step time is equal to

the maximum calculation time.

3.5 CAPABILITIES OF THE DEVELOPED FINITE DIFFERENCE SOLUTION

The developed code can be run on micro-computers and has the following

capabilities:

- Calculating the initial vertical effective stress with depth,

- Estimating the initial void ratio with depth,

- Considering variations of permeability with radius, depth, and void ratio,

- Considering variations of overconsolidated ratio with depth and radius,

- Simulating layered soil,

- Considering different set of elastic visco-plastic model and soil parameters for

each set of finite difference,

- Considering the combined effects of smear zone and creep,

- Considering elastic visco-plastic model and soil parameters as functions of time,

- Considering various permeability profiles in the smear zone in combination with

creep,

105

- Considering shear strength variation due to mandrel installation and

investigating the long term effects in combination with creep.

3.6 SUMMERY

In this chapter, the adopted elastic visco-plastic model capable of capturing

time dependant behaviour of soft soils was explained in detail. Different finite

difference methods are available in literature to solve the elastic visco-plastic

equations in combination with 1-D consolidation equation. Implicit (e.g. Crank-

Nicolson) and explicit methods, which are capable of solving parabolic equations,

were explained in depth. In order to ensure the stability and the convergence of the

explicit method, the ratio should be less than or equal . However,

the implicit Crank-Nicolson method has no such limitation. Consequently, in this

research the implicit Crank-Nicolson method has been chosen to solve the equations.

The finite difference solution for axsisymmetric equation, applying the Crank-

Nicolson method to combine the elastic visco-plastic and 1-D consolidation

equations were systematically presented. The developed MATLAB code was

explained in detail. The developed MATLAB code can be run on micro-computers;

and it is capable of simulating layered soils, considering various initial values as

functions of depth and radius, considering time dependant soil parameters, etc.

Verification exercise is presented in the next chapter by comparing the predictions

with the soil laboratory experimental results.

106

CHAPTER FOUR

4 LABORATORY EXPERIMANTS AND VERIFICATION OF THE NUMERICAL SOLUTION

4.1 GENERAL

This chapter presents the details and results of an array of laboratory tests

carried out using oedometer and Rowe cell apparatus to verify the developed

numerical code for the axisymmetric condition. Two different sizes of Rowe cells,

with diameters of 75.3 mm and 250 mm, were used in this research. In this chapter,

the Rowe cells with the diameters of 75.3 mm and 250 mm are called small and large

Rowe cells, respectively. Oedometer tests were carried out to finalise soil mixes of

reduced permeability and intact zones. Two sets of small Rowe cell tests were

conducted on selected soil mixes to obtain the elastic visco-plastic model parameters.

Large Rowe cell test was carried out for the purpose of verification by installing a

vertical drain in the centre of the cell, and using a different mix with reduced

permeability for the area surrounding the vertical drain simulating the reduced

permeability zone. To simulate the vertical drain, a compacted sand column covered

with flexible porous geotextile installed in the centre was used. Transducers were

installed at different heights and various distances away from the vertical drain to

capture the pattern of excess pore water pressure variations during consolidation. At

the same time, the surface settlement was measured using a displacement transducer

connected to a data logger. Finally, the laboratory test results were used to verify the

developed numerical code by comparing measured and predicted surface settlements

and excess pore water pressure values at different heights and radiuses.

107

4.2 TESTING APPARATUS AND EXPERIMENTAL PROCEDURE

4.2.1 Large Rowe Cell Apparatus

The large scale Rowe cell used in this study consists of the body, the base,

and the top. The internal diameter and height of the cell are 250 mm and 200 mm,

respectively (Figure 4.1a). Nine bolts are used to fix the base and the cover to flanges

on the body. A rubber loading jack and a ‘O’ ring seal the top part; and another ‘O’

ring seals the base.

A water pressure acting on a convoluted rubber jack has been used to apply

a uniform load to the sample. To measure vertical settlement at the centre of the

sample, a spindle that is attached to the jack and passes through the cover of the cell

using an LVDT (Linear Variable Differential Transformer) was used. Two washers

seal the spindle through the centre of the rubber diaphragm on the jack to eliminate

any error in the settlement readings due to the rubber diaphragm compressing under

pressure. Nine pore water pressure transducers (PWPTs) have been installed at

various distances and depths to measure the changes in pore water pressure during

consolidation (Figure 4.1). All the transducers are connected to a PC based data

logger to continuously capture and record the test data.

A single drainage system from the top through a perforated brass disc

situated between the sample and the jacket has been designed for the Rowe cell. The

drainage outlet is via the centre of the settlement spindle and a short flexible tube

leading to a Klinger valve at the edge and on top of the cell (Figure 4.1a).

A series of pressure lines, connected to the enterprise level pressure/volume

controllers (ELDPC) that are filled with de-aired water, are used to apply pressure to

the cell (cell pressure) and to the jacket (back pressure). The de-aired water in the

ELDPC cylinder is pressurised and displaced by a piston moving in the cylinder. The

piston is actuated by a ball screw that rotates in a ball nut held by an electric motor

and gearbox that move rectilinearly on a ball slide. The ELDPC instrument is

illustrated in Figure 4.2. The applied ELDPC had a volumetric capacity (nominal) of

200 cm3 for a pressure range of 1 MPa. The resolution of the measurements for

pressure and volume were 1 kPa and 1 mm3, respectively. The pressure and volume

108

90m

m

40m

m64m

m

18m

m

Waterde-airingscrew

Porousstone

Settlement gauge support

Cell body

Cell base

Cell top

Back pressure

Drainage control valve

Cell pressuresupply line

Initial drainage &de-airing valve


O-ring seal

Diaphragm

Settlementrod

Settlement dial gauge

de-airingsystem PWPT A1

PWPT B2

PWPT A3

PWPT B4

PWPT A4

PWPT A2

PWPT B5

PWPT B1PWPT B3

de-airingsystem

Porous plate

200m

m250mm

measurements were accurate to 0.25% of the full range and 0.4% of the measured

value with +/- 500 mm3 back lash, respectively.

Figure 4.1. Large scale Rowe cell apparatus (a) schematic diagram of the cell and (b) locations of the pore pressure transducers at the base of the cell (after Parsa-Pajouh

2014)

The pressure was captured by an integral solid state transducer. To enable

the controller to seek a target pressure or step to a target volume change, control

(a)

(b)

PWPT B2

PWPT B1

PWPT B5

PWPT B3

PWPT B4

27mm

52mm

75mm

102mm

109

algorithms were built into the on-board micro-controller. The change in volume was

measured by counting the steps of the incremental motor. The instrument was

controlled via PC based software.

Figure 4.2. A photographic view of the GDS pressure/volume controller device

Figure 4.3. Infinite volume controller instrument

Using a single pressure/volume controller, there is a need to manually fill or

empty the controller once the volumetric capacity of the barrel has reached either

100% full or 100% empty, respectively. To solve this problem in this study, two

110

parallel pressure/volume controllers (primary and secondary) connected to an infinite

volume controller (IVC) device (Figure 4.3) were used for each pressure line. The

IVC was designed to eliminate constraints from the volume such that in a test, fluid

can flow continuously under pressure control or volume control. The IVC system

automatically switches between the two controllers when one runs out of volume.

Figure 4.4. Schematic diagram of Rowe cell set-up (after Parsa-Pajouh 2014)

One controller, called the primary/master controller was used to provide

pressure source. At the same time the secondary/slave controller can refill and

become ready for switching when the master controller exceeds its volumetric limit.

When this happens the secondary controller takes over by providing the pressure

DL: Data logger LVDT: Vertical Displacement Transducer IVC: Definite Volume Controller CP: Cell Pressure P/V C: Pressure/Volume Controller BP: Back Pressure PWPT: Pore water Pressure Transducer PC: Computer

WaterTank

P/V CP/V C

P/V CP/V C

IVC IVC

PC

CP

BP

DLDL

DLPWPT

LVDT

111

and thus becomes the master. The primary controller becomes the slave and as a

result can refill/empty and centralise itself. When this is complete, the primary

controller resumes pressure control and becomes the master again and the

secondary controller centralises itself.

Figure 4.5. Established setup in the laboratory (large Rowe cell)

The secondary pressure controller is only temporarily used as the master

(to provide the required pressure or a change in volume) when the primary

controller exceeds its volume capacity and thus needs to refill/empty and centralise

itself. The infinite volume controller panel consists of four boxes of infinite volume

controller channel which connect the infinite volume controller panel to the

computer. GDS software controls the system and captures the data from data loggers

in every stage of loading. An elevated water reservoir is used to provide the de-aired

water. A schematic diagram of the experimental set up is shown in Figure 4.4.

Figure 4.5 shows the setup established in the laboratory.

Water tank

Rowe cell

Pressure/Volume

Controller (PVC)

Pore Water Pressure

Transducers (PWPT)

Data Logger (DL)

Infinite Volume

Controller (IVC)

112

4.2.2 Material Properties

4.2.2.1 Soil samples

A number of reconstituted clay samples were prepared by mixing the

following materials to select the reduced permeability zone soil (with reduced

horizontal permeability) and the intact zone soil to carry out the tests:

- Q38 kaolinite

- ActiveBond23 bentonite

- Fine sand

Kaolinite and bentonite were selected because they are common artificial

clays with significantly different properties. Based on Casagrande method, the

kaolinite samples had an average liquid limit of 50% in comparison with 340% for

the bentonite sample. According to AS 1289.3.4.1, the shrinkage limits of kaolinite

and bentonite samples were 9% and 35%, respectively. Table 4.1 summarises the

properties of the clay samples used in this study.

Table 4.1. Properties of the adopted soil samples in this study

Soil Type Liquid Limit

(%)

Plastic Limit

(%)

Linear Shrinkage

(%)

USCS

Symbol

Q38 kaolinite 49.5 27.5 9 CL

ActiveBond23

Bentonite 340 50 35 CH

Based on Nelson and Miller (1997) classification, the adopted kaolinite and

bentonite clays have medium and very high expansion or shrinkage potentials,

respectively. ActiveBond 23 is a pure form of bentonite, which is plastic and

impermeable, and has a high absorbing and swelling capacity, as well as being highly

viscous when suspended in water. Generally, Bentonite is used in constructing

diaphragm walls, piling, tunnelling and sealing dams. The high swelling properties of

bentonite in exposure to water facilitates sealing porous soils and dams. Q38

113

kaolinite clay is a dry milled creamy white kaolin China clay. Kaolinite is one of the

most abundant minerals in soil, and as such is often encountered in on-site

conditions. Kaolinite is formed by the breakdown of feldspar, which is induced by

water and carbon dioxide, and is often formed by the alteration of aluminium silicate

minerals in a warm and humid environment (Craig, 2000; Murray, 1999). Uniformly

graded sand (SP) was used for two purposes: (i) to be added to clay materials to

simulate the in situ clayey soils that usually contain fine sand, and (ii) represent the

drain material. The grain size distribution curve of the utilised sand is illustrated in

Figure 4.6, and some important sizes are shown in Table 4.2.

Figure 4.6. Grain size distribution curve for vertical drain sand

Table 4.2. Important sizes for vertical drain sand

Symbol Grain size (mm)

D10 0.24

D30 0.4

D60 0.55

Note: D10 is the effective particle size (the grain diameter at 10% passing), D30 is the grain diameter at 30% passing, and D60 is the grain diameter at 60% passing

0

10

20

30

40

50

60

70

80

90

100

0.001 0.01 0.1 1 10

Prec

ent p

assi

ng (%

)

Grain size (mm)

Clay Silt Sand Gravel

114

Table 4.3 shows three mix designs for the samples. The Australian Standard

(AS 1289.3.5.2) was used to determine the plastic limit (PL) and liquid limit (LL) of

the mixtures, which are illustrated in Table 4.4. The specimens were thoroughly

mixed with a water content that was 1.4-1.8 times the liquid limit (LL) and kept in a

closed container for couple of days to ensure full saturation and homogeneity. The

properties of the reconstituted clay samples are shown in Table 4.4.

Table 4.3. Mix design for the reconstituted samples

Soil Reference Number Q38 Kaolinite (%) ActiveBond 23

Bentonite (%) Fine sand (%)

S1 70 15 15

S2 68 17 15

S3 65 20 15

Table 4.4. Properties of the reconstituted samples

Soil Reference Number S1 S2 S3

Water Content, w (%) 120 120 120

Liquid Limit, LL (%) 67 70 87

Plastic Limit, PL (%) 27 29 34

Plasticity Index, PI 40 41 43

4.2.2.2 Consolidation tests on reconstituted samples

Materials for reduced permeability and intact zones have been selected from

the reconstituted samples after a number of oedometer tests were carried out. A

preconsolidation pressure of 20 kPa was applied to samples before consolidation test

was conducted. For this purpose, three cylinders were filled with the reconstituted

samples (Table 4.3) and submerged in buckets of water to maintain their saturation

condition. The pre-consolidation process is shown in Figure 4.7. Once pre-

115

consolidation was completed, samples were taken from the soils using the oedometer

ring (Figure 4.8).

A conventional oedometer test was conducted based on the Australian

Standard (AS 1289.6.6.1) to determine the coefficient of permeability for each

sample by applying five stages of loading, including: 12.5 kPa, 25 kPa, 50 kPa, 100

kPa, and 200 kPa (Figure 4.9). The soil samples were 20 mm thick by 50 mm in

diameter, respectively. The settlement and pore water pressure data were collected

continuously for 24 hours for each load increment using a data logger.

Figure 4.7. Pre-consolidation process prior to the oedometer test; (a) cylinder contacting reconstituted sample and (b) samples under pre-consolidation pressure

The data obtained from the oedometer tests were analysed to calculate the

permeability of the reconstituted samples. Table 4.5 shows the permeability of the

samples under a surcharge of 20 kPa. According to Taylor (1948), the variation of

permeability ( ) with void ratio ( ) for the clays can be estimated using the

following equation:

( 4.1)

where is the permeability change index, and are the initial

coefficient of permeability and the void ratio, respectively. The variations of

permeability against the void ratio, for all samples, are shown in Figures 4.10 to 4.12.

(a) (b)

116

Figure 4.8. Preparing the samples for the oedometer test, (a) placing the oedometer ring, (b) cutting the extra top part, (c) cutting the extra bottom part, and (d) the final

sample

Figure 4.9. Consolidation test, (a) placing the prepared sample and (b) oedometer apparatus connected to the data logger

(b) (a)

(c) (d)

(a) (b)

117

Figure 4.10. Variation of permeability against void ratio (sample S1)


0.4

0.6

0.8

1.0

1.2

1.4

1.6

1E-10 1E-09 1E-08 1E-07

Void

ratio

(e)

Permeability (m/s)

Q38 Kaolinite = 70% ActiveBond 23 Bentonite = 15% Fine sand = 15%

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1E-10 1E-09 1E-08 1E-07

Void

ratio

(e)

Permeability (m/s)


118


Table 4.5. Permeability of mixtures (Surcharge = 20 kPa)

Soil reference number S1 S2 S3

Permeability (m/s)

Void ratio (e)

In this study, the permeability ratio ( ) of 4 and an extent ratio ( )

of 3 were desired to conduct the consolidation test assisted with a vertical drain.

Samples S1 and S3 were chosen as the soils for the intact and reduced permeability

zones, respectively, to meet the permeability ratio criteria ( ).

Table 4.6 indicates the properties (permeability and extent) of the intact zone, the

reduced permeability zone, and the vertical drain.

Table 4.6. Properties of the intact zone, the reduced permeability zone, and drain

Area Intact zone Reduced

permeability zone

Vertical drain

Selected Soil S1 S3 Uniformly graded sand (SP)

Radius (mm) 125 33 11

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1E-10 1E-09 1E-08

Void

ratio

(e)

Permeability (m/s)


119

4.2.2.1 Small Rowe cell tests on reconstituted samples

Two sets of Rowe cell tests were conducted on reconstituted samples S1 and S3 for

the applied stresses in the range of 20 kPa to 800 kPa to calculate the soil properties

of intact and reduced permeability zones. The internal diameter and height of the cell

are 50 mm and 40 mm, respectively (Figure 4.13).

