Finite element modeling of peaty soft ground preconsolidated by vertical drains under vacuum – surcharge preloading

determination of the length of drainage path of the vertical drain l of the axisymmetric unit cell in each soil layer of a multi-layer subsoil, where the vertical drain driven through i

Finite element modeling of peaty soft ground preconsolidated by vertical

drains under vacuum-surcharge preloading

真空圧密工法適用下の泥炭性軟弱地盤の有限要素解析モデルに関する研究

Tuan Anh TRAN

A thesis submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Examination Committee: Prof Toshiyuki Mitachi

Prof Seiichi Miura Prof Yoshiaki Fujii

Doctoral Thesis No

Division of Structural and Geotechnical Engineering

Graduate School of Engineering, Hokkaido University

December, 2007

Within few weeks, I will be leaving Japan, saying good-bye to Sapporo, Hokudai, Professors, Lecturers, and good friends of mine The moment of separation is always a hard time for anyone, but any enjoyable game always must have an end For me, the five-year study here has created memorable memories and invaluable lessons As a last sentence to Japan, I would like to sincerely thank Japan and Japanese for giving me an invaluable chance of five-year study here

The first Japanese person, I would like to mention, is the Head of Soil Mechanics Laboratory, (Hokkaido University) Prof Toshiyuki Mitachi, an exemplary and scholarly teacher, who is the advisor of mine Without his enthusiastic instruction, constant feed-back, continuous attention, protection, and insightful discussions whenever, wherever, I would be unable to exist in Japan until this time, and this thesis would be unable to reach this level as well I am deeply indebted to him His help will never be forgotten

The next is JICA (Japan International Cooperation Agency), the agency has kindly provided me the three-year PhD scholarship under AUN/SEED-Net Project (ASEAN University Network/Southeast Asia Engineering Education Development Network Project) I owe a deep gratitude to them

The second person is Prof Tamon Ueda, who has asked JICA office in Chulalongkorn University, in Bangkok, for providing me this scholarship His kind help is greatly appreciated

The third person is Associate Prof Hiroyuki Tanaka I would like to thank him very much for his valuable comments and helpful discussion over the years In addition, his kindness of bringing me to several field trips in Kansai Airport, other places in Osaka, and in the suburbs of Sapporo City, is gratefully acknowledged

The fourth person is Ex-associate Prof Satoru Shibuya, and now being Professor at Geotechnical Engineering Laboratory at Kobe University I still remember valuable discussions, first advices from him at the first period of time when I came here Especially, his excellent lectures concerning physical and mechanical properties of geomaterials, during the time he had been here, have indeed opened my mind about soil properties I would like to sincerely thank him for all those things

I also wish to express my gratitude to the members of PhD thesis examination committee, Prof Seiichi Miura and Prof Yoshiaki Fujii Their thoughtful comments and advice are gratefully appreciated

I would also like to thank Dr Nobutaka Yamazoe very much for teaching me FEM, programming, and providing me his FORTRAN source code program I will always keep in mind that generous help Another one, I wish to thank, is Fumihiko Fukuda sensei, the research associate in my laboratory, who has given advices, assistance, and sympathy to me regarding difficulties in my research as well as during the time I have worked at Lab

I also wish to extend my appreciation to Mr Kozai, Mr Hasegawa, Mr Mitchi, Mr Nishida, Mr Tsutsumi, and Mr Oka They have helped me very much in various aspects of life, research, and learning many things of Japanese culture I am very grateful to them

With regard to documentary procedures required by Japanese Geotechnical Society as well as other procedures related to me required by Kyomu (Educational administration office), special thanks are due to enthusiastic help of Kudo-san and Utsugi-san, who are the technician and the secretary, respectively, of my Lab, in those works

As for other students of my Lab, thanks also go to friendly atmosphere and open mind that they have shown to me over the years of my study here

In addition, I also wish to thank other people, whose name have not been mentioned here, but have helped indirectly in various aspects of my study from Master to PhD courses in Hokudai I wish they would know that their help is appreciated at all time

Finally, the deepest gratitude and appreciation are due to the beloved parents, lovely sisters of mine, who have great faith in, indefatigably supported and encouraged me during the past five years

Sapporo, the last winter of mine, 12/2007

Tuan Anh TRAN

博士の専攻分野の名称 博士（工学） 氏名 : Tuan Anh TRAN

学 位 論 文 題 名 Title of dissertation submitted for the degree

Finite element modeling of peaty soft ground preconsolidated by vertical drains

under vacuum-surcharge preloading

真空圧密工法適用下の泥炭性軟弱地盤の有限要素解析モデルに関する研究

The plane-strain simulation method of soft ground incorporating vertical drains under surcharge preloading is usually adopted, because this method can save time and does not require a high specification computer as that of 3D simulation method

vacuum-However, the performance of a vertical drain in the field is close to an axisymmetric unit cell Therefore, this thesis developed a conversion method to convert from an axisymmetric unit cell to an equivalent plane strain unit cell under vacuum surcharge preloading condition Specifically, the analytical models for the axisymmetric and plane-strain unit cells and their analytical solutions are developed in this thesis After that, a conversion method that can make the degree of consolidation of the plane strain unit cell equal to that of the axisymmetric cell is proposed

In Chapter 1, background, research objectives and scope of research of this thesis are described

determination of the length of drainage path of the vertical drain (l) of the axisymmetric unit cell in

each soil layer (of a multi-layer subsoil, where the vertical drain driven through) is not required; besides, the proposed equivalent plane strain unit cell of this method does not need the inclusion of the plane-strain smear zone Therefore, the proposed method is simpler and more convenient than Indraratna et al (2005) method, which is the newest conversion method available on the world

In Chapter 4, the proposed method was verified by FEM for 3 ideal cases of subsoil: (1) one homogeneous clay layer, (2) two clay layers, and (3) three soil layers which is clay-sand-clay sandwich subsoil that can be found in many places in Japan The verification results showed that the proposed method yielded excellent agreement between the axisymmetric and plane strain cells in all the cases of subsoil In particular, the results indicated that the effects of both well resistance (in the vertical drain) and smear zone (around the vertical drain) are satisfactorily modeled by the proposed plane strain unit cell

Besides, FE analyses of two cases of subsoil (2) and (3) showed that Indraratna et al (2005) method could not produce matching results if the drain had high well-resistance and is installed in the subsoil having multi layers; moreover, in some cases, the converted permeability of their plane strain unit cell could even become negative