Figure 4.13. Schematic diagram of the small Rowe cell apparatus

The following steps were conducted to prepare the small Rowe cell and

conduct the tests: (i) preparing the reconstituted sample by mixing the right portions,

(ii) adding water to the mix to reach the saturated level, (iii) applying a vacuum

pressure to the water tank for the de-airing process, (iv) connecting the pore pressure

transducer, pressure lines, drainage and de-airing pipes, and data transfer cables

according to the illustrated diagram in, (v) running the software to check the

functionality of pressure controllers and data loggers, (vi) de-airing process of the

pore pressure transducers, (vii) filling the Rowe cell with soil sample and levelling

the surface of sample (Figure 4.14a), (viii) placing the porous plate on top of the

sample and fixing the top cap (Figure 4.14b and Figure 4.14c), and (ix) applying a

Cell top

Cell base

Cell pressuresupply line

Drainage control valve

O-ring seal

Diaphragm

Settlement dial gauge

Cell body

Settlementrod

Back pressure


PWPT

de-airingsystem

Porous plate

75.3 mm

Watterde-airing

screw

Porousstone

Settlement gauge support

56 m

m

120

pressure of 110 kPa and a back pressure of 100 kPa for 24 hours to ensure full

saturation (Figure 4.14d). Figure 4.15 shows the setup established in the laboratory.

Figure 4.14. Testing procedure, (a) filling the Rowe cell with soil sample and levelling the surface of sample, (b) placing the porous plate on top of the sample, (c)

fixing the top cap, and (d) Applying a pressure to ensure full saturation

To ensure the saturation, the B-check stage was operated. With the applied

stress increment of 10 kPa, the B-value was calculated as the ratio of the increase of

excess pore water pressure to the stress increment at PWP measurement point. The

B-Check values for Rowe-cell test on Sample S3 and S1 were 0.98 and 0.97,

respectively. Loading stages were conducted in series by increasing the cell pressure

and maintaining the constant back pressure. The details of the loading stages for

Sample S1 and S3 are presented in Table 4.7 and Table 4.8, respectively. The

settlement and pore water pressure data captured continuously by an LVDT at the top

and a pore water pressure transducer at the bottom of the cell. Figures 4.16 to 4.21

and Figures 4.22 to 4.27 show Rowe cell test results carried out on reconstituted soil

samples S1 and S3, respectively.

(a) (b)

(c) (d)

121

Figure 4.15. Established setup in the laboratory (small Rowe cell)

Table 4.7. Details of loading stages using small Rowe cell (Sample S1)

Loading

stage

(kPa)

Loading

condition

Target

effective

stress (kPa)

Back

pressure

(kPa)

Initial cell

pressure

(kPa)

Final cell

pressure

(kPa)

Duration

(days)

25 Loading 25 50 60 75 14

50 Loading 50 50 75 100 20

200 Loading 200 50 100 250 20

400 Loading 400 50 250 450 20

200 Unloading 200 50 450 250 1

100 Unloading 100 50 250 150 1

50 Unloading 50 50 150 100 3

25 Unloading 25 50 100 75 1

50 Reloading 50 50 75 100 2

100 Reloading 100 50 100 150 2

200 Reloading 200 50 150 250 2

400 Reloading 400 50 250 450 2

800 Reloading 800 50 450 850 21

Pressure/Volume

Controller (PVC)

Data Logger (DL)

Rowe cell

122

Table 4.8. Details of loading stages using small Rowe cell (Sample S3)

Loading

stage

(kPa)

Loading

condition

Target

effective

stress (kPa)

Back

pressure

(kPa)

Initial cell

pressure

(kPa)

Final cell

pressure

(kPa)

Duration

(days)

20 Loading 20 100 110 120 14

50 Loading 50 100 120 150 21

100 Loading 200 100 150 250 21

200 Loading 400 100 250 300 21

400 Loading 200 100 300 500 24

800 Loading 100 100 500 900 24

50 Unloading 50 100 900 150 3

100 Reloading 25 100 150 200 1

200 Reloading 50 100 200 300 1

400 Reloading 100 100 300 500 2

800 Reloading 200 100 500 900 2

Figure 4.16. Consolidation test results on reconstituted sample S1 (loading)

0

10

20

30

40

50

60

0.1 1 10 100 1000 10000 100000

Verti

cal s

train

(%)

Time (min)

5-25kPa25-50kPa50-200kPa200-400kPa

Applied vertical effective stress range

123

Figure 4.17. Excess pore water pressure measurement on reconstituted sample S1 (loading)

Figure 4.18. Consolidation test results on reconstituted sample S1 (unloading)

0

20

40

60

80

100

120

140

160

0 1000 2000 3000 4000

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (min)

5-25kPa

25-50kPa

50-200kPa

200-400kPa

51

52

53

54

55

56

57

58

0.1 1 10 100 1000 10000

Verti

cal s

train

(%)

Time (min)

400-200kPa200-100kPa100-50kPa50-25kPa



124

Figure 4.19. Excess pore water pressure measurement on reconstituted sample S1 (unloading)

Figure 4.20. Consolidation test results on reconstituted sample S1 (reloading)

-25

-20

-15

-10

-5

00 500 1000 1500 2000

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (min)

400-200kPa

200-100kPa

100-50kPa

50-25kPa

53

55

57

59

61

63

0.1 1 10 100 1000 10000 100000

Verti

cal s

train

(%)

Time (min)

25-50kPa50-100kPa100-200kPa200-400kPa400-800kPa



125

Figure 4.21. Excess pore water pressure measurement on reconstituted sample S1 (reloading)

Figure 4.22. Consolidation test results on reconstituted sample S3 (loading)

0

50

100

150

200

250

300

0 500 1000 1500 2000

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (min)

25-50kPa

50-100kPa

100-200kPa

200-400kPa

400-800kPa

0

10

20

30

40

50

60

70

80

90

0.1 1 10 100 1000 10000 100000

Verti

cal s

train

(%)

Time (min)

6-20kPa 25-50kPa

50-100kPa 100-200kPa

200-400kPa 400-800kPa



126

Figure 4.23. Excess pore water pressure measurement on reconstituted sample S3 (loading)

Figure 4.24. Consolidation test results on reconstituted sample S3 (unloading)

0

20

40

60

80

100

120

140

160

180

200

0 2000 4000 6000 8000

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (min)

6-20kPa

25-50kPa

50-100kPa

100-200kPa

200-400kPa

400-800kPa

46

47

48

49

50

51

0.1 1 10 100 1000 10000

Verti

cal s

train

(%)

Time (min)

800-50kPa



127

Figure 4.25. Excess pore water pressure measurement on reconstituted sample S3 (unloading)

Figure 4.26. Consolidation test results on reconstituted sample S3 (reloading)

-80

-70

-60

-50

-40

-30

-20

-10

00 1000 2000 3000 4000

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (min)

800-50kPa

46

47

48

49

50

51

52

0.1 1 10 100 1000 10000

Verti

cal s

train

(%)

Time (min)

50-100kPa 100-200kPa

200-400kPa 400-800kPa



128

Figure 4.27. Excess pore water pressure measurement on reconstituted sample S3 (reloading)

To calculate soil parameters, curve fitting procedure proposed by Yin

(1999) was applied. can be obtained as the slope of best fitting curves for the

unloading-reloading points. Referring to Figure 4.28 and Figure 4.29, is equal to

0.010, and 0.011 for intact and reduced permeability zones, respectively.

The next step is to define the model parameters of the reference time line.

The reference time line was obtained by curve fitting of the end of primary

consolidation data points of normally consolidated stresses (Yin 1999). Considering

the fact that the excess pore water pressure is measured during the tests, the end of

primary consolidation is selected when the excess pore water pressure is smaller than

2 kPa. End of primary consolidation data points of 50 kPa to 800 kPa, representing

normally consolidated behaviour, were obtained to define and . The slope of

the best fit line on loading line in Figures 4.28 and 4.29 are the values of , and

the intercept of the best fit line and the horizontal axis is the value of , assuming

that . Thus, for intact and reduced permeability zones are 0.081 and

0.092, respectively. While, for intact and reduced permeability zones are 0.5 kPa

and 4.6 kPa, respectively.

0

20

40

60

80

100

120

140

0 500 1000 1500 2000

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (min)

50-100kPa

100-200kPa

200-400kPa

400-800kPa


129

Figure 4.28. Variation of void ratio versus effective vertical stress (Sample S1)

Figure 4.29. Variation of void ratio versus effective vertical stress (Sample S3)

30

35

40

45

50

55

60

65

10 100 1000

Verti

cal s

train

(%)

Vertical effective stress (kPa)

Best fits

Laboratory measurements

20

25

30

35

40

45

50

55

10 100 1000

Verti

cal s

train

(%)


Laboratory measurements

Best fits



130

( 4.2)

In order to determine the model parameters in the nonlinear creep function,

Equation ( 4.2) was rearranged to the form of Equation ( 4.3).

( 4.3)

where is the strain increase due to creep only, excluding any

instantaneous strain and is the creep time corresponding to . was chosen in

advance as minutes (57600 s) and minutes (115700 s) for intact

and reduced permeability zones, respectively. Substituting values (Equation

( 3.3)) in Equation ( 4.3), corresponding to each effective stress were obtained.

The calculated values of and are tabulated in Table 4.9 and Table 4.10 for

intact and reduced permeability zones, respectively. For instance, the predicted creep

strains are compared to measured values for and presented in Figures

4.30 and 4.31.

Table 4.9. The calculated values of and at different vertical effective stress for intact zone (Sample S1)

Vertical effective

stress (kPa)

50 0.0080 0.27

200 0.0040 0.14

400 0.0037 0.05

800 0.0015 0.01

131

Table 4.10. The calculated values of and at different vertical effective stress for intact zone (Sample S3)

Vertical effective

stress (kPa)

50 0.0150 0.40

100 0.0085 0.35

200 0.0055 0.29

400 0.0011 0.23

800 0.0008 0.18

Figure 4.30. Comparison between predicted numerical creep strain and laboratory measurements at 800 kPa (sample S1)

0.000

0.001

0.002

0.003

0.004

0.005

0 5000 10000 15000 20000 25000 30000Time (min)

Test results at 800kPa

Predictions at 800kPa

Reconstituted sample S1

Cre

ep st

rain

()

132

Figure 4.31. Comparison between predicted numerical creep strain and laboratory measurements at 800 kPa (sample S3)

Combining the results obtained from Table 4.9 and Table 4.10, values as

functions of vertical effective stress for intact and reduced permeability zones are

presented in Figure 4.32 and Figure 4.33, respectively.

Figure 4.32. Changes of versus vertical effective stress for reconstituted

sample S1

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0 2000 4000 6000 8000 10000 12000

Time (min)

Test results at 800kPa

Predictions at 800kPa

0.000

0.002

0.004

0.006

0.008

0.010

10 100 1000




Cre

ep st

rain

()

133

Figure 4.33. Changes of versus vertical effective stress for reconstituted sample S3

Table 4.11, Figures 4.34 and 4.35 summarise the elastic visco-plastic (EVP)

model parameters for the soil samples tested (i.e. reconstituted samples S1 and S3).

Figure 4.34. Time dependant stress-vertical strain relationship for reconstituted sample S1

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

10 100 1000


0

10

20

30

40

50

60

70

0.1 1 10 100 1000

Verti

cal s

train

(%)

Stress (kPa)


0

Limit time line

134

Figure 4.35. Time dependant stress-vertical strain relationship for reconstituted sample S3

Table 4.11. Elastic visco-plastic model parameters for soil samples S1 and S3

Sample Soil type ( ) (s) (kPa)

S1 Intact zone 0.081 0.010 1.57 10.4 57600 0.5 1

S3 reduced

permeability zone 0.092 0.011 1.55 10.6 115200 4.6 1

Permeability of the soil samples at each loading stage in the small Rowe cell

test has been calculated using . For this purpose, the Casagrande

method (Casagrande and Fadum 1940) has been used to obtain the coefficient of the

consolidation ( ). Permeability variations against the void ratio for both samples are

calculated based on small Rowe cell test results and depicted in Figure 4.36 and

Figure 4.37.

0

10

20

30

40

50

60

70

0.1 1 10 100 1000

Verti

cal s

train

(%)

Stress (kPa)

Limit time line Limit time line

135



0.0

0.1

0.2

0.3

0.4

0.5

0.6

1E-11 1E-10 1E-09

Void

ratio

(e)

log k (m/s)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1E-11 1E-10

Void

ratio

(e)

log k (m/s)



136

Figure 4.38. Comparison between predicted numerical settlements and laboratory measurements for small Rowe cell (sample S1)

Figure 4.39. Comparison between predicted numerical settlements and laboratory measurements for small Rowe cell (sample S3)

30

35

40

45

50

55

60

65

70

75

80

0.1 1 10 100 1000 10000 100000

Verti

cal s

train

(%)

Time (min)

50kPa (measurement)200kPa (measurement)400kPa (measurement)800kPa (measurement)50kPa (predictions)200kPa (predictions)400kPa (predictions)800kPa (predictions)

15

25

35

45

55

65

75

0.1 1 10 100 1000 10000 100000

Verti

cal s

train

(%)

Time (min)

50kPa (measurement)100kPa (measurement)200kPa (measurement)400kPa (measurement)800kPa (measurement)50kPa (prediction)100kPa (prediction)200kPa (prediction)400kPa (prediction)800kPa (prediction)

137

To verify the determined model parameters, small Rowe cell tests were

simulated with the developed numerical code for 1D situation. Settlement predictions

are presented in Figure 4.38 and Figure 4.39 for Samples S1 and S3, respectively.

Referring to Figure 4.38 and Figure 4.39, numerical predictions and soil laboratory

measurement are in good agreement, thus the determined model parameters are

reasonable.

4.2.3 Preparation of large Rowe cell and initial sample

Large Rowe cell apparatus was prepared for the consolidation test by

conducting the following steps: (i) applying a vacuum pressure to the water tank for

the de-airing process, (ii) connecting the pore pressure transducers, pressure lines,

drainage and de-airing pipes, and data transfer cables according to the illustrated

diagram in Figure 4.4, (iii) running the software to check the functionality of

pressure controllers, infinite volume controllers, and data loggers, (iv) de-airing

process of the pore pressure transducers, and (v) levelling the main body of the Rowe

cell.

Following the initial preparation of the Rowe cell, two thin PVC and brass

pipes with diameters of 66 mm and 22 mm ( ), respectively, were located at

the centre of the cell at the base to act as the reduced permeability zone and vertical

drain boundaries, respectively. The pipe for the vertical drain was covered with a

filter paper to prevent the intact zone material from mixing with the sand used in the

vertical drain. The pipes and a cross sectional view of the cell are shown in

Figure 4.40.

The area between the circumference of the cell and the boundary of the

reduced permeability zone was filled with a mixture of 70% kaolinite, 15% bentonite

and 15% sand, and a water content of 120% (sample S1 from Table 4.3). The area

between the two pipes (the reduced permeability zone) was filled with slurry that was

prepared previously, based on the mixture and design of the sample S3 from

Table 4.3. Geotextiles were used to cover both areas after being levelled. To stabilise

them and minimise any chance of sliding, extra blocks were placed on top of the

pipes and then steel rings were located on the surfaces of intact and reduced

138

permeability zones. The sample then was left in this condition for 24 hours for

stabilisation (Figure 4.41).

Figure 4.40. Placing of PVC and brass pipes as the reduced permeability zone boundary and the vertical drain border, (a) top view, (b) side view and (c) a typical

cross section of the Rowe cell

(b) (a)

(c)

Inner pipe (Boundary of vertical drain)

Outer pipe (Boundary of reduced permeability zone) Filter

Disturbed zone

Vertical drainIntact zone

125m

m

33mm

11m

m

Cell circumference

Outer pipe

Inner pipe

139

Figure 4.41. Sample placement, (a) filling the intact area (intact zone) with the prepared soil and (b) the setup after placing PVC and Brass pipes as the reduced

permeability zone boundary and vertical drain border

4.3 PRE-CONSOLIDATION PROCESS AND PREPARATION OF THE FINAL SAMPLE W

To stabilise the intact and reduced permeability zones, an initial pre-

consolidation pressure of 20 kPa was applied before installing the drain.

Preconsolidation pressure was applied by placing a number of manufactured 20 mm

thick steel rings on top of each zone. Each day one ring (1.8 kPa) was added to

obtain the required preconsolidation pressure. To allow water to drain from the top

surface during preconsoldiation, the first ring of each set was designed with some

radial grooves. To keep the sample saturated, the level of water was checked

continuously. The pre-consolidation process is shown in Figure 4.42.

(b) (a)

140

Figure 4.42. Rig set up, (a) geotextile filters, (b) pre-consolidation loading rings, (c) the first two loading rings with drainage grooves and holes, (d) placing of the first

loading ring and (e) full loading condition

After completion of the preconsolidation process, to prepare the sample for

the main consolidation test, the following steps were carried out: (i) all but the first

loading ring was removed, (ii) sand was poured into the inner pipe to act as a vertical

drain and to compact the sand (Figure 4.43a), (iii) the outer pipe was removed

(Figure 4.43b), (iv) the inner pipe was removed and an extra part of the filter paper

was cut (Figures 4.44a and 4.44b), (v) the top surface was levelled (Figure 4.45a),

(vi) a geotextile filter and a porous metal plate were placed on the top surface

(Figure 4.45b), (vii) the cell was filled with water (Figure 4.46a), (viii) a porous

stone was placed at the centre of the cell to allow water to drain through the

settlement rod, (ix) the top of the cell, and the diaphragm and settlement rod were

(a) (b)

(c) (d) (e)

141

(a) (b)

(a) (b)

placed in position (Figure 4.46b), and (x) the top part of the cell was bolted to the

body.

Figure 4.43. Testing procedures, (a) Pouring the vertical drain material and (b) Pulling out the outer pipe

Figure 4.44. Testing procedures, (a) pulling out the inner pipe and (b) cutting the extra part of the filter paper

142

(a) (b)

(a) (b)

Figure 4.45. Testing procedures, (a) levelling the top surface and (b) placing the geotextile on top surface

Figure 4.46. Testing procedures, (a) filling the cell with water and (b) placing the cell top

4.3.1 Initial drainage and de-airing of the Rowe cell system

To fill the geo-membrane diaphragm with de-aired water and carrying out a

de-airing process based on the diagram, shown in Figure 4.47, the cell and back

pressures were set at 110 kPa and 100 kPa, respectively. Water was pumped into the

geo-membrane diaphragm through the valve V1 to increase the volume of the

diaphragm by applying 110 kPa of pressure. Simultaneously, the de-airing screw on

143

Water Tank

Drainage &de-airing valve

Drainage &de-airing valve

De-airingscrew

A F

G

D

E

Cell top

Cel

l pre

ssur

e

Diaphragm

Cell body

Settlement rod

Porous plate Watter

B

C

1.0m

IVCIVC

Bac

k pr

essu

re

P/V

C

P/V

C

P/V

C

P/V

C

De-airing valve De-airing valve

V1

V2

O

V4 V5

V3

top of the cell allows the trapped air in the diaphragm to escape, after which the bolts

were tightened again.

Figure 4.47. Schematic diagram of the de-airing process (after Parsa-Pajouh 2014)

By increasing the volume of the diaphragm during filling, the air trapped

between the diaphragm and the body of the cell was drained out from value V2. By

closing valve V2 and opening valves V1 and V3, the trapped air escaped through the

outlet point O, passing the path DEO (i.e. connecting points D, E, and O). By

opening valve V4, water in the tank was discharged through the connecting points A,

B, C, D, E, and O. In this way, the air trapped in the pipe between points B and C

was drained from the outlet point O. Valves V3 and V4 were closed after this stage

was completed.

The de-airing valves installed on the infinite volume controller (IVC) were

used to drain the trapped air in the pressure lines and controller devices. By closing

valves V1 and V2 and opening V4 and V5, water flows from the tank to the path and

144

pushes the trapped air through the de-airing valves. This procedure was repeated

numerous times to ensure that the system was totally de-aired. It should be

mentioned that any remaining air bubbles may cause errors in settlement and pore

water pressure measurements.

4.3.2 Vertical drain assisted consolidation test procedure

Although it was presumed that the soil sample was fully saturated as slurry

with a moisture content well above the liquid limit was used, a pressure of 110 kPa

and a back pressure of 100 kPa were applied for 24 hours to ensure full saturation.

The criterion for a fully saturated condition and obtaining a pore pressure coefficient

( ) of more than 95%, was reached. In this study, five loading, one unloading,

and three reloading stages were applied to carry out the vertical drain assisted

consolidation tests. The cell and back pressures applied in each stage of loading are

tabulated in Table 4.12. The large Rowe cell setup is shown in Figure 4.5.

Table 4.12. Details of consolidation loading stages

Loading stage Applied effective pressure (kPa)

Cell pressure (kPa)

Back pressure (kPa)

Loading duration

(day) 1 25 225 200 5 2 50 250 200 14 3 100 300 200 28 4 200 400 200 65 5 400 600 200 72 6 50 250 200 85 7 100 300 200 4 8 200 400 200 3.5 9 400 600 200 4.5

The loads mentioned above were applied instantaneously and then maintained for the

duration mentioned in Table 4.12, before moving on to the next stage.

145

4.4 RESULTS AND DISCUSSION

As expressed earlier, the large Rowe cell was filled with the reconstituted

soil samples (Table 4.3) to evaluate the intact region and the reduced permeability

zone with a circular sand drain at the centre. A consolidation test was carried out

following the pre-consolidation process (i.e. 20 kPa). The water pressure inside the

membrane diaphragm that had been placed on top of the preconsolidated sample was

used to apply the consolidation surcharges. The time dependent vertical

displacements of the sample were captured using GDSLab software, with an LVDT

(Linear Variable Differential Transformer) transducer. To monitor the dissipation of

excess pore water pressure during consolidation, a number of pore pressure

transducers were used.

Figure 4.48 shows the instrumentation plan for the 250 mm Rowe cell.

Figure 4.49 to 4.52 show the comparison of the excess pore water pressure

predictions and laboratory measurement at PWPT B2 to B5, respectively. According

to Figures 4.49 to 4.52, the excess pore water pressure measurements and numerical

predictions are in a good agreement. It was observed that the excess pore water

pressure increased almost instantly after increasing the surcharge, but this rise in the

excess pore water pressure was slightly less than the increase in the applied pressure.

During the dissipation process, the further the distance of PWPT is from the vertical

drain, the higher the value of excess pore water pressure. For example, one day after

increasing the surcharge from 200 kPa to 400 kPa, soil laboratory measurements

show 58.04 kPa, 88.50 kPa, 115.04 kPa, and 117.98 kPa for PWPT B2 to B5,

respectively. While, numerical analysis predicted 43.02 kPa, 68.55 kPa, 83.01 kPa,

and 95.05 kPa for PWPT B2 to B5, respectively. In other words, PWPT B2 with the

shortest distance ( ) from vertical drain, one day after increasing surcharge

from 200 kPa to 400 kPa, shows 51% and 55% less excess pore water pressure in

comparison with PWPT B5 with the longest distance ( ) from vertical

drain for laboratory measurements and numerical predictions, respectively.

146

Figure 4.48. Schematic diagram of the instrumentation plan, (a) the cross section of bottom of the Rowe cell and (b) plan view of the body of Rowe cell (after Parsa-

Pajouh 2014)

(a)

(b)

75m

m

102mm

52mm

27mm

Disturbed zone boundaryVertical drain boundary

PWPT B3

PWPT B4

PWPT B5

PWPT B2

PWPT B1

LVDT 1

PWP A1

PWP B2

PWP A3

PWP B4

PWP A4

PWP A2

PWP B5

PWP B1PWP B3

90m

m

40m

m64m

m

18m

m

147

Figure 4.49. Comparison of the excess pore water pressure predictions and laboratory measurement versus time throughout loading at PWPT B2


0

50

100

150

200

250

300

350

400

450

0

50

100

150

200

250

300

350

400

450

0 25 50 75 100 125 150 175

Surc

harg

e (k

Pa)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)

Prediction

Soil laboratory measurement(PWPT B2)

Surcharge (kPa)

0

50

100

150

200

250

300

350

400

450

0

50

100

150

200

250

300

350

400

450

0 25 50 75 100 125 150 175

Surc

harg

e (k

Pa)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)

Prediction

Soil laboratory measurement(PWPT B3)Surcharge (kPa)

148



0

50

100

150

200

250

300

350

400

450

0

50

100

150

200

250

300

350

400

450

0 25 50 75 100 125 150 175

Surc

harg

e (k

Pa)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)

Prediction

Soil laboratory measurement(PWPT B4)Surcharge (kPa)

0

50

100

150

200

250

300

350

400

450

0

50

100

150

200

250

300

350

400

450

0 25 50 75 100 125 150 175

Surc

harg

e (k

Pa)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)

Prediction

soil laboratory measurement(PWPT B5)Surcharge (kPa)

149

Figures 4.53 to 4.56 depict the comparison of the predicted excess pore

water pressure (adopting the developed finite difference code) and laboratory

measurement throughout unloading and reloading at PWPT B2 to B5, respectively.

Referring to Figure 4.53 to Figure 4.56, the excess pore water pressure laboratory

measurements and numerical predictions are in a good agreement. As it was

observed, the excess pore water pressure dropped almost instantly after decreasing

the surcharge from 400 kPa to 50 kPa. Throughout the excess pore water pressure

dissipation, larger distance of PWPT from the vertical drain results in the lower value

of excess pore water pressure after unloading. For instance, the developed code

predicted -24.98 kPa, -37.96 kPa, -45.05 kPa, and -57.56 kPa and soil laboratory

measurements show -11.45 kPa, -13.61 kPa, -18.75 kPa, and -19.18 kPa for PWPT

B2 to B5 one day after decreasing the surcharge from 400 kPa to 50 kPa. In other

words, PWPT B2 with the shortest distance ( ) from vertical drain shows

40% and 57% less excess pore water pressure in comparison with PWPT B5 with the

longest distance ( ) from vertical drain for laboratory measurements and

numerical predictions, respectively. Referring to Figure 4.53 to Figure 4.56, there are

some disparities in excess pore water pressure predictions and measurements which

may be due to Mandel-Cryer effect. Mandel-Cryer effect has been proposed to

explain this delayed excess pore water pressure dissipation response. Schiffman et al.

(1969) expressed Mandel-Cryer effect as the increase in total stress, which is caused

by the volumetric strain compatibility.

150

Figure 4.53. Comparison of the excess pore water pressure predictions and laboratory measurement versus time throughout unloading and reloading at PWPT B2

Figure 4.54. Comparison of the excess pore water pressure predictions and laboratory

measurement versus time throughout unloading and reloading at PWPT B3

-350

-250

-150

-50

50

150

250

350

450

-350

-250

-150

-50

50

150

250

350

450

175 200 225 250 275

Surc

harg

e (k

Pa)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)

Prediction

Soil laboratory measurement (PWPT B2)

Surcharge (kPa)

-350

-250

-150

-50

50

150

250

350

450

-350

-250

-150

-50

50

150

250

350

450

175 200 225 250 275

Surc

harg

e (k

Pa)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)

Prediction


surcharge (kPa)

151



-350

-250

-150

-50

50

150

250

350

450

-350

-250

-150

-50

50

150

250

350

450

175 200 225 250 275

surc

harg

e (k

Pa)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)

Prediction


Surcharge (kPa)

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harg

e (k

Pa)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)

Prediction

soil laboratory measurement (PWPT B5)

Surcharge (kPa)

152

Figure 4.57. Comparison of the excess pore water pressure predictions and laboratory measurement versus time throughout loading at PWPT A1

Figures 4.57 to 4.60 show the comparison of the excess pore water pressure

predictions and laboratory measurement versus time throughout loading for PWPT

A1 to A4, respectively. Excess pore water pressure increases rapidly at the beginning

of each stage due to applied surcharge. According to Figures 4.57 to 4.60, excess

pore water pressure values predicted by developed code are in good agreement with

measured values in laboratory. Obviously, the higher the position of PWPT (Closer

to the geotextile on top surface), the lower the values of excess pore water pressure.

For instance, one day after applying the surcharge from 200 kPa of 400 kPa the

values of excess pore water pressure reached 152.52 kPa, 112.06 kPa, and 65.46 kPa

in laboratory measurements and 163.41 kPa, 89.96 kPa, and 59.94 kPa in numerical

predictions for PWPT A1 to A3, respectively. That is, PWPT A3 with the highest

position ( ) shows 57% and 63% less values of excess pore water pressure

in comparison with PWPT A1 with lowest position ( ) for laboratory

measurement and numerical predictions, respectively. Referring to Figure 4.60,

PWPT A4 did not capture any value of excess pore water pressure for the last two

stages since soil settled enough to leave the PWPT out of soil sample.

0

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harg

e (k

Pa)

Exce

ss p

ore

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er p

ress

ure

(kPa

)

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Prediction

Soil laboratory measurement(PWPT A1)Surcharge (kPa)

153



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Prediction


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ss p

ore

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er p

ress

ure

(kPa

)

Time (day)

Prediction


154


Figures 4.61 to 4.63 present the comparison of the excess pore water

pressure values predicted by developed code and measured in soil laboratory

throughout unloading and reloading for PWPT A1 to A3, respectively. Excess pore

water pressure drops and increases rapidly due to unloading and reloading,

respectively, at each stage. Referring to Figures 4.61 to 4.63, numerical predictions

and soil laboratory measurements are in good agreement. As expected, larger

distance from drain (the geotextile on top surface) results in larger value of excess

pore water pressure. For example, one day after decreasing the surcharge from 400

kPa to 50 kPa, soil laboratory measurements showed -20.1 kPa, -13.4 kPa, and -7.84

kPa and developed code predicted -49.64 kPa, -35.9 kPa, and -25.05 kPa for PWPT

A1 to A3, respectively.

Figures 4.64 to 4.67 depict the measurement of the excess pore water

pressure variations at the base and different heights of large Rowe cell. As observed,

the maximum increase of the excess pore water pressure at the base and in different

heights are lower than the applied stress increment. Considering the fact that the soil

has high initial water content and was stored in water before testing, the observed

pore water pressure responses may not be explained by the degree of saturation. The

issue may be induced by the stiffness of the pore water pressure measurement device

(Robinson 1999 and Whitman et al. 1961). A partial drainage of pore water from the

0

20

40

60

80

100

120

0

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0 10 20 30 40

Surc

harg

e (kP

a)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)

Prediction


155

base of the soil sample may be allowed by the stiffness of the pore water pressure

measurement system. As discussed by Charlie (2000), throughout the consolidation

test, water may flow from soil into or out the measurement system which induce the

change of the drainage condition of the impervious base of the soil sample.

Figure 4.61. Comparison of the excess pore water pressure predictions and laboratory measurement versus time at PWPT A1


-350

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e (k

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ss p

ore

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ress

ure

(kPa

)

Time (day)

Prediction

Soil laboratory measurement (PWPTA1)Surcharge (kPa)

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ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)

Prediction

Soil laboratory measurement (PWPTA2)Surcharge (kPa)

156


Figure 4.64. Measured excess pore water pressure at transducers located on the bottom of the cell (loading)

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e (k

Pa)

Exce

ss p

ore

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er p

ress

ure

(kPa

)

Time (day)

Prediction

Soil laboratory measurement (PWPT A3)

Surcharge (kPa)

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e (k

Pa)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)





Surcharge (kPa)

157

Figure 4.65. Measured excess pore water pressure at transducers located on the bottom of the cell (unloading and reloading)

Figure 4.66. Measured excess pore water pressures from transducers located on the sides of the cell (loading)

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Pa)

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ss p

ore

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ress

ure

(kPa

)

Time (day)





Surcharge (kPa)

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Surc

harg

e (k

Pa)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)

Soil laboratory measurement (PWPT A1)Soil laboratory measurement (PWPT A2)Soil laboratory measurement (PWPT A3)Soil laboratory measurement (PWPT A4)Surcharge (kPa)

158

Figure 4.67. Measured excess pore water pressures from transducers located on the sides of the cell (unloading and reloading)

Figure 4.68 and Figure 4.69 show the variations of excess pore water

pressures with vertical distance from the bottom of the impermeable boundary and

the radial distance from the centre of the drain, respectively. Figure 4.68 shows that

the excess pore water pressure changed inversely with the vertical distance from the

impermeable base, i.e., the excess pore water pressure increased when the distance

from the impervious bottom boundary decreased. The longer vertical distance from

the top drainage boundary resulted in a higher remaining excess pore water pressure.

For instance, the excess pore water pressure measurement at PWPT A1 ( )

was 140.2 kPa after 105 days of consolidation, but it was reduced by 77% to 32.2

kPa at PWPT A3 ( ). It should be mentioned that predicted excess pore

water pressure value reduced by 64% from 128.13 kPa to 46.34 kPa.

Figure 4.69 shows that the excess pore water pressure followed an

incremental trend when the radial distance from the drain increased. For example,

105 days after consolidation, increasing the radial distance from 27 mm (PWPT B2)

to 102 mm (PWPT B5) resulted in 164% rise in the excess pore water pressure

corresponding to change from 44.07 kPa to 116.02 kPa. This increase is 127% and a

change from 34.28 kPa to 77.88 kPa for numerical predictions. According to Figures

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450

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175 200 225 250 275

Surc

harg

e (k

Pa)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (day)

Soil laboratory measurement (PWPT A1)Soil laboratory measurement (PWPT A2)Soil laboratory measurement (PWPT A3)Soil laboratory measurement (PWPT A4)Surcharge (kPa)

159

4.68 and 4.69, there are some disparities between measured and calculated values for

excess pore water pressure. As mentioned by Robinson (1999), the pore water

pressure measured throughout consolidation tests reaches its maximum value (equal

to pressure increment) not immediately after the pressure application but after some

finite time. Moreover, the maximum value is usually less than the applied pressure

increment. It has been shown that these effects can be related to the relationship

between volumetric compliance of the pore pressure measuring system and the

volume compressibility of the soil skeleton.