In Chapter 5, by using the proposed conversion method, a FE 2D simulation method of a full-scale vacuum-embankment on soft ground is also developed The comparison between the simulated results and the field-observed data in each soil layer, beneath the full-scale vacuum-embankment (constructed

in Kushiro, Hokkaido, Japan) showed good agreement in terms of vertical, horizontal displacements

and excess pore water pressure (PWP) This confirmed the validity of the proposed conversion method

as well as the simulation method

In addition, the verification of Shinsha et al (1982) and Chai et al (2001) methods via that full-scale embankment is also conducted It is noted that Shinsha et al (1982) method is being widely used in Japan, and Chai et al (2001)’s method is one of the newest methods being used on the world

The FE analysis results indicated that Shinsha et al (1982) method could produce large overprediction

in terms of vertical displacement and negative excess PWP, albeit it could produce good prediction in terms of horizontal displacement But, this good prediction is attributed to the overprediction of negative excess PWP, which could create computed high isotropic compression within the vertical drain-improved zone beneath the embankment

Besides, the FE analysis revealed that Chai et al (2001) method could turn out significant underprediction related to negative excess PWP in some soil layers, but it could produce relatively good prediction in terms of vertical displacement in each soil layer Regarding horizontal displacement, this method could give relatively good prediction at the time of just before filling the embankment, but

it significantly overpredicted the field data after filling

In Chapter 6, summary and conclusions of this thesis are presented

Finally, recommendations for further research are provided in Chapter 7

Title page i

Acknowledgement ii

Abstract iv

Table of contents vi

CHAPTER 1: INTRODUCTION 1 1.1 Background 1

1.2 Objectives and scope of the research 2

1.3 Thesis outline 2

REFERENCES OF CHAPTER 1 3

CHAPTER 2: REVIEW OF AVAILABLE CONVERSION METHODS 8 2.1 Available conversion methods under conventional surcharge preloading condition 8

2.2 Available conversion methods under vacuum-surcharge preloading condition 9

REFERENCES OF CHAPTER 2 12

CHAPTER 3: THE PROPOSED CONVERSION METHOD 13 3.1 Analytical models of the axisymmetric and plane-strain unit cells under vacuum-surcharge preloading 13

3.2 Analytical solution for the axisymmetric unit cell (modified by the writer after Indraratna et al (2005)) 14

3.3 The proposed analytical solution for the plane strain unit cell without smear zone 14

3.4 The proposed conversion expression of permeability for the plane strain unit cell 15

3.5 The way to use the proposed conversion method 16

REFERENCES OF CHAPTER 3 17

CHAPTER 4: VERIFICATION OF THE PROPOSED METHOD VIA FINITE ELEMENT METHOD 18 4.1 Outline 18

4.2 Detailed description of the case of one clay layer 19

4.2.1 Boundary conditions and computed cases 20

4.3 Detailed description of the case of two clay layers 21

4.4 Detailed description of the case of clay-sand-clay sandwich 22

4.5 Results and discussion 24

4.5.1 In the case of one homogeneous clay layer 24

4.5.2 In the case of two clay layers 24

4.5.3 In the case of clay-sand-clay sandwich 25

4.6 Conclusions of chapter 4 26

REFERENCES OF CHAPTER 4 27

CHAPTER 5: APPLICATION OF THE PROPOSED METHOD, SHINSHA ET AL (1982) METHOD, AND CHAI ET AL (2001) METHOD TO A FULL-SCALE TEST VACUUM-EMBANKMENT 42 5.1 The full-scale test embankment and soil profile 42

5.2 FE simulation of the test embankment 44

5.2.1 Soil models used 44

5.2.1.1 Outline of critical state models 44

5.2.1.2 Cam-Clay model 44

5.2.1.2.1 The Cam-clay flow rule 48

5.2.1.3 Modified Cam-Clay model 50

5.2.1.4 Drucker-Prager model 52

5.2.2 FE mesh and element types 54

5.2.2.1 The proposed method .54

5.2.2.2 Shinsha et al (1982) method 55

5.2.2.3 Chai et al (2001) method .55

5.2.3 Simulation of surcharge loading 55

5.2.4 Simulation of vacuum preloading 56

5.2.4.1 The proposed method 56

5.2.4.2 Shinsha et al (1982) method 56

5.2.4.3 Chai et al (2001) method 56

5.3 Determination of soil properties for FE simulation 57

5.3.1 Determination of parameters for soil models 57

5.3.1.1 The relationship between M and Su (undrained shear strength) 57

5.3.1.2 Su (undrained shear strength) 59

5.3.1.3 K0 (coefficient of earth pressure at rest) 59

5.3.1.4 p’c (size of yield locus) 60

5.3.1.5 Γ (specific volume intercept of CSL when p’ = 1.0 kN/m2 ) 61

5.3.1.6 ν (Poisson’s ratio) 62

5.3.1.7 λ (compression index) and κ (swelling index) 62

5.3.1.8 Parameters for sand layers 62

5.3.1.8.2 E (elastic modulus) 62

5.3.2 Determination of permeability for soil and vertical drain elements 63

5.3.2.1 The proposed method 63

5.3.2.2 Shinsha et al (1982) method 64

5.3.2.3 Chai et al (2001) method 65

5.3.3 Input-data table of soil parameters 66

5.3.3.1 Input data based on Lab & In-situ tests and Interpolation (LII) 66

5.3.3.2 Input data based on Back Analysis (BA) 66

5.4 Results and discussion 69

5.4.1 Simulated and observed excess pore water pressures 69

5.4.1.1 The proposed method 69

5.4.1.2 Shinsha et al (1982) method and Chai et al (2001) method 69

5.4.2 Simulated and observed vertical displacements 69

5.4.2.1 The proposed method 69

5.4.2.2 Shinsha et al (1982) method and Chai et al (2001) method 69

5.4.3 Simulated and observed horizontal displacements 70

5.4.3.1 The proposed method 70

5.4.3.2 Shinsha et al (1982) method and Chai et al (2001) method 70

5.5 Conclusions of chapter 5 71

REFERENCES OF CHAPTER 5 79

CHAPTER 6: SUMMARY AND CONCLUSIONS 80 CHAPTER 7: RECOMMENDATION FOR FURTHER RESEARCH 82 APPENDIX A: Formulation of the analytical solution for the average excess pore pressure of the axisymmetric unit cell 83

APPENDIX B: Formulation of the analytical solution for the average excess pore pressure of the plane strain unit cell 87

APPENDIX C: Figures of comparison between FEM results of the axisymmetric unit cell and those of the plane strain unit cells in the case of k1=1 90

LIST of SYMBOLS 92

in soft clayey subsoil can be achieved; in short, we call this preloading technique vacuum-surcharge preloading