Figure 4.68. Variations of excess pore water pressures with the vertical distance from the bottom of the impermeable boundary

10

20

30

40

50

60

70

0 25 50 75 100 125 150 175 200

Verti

cal d

ista

nce

from

the

botto

m

impe

rmea

ble

boun

dary

of t

he c

ell (

mm

)

Excess pore water pressure (kPa)

Consolidation time=105days (measurement)Consolidation time=107days (measurement)Consolidation time=109days (measurement)Consolidation time=111days (measurement)Consolidation time=105days (Prediction)Consolidation time=107days (Prediction)Consolidation time=109days (Prediction)Consolidation time=111days (Prediction)

160

Figure 4.69. Variations of excess pore water pressures with the radial distance from the centre of the drain

Figure 4.70 depicts creep coefficient ( ) variation with time at the

location of PWPT B5 ( . Evidently, the creep coefficient changes with

both time and vertical effective stress. In other words, lower effective vertical

stresses (or higher excess pore water pressures) result in higher creep coefficients.

For instant, the creep coefficient drops from to (4%

reduction) throughout loading from 100 kPa to 200 kPa or the creep coefficient

increased by 4% throughout unloading from 400 kPa to 50 kPa. Although the creep

coefficient ( ) changes slightly after dissipation of excess pore water pressure,

significant change in the creep coefficient occurs during excess pore water pressure

dissipation. Figure 4.71 shows the creep strain limit prediction at the location of

PWPT B5 ( . Similar to the creep coefficient, the creep strain limit is

also inversely related to the effective vertical stress. As a result, the more the

effective stress the lower the creep strain limit. For example, the creep strain limit

drops from 0.25 to 0.18 (28% reduction) throughout loading from 100 kPa to 200

kPa or the creep strain limit increased by 244% throughout unloading from 400 kPa

to 50 kPa. Referring to Figure 4.71, although creep strain limit a function of effective

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70 80 90 100 110

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Radial distance from drain (mm)

Consolidation time=105days (measurement)Consolidation time=107days (measurement)Consolidation time=109days (measurement)Consolidation time=111days (measurement)Consolidation time=105days (Prediction)Consolidation time=107days (Prediction)Consolidation time=109days (Prediction)Consolidation time=111days (Prediction)

161

stress and time the significant part of creep strain limit variations occurs during

excess pore water pressure dissipation.

Figure 4.70. Predicted creep coefficient ( ) values versus time

Figure 4.71. Creep strain limit values predictions versus time

0

50

100

150

200

250

300

350

400

450

0.012

0.013

0.014

0.015

0.016

0 50 100 150 200 250

Surc

harg

e (k

Pa)

Cre

ep c

oeffi

cien

t

Time (days)

Creep coefficient

Surcharge (kPa)

0

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100

150

200

250

300

350

400

450

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 50 100 150 200 250

Surc

harg

e (k

Pa)

Cre

ep st

rain

lim

it

Time (days)

Creep limit

Surcharge (kPa)

0.

0.

()

0()

162

Figures 4.72 and 4.73 present the comparison between the predicted time

dependent settlements and the laboratory measurements throughout loading and

unloading/reloading, respectively. As mentioned earlier, to capture the reported

laboratory settlement values, the GDSLab software, with an LVDT (Linear Variable

Differential Transformer) transducer was used. Referring to Figures 4.72 and 4.73,

laboratory measurements and numerical predictions are in good agreement. It should

be mentioned that there are some disparities between soil laboratory measurement

and numerical predictions on early stages of the test which can be the result of

disturbance caused by the removal of the pipes surrounding vertical drain and

reduced permeability zone.

To avoid the infinity compression of the soil, the nonlinear creep function

was proposed by Yin (1999). Consequently, the creep coefficient ( ) in the nonlinear

creep function is calculated based on the initial creep coefficient ( ) and creep strain

limit ( ) and the equivalent time ( ) (Equation (3.7)). Even though the initial

creep coefficient ( ) remains constant after the completion of excess pore water

dissipation, the creep coefficient ( ) continuously decreases with time. The creep

strain rate gradually decreases when the maximum effective stress is almost

established by completion of the dissipation of the excess pore water pressure. After

the completion of excess pore water dissipation the compression is allowed to

continue under the constant effective stress. As a result, the equivalent time ( )

increases and the equivalent time line moves toward the limit time line.

Consequently, the associated creep strain rate and creep coefficient decrease which

explain the variations of the corresponding creep strain rate and creep coefficient in

Figures 4.70 and 4.71. Thus, it can be observed in the settlement curves (Figures 4.72

and 4.73) that the slope of settlement keeps decreasing after the completion of the

excess pore water pressure dissipation.

163

Figure 4.72. Comparison between predicted numerical settlements and laboratory measurements (loading)

Figure 4.73. Comparison between predicted numerical settlements and laboratory measurements (unloading and reloading)

0

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10

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60

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Surc

harg

e (k

Pa)

Settl

emen

t (m

m)

Time (day)

Prediction

Soil laboratory measurement

Surcharge (kPa)

0

50

100

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45054

55

56

57

58

59

60

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Surc

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Pa)

Settl

emen

t (m

m)

Time (day)

Prediction

Soil laboratorymeasurement

164

4.5 SUMMARY

In this chapter, the measured results of a fully instrumented large scale

Rowe cell rig as well as a small Rowe cell apparatus were used to verify the

developed numerical code. Staged uniform loads were applied to the sample using a

water pressure system acting on a convoluted rubber jack. To monitor the variations

of settlement and excess pore pressure dissipation during consolidation, an LVDT

and pore pressure transducers were used, respectively. To conduct oedometer tests to

select the proper mixtures to be used in the Rowe cell, a number of samples were

prepared by mixing various percentages of Q38 kaolinite, ActiveBond23 bentonite,

and uniformly graded sand (SP). According to the results of the oedometer tests, two

samples, meeting the criteria of were selected as the intact (sample S1)

and reduced permeability (sample S3) zones materials. Two sets of tests were carried

out on the selected intact and reduced permeability zones samples with the small

Rowe cell to calculate the soil properties. To verify the calculated soil parameters,

small Rowe cell test were simulated with the developed code.

Following the Rowe cell preparation, to simulate the reduced permeability

zone and vertical drain boundaries, 66 mm and 22 mm diameter pipes ( ),

respectively, were placed in the centre of the cell at the base. The area between the

boundary of the vertical drain and reduced permeability zones was filled with the

prepared slurry for this zone (sample S3). While, the area between the circumference

of the cell and the boundary of the reduced permeability zone was filled with the

slurry that was prepared on the selected mix design for this zone (sample S1). The

reconstituted sample was then pre-consolidated under a 20 kPa load by placing a

number of 20 mm thick steel rings on top of each other. After completion of

preconsolidation process, fine sand was poured inside the central pipe to form the

vertical drain and then both pipes were pulled out. To provide a one-way drainage

condition, a geotextile filter and a porous metal plate were located on the top surface

of the sample while no drainage was allowed from the base. The whole system was

de-aired as the final stage of the sample preparation process. Five loading, one

unloading, and three reloading stages were applied to conduct the PVD assisted

consolidation tests.

165

Two small Rowe cell test results have been used to calculate soil

parameters. As explained earlier, the creep strain limit is calculated by applying

Equation ( 3.3). The initial creep coefficient is calculated by substituting Equation

( 3.3) in Equation ( 4.3). As presented, the creep strain limit and the creep coefficient

are reversely related to vertical effective stress and time. Consequently, the creep

strain limit and the creep coefficient will be lower if the time or the vertical effective

stress are greater.

The developed numerical code was used to simulate the consolidation test.

Laboratory measurements were applied to evaluate the validity of the developed

numerical model. The numerical results show that the proposed finite difference

procedure incorporating the elastic visco-plastic soil behaviour is appropriate for the

consolidation analysis of preloading with vertical drains. Referring to Figures 4.72

and 4.73, the laboratory settlement measurements and the numerical predictions are

in a good agreement. As observed in the settlement curves (Figures 4.72 and 4.73),

the slope of settlement keeps reducing after the completion of the excess pore water

pressure dissipation. This is due to the fact that after completion of excess pore water

pressure dissipation and continuation of the compression under the constant effective

stress the equivalent time ( ) increases and the equivalent time line moves toward

the limit time line. Consequently, the corresponding creep strain rate and creep

coefficient decrease (Figures 4.70 and 4.71).

As presented in Figures 4.49 to 4.63, the excess pore water pressure

measurements and predictions are reasonably in a good agreement. There are some

disparities between measured and calculated values for excess pore water pressures.

The disparities can be due to Mandel-Cryer effect, which has been proposed to

explain this delayed excess pore water pressure dissipation response. Schiffman et al.

(1969) expressed the Mandel-Cryer effect as the increase in total stress, which is

caused by the volumetric strain compatibility. Furthermore, as mentioned by

Robinson (1999), the maximum value of excess pore water pressure value is usually

less than the applied pressure increment, which can be related to the relationship

between volumetric compliance of the pore pressure measuring system and the

volume compressibility of the soil skeleton.

166

CHAPTER FIVE

5 EFFECTS OF SOIL DISTURBANCE DUE TO PVD INSTALLATION ON LONG TERM GROUND

BEHAVIOUR

5.1 GENERAL

The installation of PVDs speeds up the dissipation of excess pore pressure

generated throughout preloading by means of reducing the drainage path within the

ground. Drainage path reduction hastens the consolidation process thus increases the

strength and stiffness of soft clayey soils. However, the installation of prefabricated

vertical drains (PVDs) disturbs the soil around the drain, which results in a zone of

reduced hydraulic conductivity and shear strength. This reduction in hydraulic

conductivity and shear strength induced by soil disturbance can remarkably affect the

excess pore water pressure dissipation rate and the creep coefficient. The extent of

the soil shear strength and hydraulic conductivity changes in the disturbed zone

versus the distance from the vertical drain has not been identified with certainty and

so far there is no comprehensive or standard method for measuring these

characteristics. Although some efforts have been made to simulate a gradual

variation of the hydraulic conductivity with radius (Madlav et al. 1993, Chai et al.

1997, Hawlader et al. 2002), it has never been a straight forward task to quantify the

disturbing effects.

In this chapter, two case studies of vertical drains assisted preloading were

numerically simulated to investigate the effects of soil disturbance induced by the

installation of vertical drains. Different variations of the overconsolidation ratio and

the hydraulic conductivity in the disturbed zone in combination with time dependant

behaviour of soft soils are considered. The effects of overconsolidation ratio and the

hydraulic conductivity profiles on the settlement, the excess pore water pressure

167

dissipation rate, the visco-plastic strain rate, and the creep strain limit are

investigated and discussed.

5.2 VÄSBY TEST FILL CASE STUDY

The Swedish Geotechnical Institute (SGI) designed and built Väsby test fill

in 1945 near Uplands Väsby village, 30km north of Stockholm on the east coast of

Sweden to study the long-term behaviour of Swedish clays and the suitability of the

site for construction of an airport. Three test fills including one with vertical drains

and two without vertical drains were built in the Väsby test field as shown in

Figure 5.1. The Test Areas I and II were constructed without paper drains by placing

fill to a height of 2.5 m and 0.3 m, respectively. Test Area III consisted of installing

paper drains in the clay soils and placing fill to a height of 2.5 m. The test Area III

with vertical drains has been selected to be discussed in this study.

As reported by Chang (1969 and 1981), work on the Test Area III, which

measured 30×30m, was begun in 1945. The clay layer (Väsby post glacial clay)

underlying this area was nearly 10 m thick. Prefabricated vertical drains installation

depth and spacing were 5 m and 0.7 m, respectively. Referring to Figure 5.2,

Settlement monitoring devices were established at the surface of the scraped ground

and at depths of 3.8 m and 5 m. The gravel fill, with the unit weight of 17kN/m3, was

located to the height of 2.5 m on the top of the soft clay. A 0.8-m layer of gravel fill

was removed half a year later (i.e. 182 days) after completion of the surcharge

placement.

Eight consolidometer tests using conventional oedometer cells were carried

out on undisturbed specimens of the Väsby post glacial clay deposit by Chang (1969)

(Figure 5.3) for the applied stresses in the range of 5 kPa to 160 kPa. The required

soil properties can be calculated adopting the developed procedure explained by Yin

(1999) employing laboratory test results, and the adopted soil parameters are

reported in Table 5.1 and Figure 5.4 to 5.8. Referring to Figure 5.5, in this case

study, the ratio of can be calculated as a function of the vertical effective stress for

Väsby post glacial clay deposit ( ).

168

Stjarnborgsbanan

Figure 5.1. Väsby test field (after Chang 1981)

Area I No Drain

Area II No Drain

Area III Paper Drain

N

0

Saln

aban

an

10 20 30m Scale

169

Figure 5.2. Soil profile beneath the Väsby test fill

Figure 5.3. Consolidation test results on Väsby post glacial clay samples for vertical stresses between 5 kPa and 160 kPa

Data obtained from Chang (1969) and (1981)

SP1

30 m North South

0.0

-11.2

-14.2 Bedrock

Glacial varved clay

Väsby post glacial clay

Fill material ( )

-3.8

SP3 -5.0

-12.2 -9.8

0.0

Scale

0 m m10

SP2

+2.5 (before surcharge removal) +1.7 (after surcharge removal)

PVDs

0

10

20

30

40

50

60

70

10 100 1000 10000 100000

Verti

cal s

train

(%)

Time (s)

0 to 5 kPa5 to 10 kPa10 to 20 kPa20 to 30 kPa30 to 45 kPa45 to 67.5 kPa67.5 to 105 kPa105 to 160 kPa

Data obtained from oedometer tests on undisturbed Väsby

post glacial clay samples reported by Chang (1969)

170

0

0.01

0.02

0.03

0.04

0.05

0.06

10 100

Vertical effective stress (σz') (kPa)

Figure 5.4. Time dependant stress-vertical strain relationship for Väsby post glacial clay

Figure 5.5. Changes of versus vertical effective stress

0

20

40

60

80

100

1 10 100 1,000Ve

rtica

l stra

in (%

)Stress (kPa)

Limit time line

171

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1E-11 1E-10 1E-09 1E-08

Void

ratio

(e)

Permeability (k) (m/s)

Table 5.1. Adopted soil properties for Väsby post glacial clay

(m/s) (kN/m3)

0.0281 0.388 12000 35 0.785 2.95 13.1 - 14.5

Figure 5.6. Permeability changes versus void ratio

Figure 5.8 illustrates variation of the permeability coefficient with the void

ratio for the undisturbed sample of Väsby Post glacial clay deposit adopted in this

study applying . is calculated using Casagrande method

(Casagrande and Fendum 1940). To investigate the influence of the permeability

variation pattern in the disturbed zone on the numerical predictions, all permeability

profiles, reported in Figure 5.7, have been considered. Various available permeability

variation equations are tabulated in Table 5.2. Table 5.3 and Figure 5.8 summarise

the adopted parameters and pattern to simulate different permeability profiles in the

disturbed zone.

172

(a) (b)

Figure 5.7. Cross section of the disturbed zone surrounding a vertical drain, (a) two zones hypothesis, (b) three zones hypothesis

Soil deformation during consolidation greatly depends on the initial state

( ) and boundary conditions. The thickness of the simulated soil deposit, which

has overconsolidation ratios of 1.22 to 1.27, with both radial and vertical drainage

systems is 5m with a drainage blanket on top (refer to Equation ( 3.41) and

Figure 3.8). Time steps are assumed to be one tenth of a day with a total of 75,000

steps, representing 20 years. Initial values of the excess pore water pressures ( ),

vertical effective stresses ( ), and vertical strains ( ) are assumed to be 0,

, and 0, respectively.

Intact Zone Intact Zone

Smear Zone Transition Zone

Smear Zone

Vertical Drain

Vertical Drain

R R

1

1

1

1

1 1

Case A

Case B

Case C Case F

Case E

Case D

173

Table 5.2. Various available permeability variation equations

Case Suggested equations for permeability in disturbed zone Reference

Case A Barron (1948)

Holtz and Holm

(1973)

Case B Rujikiatkamjorn

and Indraratna

(2009)

Case C

Walker and

Indraratna (2006)

Case D

Madhav et al.