It is observed that, in vacuum-surcharge preloading, the vacuum pressure propagates along the vertical drains, and therefore, a nearly isotropic compression zone is created within the subsoil zone beneath the embankment; therefore, this technique can enhance the stability of the subsoil during consolidation under the embankment

To illustrate this method, main in-situ instruments and mechanism of the method are depicted by Figs 1-1, 1-2, and 1-3

From those figures, it is easily to see that vacuum-surcharge preloading method has characteristics being much more advantageous than conventional surcharge preloading method; those advantages are:

• Isotropic consolidation eliminates the risk of failure of embankment

• Significant time savings over other consolidation methods Loading and construction can proceed as early as three weeks after process has started

Regarding the effect of a vertical drain on the drainage of the soft ground surrounding it, it is widely agreed that the performance of a vertical drain under conventional embankment (or conventional surcharge) can be represented by an axisymmetric unit cell (Barron 1948, Hansbo 1981) as shown in Fig 1-4 Therefore, to simulate the performance of vertical drains under embankment by FEM, we

have to make a very large 3D full-scale simulation, in which a lot of cubic elements have to be employed (Chai et al 1995, Indraratna and Redana 2000) As a result, the time needed for computation becomes very long, and a very strong specification computer is needed

On the contrary, if we assume that the performance of a vertical drain can be equivalently represented

by a plane strain unit cell (see Figs 1-4 and 1-5), then an equivalent full-scale plane strain simulation

of soft ground improved by vertical drains under embankment can be made by using this plane strain unit cell (see Fig.1-6) Consequently, the time needed for computing the full-scale plane strain simulation is much shorter than that needed in a full-scale 3D simulation In fact, Chai et al (1995) and Indraratna and Redana (2000) confirmed the feasibility of this idea

1.2 Objectives and scope of the research

Considering the situation mentioned above, the author conducted the research with the objectives and scope as follows

[1] Developing a conversion method to convert from an axisymmetric unit cell to a plane strain unit cell under vacuum-surcharge preloading condition

[2] Proposing a 2D simulation method for a full-scale soft ground incorporating vertical drains under vacuum-embankment preloading

[3] Validating the proposed conversion method by FEM for the axisymmetric and plane strain unit cells in 3 cases: (1) 1 homogeneous clay layer; (2) 2 clay layers; and (3) clay-sand-clay sandwich, in which a sand layer is sandwiched between two clay layers

[4] Verifying the proposed conversion method via a full-scale test vacuum-embankment, in which a comparison between the computed results and the observed data is conducted

[5] Verifying Shinsha et al (1982) and Chai et al (2001) conversion methods via the full-scale test vacuum-embankment mentioned above In these methods, Shinsha et al (1982)’s one is being widely used in Japan, and Chai et al (2001)’s method is one of the newest methods being used on the world

1.3 Thesis outline

Chapter 1 introduces the background knowledge to help the reader understand the content of the thesis

In addition, this chapter also provides the objectives and scope of the research Finally, at the end of this chapter, a thesis outline and its overall flowchart are provided (see Fig.1-7)

Chapter 2 provides a comprehensive review of the available conversion methods, which is being widely used all over the world

Chapter 3 introduces the development of the proposed conversion method under vacuum-surcharge preloading condition In this chapter, the analytical models proposed for the axisymmetric and plane-strain unit cells and their analytical solutions are presented After that, a conversion method that can make the degree of consolidation of the plane strain cell equal to that of the axisymmetric cell is developed

Chapter 4 verifies the proposed conversion method for 3 ideal cases of subsoil: (1) one homogeneous clay layer, (2) two clay layers, and (3) three soil layers which is a clay-sand-clay sandwich subsoil that can be found in many places in Japan In this chapter, the verification of Indraratna et al (2005) method is also conducted for cases (2) and (3), because this method is the newest conversion method which exists on the world After all, the comparison of merits and demerits between Indraratna et al (2005) method and the proposed method is provided

Chapter 1: Introduction

Chapter 5 presents the application of the proposed method to a full-scale test embankment in Kushiro City, Hokkaido, Japan Besides, the verification of Shinsha et al (1982) method, which is being widely used in Japan, and Chai et al (2001) method, which is one of the newest conversion methods existing on the world, via that full-scale test embankment is conducted Especially, the merits and demerits of the proposed, Shinsha et al (1982), and Chai et al (2001) methods are elucidated through comparing the computed results of each method with the field-measured data

Chapter 6 provides the summary and conclusions of the research of this thesis

Chapter 7 recommends potential topics for further research

REFERENCES OF CHAPTER 1

[1] Barron RA (1948) Consolidation of fine-grained soils by drain wells ASCE Transactions; 113:

718-754

[2] Hansbo S (1981) Consolidation of fine-grained soils by prefabricated drains In: Proceedings

of 10 th International Conference on Soil Mechanics and Foundation Engineering, Stockholm,

Sweden; Vol 3 p 677-682

[3] Chai JC, Miura N, Sakajo S, and Bergado DT (1995) Behavior of vertical drain improved

subsoil under embankment loading Soils and Foundations; 35(4): 49-61

[4] Indraratna B and Redana IW (2000) Numerical modeling of vertical drains with smear and

well resistance installed in soft clay Canadian Geotechnical Journal; 37: 132-145

[5] Chai JC, Shen SL, Miura N, and Bergado DT (2001) Simple method of modeling PVD

improved subsoil Journal of Geotechnical and Geoenvironmental Engineering, ASCE; 127(11):

965-972

[6] Shinsha, H., Hara, H., Abe, T., and Tanaka, A (1982): Consolidation settlement and lateral

displacement of soft ground improved by sand drains, Tsuchi-to-Kiso, Japanese Geotechnical

Society, 30 (5), 7-12 (in Japanese)

[7] Indraratna B, Rujikiatkamjorn C, and Sathananthan I (2005) Analytical and numerical

solutions for a single vertical drain including the effects of vacuum preloading Canadian Geotechnical Journal; 42: 994-1014

Vertical drain

A

(arrows in the circle indicate flow direction of water)

Influence zone Vacuum pump

Geomembrane

Fig 1-1 Cross section and plan view of subsoil improved by vertical drains beneath embankment

a future highway Vacuum preloading

Fig 1-2 Some examples of application of vacuum preloading to soft ground

A future highway

Soft ground was improved

Vacuum pressure released

EMBANKMENT

Soft ground was improved

STAGE 1: Installing PVDs (prefabricated vertical drain), impervious membrane, and

vacuum pump system

STAGE 2: Pumping vacuum Isotropic consolidation is established in soft ground

STAGE 3: Soil embankment is stacked up Vacuum pressure is still remained

FINAL STAGE: After some months, vacuum pressure is released

Soft ground is fully consolidated

Fig 1-4 Axisymmetric unit cell and its equivalent plane strain unit cells

Fig 1-5 Conversion from the axisymmetric performance to the plane-strain performance