(1993)

Case E

Onoue et al.

(1991)

Case F

Basu et al. (2006)

Table 5.3. Fitting parameters for disturbed zone permeability profile for Cases A-F

0.25 0.6 0.75 0.15 0.05 0.1

174

Figure 5.8. Variations of initial permeability profile for Cases A to F

5.2.1 RESULTS AND DISCUSSION

Figure 5.9 shows the excess pore water pressure predictions at the depth of

2.6m and in the middle of two vertical drains for different soil hydraulic conductivity

profiles including Cases A to F. Comparing Figure 5.8 and Figure 5.9, it is evident

that a lower average permeability in the disturbed zone results in a slower excess

pore water pressure dissipation rate. For example, Cases A and C with the lowest and

highest average disturbed zone permeability (

), result in the highest and lowest remaining excess pore water pressures during

the consolidation process, respectively. In addition, Cases D and F with comparable

disturbed zone permeability values (Figure 5.8) demonstrate similar pattern of excess

pore water pressure dissipation. As shown in Figure 5.9, while the test embankment

is being built, the excess pore water pressure keeps increasing and reaches its

maximum value after 25 days, irrespective of the choice of the disturbed zone

permeability profile. However, it can be observed that the maximum excess pore

water pressure significantly depends on the choice of the permeability variation in

the disturbed zone.

As expected, the higher the average disturbed zone permeability is, the

lower the maximum values of excess pore water pressure in the soil profile will be.

0.00

0.25

0.50

0.75

1.00

0 20 40 60 80 100 120 140

Radial distance (mm)

Case ACase BCase CCase DCase ECase F

0 Disturbed zone boundary

Vertical drain location

175

For example, the maximum predicted excess pore water pressure values for Cases A

to F, occurring at the end of fill placement period, are 36.5 kPa, 32.1 kPa, 27.9 kPa,

35.2 kPa, 30.7 kPa, and 34.9 kPa, respectively. In other words, Case A with constant

initial disturbed zone permeability overestimates the maximum excess pore water

pressure by 31% in comparison to Case C. In addition, as illustrated in Figure 5.9,

there is a rapid decrease in the excess pore water pressure induced by the partial

removal of the fill material. This immediate reduction in the excess pore water

pressure is approximately 13.5 kPa equal to the removal of 0.8m of the gravel fill

material with the unit weight of 17 kN/m3. Since the remaining excess pore water

pressures in the soil deposit at the time of partial removal of the embankment were

less than the removed fill material, negative excess pore water pressures were

generated. Soon after completion of the unloading, the excess pore water pressures

increased aiming to reach the equilibrium followed by dissipation.

Figure 5.9. Excess pore water pressure values predicted by developed code versus time for Cases A to F

Figure 5.10 shows the variations of excess pore water pressure with time for

Case A. Obviously; excess pore water pressure is inversely related to radius. As

explained earlier, during the construction of the embankment, excess pore water

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-5

0

5

10

15

20

25

30

35

40

0.1 1 10 100 1000 10000

Emba

nkm

ent h

eigh

t (m

)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (days)

Case ACase BCase CCase DCase ECase FEmbankment height

Depth = 2.6m In the middle of two vertical drains

176

pressure increases and reaches its maximum value (i.e. 25 days) and then starts

decreasing. Figure 5.11 shows the comparison between excess pore water pressure

values just before unloading for Cases A to F at the depth of 2.6m. Obviously, excess

pore water pressure is directly related to the distance from the vertical drain. In

addition, the lower the average disturbed zone permeability is the higher the value of

excess pore water pressure value in the soil profile will be. For example, Case A with

lower initial average permeability shows 43% less excess pore water pressure in

comparison to Case C, which has the highest initial average permeability.

Figure 5.12 depicts the creep coefficient ( ) variation with time at depth

of 2.6m in the middle of two vertical drains. Evidently, the creep coefficient changes

with both time and choice of the disturbed zone permeability profile. For instant, the

creep coefficient drops from 0.0505 to 0.0423 (16.2% reduction) between 10 days

and 100 days, respectively (Case A). Since the creep coefficient ( ) is inversely

related to the effective vertical stress ( ) as shown in Figure 5.5, lower effective

vertical stresses (or higher excess pore water pressures) cause higher creep

coefficients. For example, as reported in Figure 5.12, Case A (the lowest averaged

permeability in the disturbed zone) and Case C (the highest average permeability in

the disturbed zone) (Figure 5.8), result in the highest and the lowest creep

coefficients ( ), respectively. As shown in Figure 5.12, for the reported cases,

influence of time on the creep coefficient is more pronounced than the effect of

choice of disturbed zone permeability profile. Moreover, as observed in Figure 5.12,

there is a slight increase in the creep coefficient due to the partial removal of the fill

material followed by a marginal increase with time, similar to the excess pore water

pressure dissipation pattern reported in Figure 5.9.

177

Figure 5.10. Variations of excess pore water pressure with time for Case A

Figure 5.11. Variations of the excess pore water pressure values just before unloading ( ) for Cases A to F

-5

0

5

10

15

20

25

30

35

40

0 50 100 150 200 250 300 350 400

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)


Initial (0 day)After 10 days (during embankment construction)After 25 days (full embankment construction)After 50 daysAfter 182 days (just before unloading)

Depth = 2.6m

0

2

4

6

8

10

12

14

0 100 200 300 400

Exce

ss p

ore

wat

er p

ress

ure

just

be

fore

unl

oadi

ng (k

Pa)



Depth = 2.6m

178

Figure 5.12. Predicted creep coefficient ( ) values versus time for Cases A to F

Figure 5.13 shows variations of the predicted creep strain limit with time

and choice of the disturbed zone permeability profile. Evidently, variations of the

creep strain limit with time are more notable than with the permeability profile. For

example, there is 67% reduction in the creep strain limit (from 2.33 to 0.77) from 10

days to 100 days (Case A), while this reduction is only up to 7.7% due to the choice

of the permeability profile. Since, similar to the creep coefficient, the creep strain

limit is also inversely related to the effective vertical stress (see Equations ( 3.3) and

( 3.4)), the lower equivalent permeability in the disturbed zone (resulting in a higher

excess pore water pressures) leads to higher creep strain limit at any given time. In

addition, unloading due to the partial removal of the embankment contributes to a

slight increase in the creep strain limit followed by a gradual decrease similar to the

creep coefficient variations.

Referring to Figure 5.9, due to the simultaneous and professional increase in

the total stresses and excess pore water pressures keep in the early stages of loading,

the effective vertical stresses remain nearly unchanged. Thus, the creep strain limit

and the creep coefficient (Figures 5.12 and 5.13), which are inversely proportional to

the effective vertical stresses, remain unchanged in the early stages of loading.

0.040

0.042

0.044

0.046

0.048

0.050

0.1 1 10 100 1000 10000

Cre

ep c

oeffi

cien

t

Time (days)


Depth = 2.6m

In the middle of two vertical drains

Partial embankment removal

0

()

Full embankment constructed

179

Figure 5.13. Creep strain limit values predicted by the developed code versus time for Cases A to F

Figures 5.14 and 5.15 present the comparison between the predicted time

dependent settlement and the field measurements for depths of 5m and 3.8m,

respectively. To calculate the reported field settlement values, the in situ settlement

records at the settlement plate SP1 (located at the ground surface) and settlement

plate SP2 (located at the depth of 5m) were subtracted from the settlement records

SP3 located 5m deep (see Figure 5.2). Numerical results show although settlement

predictions converge after a very long time (infinity), there are significant differences

due to the choice of permeability variations in the disturbed zone. The predicted

ground surface settlements just before the partial removal of the embankment for

Cases A to F are 501 mm, 618 mm, 733 mm, 541 mm, 659 mm, and 552 mm,

respectively, while, the in situ measurement is 545 mm. In other words, there is up to

46% difference in the predicted settlement after 182 days (just before unloading) due

to the variation in the permeability profile in the disturbed zone. This difference for

the settlement at depth 3.8 m (SP2-SP3) is 55%.

0.65

0.85

1.05

1.25

1.45

1.65

1.85

2.05

2.25

2.45

1 10 100 1000 10000

Cre

ep li

mit

stra

in

Time (days)


Depth = 2.6m

In the middle of two vertical drains


2.

2.(

) Full embankment constructed

180

Figure 5.14. Comparison of the settlement predictions for Cases A to F and the field measurements at the ground surface

Figure 5.15. Comparison between the settlement predictions for Cases A to F and the field measurements at 3.8 m depth

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.1 1 10 100 1000 10000

Settl

emen

t (m

)

Time (days)

Case ACase BCase CCase DCase ECase FField Measurement after Chang (1969)



0.00

0.05

0.10

0.15

0.20

0.25

0.1 1 10 100 1000 10000

Settl

emen

t (m

)

Time (days)

Case ACase BCase CCase DCase ECase FField Measurement after Chang (1969)



181

Based on the settlement predictions and measurements reported in

Figure 5.14 and Figure 5.15, although the settlement rate decreases significantly after

unloading, settlement continues increasing due to visco-plastic deformation of the

soil while insignificant excess pore water pressures are remaining. Comparing the

measurements and predictions reported in Figure 5.14 and Figure 5.15 for this

particular case study, numerical analysis predictions adopting Cases E and B with

bilinear or linear variation profile of the initial permeability in the disturbed zone

with radius, are in a reasonable agreement with the field measurements particularly

after unloading.

Figure 5.16. Comparison between post construction settlement prediction for Cases A to F and the field measurement at the ground surface

Assuming the post construction settlement being the settlement occurring

after the partial removal of the embankment, Figure 5.16 shows the post construction

settlement for 20 years. Evidently, the post construction settlement, which is mainly

due to the viscous creep deformation, is influenced significantly by the permeability

variations in the disturbed zone. Figure 5.17 shows the required time to achieve 500

m of the ground surface settlement for all cases. Evidently, numerical results show

that Case C with the highest average initial disturbed zone permeability needs 61%

less time to achieve 500mm settlement compared to Case A with the lowest average

initial disturbed zone permeability. Consequently, selecting the permeability profile

affects the decision of the proper time to remove the embankment considerably.

0.00

0.05

0.10

0.15

0.20

0.25

1 10 100 1000 10000

Post

con

stru

ctio

n se

ttlem

ent (

m)

Time (days)


182

Figure 5.17. The required time to achieve 500 mm of settlement for Cases A to F at the ground surface

Figure 5.18. Variations of permeability profile versus time for Case A

0

20

40

60

80

100

120

140

160

180

200

Case A Case B Case C Case D Case E Case F

Tim

e to

ach

ieve

500

mm

of s

ettle

men

t (d

ays)

0

1E-10

2E-10

3E-10

4E-10

5E-10

6E-10

7E-10

0 100 200 300 400

Hor

izon

tal p

erm

eabi

lity

(m/s

)

Vertical distance (mm)

Initial (0 day)10 days50 days100 days500 days1000 days2000 days5000 days


Depth = 2.6m

183

Figure 5.19. Variations of permeability ratio with time in disturbed zone for Case A

Figures 5.18 and 5.19 show the variations of the permeability profile with

time adopting Case A as the initial permeability profile. It can be concluded that the

permeability coefficient, the permeability ratio ( ) and the permeability variation

pattern change with time. Obviously, while the excess pore water pressure in

dissipating, the excess pore water pressure dissipation rate and consequently the void

ratio reduction rate decrease with the radius (distance from the vertical drain).

Therefore, the permeability decreases with time, but increases with radius. Thus, the

pattern of the soil hydraulic conductivity variations does not keep similar to the

initial pattern, in the process of consolidation and continuous ground settlement. For

example, as Figures 5.18 and 5.19 illustrate, although the uniform initial permeability

coefficient and ratio were assumed for the disturbed zone, non-uniform/nonlinear

variation exists during the consolidation process. According to Figure 5.19, within

the disturbed zone, while the excess pore water pressure is dissipating, the

permeability ratio is decreasing following a direct relationship with radius. After

excess pore water pressure dissipation, the void ratio reduction rate slows down

(Figure 5.12) and consequently, the permeability ratio starts increasing (Figure 5.19).

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0.21

0.23

0.25

0 50 100 150Radial distance (mm)

Initial (0 day)10 days50 days100 days500 days1000 days2000 days5000 days

Disturbed zone boundary Vertical drain location

Depth = 2.6m

184

5.3 SKÅ-EDEBY TEST FILL CASE STUDY

Skå-Edeby is located on an island about 25 kilometres west of Stockholm.

According to Hansbo (1960), four circular test fills were built to study the

consolidation of soft clays and the effect of vertical drains on the rate of

consolidation (Figure 5.20). In 1972 another circular test fill was built to investigate

the field performance of vertical drains. Test Area I was divided into three

subsections with vertical drains spacing of 0.9 m, 1.5 m, and 2.2 m. Test Areas II, III

and IV are 35 m in diameter while Test Area V is 31 m in diameter. In Test Area II

and III the sand drains spacing was 1.5 whereas in Area V Geodrains were installed

at 0.9 m spacing. No drains were installed in Area IV. Geodrains were installed using

sounding rods. The sand drains were of displacement type and were 0.18 m in

diameter. All vertical drains were installed in triangular patterns with an average

length of 12 m. In this study, the subsection A of Test Area I with 1.5 m as vertical

drains spacing is selected for investigation (Figure 5.20).

The clean sandy gravel fill was placed in three successive lifts at a unit

weight of 18 kN/m3 to the height of 1.5 m. The clay deposit (glacial clay) underlying

this area was approximately 10 m thick. Settlement monitoring devices were

established at the surface of the scraped ground and at the depth of 5 m, 7.5 m and 10

m. The piezometers were placed at the centre of triangles at the depth of 2.5 m, 5 m

and 9 m (Figure 5.21). As reported by Lo (1991), Figures 5.22 and 5.23 show in-situ

measurement of the initial void ratio profile versus time and preconsolidation

pressure versus depth, respectively.

185

Figure 5.20. Skå-Edeby test field (After Hansbo 1960)

Figure 5.21. Soil profile beneath the Skå-Edeby test fill

1.5m

10m

Glacial clay

SP1

SP4

SP2

SP3

0.0

-2.5

-5.0

-10.0

+1.5

A

B C

0 50 100m

Scale

Area I (Sand Drain)

Sampling Site Holm and Holtz (1977)

Area II (Sand Drain)

Area III (Sand Drain)

Area V (Geodrain)

Area IV (No Drain)

A: 0.9 m Drain Spacing B: 1.5 m Drain Spacing C: 2.2 m Drain Spacing

N

186

Figure 5.22. Initial void ratio profile versus depth

Figure 5.23. Preconsolidation pressure profile versus depth

As reported by Hansbo (1960), 27 incremental loading oedometer tests were

conducted in 1957 for all Test Areas applying specimens 60.5 mm in diameter and

20 mm in height. Samples were taken at 2, 5, 8m below the ground surface and rarely

at 10 m depths by SGI IV sampler (SGI IV sampler was easy to handle but made

some disturbance). According to Hansbo (1960), adequate time was allowed for

0

2

4

6

8

10

1 1.5 2 2.5 3

Dep

th (m

)

Initial void ratio

0

2

4

6

8

10

12

14

0 20 40 60 80 100

Dep

th (m

)

Preconsolidation pressure (kPa)

: Preconsolidation pressure

z: depth (m)

Data obtained from Hansbo (1960)

187

primary consolidation to complete (initially a time allowance of 3 days, occasionally

even more was applied).

Hansbo (1960) reported four additional consolidation test results on samples

taken by SGI VIII, which provides better samples, at depths of 3, 4, 6, and 7 m with

the test durations of 24 hours. Consequently, the effects of sample disturbance due to

coring were reduced. Additionally, the sensitivity was obtained by the field vane

shear test (FVST) and unconfined compression tests, and fall cone tests (FCT). The

sensitivity, the ratio between the undrained shear strengths of the undisturbed and

remoulded states of the soil, is considered to estimate the strength loss, as the

disturbance of the soil increases (Mitchell and Houston 1969). For the soil profile at

Skå-Edeby, the sensitivity increases from 5 in the upper layers to 15 in the lower

layers near the bottom. There were minor differences of the sensitivity obtained by

the FVST and FCT except at the top 1 m.