2D cell with smear zone 2D cell without smear zone

Axisymmetric cell

De

Chapter 1: Introduction

Chapter 1 INTRODUCTION

REVIEW OF AVAILABLE CONVERSION METHODS

1 One homogeneous clay layer

2 Two clay layers

3 Clay-sand-clay sandwich

Chapter 5 APPLICATION OF THE PROPOSED METHOD, SHINSHA ET AL (1982), AND CHAI ET AL (2001) METHODS TO A FULL-SCALE TEST VACUUM-EMBANKMENT

SUMMARY AND CONCLUSIONS

Chapter 6

RECOMMENDATION FOR FURTHER RESEARCH

Chapter 7

Fig 1-7 Flowchart of the thesis outline

Fig 1-6 The performance of the vertical drain system in the 2D FE simulation

(a) Cross section A-A

(arrows indicate flow direction of water) (b) Plan view of plane strain cells

Chapter 2

Review of Available Conversion Methods

“Good judgement comes from experience Experience comes from bad judgement”

2.1 Available conversion methods under conventional surcharge preloading condition

Shinsha et al (1982) and Zeng et al (1987) have involved matching the average degree of consolidation of 50% of an axisymmetric unit cell with that of a plane strain unit cell By doing so, together with the assumption that the effect of the well resistance is insignificant, they obtained the following conversion expression of permeability

ha ha hp

hp

k

R T k

B

= (2.1-1)

where k hp is horizontal permeability coefficient of the plane strain unit cell; k ha is horizontal

permeability coefficient in undisturbed zone of axisymmetric unit cell; B is half-width of the plane strain unit cell; R is radius of the influence zone; T ha and T hp are dimensionless time factors for horizontal drainage in axisymmetric unit cell and in plane strain unit cell, respectively It should be noted that this method produces matching results at the average degree of consolidation of 50 % only;

it does not then apply for other degree of consolidation, because the ratio Thp/Tha of other degree of consolidation will change In addition, in the case of the well resistance were significant, the degree of consolidation would vary with depth, and therefore this method would become invalid

Sekiguchi et al (1986) proposed a macro-element for modeling the vertical drains in plane strain finite element analysis In this method, the influence zone of a unit cell is modeled by a plane strain macro-element in which the drain is at the center And, only one unknown of average excess pore water pressure of a unit cell is computed at the center of the element; besides, the drain is assumed to be no well resistance Furthermore, not all of the available commercial software in geotechnical engineering incorporated this special element; therefore, a simple conversion method that can apply to all of

Chapter 2: Review of Available Conversion Methods

conventional geotechnical software to model the behavior of vertical drain in plane strain condition is needed

In 1992, Hird et al introduced an analytical model for a plane strain unit cell under conventional surcharge condition Based on this analytical model, he proposed an analytical solution for the average excess pore pressure of this plane strain unit cell Equating this solution with the solution proposed by Hansbo (1981) for an axisymmetric unit cell, he obtained the conversion expressions of either geometry or permeability for the plane strain unit cell, namely if the plane strain permeability is chosen equal to the undisturbed permeability of the axisymmetric unit cell, then the matching, which

he named it geometric matching, can be achieved In this method, the effect of well resistance is matched independently If choosing B = R then the matching, which is called permeability matching,

can also be obtained In this way, the effect of well resistance is again matched independently

Aiming at “well resistance matching”, Chai et al (1995) based on the solutions of Hird et al (1992)

and Hansbo (1981) to propose a conversion expression of both permeability and geometry for a plane strain unit cell under conventional surcharge condition Regarding this conversion expression, we think that only if khp were chosen equal to kha would the expression be valid That is because this expression was developed on the basis of Eq (9) in the paper of Hird et al (1992), in which Eq (9) is based on the assumption that khp = kha Alternatively, they proposed that if choosing B=R and

kha=khp=kh the equivalent discharge capacity of the drain wall of the plane strain unit cell can be determined by Eq (8) in their paper (Chai et al 1995)

In 1997, based on Hird et al (1992) analytical model for a plane strain unit cell without smear zone, Indraratna and Redana proposed another analytical plane strain model, in which the smear zone is incorporated into the plane strain unit cell Subsequently, they offered an analytical solution for the average excess pore pressure of their plane strain cell By equating this solution with that of Hansbo (1981) for the axisymmetric cell, they obtained a conversion expression of permeability for their plane strain cell

2.2 Available conversion methods under vacuum-surcharge preloading condition

In 2004, Indraratna et al presented a paper, in which they conducted a plane-strain numerical modeling of an embankment stabilized with vertical drains subjected to vacuum preloading In this numerical modeling, they used the conversion method from the axisymmetric unit cell to the plane strain unit cell proposed by Indraratna and Redana (2000)

However, we observed that the conversion method of Indraratna and Redana (2000) is merely developed under conventional surcharge preloading condition, i.e there is not any boundary condition related to vacuum preloading

In addition, it is well known that the distribution of vacuum pressure along the vertical drain decreases from the top to the bottom (Indraratna et al 2005) as shown in Fig 2-1a On the contrary, the boundary condition of excess pore pressure along the vertical drain, in the method of Indraratna and Redana (2000), is that the excess pore pressure is merely equal to zero at the top of the vertical drain Therefore, there is not a clear and convincing theoretical basis to prove that the method of Indraratna and Redana (2000) are still able to apply to the case of vacuum-surcharge preloading

For these reasons, in 2005, Indraratna et al published a paper, in which they proposed another conversion method under a clearer condition being surcharge loading combined with vacuum preloading based on a more logical theoretical basis in comparison with that of Indraratna and Redana (2000) In this paper, they developed two analytical models for the axisymmetric cell and the plane strain cell Based on these two models, they proposed two solutions for development of the average excess pore water pressure with time within these two cells

Afterwards, by equating these two solutions, they obtained a conversion expression of permeability

from the axisymmetric unit cell to the plane strain unit cell As a result, by using this conversion expression and their plane strain cell, a full-scale plane strain modeling of soft ground improved by vertical drains under vacuum-surcharge preloading can be made

(a) Analytical model of the axisymmetric unit cell

qwp

Plane-strain undisturbed zone

dy = 1

wpq

sb

B

wb

Plane-strain smear zone

Chapter 2: Review of Available Conversion Methods

We see that, in their method, the inclusion of the plane strain smear zone in their plane strain unit cell

is deemed to be needed (see Figs 1-4 and 2-2a), and the determination of the length of drainage path of

the drain (l) of their axisymmetric unit cell in each soil layer of a multi-layer subsoil (where the

vertical drain is driven through) is required (see Fig 2-1a)