Figure 5.24. Consolidation tests results on Skå-Edeby glacial clay samples for vertical stresses between 21.6 kPa and 338.3 kPa

0

5

10

15

20

25

30

35

40

0.1 1 10 100 1000 10000

Verti

cal S

train

(%)

Time (min)

0 - 21.6kPa21.6 - 41.2kPa41.2 - 80.4kPa80.4 - 166.7kPa166.7 - 338.3kPa

Data obtained from oedometer tests on undisturbed Skå -

Edeby glacial clay samples reported by Hansbo (1960)

188

Figure 5.24 shows five consolidometer test results carried out on

undisturbed specimens of the clay deposit, which were taken by SGI VIII, reported

by Hansbo (1960) in the stress range of 21.6 kPa to 338.3 kPa. The required soil

properties can be calculated adopting developed procedure explained by Yin (1999)

employing laboratory test results, and the adopted soil parameters are reported in

Table 5.4 and Figure 5.25 to Figure 5.27. According to Yin (1990), parameter is

related to the conventional compression index ( or ) and is related

to the conventional recompression index (unloading and reloading data) (

or ). As Yin and Graham (1989) stated, can be considered as the time

corresponding to the end of excess pore water pressure dissipation process (i.e.

conventional end of primary consolidation time ) for the small laboratory

sample. Once is selected, then would be the corresponding stress on the

reference time line, when . Parameter is the initial creep coefficient

corresponding to time when the excess pore water pressure has been dissipated

( or where is the tangential secondary compression index

when for the laboratory sample). It should be noted that is highly

dependent on the level of the applied effective stress, thus it changes during the

excess pore water pressure dissipation process.

Table 5.4. Adopted soil properties for Skå-Edeby glacial clay

(kN/m3)

0.0233 0.1179 36500 25 0.9085 2.02 15.1

189

Figure 5.25. Time dependant stress-vertical strain relationship for Skå-Edeby glacial clay

Figure 5.26. Changes of versus vertical effective stress

0

10

20

30

40

50

60

70

80

90

100

1 10 100 1,000Ve

rtica

l stra

in (%

) Stress (kPa)

0.0060

0.0070

0.0080

0.0090

0.0100

0.0110

0.0120

0.0130

0.0140

10 100 1000

Vertical effective stress (σz') (kPa)

Skå-Edeby glacial clay sample

( )

Limit time line

190

Figure 5.27. Permeability changes versus void ratio

As stated earlier, the mechanical installation of vertical drains in the ground

inevitably causes disturbance of the surrounding soil due to induced shear strains. As

stated by Baligh (1985) and Whittle and Aubeny (1993), the induced shear strain is a

function of the radial distance and the diameter of the mandrel. Shear strains caused

by vertical drains installation contribute to the reduction of the shear strength and as

a result the overconsolidation ratio of the soil. For example, as reported by

Massarsch (1976) and Bozozuk et al. (1978), the disturbance due to installation of

vertical drains can cause up to 40% and 35% drop in the shear strength of Swedish

clay near Gothenburg and sensitive marine clay in eastern Canada, respectively. The

extent of the soil overconsolidation ratio changes in the disturbed zone versus the

distance from the vertical drain has not been identified with certainty and so far there

is no comprehensive or standard method for measuring these characteristics.

0.00

0.50

1.00

1.50

2.00

2.50

1.E-11 1.E-10 1.E-09

Void

ratio

(e)

Permeability (k) (m/s)

191

Figure 5.28. Cross section of the disturbed and transition zones surrounding a

vertical drain

Case A

Case B

Case C

Case D

Case E

Intact Zone

Transition Zone

Disturbed zone Vertical Drain

R

192

In general, similar to possible permeability ratio variation patterns proposed

by Barron (1948), Rujikiatkamjorn and Indraratna (2009), Madhav et al. (1993),

Onoue et al. (1991), and Basu et al. (2006), two broad concepts are suggested to

define the characteristics of the soil surrounding the drain: (i) two zones hypothesis,

consisting of the intact zone surrounding the disturbed zone adjacent to the vertical

drain and (ii) three zones hypothesis, comprised of the undisturbed zone surrounding

the transition zone and the disturbed zone in the immediate vicinity of the vertical

drain. In this research, various patterns capturing the variations of the initial

overconsolidation ratio and the shear strength of soil in the disturbed region as

illustrated in Figure 5.28 and Table 5.5 have been adopted in the numerical analysis

and for comparison and discussion. Table 5.6 and Figure 5.30 summarise the adopted

parameters and pattern to simulate different OCR profiles in the disturbed zone.

Moreover, Figure 5.29 illustrates variation of the permeability coefficient with the

void ratio for the undisturbed sample of Skå-Edeby glacial clay deposit adopted in

this study.

Table 5.5. Various overconsolidation ratio and normalized shear strength variation equations

Case OCR variation Normalized shear strength

variation

Radius

Case A

Case B

Case C

Case D

Case E

*

193

Figure 5.29. Permeability profile in the disturbed and transition zones for all cases

Figure 5.30. Variations of overconsolidation ratio profile for Cases A to E at depth of 2.5m

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 100 200 300 400 500 600 700 800 900

Radial distance from drain (mm)

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

0 10 20 30 40 50 60 70 80 90

Ove

rcon

solid

atio

n ra

tio (O

CR

)


Case ACase BCase CCase DCase E

0


Disturbed and transition zones boundary

Disturbed and transition zone boundary Vertical drain location

194

Table 5.6. Fitting parameters for disturbed zone overconsolidation ratio profile for Cases A to E

1 0.75OCR 0.09 0.12 0.18

Soil deformation during consolidation greatly depends on the initial state

( ) and boundary conditions. The thickness of the simulated soil deposit is 10m

with a drainage blanket on top (refer to Equation ( 3.41) and Figure 3.8). Initial values

of vertical strains ( ) have been calculated applying Equation ( 3.40). Time steps are

assumed to be one tenth of a day with a total of 24000 steps, representing 6.5 years.

Initial values of the excess pore water pressures ( ), and vertical effective stresses

( ) are assumed to be 0, and , respectively.

5.3.1 RESULTS AND DISCUSSION

Figures 5.31 to 5.33 show the comparison between the excess pore water

pressure predictions and the field measurements in the middle of two vertical drains

and at the depth of 2.5 m, 5 m and 9 m, respectively, for different soil

overconsolidation ratio profiles including Cases A to E (Figure 5.28 and Table 5.5).

According to Figures 5.31 to 5.33, the excess pore water pressure field measurements

and prediction patterns are in a good agreement. While the test embankment is being

built, the excess pore water pressure keeps increasing and reaches its maximum value

after 62 days, irrespective of the choice of the disturbed and transition zones

overconsolidation ratio profiles. However, it can be observed that the maximum

excess pore water pressure notably depends on the choice of the OCR variation in the

disturbed and transition zones.

195

Figure 5.31. Comparison of the developed code excess pore water pressure predictions for cases A to E and the field measurements at depth of 2.5m

Figure 5.32. Comparison of the developed code excess pore water pressure predictions for cases A to E and the field measurements at depth of 5m

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0

5

10

15

20

25

0 100 200 300 400 500 600 700

Emba

nkm

ent h

eigh

t (m

)

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (days)

Case A (Predictions)Case B (Predictions)Case C (Predictions)Case D (Predictions)Case E (Predictions)Field measurement after Hansbo (1960)Embankment height

0

2

4

6

8

10

12

14

16

18

20

22

0 100 200 300 400 500 600 700

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (days)

Case A (Predictions)Case B (Predictions)Case C (Predictions)Case D (Predictions)Case E (Predictions)Field measurement after Hansbo (1960)

Depth = 2.5 m In the middle of two vertical drains

Depth = 5 m In the middle of two vertical drains

196

Figure 5.33. Comparison of the developed code excess pore water pressure predictions for cases A to E and the field measurements at depth of 9m

As expected, the higher the average disturbed and transition zones OCR and

shear strength are, the lower the maximum values of excess pore water pressure in

the soil profile will be. For example, the maximum predicted excess pore water

pressure values at the depth of 2.5m for Cases A to E, occurring at the end of fill

placement period, are 15.99 kPa, 18.39 kPa, 16.84 kPa, 17.60 kPa, and 16.16 kPa,

respectively (Figure 5.31). In other words, Case B with constant initial disturbed and

transition zones overconsolidation ratio overestimates the maximum excess pore

water pressure by 15% in comparison to Case A. This difference for the excess pore

water pressure at the depth of 5m and 9m (Figure 5.32 and Figure 5.33) are 12.5%

and 10.4%, respectively. Referring to Figure 5.31 to Figure 5.33, since the initial

overconsolidation ratio values are higher near the ground surface, excess pore water

pressure difference caused by the soil disturbance (OCR reduction) decreases by

depth (i.e. reduced from 15% at the depth of 2.5 m to 10.4% at the depth of 9m).

As observed in Figures 5.31 to 5.33, there are some discrepancies between

the field measurements and predictions at the end of loading which may be the result

of the assumed initial stress state, adopted soil constitutive models, and soil

anisotropy. It should be mentioned that many field measurements show that the

excess pore pressure values do not decrease immediately at the end of loading or

0

2

4

6

8

10

12

14

16

18

20

0 100 200 300 400 500 600 700

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)

Time (days)

Case A (Predictions)Case B (Predictions)Case C (Predictions)Case D (Predictions)Case E (Predictions)Field measurement after Hansbo (1960)

Depth = 9m In the middle of two vertical drains

197

construction as expected which may be due to the increase in total stress, caused by

the volumetric strain compatibility (Seah and Koslanant 2003; Leoni et al. 2008;

Fatahi et al. 2013). This delayed dissipation significantly contributes to the observed

disparities between the measured and predicted values (Crook et al. 1984; Conlin and

Maddox 1985; Kabbaj et al. 1988; Rowe and Li 2002). Mandel-Cryer effect has been

proposed to explain this delayed excess pore water pressure dissipation response.

Schiffman et al. (1969) expressed Mandel-Cryer effect as the increase in total stress,

which is caused by the volumetric strain compatibility. As mentioned earlier, soil

disturbance causes a lower shear strength and overconsolidation ratio, which results

in a lower initial strain in the soil profile. Obviously, the disturbed zone with lower

shear strength and lower initial strain values settles more under the same loading.

Figure 5.34. Variations of excess pore water pressure with time for Case B

Figure 5.34 shows the variations of the excess pore water pressure with time

for Case B assuming normally consolidated soil in the disturbed zone. Obviously, the

excess pore water pressure dissipation rate is inversely related to the radial distance

from the drain. As explained earlier, during the construction of the embankment, the

-2

0

2

4

6

8

10

12

14

16

18

0 100 200 300 400 500 600 700

Exce

ss p

ore

wat

er p

ress

ure

(kPa

)


Initial (0 day)

After 10 days (during embankment construction)

Ater 62 days (full embankment construction)

After 200 days

After 365 days

Depth = 2.5 m


198

excess pore water pressure increases and reaches to its maximum value after 62 days

(end of embankment construction) and then starts decreasing.

Figure 5.35 shows the comparison between the excess pore water pressure

values at the end of loading stage for Cases A to E at the depth of 2.5m. Obviously,

the excess pore water pressure is directly related to the distance from the vertical

drain. In addition, less disturbance (i.e. a higher average OCR) results in lower

values of the excess pore water pressure in the soil profile. For example, Case A

with lowest average disturbance shows 15.9% less excess pore water pressure in

comparison to Case B, representing the highest disturbance.

Figure 5.35. Variations of the excess pore water pressure values at the end of loading ( ) for Cases A to E

According to Figure 5.31 to Figure 5.35, the excess pore water pressure in

the disturbed zone decreases (the effective stress increases) by time throughout the

consolidation process, while effective stress changes are slightly different for Cases

A to E. As a result, since soil strength is a function of the effective stress, the shear

strength gradually increases by time during consolidation. Additionally, the void

ratio decreases further due to viscous effects (creep), which causes an increase in

preconsolidation pressure. Consequently, referring to Table 5.5, soil strength

increases due to reduction of void ratio over time during consolidation. By and

0

2

4

6

8

10

12

14

16

18

0 100 200 300 400 500 600 700

Exce

ss p

ore

wat

er p

ress

ure

at th

e en

d of

load

ing

(kPa

)


Case A

Case B

Case C

Case D

Case EDepth = 2.5 m t = 62 days


199

large, although strength gain with time during consolidation, caused by the excess

pore water pressure and void ratio reduction, has been captured, other possible

effects contributing to strength gain with time such as thixotropy, and natural

cementation due to ground water and soil chemistry have not been included in the

analysis.

Figure 5.36. Predicted creep strain limit ( ) values versus radial distance at the end of loading ( ) for Cases A to E

Figures 5.36 and 5.37 depict predicted creep strain limit ( ) values and

predicted visoc-plastic strain rates ( ), respectively, versus radial distance at the end

of loading (i.e. 62 days) for Cases A to E at the depth of 2.5m. Comparing

Figure 5.35 and Figure 5.36, higher rate of visco-plastic strain rate ( ) due to

reduced overconsolidation ratio results in lower creep strain limit. As observed, Case

B with the highest degree of disturbance (i.e. lowest OCR) shows up to 85.7% higher

visco-plastic strain rate than Case A, which has the lowest degree of disturbance

(highest average OCR). Soil profile with a lower initial strain and similar horizontal

permeability settles more, which induces a lower creep strain limit (i.e. Case A with

lower degree of disturbance shows the highest value of creep strain limit in vicinity

of drain and the lowest value of creep strain limit at further distance from vertical

0.64

0.66

0.68

0.70

0.72

0.74

0.76

0 100 200 300 400 500 600 700

Cre

ep st

rain

lim

it


Case A

Case B

Case C

Case D

Case EDepth = 2.5 m

0

()


200

drains) (Figures 5.36 and 5.37). Referring to Figure 5.37, at the end of loading when

insignificant excess pore water pressure dissipation has occurred, outside the

disturbed and transition zones, stress-strain situation of soil is equal or close to limit

time line which results in small values of visco-plastic strain rate (Equation ( 3.10)).

Figure 5.38 shows visco-plastic strain rate ( ) values versus radial distance

at 200 days for cases A to E. As expected, visco-plastic strain rate is inversely related

to the distance from the vertical drain. Furthermore, the less the disturbance (i.e.

higher average OCR) is, the lower the value of visco-plastic strain rate. For example,

Case A with lower degree of disturbance shows 12% less visco-plastic strain rate in

comparison to Case B, which has the highest degree of disturbance. Referring to

Figures 5.38 to 5.41, lower values of visco-pastic strain rate results in lower values of

settlement.

Figure 5.37. Predicted visco-plastic strain rate ( ) values versus radial distance at the end of loading ( ) for Cases A to E

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

3.5E-03

0 100 200 300 400 500 600 700

Vis

co-P

last

ic se

ttlem

ent r

ate

()


Case A

Case B

Case C

Case D

Case E

Depth = 2.5 m


201

Figure 5.38. Predicted visco-plastic strain rate ( ) values versus radial distance at 200 days for Cases A to E

Figure 5.39. Comparison of the settlement predictions for Cases A to E and the field measurements at the ground surface

6.0E-05

7.0E-05

8.0E-05

9.0E-05

1.0E-04

1.1E-04

1.2E-04

1.3E-04

1.4E-04

1.5E-04

0 100 200 300 400 500 600 700

Vis

co-P

last

ic se

ttlem

ent r

ate

()


Case A

Case B

Case C

Case D

Case E

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0 500 1000 1500 2000 2500

Settl

emen

t (m

)

Time (days)

Case A (Predictions)

Case B (Predictions)

Case C (Predictions)

Case D (Predictions)

Case E (Predictions)

Field measurement after Hansbo (1960)



Depth = 2.5 m

202

Figures 5.39 to 5.41 present the comparison between the predicted time

dependent settlement and the field measurements for ground surface and depths of

2.5 m and 5 m, respectively. To calculate the reported field measurement values, the

in situ settlement records at the settlement plate SP1 (located at the ground surface),

settlement plate SP2 (located at the depth of 2.5 m), and settlement plate SP3

(located at the depth of 5 m) were subtracted from the settlement records of SP4

located 10m deep (Figure 5.21).

Numerical results show that there are notable differences in the settlement

predictions due to the choice of overconsolidation ratio variations in the disturbed

and transition zones. The predicted ground surface settlements after 1000 days for

Cases A to E are 838 mm, 925 mm, 884 mm, 905 mm, 864 mm, respectively, while,

the in situ measurement is 908.6 mm. In other words, there is up to 10.4% difference

in the predicted settlement after 1000 days due to the variation in the

overconsolidation ratio profile in the disturbed and transition zones. This difference

for the settlement at depth 2.5 m (SP2-SP4) and 5m (SP3-SP4) are 8.1% and 6%,

respectively. Referring to Figures 5.39 to 5.41, since the initial overconsolidation

ratio values are higher closer to the ground surface, settlement difference induced by

the soil disturbance (OCR reduction) decreases by depth (i.e. reduced from 10.4% for

ground surface to 6% for the depth of 5 m).