However, in our experience, the inclusion of plane strain smear zones in plane strain finite element simulation increases the number of elements and material parameters In particular, for a full-scale simulation including many soil layers and a large number of small vertical drain elements, the number

of elements and the material parameters for the plane strain smear zones become very large

In addition, we observed that, in their theory, the length of drainage path of the drain (l) is the length

from the bottom to the top of the vertical drain in the axisymmetric unit cell And, in this axisymmetric unit cell, they defined that the pore water pressure at the bottom of the vertical drain reaches the maximum value (i.e the derivative of the pore-water-pressure function with respect to depth z at this point is equal to zero), and the pore water pressure at the top of the vertical drain is equal to the vacuum presure applied to the top of the vertical drain (-p0)

In reality, the soft ground usually has many soil layers Therefore, we see that the derivative of the pore-water-pressure function of the vertical drain, with respect to depth z, at the bottom boundary of each soil layer is usually not equal to zero except the case that the bottom boundary of the lowest soil

layer is undrained Consequently, determination of the length of drainage path of the drain (l) of the

vertical drain, according to their theory, in each soil layer is very difficult

In 2006, Chai et al presented a plane-strain numerical modeling of a soft ground improved by vertical drains under embankment supported by vacuum preloading In their numerical modeling, they used a conversion method from the axisymmetric cell to the plane strain cell proposed by Chai et al (2001)

In this conversion method, the horizontal and vertical permeability of the smear and undisturbed zones

of the axisymmetric cell will be theoretically converted to the equivalent vertical permeability of the equivalent zone of the plane strain cell It is observed that, this method is developed under the condition of conventional surcharge loading, and the boundary condition of excess pore pressure of the vertical drain defined in this method is that the excess pore pressure at the top of the drain is equal to zero, only Therefore, as mentioned above, it would not be logical and convincing enough to say that this method could be still applied well to the case under vacuum-surcharge preloading condition

As also mentioned above, usually the subsoil has many layers, and in some cases, there is a sandy silt layer having high permeability exists just below an upper clay layer having low permeability In this case, the vacuum pressure propagates from the vertical drain to the surrounding soil in the sandy silt layer will be much faster than the propagation of the vacuum pressure in the surrounding soil of the upper clay layer This means that the vacuum pressure in the sandy silt will reach the maximum value much sooner than that in the upper clay layer

However, in Chai et al (2001) method, their plane strain unit cell only has a vertical permeability and has no drain-wall Therefore, even though the lower soil layer has a much higher permeability than that of the upper clay layer, and if the vacuum pressure is applied to the top of the cell, then the vacuum pressure would only able to propagate gradually from the upper layer to the lower layer in their plane strain cell This means that, in their plane strain cell, there is no way to let the vacuum pressure in the lower clay layer reach the maximum value sooner than that in the upper clay layer Therefore, we think that the method of Chai et al (2001) is totally unable to be applied to this case In fact, in the paper of Chai et al (2006), it is observed that the measured excess pore water pressures in almost all of soil layers were not well simulated by their method

Considering the situation mentioned above, this study proposes a more convenient conversion method from an axisymmetric unit cell to a plane strain unit cell under vacuum-surcharge preloading condition

In particular, the proposed conversion method has the following advantages: (1) The determination of

the length of drainage path of the drain (l) of the vertical drain in each soil layer of a multi-layer

subsoil (where the vertical drain is driven through) is not required, and (2) the plane strain unit cell of this method does not require the inclusion of the plane strain smear zone (see Figs 1-4, 2-1b, and 2-2b) (3) The method can be applied well to the subsoil in which the lower soil layer has a higher permeability than that of the upper soil layer

REFERENCES OF CHAPTER 2

[1] Chai JC, Miura N, Sakajo S, and Bergado DT (1995) Behavior of vertical drain improved

subsoil under embankment loading Soils and Foundations; 35(4): 49-61

[2] Chai JC, Shen SL, Miura N, and Bergado DT (2001) Simple method of modeling PVD

improved subsoil Journal of Geotechnical and Geoenvironmental Engineering, ASCE; 127(11):

965-972

[3] Chai JC, Carter JP, and Hayashi S (2006) Vacuum consolidation and its combination with embankment loading Canadian Geotechnical Journal; 43: 985-996

[4] Hansbo S (1981) Consolidation of fine-grained soils by prefabricated drains In: Proceedings

of 10 th International Conference on Soil Mechanics and Foundation Engineering, Stockholm,

Sweden; Vol 3 p 677-682

[5] Hird, C.C., Pyrah, I.C., and Russel, D (1992): Finite element modeling of vertical drains

beneath embankments on soft ground Géotechnique, 42 (3), 499-511

[6] Indraratna, B., and Redana, I.W (1997): Plane strain modeling of smear effects associated

with vertical drains Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 123

(5), 474-478

[7] Indraratna B and Redana IW (2000) Numerical modeling of vertical drains with smear and

well resistance installed in soft clay Canadian Geotechnical Journal; 37: 132-145

[8] Indraratna B, Bamunawita C, and Khabbaz H (2004) Numerical modeling of vacuum preloading and field applications Canadian Geotechnical Journal; 41: 1098-1110

[9] Indraratna B, Rujikiatkamjorn C, and Sathananthan I (2005) Analytical and numerical

solutions for a single vertical drain including the effects of vacuum preloading Canadian

Geotechnical Journal; 42: 994-1014

approach to analyzing the plane strain behavior of soft foundation with vertical drains Proc 31 st Symposium of JSSMFE, 111-116 (in Japanese)

[11] Shinsha, H., Hara, H., Abe, T., and Tanaka, A (1982): Consolidation settlement and lateral

displacement of soft ground improved by sand drains, Tsuchi-to-Kiso, Japanese Geotechnical

Society, 30 (5), 7-12 (in Japanese)

[12] Zeng, G X., Xie, K H., and Shi, Z Y (1987): Consolidation analysis of sand drained ground

by FEM Proc 8 th Asian Regional Conference on SMFE, Kyoto, 1, 139-142

Chapter 3: The Proposed Conversion Method

Chapter 3

The Proposed Conversion Method

“Imagination is more important than knowledge ”

Albert Einstein (1879 - 1955)

To find the conversion expression of permeability from the axisymmetric unit cell to the plane strain unit cell without smear zone, under vacuum-surcharge preloading condition, I conducted the mathematical formulation as follows

Firstly, based on the analytical model of Indraratna et al (2005) for the axisymmetric unit cell, I used analytical mathematics to find a function of development of excess pore water pressure with time at any given depth z within this cell

Subsequently, I built another analytical model for the plane strain unit cell having no smear zone; this model is simpler than that added the plane-strain smear zone of Indraratna et al (2005) After that, based on mathematics, I found the function of development of excess pore water pressure with time, also at any given depth z, for this cell

Finally, by equating two obtained functions of excess pore water pressure of these two cells, I found the conversion expression of permeability from the axisymmetric unit cell to the plane strain unit cell without smear zone under vacuum-surcharge preloading condition

3.1 Analytical models of the axisymmetric and plane strain unit cells under vacuum-surcharge preloading

In general, these two models are depicted in Fig 2.1 In these figures, σ1 denotes the surcharge; p0 is the vacuum pressure applied to the top of the drain as well as the drain wall; k1 is the maintaining factor of vacuum pressure (0 ≤ k1 ≤1); z is depth; l is the length of drainage path of the drain as well as

the length of the unit cell; rw is equivalent drain radius; rs is smear zone radius; R is equivalent radius

of the influence zone; bw is half-width of the drain-wall; B is half-width of the plane strain unit cell

Main assumptions of these models are

1 The soil within the cell is fully saturated and homogeneous

2 The permeability of the soil is assumed to be constant during consolidation

3 The vertical flow within the soil of the relatively long unit cell is insignificant, i.e it is

assumed that only radial flow occurs within the soil

4 Equal strain hypothesis of Kjellman (1948) is followed, i.e the horizontal sections of the

axisymmetric and plane strain unit cells remain horizontal during the consolidation process

5 The displacement at outer boundaries of the vertical drain and the cell are fixed in horizontal

direction, i.e only vertical displacement is allowed at these boundaries

6 Darcy’s law is considered to be valid, and the solutions are based on the Darcy’s law,

7 The change in volume corresponds to the change in void ratio, and coefficient of volume

compressibility, m v, is constant during consolidation process

8 Indraratna et al (2005) assumption about the loss of vacuum pressure along the vertical drain

is employed, i.e the vacuum loss is a linear increase with depth (see Fig 2.1), in which – p0 is

the vacuum pressure at the top of the drain, and – k1p0 is the corresponding value at the bottom

of the drain

3.2 Analytical solution for the axisymmetric unit cell (modified by the present author after

Indraratna et al (2005))

Based on the analytical model of Indraratna et al (2005), the author constructed the following

analytical function of the pore water pressure within the axisymmetric unit cell, at any given depth z

=

l

z k p

T l

z k p

u

za

8exp)

1(

sa ha a

a za

q

k z l z s

k

k s

n

) 2 ( 4

3 ln

r

r

where k ha and ksa are horizontal permeability coefficients of the axisymmetric unit cell in undisturbed

zone and in smear zone, respectively; qwa is discharge capacity of the drain; T ha is dimensionless time

factor for horizontal drainage in axisymmetric unit cell; z is depth; l is the length of drainage path of

the drain of the cell; R is radius of the axisymmetric cell; rs and rw are the radii of the smear zone and

the drain, respectively

and 2 2

4

t m

k R

t C T

w v

ha ha

where Cha is coefficient of consolidation for horizontal drainage in axisymmetric case; mv is coefficient

of volume compressibility for one-dimensional compression; t is time; γw is unit weight of water

3.3 The proposed analytical solution for the plane strain unit cell without smear zone

Chapter 3: The Proposed Conversion Method

On the basis of the proposed analytical model of the plane-strain unit cell, the following solution for

the average excess pore water pressure, at a given depth z, was developed

=

l

z k p

T l

z k p

u

zp

8exp)

1(

k wp

hp hp

hp

m

k B

t B

t C T

in which k hp is horizontal permeability coefficient in the equivalent zone of the plane strain cell; qwp is

discharge capacity of the drain-wall; T hp is dimensionless time factor for horizontal drainage in the

plane strain cell; Chp is coefficient of consolidation for horizontal drainage in the plane strain cell; B is

the half-width of the plane strain cell

3 4 The proposed conversion expression of permeability for the plane strain unit cell

By adapting the matching procedure of Hird et al (1992) under conventional surcharge preloading

condition for the proposed plane-strain cell under vacuum-surcharge preloading condition, the

following steps are conducted

Equating Eq (1) and Eq (4), the following equation can be obtained

zp hp

za

ha T T

za

ha B

k R

4

3ln3

2

z lz q

B k k q

R k k R k s k

k s

n B

k

wp

ha hp wa

ha hp hp

a sa ha a

wa

ha hp

q

B k k q

R k

then, the effect of well resistance is matched independently, as follows

And, by using Eq (10) as a condition for Eq (9), the conversion expression of permeability for the

equivalent plane strain unit cell can be obtained

a sa ha

a a

ha hp

s k

k s

n

k R

B k

ln4

3ln3

2

2 2

+

−

In conclusion, the proposed method to convert from the axisymmetric unit cell to the plane strain unit

cell under vacuum-surcharge preloading condition includes the combined use of both Eqs (11) and

(12)

3 5 The way to use the proposed conversion method

In practice, parameters R, na, sa, and kha are available Besides, we can obtain the ratio kha/ksa either

from laboratory test or from in-situ test But, usually, we have to assume this ratio, because, this ratio

is usually not available in construction practice of embankment due to time and cost Afterwards, we

must choose a value of B so that it is convenient for creating a 2D FE mesh at full-scale Subsequently,

by inputting all those values into Eq (12), we can obtain the horizontal permeability coefficient for the

equivalent zone between two vertical drains in the full-scale plane strain simulation

In the next step, usually, the information on discharge capacity, qwa, of the vertical drain is available;

therefore, by inputting parameters qwa, the previously chosen value of B, and R into Eq (11), we can

get the value of qwp for the drain-wall of the plane strain cell

Finally, inputting this qwp into the following equation, one can obtain kwp for the drain-walls in the

full-scale plane strain simulation

w

wp wp

b

q k

2

In order to make clear the difference between the proposed conversion expression of permeability and

that of Indraratna et al (2005), I made the Table 3.1 as follows

Table 3.1 Comparison of the conversion expressions of permeability between Indraratna et al

(2005) method and the proposed method under vacuum-surcharge preloading condition

The conversion expression of permeability

proposed by Indraratna et al [12]