It is worthy of note that settlement including creep of normally consolidated

soft soil is more than settlement of the same soil but being overconsolidated or

heavily overconsolidated (Feng 1991; Mesri and Feng 1991; Yin 2006).

Furthermore, referring to Figures 5.39 to 5.41, as time increases effects of

overconsolidation ratio variations are more notable due to the fact that the

contribution of the time dependant viscous deformation of the soil becomes more

pronounced.

It should be mentioned that in this research a vertical drain in an

axisymmetric condition at the centre of the embankment was simulated. Thus, the

consolidation equation is 1D, and only vertical deformation at the centre of the

embankment can be captured. Obviously, long term settlement induced by creep and

lateral deformation of soft soil can influence the behaviour of the embankment. To

capture lateral deformation, the actual embankment should be simulated. As stated

by Fatahi et al. (2013), the soil settlement, excess pore water pressure and lateral

203

deformations under the embankment increase when the soil creep rate (e.g., creep

ratio) increases. Although the soil creep causes higher settlement of the ground, as a

result of the increased pore water pressure, the factor of safety against instability of

the embankment decreases, while the creep ratio increases. In addition, it was

reported that during consolidation, the factor of safety against embankment

instability increases unless during fill placement, which decreases. Furthermore,

Fatahi et al. (2013) concluded that the rate of increase in factor of safety with time

after end of construction decrease with the creep coefficient.

Figure 5.40. Comparison of the settlement predictions for Cases A to E and the field measurements at 2.5m depth

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 500 1000 1500 2000 2500

Settl

emen

t (m

)

Time (days)

Case A (Predictions)

Case B (Predictions)

Case C (Predictions)

Case D (Predictions)

Case E (Predictions)

Field measurement After Hansbo (1960)


Depth = 2.5 m

204

Figure 5.41. Comparison between the settlement predictions for Cases A to E and the field measurements at 5m depth

5.4 SUMMARY

In this chapter, two vertical drain assisted preloading case studies were

explained in detail. The numerical finite difference solution adopting an elastic

visco-plastic model with nonlinear creep function incorporated in the consolidation

equations was used to simulate the selected case studies investigating the effects of

soil disturbance on time dependant behaviour of soft soils. Installation of vertical

drains induces soil disturbance decreasing the in situ shear strength and horizontal

hydraulic conductivity in the vicinity of drains. The soil disturbance causes a slower

rate of excess pore water pressure dissipation and a higher rate of deformation than

what would be expected in the absence of the disturbance. Assessing the degree of

change in the shear strength and the hydraulic conductivity in the disturbed zone to

be used in the design procedure is a challenging task.

Väsby test fill located 30 km north of Stockholm on the east coast of

Sweden was the first case study to investigate the effect of soil disturbance on the

hydraulic conductivity reduction and consequently the excess pore water pressure

dissipation and the deformation rate. According to available literature, six possible

profiles of the initial hydraulic conductivity in the disturbed zone (Cases A, B, C, D,

205

E, F) have been considered in this study. Different initial hydraulic conductivity

profiles in the disturbed zone result in various values of excess pore water pressure

and effective vertical stresses at any given time in the soil profile. Consequently, the

induced changes in the vertical effective stresses not only influence the consolidation

process but also influence the creep coefficient and the creep strain limit resulting in

different settlement rates at any given time. Consequently, the initial hydraulic

conductivity selection in the disturbed zone has a significant effect on selecting

unloading time and therefore on the post construction settlement. It is noted that

since permeability is a function of void ratio, the assumed initial hydraulic

conductivity profile does not remain the same during the consolidation process. In

other words, the permeability ratio and its variation pattern in the disturbed zone is

significantly time dependant. It is worthy of note that the creep coefficient and the

creep strain limit, regardless of the initial hydraulic conductivity selection, change

during loading and unloading and also during excess pore water pressure dissipation.

The creep coefficient and the creep strain limit are functions of the effective vertical

stress and time. Consequently, during loading and unloading and during the excess

pore water pressure dissipation period, the effective vertical stress changes result in

creep coefficient and creep strain limit variations.

Skå-Edeby test fill located on an island about 25 km west of Stockholm was

the first case study to investigate the effect of soil disturbance on shear strength, as a

function of overconsolidation ratio, reduction and consequently excess pore water

pressure dissipation and deformation rate. Five possible profiles of the

overconsolidation ratio in the disturbed and transition zones (Cases A, B, C, D, E)

have been considered. Reported laboratory and field measurements data, taken from

Skå-Edeby, a test fill in Sweden, containing soft soil deposit improved with

prefabricated vertical drains and preloading with staged loading and unloading

process, have been discussed and compared with the numerical predictions in this

study. Different OCR profiles in the disturbed and transition zones result in various

visco-plastic strain rates and creep strain limits. Consequently, the induced changes

in visco-plastic strain rate and creep strain limit influence the settlement at any given

time. Effects of reduced shear strength in the disturbed zone on the predicted

settlements are more evident in long-term due to more creep contribution as time

increases. Therefore, the selection of the OCR profile in the disturbed and transition

206

zones has important effects on determining the unloading time and thus the post

construction settlement. It can be noted that although permeability variations have

more effects on the settlement in comparison to shear strength variation, shear

strength variations still have notable effects.

The numerical results show that the proposed finite difference procedure

incorporating elastic visco-plastic soil behaviour is appropriate for the consolidation

analysis of preloading with vertical drains. The proposed solution can readily be used

for layered soil deposits, time dependent loading and unloading, while considering

combined effects of soil disturbance effects and visco-plastic behaviour.

207

CHAPTER SIX

6 CONCLUSIONS AND RECOMMANDATIONS

6.1 SUMMARY

After brief introduction in Chapter 1, an introduction and literature review

of time dependant behaviour of soft soils improved with vertical drain assisted

preloading were presented in Chapter 2. This chapter reviewed the creep mechanisms

and constitutive models proposed to simulate the behaviour of soils. Furthermore,

two broad concepts proposed by researchers to estimate the time dependant

deformation of soft soils namely Hypotheses A and B have been presented. One of

the main settlement calculation approaches supported by Hypothesis A, the creep

ratio concept, has been discussed. In addition, the constitutive models based on

Hypothesis B presented in literature have been divided into three categories namely

empirical models, rheological models, and general stress-strain-time models and

explained. Moreover, preloading assisted vertical drains as one of the most common

methods of improving soil properties have been explained in details. Furthermore,

the analytical and numerical methods presented in literature to study the long term

settlement (creep) of soft soils improved with preloading and vertical drains were

reviewed. In Chapter 3, the numerical solution incorporating the elastic visco-plastic

model with nonlinear creep function in combination with the consolidation equations

were presented. Chapter 4 presented an experimental investigation where a large

Rowe cell was used to simulate the combined vertical and radial consolidation

processes (with vertical drain) by introducing a zone of reduced permeability

surrounding the vertical drain. The laboratory results were used to validate the

developed numerical solution by comparing the measured and predicted settlement

and excess pore water pressures. In Chapter 5, two case studies were used to

investigate the effects of various shear strength and permeability profiles on time

dependant behaviour of soft soils. Skå-Edeby test fill was used to study the effects of

208

shear strength reduction induced by installation of vertical drains on time dependant

settlement (creep) of soft soils. Different variations of permeability profile in

disturbed zone have been applied to Väsby test fill for the purpose of studying the

effects of disturbance on time dependant settlement of soft soils.

6.2 CONCLUSIONS

To speed up consolidation rate and improve the strength of soft soil, vertical

drains assisted preloading system has been largely applied as a ground improvement

technique for road and railway projects. The installation of vertical drains disturbs

the soil near the drains, reduces the permeability and shear strength of the smear

zone, and retards the rate of consolidation quite significantly. Numerous methods are

presented in literature to study the long term settlement of soft soils capturing the

visco-plastic behaviours of soils. However, the combined effects of disturbed zone

and the visco-plastic behaviour of the soil have not been addressed in the literature.

In this research, to investigate the effects of soil disturbance induced by the

installation of vertical drains on time dependant performance of soft soil deposits, the

elastic visco-plastic model consisting of a nonlinear creep function and creep strain

limit, developed by Yin and Graham (1990) and Yin (1999), has been incorporated in

the consolidation equation. The assumptions to obtain the governing equations are:

(i) the soil is fully saturated, (ii) water and soil particles are incompressible, (iii)

Darcy’s law is valid and (iv) strains are small. Evidently, when the soil comprises of

horizontal layers with thickness (or the length of vertical drains) much less than the

dimensions of the preloading area, or for the points located at the centre of the

embankment, the average strain or deformation of the soil can be calculated using 1D

(vertical) deformation assumption reasonably accurate. As expressed in the literature,

regardless of the pressure, the void ratio for a particular soil can reach to a minimum

value. Thus, the compression may end under the final effective stress when the

ultimate equilibrium inside the soil structure is reached or when almost no void exists

inside the clay mass. Since it is not feasible to carry out the tests for a very long

duration approaching infinity, creep strain limit measurement is not an easy task.

Consequently, it can be assumed that the limit strain can be reached when the volume

of voids within the soil approaches zero under the applied stress at the infinity time.

209

Yin et al. (2002) proposed that the creep strain limit may be estimated based on the

initial void ratio ( ). However, the author believes that is an

overestimation of the creep strain limit since it consists of the conventional

consolidation volume change induced by hydrodynamic excess pore water pressure

dissipation as well as the creep. Thus, the soil void ratio at a certain effective stress

on the reference time line should be used to define the creep strain limit for a

particular applied effective stress ( ).

For the finite difference solution, Crank-Nicolson method has been applied

to solve the 2-D axisymmetric consolidation equations incorporating the elastic

viscopleastic behaviour of soils. In this method, two steps have been used in partial

differentials of pore water pressure over distance to stabilise the process quicker.

Furthermore, in order to ensure stability and convergence in the explicit method, the

time step function (Equation (3.23)) must be less than 0.5. However, the implicit

Crank-Nicolson method has no such limitation. The finite difference solution has

been developed as a MATLAB code. The developed code can be run on micro-

computers and it is capable of: (i) calculating initial vertical effective stress with

depth, (ii) estimating the initial void ratio with depth, (iii) considering variations of

permeability with radius, depth, and void ratio, (iv) considering variations of

overconsolidated ratio with depth and radius, (v) simulating layered soil, (vi)

considering different set of elastic visco-plastic model and soil parameters for each

set of finite difference, and (vii) considering elastic visco-plastic model and soil

parameters as functions of time.

To verify the developed numerical code, a fully instrumented large Rowe

cell apparatus was used. To select the proper mixtures to be used in the Rowe cell, a

number of samples were prepared by mixing various percentages of Q38 kaolinite,

ActiveBond23 bentonite, and uniformly graded sand (SP). Oedometer tests were

carried out to select the samples, meeting the criteria of for the intact

(sample S1) and reduced permeability (sample S3) zones materials. Two small Rowe

cell test results were used to calculate soil parameters. To calculate soil parameters

curve fitting procedure proposed by Yin (1999) was applied. The creep strain limit

and the creep coefficient are reversely correlated to the vertical effective stress and

210

time. Accordingly, the more the time or the vertical effective stress is, the lower the

creep strain limit and creep coefficient will be.

Comparison of numerical predictions with laboratory measurements has

revealed that the proposed finite difference solution incorporating elastic visco-

plastic soil behaviour is appropriate for the consolidation analysis of preloading with

vertical drains. Comparison of the results indicated that the laboratory measurements

and the predictions for settlement were in a good agreement. It should be mentioned

that there have been some disparities between soil laboratory measurement and

predictions on early stages of the test, which may be the result of disturbance induced

by the removal of the pipes surrounding vertical drain and the reduced permeability

zone. Moreover, the excess pore water pressure measurements and the numerical

predictions have been in a reasonable agreement. As expected, during the dissipation

process, the further the distance of the pore water pressure transducers (PWPT) is

from the vertical drain, the higher the value of excess pore water pressure. For

instance, one day after increasing the surcharge from 200 kPa to 400 kPa, PWPT B2

with the shortest distance ( ) from the vertical drain showed 51% and 55%

less excess pore water pressure in comparison with PWPT B5 with the longest

distance ( ) from vertical drain based on laboratory measurement and

numerical predictions, respectively. Evidently, the higher the position of PWPT

(closer to the drainage layer on top surface), the lower the values of excess pore

water pressure. For instance, one day after increasing the surcharge from 200 kPa to

400 kPa, PWPT A3 at the highest position ( ) measured 57% and 63% less

values of excess pore water pressure in comparison with PWPT A1 with lowest

position ( ) for laboratory measurement and numerical predictions,

respectively.

According to Figures 4.70 and 4.71, the creep coefficient and creep strain

limit vary with both time and vertical effective stress. In other words, lower vertical

effective stresses (or higher excess pore water pressures) induce higher creep

coefficients and creep strain limit. For example, during loading stage from 100 kPa

to 200 kPa, which was carried out using the large Rowe cell, the creep coefficient

and the creep strain limit dropped by 4% and 28%, respectively. On the other hand,

the creep coefficient and the creep strain limit increased by 4% and 224%,

respectively, during unloading from 400 kPa to 50 kPa.

211

To investigate the effect of soil disturbance and hydraulic conductivity

reduction on the excess pore water pressure dissipation and the deformation rate,

Väsby test fill was selected as the first case study. Väsby test fill was located 30km

north of Stockholm on the east cost of Sweden. In this study, six possible profiles of

the initial hydraulic conductivity in the disturbed zone (Cases A, B, C, D, E, F in

Figure 5.7) have been considered. Various initial hydraulic conductivity profiles in

the disturbed zone induce different values of excess pore water pressures and

effective vertical stresses at any given time in the soil profile. Consequently, not only

the consolidation process but also the creep coefficient and the creep strain limit are

influenced by the induced changes in the vertical effective stresses. As a result,

different settlement rates are observed corresponding to various permeability profiles

in the disturbed zone at any given time. Accordingly, the initial hydraulic

conductivity selection in the disturbed zone has a significant effect on selecting

unloading time and therefore on the post construction settlement.

It is noted that since permeability is a function of void ratio, the hydraulic

conductivity profile does not remain the same during the consolidation process.

Obviously, permeability ratio and its variation pattern in the disturbed zone is

considerably time dependant. It should be mentioned that regardless of the initial

hydraulic conductivity selection, the creep coefficient and creep strain limit vary

throughout loading and unloading processes while the excess pore water pressure

variation occurs.

To study the effects of the reduced shear strength in the disturbed zone on

the ground settlement and excess pore water pressure response, Skå-Edeby test fill

case study was used. Skå-Edeby test fill was located on an island about 25 kilometres

west of Stockholm. Skå-Edeby test fill contained soft soil deposit improved with

prefabricated vertical drains and preloading. Five possible profiles of the

overconsolidation ratio (shear strength) in the disturbed and transition zones (Cases

A, B, C, D, E in Figure 5.28) have been considered. In this study, reported laboratory

and field measurements and staged loading and unloading processes have been

discussed and compared with the numerical predictions.

Various OCR profiles in the disturbed and transition zones cause different

visco-plastic strain rates and creep strain limits. Therefore, the induced variations in

the visco-plastic strain rate and the creep strain limit affect the deformation. As time

212

increases, reduced shear strength has more evident effects on the predicted

settlements in the disturbed zone due to more creep contribution. Consequently, the

unloading time and also the post construction settlement are considerably influenced

by the selection of OCR profile in the disturbed and transition zones. It should be

noted that even though permeability variations have more effects on deformation

compare to shear strength variations; shear strength variations, still have notable

effects.

Installation of prefabricated vertical drains disturbs the surrounding soil,

which results in hydraulic conductivity and shear strength reduction. Different

variations of the hydraulic conductivity and the shear strength profiles in the

disturbed zone result in varying excess pore water pressure dissipation rate and

consequently varying the effective vertical stresses in the soil profile. Thus, the creep

coefficient and the creep strain limit are notably influenced inducing significant

changes in the predicted settlement rate.

Soil settlement may end under the final effective stress when the ultimate

equilibrium inside the soil structure is reached or when almost no void exists inside

the clay mass. As a result, assuming the creep coefficient to be constant may not be

realistic. Consequently, the creep coefficient ( ) and visco-plastic settlement rate ( )

should be considered to decrease with time (Figures 4.70, 5.12, 5.37, and 5.38). In

other words, an appropriate visco-plastic model should introduce a limit to settlement

with time dependant creep coefficient to predict the long term settlement accurately.

It is recommended that practicing engineers consider the effects of soil

disturbance (shear strength and permeability reduction) in the vicinity of vertical

drains combined with soil creep by applying a nonlinear elastic visco-plastic

constitutive model, while investigating the long term performance of embankments

built on the soft soil improved with prefabricated vertical drains assisted preloading.

6.3 RECOMMENDATIONS FOR FUTURE RESEARCH

This research may be further expanded by conducting the following studies:

Considering 3D consolidation equations allowing the stress distribution with

depth and horizontal distance as well as three dimensional deformations in the

213

numerical solution which results in more accurate calculation of excess pore

water pressure values and total settlement.

Carrying out a large Rowe cell test with the reduction in the over consolidation

ratio (OCR) inside the disturbed zone. Large Rowe cell can be divided into two

parts of disturbed and intact zones. The artificial disturbed zone can be

preloaded (preconsolidated) to a lower pressure in comparison to the intact zone.

The developed numerical model can be further verified against the test results.

Including well resistance and discharge capacity combined with soil creep to

investigate the long term performance of improved clay considering the delayed

excess pore water pressure dissipation influencing the soil creep. The deep

installation of vertical drains will increase well resistance. The well resistance

factor is generally less significant than the drain spacing and the smear effects.

However, the well resistance of long PVDs is reasonably substantial and affects

the excess pore pressure distribution and induces clogging in the drainage

system.

Including the nonlinear creep function as a subroutine in well-established

numerical software (e.g. FISH in FLAC and UMAT in ABAQUS) so the model

will be more versatile for different and more complex geometries.

Conducting further laboratory tests with several vertical drains to investigate the

optimum prefabricated vertical drains spacing.

Simulating the actual embankment using finite element method or finite

difference method to conduct a quantitative evaluation of the influence of creep

effects on embankment behaviour. It should be noted that lateral deformations

are more critical than vertical deformations near the toe of the embankment.

Considering the effects of overlapping disturbed zones to investigate the changes

in soil parameters (e.g. permeability and shear strength). Overlapping disturbed

zones might changes the soil parameters further than one individual disturbed

zone.

Conducting field and laboratory tests to establish the real variation of the

overconsolidation ratio in the disturbed zone.

Conducting numerical analysis considering large strain formulations resulting in

more accurate settlement predictions incorporating creep.

214

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APPENDIX A

DEVELOPED MATLAB CODE FOR 2D AXISYMMETRIC EQUATION

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clc, clear all % % generating mesh and calculating time steps % deltaz = z / (n - 1); % for i=1:m-1 if i < m-1 % deltar(i) = 0.01; % elseif i == m-1 % deltar(i) = 0.011; % end end % % for i=1:m-1 % if i == 1 % dist(i) = 0.0011; % elseif i > 1 % dist(i) = dist(i - 1) + deltar(i); % end end % time(1) = 0; % % for t = 1:q-1 % deltat(t) = 1e-3; % time(t + 1) = time(t) + deltat(t); end % % for t = 1:q if t <= 100 % deltats(t) = (400000 / 100) * t; % elseif t > 100 && t <= 72999 % deltats(t) = 400000; % elseif t > 72999 && t <= 73100 % deltats(t) = 400000 - (350000 / 100) * (t - 72999); % elseif t > 73100 && t <= 157999 % deltats(t) = 50000; %

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elseif t > 157999 && t <= 159050 % deltats(t) = 50000 + (50000 / 50) * (t - 157999); % elseif t > 159050 && t <= 161999 % deltats(t) = 100000; % elseif t > 161999 && t <= 162050 % deltats(t) = 100000 + (100000 / 50) * (t - 161999); % elseif t > 162050 && t <= 165499 % deltats(t) = 200000; % elseif t > 165499 && t <= 165550 % deltats(t) = 200000 + (200000 / 50) * (t - 165499); % elseif t > 165550 % deltats(t) = 400000; % end end % % for t = 1:q % if t == 1 % deltau(t) = deltats(t); % else % deltau(t) = deltats(t) - deltats(t-1); end end % % for j = 2:n-1 for i = 2:m-1 A(i,j) = (i-1) + (j-2) * (m-2); end end % % for j = 1:n eszi(j) = (j - 0.5) * (gssat - gw) * deltaz; end % % % calculating initial values % for t = 1:q-1 % % % if t == 1

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pwp(1,1:n,2) = 0; % % for i = 1:m for j = 1:n % ei = 1.55; % et(i,j,2) = ei; % ez(i,j,1) = 0.0; % end % end % % % % for i = 1:m for j = 1:n % pwp(i,j,1) = deltau(t); % end end % % for i = 1:m for j = 1:n % ezref(i,j) = lov * log(((eszi(j) + deltats(t)) - pwp(i,j,1)) / esz0); % eref(i,j) = - (1 + (ei / (ez(i,j,1) - ezref(i,j)))) / ... (1 - 1 / (ez(i,j,1) - ezref(i,j))); % crli(i,j) = eref(i,j) / (1 + ei); % end end % % for i = (rsmear + 1):m for j = 1:n % kpr(i,j,2) = 10 ^ ((ei - kcon) / ck) * 86400; % kpz(i,j,2) = 0.1 * kpr(i,j,2); end end % % for i = 1:rsmear for j = 1:n % kpr(i,j,2) = kpr((rsmear + 1),j,2) / ksmear; % kpz(i,j,2) = kpr(i,j,2);

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end end % % elseif t > 1 % pwp(1,1:n,1:2) = 0; % pwp(1:m,1,1:2) = 0; % % for i = 1:m for j = 1:n % pwp(i,j,1) = pwp(i,j,1) + deltau(t); % end end % % for i = 1:m for j = 1:n et(i,j,1) = et(i,j,2); et(i,j,2) = 0; end end % % calculating pore water pressure on each time step % % for i = 1:m for j = 1:n % ezref(i,j) = lov * log(((eszi(j) + deltats(t)) - pwp(i,j,1)) / esz0); % et(i,j,2) = et(i,j,1) - (ez(i,j,2) - ez(i,j,1)) * (1 + ei); % % eref(i,j) = - (1 + (et(i,j) / (ez(i,j,2) - ezref(i,j)))) / ... (1 - 1 / (ez(i,j,2) - ezref(i,j))); % crli(i,j) = eref(i,j) / (1 + ei); % end end % % for i = 1:m for j = 1:n ez(i,j,1) = ez(i,j,2); ez(i,j,2) = 0; end end % % for i = 1:m for j = 1:n

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kpr(i,j,1) = kpr(i,j,2); kpr(i,j,2) = 0; end end % % for i = 1:m for j = 1:n % kpr(i,j,2) = kpr(i,j,1) * 10 ^ ((et(i,j,2) - et(i,j,1)) / ck); % if i <= (rsmear) % kpz(i,j,2) = kpr(i,j,2); % elseif i > (rsmear) % kpz(i,j,2) = 0.1 * kpr(i,j,2); % end end end end % % % % for j = 1:n mv(1,j) = kov / ((eszi(j) + deltats(t)) - pwp(1,j,1)); % % % if (lov * (log(((eszi(j) + deltats(t)) - pwp(1,j,1)) / esz0)) +... (eref(1,j) / (1 + ei)))... < ... ez(1,j,1) % % % saayov(1,j) = 0; % ezlvp(1,j) = 0; % g(1,j) = 0; % else saayov(1,j) = -0.002 * log(((eszi(j) + deltats(t)) - pwp(1,j,1)) / 1000)... + 0.0166; % ezlvp(1,j) = (eref(1,j) / (1 + ei)); % g(1,j) = (saayov(1,j) / t0) * ... ((1 + ((ez0ep + lov * log(((eszi(j) + deltats(t)) - pwp(1,j,1))... / esz0) - ez(1,j,1)) / ezlvp(1,j))) ^ 2) * ... (exp((1 / saayov(1,j)) * ...

235

((ez0ep + lov * log(((eszi(j) + deltats(t)) - pwp(1,j,1)) / esz0)... - ez(1,j,1))) / ... (1 + (ez0ep + lov * log(((eszi(j) + deltats(t)) - pwp(1,j,1))... / esz0) - ez(1,j,1)) / ezlvp(1,j)))); % end % ez(1,j,2) = ez(1,j,1) - mv(1,j) * pwp(1,j,2) + mv(1,j) *... pwp(1,j,1) + g(1,j) * deltat(t); end % % for j = 2:n-1 for i = 2:m-1 % mv(i,j) = kov / ((eszi(j) + deltats(t)) - pwp(i,j,1)); % % if (lov * (log(((eszi(j) + deltats(t)) - pwp(i,j,1)) / esz0)) +... (eref(i,j) / (1 + ei)))... < ... ez(i,j,1) % g(i,j) = 0; % saayov(i,j) = 0; % ezlvp(i,j) = 0; % else saayov(i,j) = - 0.002 * log10(((eszi(j) + deltats(t)) - pwp(i,j,1)) / 1000)... + 0.0166; % ezlvp(i,j) = (eref(i,j) / (1 + ei)); % g(i,j) = (saayov(i,j) / t0) * ... ((1 + ((ez0ep + lov * log(((eszi(j) + deltats(t)) - ... pwp(i,j,1))... / esz0) - ez(i,j,1)) / ezlvp(i,j))) ^ 2) * ... (exp((1 / saayov(i,j)) * ... ((ez0ep + lov * log(((eszi(j) + deltats(t)) - ... pwp(i,j,1)) / esz0)... - ez(i,j,1)) / ... (1 + (ez0ep + lov * log(((eszi(j) + deltats(t))... - pwp(i,j,1))... / esz0) - ez(i,j,1)) / ezlvp(i,j))))); % end % if i < m-1 && j < n-1 % B(((j-2)*(m-2)+(i-1)),A(i,j)) = ... (kpr(i,j,2) / (gw * (deltar(i) ^ 2))) + ... (kpz(i,j,2) / (gw * (deltaz ^ 2))) + (mv(i,j) / deltat(t));

236

% % B(((j-2)*(m-2)+(i-1)),A(i+1,j)) = ... - ((kpr(i,j,2) / gw) * ((1 / (2 * (deltar(i) ^ 2))) + (1 / (4 * ... dist(i) * deltar(i))))); % B(((j-2)*(m-2)+(i-1)),A(i,j+1)) = ... - (kpz(i,j,2) / gw) * (1 / (2 * (deltaz ^ 2))); % elseif i == m-1 && j < n-1 % B(((j-2)*(m-2)+(i-1)),A(i,j)) = ... (kpr(i,j,2) / (gw * (deltar(i) ^ 2))) + ... (kpz(i,j,2) / (gw * (deltaz ^ 2))) + (mv(i,j) / deltat(t))... - ((kpr(i,j,2) / gw) * ((1 / (2 * (deltar(i) ^ 2))) + (1 / (4 * ... dist(i) * deltar(i))))); % B(((j-2)*(m-2)+(i-1)),A(i,j+1)) = ... - (kpz(i,j,2) / gw) * (1 / (2 * (deltaz ^ 2))); % elseif i < m-1 && j == n-1 % B(((j-2)*(m-2)+(i-1)),A(i,j)) = ... (kpr(i,j,2) / (gw * (deltar(i) ^ 2))) + ... (kpz(i,j,2) / (gw * (deltaz ^ 2))) + (mv(i,j) / deltat(t)); % B(((j-2)*(m-2)+(i-1)),A(i+1,j)) = ... - ((kpr(i,j,2) / gw) * ((1 / (2 * (deltar(i) ^ 2))) + (1 / (4 * ... dist(i) * deltar(i)))))... - (kpz(i,j,2) / gw) * (1 / (2 * (deltaz ^ 2))); % elseif i == m-1 && j == n-1 % B(((j-2)*(m-2)+(i-1)),A(i,j)) = ... (kpr(i,j,2) / (gw * (deltar(i) ^ 2))) + ... (kpz(i,j,2) / (gw * (deltaz ^ 2))) + (mv(i,j) / deltat(t))... - ((kpr(i,j,2) / gw) * ((1 / (2 * (deltar(i) ^ 2))) + (1 / (4 * ... dist(i) * deltar(i)))))... - (kpz(i,j,2) / gw) * (1 / (2 * (deltaz ^ 2))); end % if (i-1) > 1 B(((j-2)*(m-2)+(i-1)),A(i-1,j)) = ... (kpr(i,j,2) / gw) * ((1 / (4 * dist(i) * deltar(i))) - (1 / ... (2 * (deltar(i) ^ 2)))); end % if (j-1) > 1 B(((j-2)*(m-2)+(i-1)),A(i,j-1)) = ... - (kpz(i,j,2) / gw) * (1 / (2 * (deltaz ^ 2))); end %

237

% C((j-2)*(m-2)+(i-1),1) = ... (kpr(i,j,2) / gw) * ((1 / (deltar(i) ^ 2)) * (pwp(i-1,j,1) - ... 2 * pwp(i,j,1) + pwp(i+1,j,1)) + (1 / (4 * dist(i) * ... deltar(i))) * (pwp(i+1,j,1) - pwp(i-1,j,1))) + ... (kpz(i,j,2) / gw) * ((1 / (2 * deltaz ^ 2)) * (pwp(i,j-1,1) -... 2 * pwp(i,j,1) + pwp(i,j+1,1))) + (mv(i,j) / deltat(t)) ... * pwp(i,j,1) + g(i,j); % % end end % % calculating settlement on each time step % B1 = inv(B); D = B1 * C; % for y = 1:(n-2) for x = 1:(m-2) pwp(x+1,y+1,2) = D((m-2)*(y-1)+x); end end % for j = 1:n pwp(m,j,2) = pwp(m-1,j,2); end % for i = 1:m pwp(i,n,2) = pwp(i,n-1,2); end % for j = 2:n-1 for i = 2:m-1 ez(i,j,2) = ez(i,j,1) - mv(i,j) * pwp(i,j,2) ... + mv(i,j) * pwp(i,j,1) + g(i,j) * deltat(t); end end % % for i = 2:m-1 ez(i,1,2) = ez(i,2,2); end % % for i = 1:m ez(i,n,2) = ez(i,n-1,2); end % % for j = 1:n ez(m,j,2) = ez(m-1,j,2); end % clear mv %

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clear g % % if t == 1 pwpd = 0; else load(['pwpd' '.mat'],(['pwpd'])) end % if (t - 1) / 50 - floor((t -1) / 50) == 0 pwpd = 0; end % for i = 1:m for j = 1:n pwpd(i,j,(t-(floor((t-1)/50)*50))) = pwp(i,j,1); end end % save(['hpwp' num2str(1+(floor((t-1)/50))) '.mat'],(['pwpd'])) save(['pwpd' '.mat'],(['pwpd'])) clear(['pwpd' '.mat'],(['pwpd'])) % for i = 1:m for j = 1:n pwp(i,j,1) = pwp(i,j,2); pwp(i,j,2) = 0; end end % % if t == 1 ezd = 0; else load(['ezd' '.mat'],(['ezd'])) end % if (t - 1) / 50 - floor((t -1) / 50) == 0 ezd = 0; end % for i = 1:m for j = 1:n ezd(i,j,(t-(floor((t-1)/50)*50))) = ez(i,j,1); end end % save(['hez' num2str(1+(floor((t-1)/50))) '.mat'],(['ezd'])) save(['ezd' '.mat'],(['ezd'])) clear(['ezd' '.mat'],(['ezd'])) % % if t == 1 saayovd = 0; else load(['saayovd' '.mat'],(['saayovd'])) end % if (t - 1) / 50 - floor((t -1) / 50) == 0 saayovd = 0;

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end % for i = 1:m-1 for j = 1:n-1 saayovd(i,j,(t-(floor((t-1)/50)*50))) = saayov(i,j); end end % save(['hsaayov'num2str(1+(floor((t-1)/50))) '.mat'],(['saayovd'])) save(['saayovd' '.mat'],(['saayovd'])) clear(['saayovd' '.mat'],(['saayovd'])) clear saayov % if t == 1 crlid = 0; else load(['crlid' '.mat'],(['crlid'])) end % if (t - 1) / 50 - floor((t -1) / 50) == 0 crlid = 0; end % for i = 1:m-1 for j = 1:n-1 crlid(i,j,(t-(floor((t-1)/50)*50))) = crli(i,j); end end % save(['hcrli' num2str(1+(floor((t-1)/50))) '.mat'],(['crlid'])) save(['crlid' '.mat'],(['crlid'])) clear(['crlid' '.mat'],(['crlid'])) clear crli % clear ezlvp % % clear kz clear kr clear eref clear e end

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