The conversion expression of permeability of the proposed method

=

2

3

24

3ln

q

k s

k

k s

n

k k

k

k

wa

ha a

a s ha

a

a

sp hp

ha

hp

π

θβα

sa ha

a a

ha hp

s k

k s

n

k R

B k

ln 4

3 ln 3

2

2 2

Chapter 3: The Proposed Conversion Method

r

r

) 1 ( 3

p p

n n

s n

)1(

1

p p

p

s s s

n n n

n

s

β

211

3

4

l n Bq

k

p wp

b

b s b

B

khp and ksp are horizontal permeability coefficients in the undisturbed and smear zones of Indraratna et

al (2005) plane-strain unit cell, respectively; in the proposed method, k hp is horizontal permeability coefficient in the equivalent zone of the proposed plane strain unit cell; bs is half-width of the plane-strain smear zone in the plane strain unit cell of Indraratna et al (2005) method (see Fig 2.2a) The meaning of other symbols was previously mentioned and is shown in Fig 2.2a and Fig A1 in Appendix A

In Table 3.1, one can see that the length of drainage path of the drain (l) and parameters associated

with plane strain smear zone such as ksp, α, and β are not required in the proposed conversion expression of permeability For this reason, it can be said that the proposed conversion of permeability

is much simpler than that of Indraratna et al (2005)

REFERENCES OF CHAPTER 3

[1] Hird, C.C., Pyrah, I.C., and Russel, D (1992): Finite element modeling of vertical drains

beneath embankments on soft ground Géotechnique, 42 (3), 499-511

[2] Indraratna B, Rujikiatkamjorn C, and Sathananthan I (2005): Analytical and numerical

solutions for a single vertical drain including the effects of vacuum preloading Canadian

Geotechnical Journal; 42: 994-1014

[3] Kjellman W (1948): Consolidation of fine-grained soils by drain wells Transaction ASCE;

113: 748-751 Contribution to the discussion

Chapter 4

Verification of the proposed method via

Finite Element Method

“The only real valuable thing is intuition”

Albert Einstein (1879 - 1955)

4.1 Outline

To verify the proposed method, the author has conducted FE analyses for 3 cases under surcharge preloading condition:

vacuum-[1] Case 1 is a homogeneous clay layer that has the thickness being 10 m

[2] Case 2 is the case having 2 clay layers, in which the upper clay layer and the lower clay layer have the thickness being 4.95 m and 5.05 m, respectively

[3] Case 3 is the case of three layers being clay-sand-clay sandwich: This case is a subsoil, in which a sand layer of 1 m thickness is sandwiched between two thick clay layers About these two thick clay layers, the upper clay layer has 10 m thickness, and the lower clay layer has 5 m thickness Further, the permeability of the upper clay layer was chosen to be equal to that of the clay layer in the case of one clay layer mentioned above And, the permeability of the lower clay layer was assumed to be equal to 1/5 value of the permeability of the upper clay layer

In general, the illustration of these three cases is shown schematically in Fig 4.1 In Fig 4.1a, k1ha and

k1sa are the horizontal permeability in the undisturbed and smear zones of the axisymmetric cell, respectively k1va and k1vsa are the vertical permeability in the undisturbed and smear zones, respectively

In Fig 4.1b, the horizontal and vertical permeability of the upper clay layer is the same as those of the homogeneous clay layer in Fig 4.1a, but the horizontal and vertical permeability of the lower clay layer is 5 times higher than those of the upper clay layer In this figure, k2ha and k2sa are the horizontal permeability in the undisturbed and smear zones; besides, k2va and k2vsa are the vertical permeability in the undisturbed and smear zones in this layer

Chapter 4: Verification of the proposed method via Finite Element Method

In Fig 4.1c, all permeability coefficients of the upper clay layer are the same as those of the one

homogeneous clay layer in Fig 4.1a And, k2ha and k2va are the horizontal and vertical permeability in

the sand layer In the lower clay layer, k3ha and k3sa are the horizontal permeability in the undisturbed

and smear zones; besides, k3va and k3vsa are the vertical permeability in the undisturbed and smear

zones in this layer

4.2 Detailed description of the case of one clay layer:

Regarding the equivalent radius of the band-shaped drain, rw, FE analyses performed by Rixner et al

(1986) and supported by Hansbo (1987) indicated that the equivalent diameter of the band-shaped

drain for use in practice can be determined by

2

b a

(14)

where a and b is the length and the width of the rectangular cross section of the band-shaped drain,

respectively

Commonly, the band-shaped drain or PVD has dimensions being 10 cm x 0.4 cm Using Eq (14), we

can obtain dw ≅ 5 cm and therefore rw ≅ 2.5 cm or 0.025 m

Concerning the smear zone radius, rs, according to Jamiolkowski and Lancellotta (1981), the diameter

of the smear zone, ds, can be in the range of 2.5dm to 3dm, where dm is the equivalent diameter of the

mandrel used for driving PVD into the soft ground Referring to the practice of vacuum-embankments

in Hokkaido, Japan, (Tran et al (2004a; 2004b)) the mandrel has an equivalent diameter dm = 12 cm,

and, in this study, the author chose ds = 3dm This leads to the smear zone diameter used in this study is

36 cm or 0.36 m; therefore, rs = 0.18 m

With regard to the equivalent radius of the influence zone, R, it is common that the PVDs are usually

driven according to a square net; therefore, the equivalent radius of the influence zone performing as

an axisymmetric unit cell needs to be calculated Logically, it can be defined to be the radius of a

circle having the same area as that of a square of the square net, in which the PVDs are driven

Therefore, the following expression is obtained

where S is the spacing between two vertical drains driven according to a square net; R is the

equivalent radius of the influence zone

The author observed that, at a vacuum-embankment in Kushiro city, Hokkaido, Japan, the PVDs were

driven according to a square net with S = 0.8 m (Tran et al 2004a) Hence, by using Eq (15), the

equivalent radius R was determined to be 0.45 m for FE analyses in this study

Regarding B for FE analyses in this study, the author made two plane strain unit cells, one has B =

0.45 m and the other one has B = 0.75 m, in which their permeability was converted by the proposed

conversion method from the axisymmetric unit cell having R = 0.45m For convenience, bw of these

two plane strain cells was chosen equal to rw (i.e 0.025 m) of the axisymmetric unit cell

Concerning well resistance of vertical drains, the author has tested the proposed conversion method

with various values of well resistance that are selected on the basis of Mesri and Lo (1991) discharge

capacity factor as follows

k l

k

q

ha wa ha

wa

where k ha is horizontal permeability coefficient in undisturbed zone of the axisymmetric unit cell; qwa

is discharge capacity of the drain; l is the length of drainage path of the drain of the cell; rw is the drain

radius; kwa is equivalent vertical permeability coefficient of the drain

Mesri and Lo (1991) reported that well resistance is considered to be insignificant if Fd is larger than 5

Therefore, the author has tested the proposed conversion method for two values of Fd, which are 0.1

(i.e very high well resistance) and 20 (i.e no well resistance) respectively

4.2.1 Boundary conditions and computed cases

Because one of the basic assumptions of the proposed analytical solutions is based on “equal strain

hypothesis”, therefore, the author has conducted FE analyses as follows

(1) All the drain, the smear and undisturbed zones of the axisymmetric cells, and both the drain-wall

and the equivalent zone of the plane strain cells converted by the proposed method were simulated

by linear elastic models having the same elastic modulus (E = 1000 KN/m2) and zero Poisson’s

ratio

(2) In the case of the axisymmetric cell, to avoid horizontal displacement from the smear zone to the

drain or vice versa, the nodes on the adjoining boundary between the drain and the smear zone are

allowed to move in the vertical direction only Similarly, the nodes on the adjoining boundary

between the drain wall and the equivalent zone of the plane strain cell are also allowed to move in

the vertical direction only

(3) An undrained rigid plate was put on the surface of each of these axisymmetric and plane strain

cells to ensure the uniform settlement at the surface of the cells (see Fig 4.3) In addition, in

accordance with the boundary condition of the analytical solutions, the vacuum pressure is applied

to the top of the drain, not to the surface of the cell; besides, all the vertical permeability of both

the axisymmetric and plane strain cells are set equal to zero This case, the author named it

VTD-ES (Vacuum at the top of the drain with equal strain)

In addition, the author carried out other FE analyses to check the applicability of the proposed method

in the case of free-strain is allowed at the surface of the cell; in this case, the undrained rigid plate in

Fig 4.3 is removed from the surface of the cells, and the vacuum pressure is applied to both the top of

the drain and the surface of the cell The author named this case VS-FS (vacuum pressure applied to

the surface with free strain)

In 1986, Rixner et al (1986) reported that, of clay soil with no or slightly developed macrofabric,

essentially homogeneous deposits, the ratio of the horizontal permeability to the vertical permeability,

kh/kv, can be in the range of 1 to 1.5 Based on this report, for the case VS-FS, the vertical permeability

of the axisymmetric and plane strain cells is chosen as follows

(1) The author assumes the ratio, kha/kva, equal to 1 for the undisturbed zone of the axisymmetric

unit cell

(2) For the smear zone of the axisymmetric unit cell, the author assumes its vertical permeability,

kvsa, is equal to the vertical permeability of the undisturbed zone, kva.

Chapter 4: Verification of the proposed method via Finite Element Method

(3) For the plane strain cells converted by the proposed method, the vertical permeability of the equivalent zone (kvp) is assumed to be equal to the vertical permeability of the undisturbed zone of the axisymmetric unit cell (kva)

In summary, in the proposed method, the author assumed that the vertical permeability coefficient kva

is equal to kha, and kvp = kva = kvsa

According to Kobayashi et al (1990), the horizontal permeability of the smear zone of the clay can be decreased to 1/5 of the horizontal permeability of the undisturbed zone Therefore, for the axisymmetric cell, the author assumed the ratio of the horizontal permeability of the smear zone to that

of the undisturbed zone, ksa/kha, to be equal to 1/5

In summary, all computed cases are illustrated in Fig 4.2, and all input parameters used for the case of one clay layer are tabulated in Table 4.1 And, the boundary conditions and meshes of the cells are shown in Fig 4.3

Note that (1) The meaning of k1 in Fig 4.2 is the maintaining factor of vacuum pressure (0 ≤ k1 ≤1); (2) In engineering practice, the value of qwa is available, and then qwp can be calculated by using Eq (11) In this study, for the purpose of investigating the effect of well resistance on each of cases computed by Indraratna et al (2005) method and by the proposed method, the discharge capacity factor Fd was firstly assumed as shown in Table 4.1, and then qwa was calculated based on Eq (16) Regarding the FEM program used, the Sage Crisp program developed by the CRISP Consortium Ltd and SAGE Engineering Ltd (1999) on the basis of consolidation theory of Biot (1941) was employed; the author used linear strain quadrilateral elements that incorporate quadratic displacement nodes together with linearly interpolated pore pressure nodes in this program (see Fig 4.4)

4.3 Detailed description of the case of two clay layers

The author observed that the conversion method proposed by Indraratna et al (2005) showed very good matching results for the case of one clay layer, in their paper Therefore, a check on the applicability of both the proposed method and Indraratna et al (2005) method for the case of 2 clay layers would be expected; for this reason, FE analyses for the case of 2 clay layers were conducted in this study

In this two-clay-layer case, the geometric parameters of the axisymmetric unit cell, R, rs, rw, were chosen the same as those of the axisymmetric unit cell in the case of one clay layer listed in Table 4.1 And, the thickness of the upper clay layer and of the lower clay layer is 4.95 m and 5.05 m, respectively In general, the boundary conditions and the mesh of the cells are shown in Fig 4.5 With regard to the soil properties and material parameters of the axisymmetric unit cell in this two-clay-layer case:

For the vertical drain, all material parameters were assumed to be the same as those in the case of one homogeneous clay layer

For the upper and lower clay layers, all soil properties were assumed to be the same as those of the case one homogeneous clay layer In this assumption, the permeability coefficients of the upper clay layer are equal to those in the case of one homogeneous clay layer, but the horizontal and vertical permeability coefficients of the lower clay layer were assumed to be 5 times higher than that of the corresponding permeability coefficients of the upper clay layer

In summary, the permeability of each soil layer assumed in this two-clay-layer case is illustrated in Fig 4.1

Note that, for the plane strain cell converted by Indraratna et al (2005) method, the author assumed the ratio of the smear zone permeability to the undisturbed zone permeability, ksp/khp, equal to 1/5 This ratio is the same as that of the axisymmetric cell On the other hand, the plane strain unit cell of the proposed method has no plane strain smear zone; therefore, such a kind of ratio is not required in the proposed plane strain unit cell

Regarding the assumption of vertical permeability, in the case of VTD-ES, of both the two clay layers, the vertical permeability in the smear and undisturbed zones of both the axisymmetric cell and Indraratna et al (2005) plane strain cell is assumed to be zero Similarly, the vertical permeability in the equivalent zone of the proposed plane strain cell is also assumed to be zero in this VTD-ES case

In the case of VS-FS, the vertical permeability of the plane strain cells was assumed as follows: (1) For the plane strain cells converted by Indraratna et al (2005) method, for the upper and lower clay layers, both the vertical permeability of the smear zone, kvsp, and of the undisturbed zone, kvp, are equal to the vertical permeability of the undisturbed zone, kva, of the corresponding layer in the axisymmetric cell This means that kvsp = kvp = kva for each of the upper and lower clay layers

(2) For the plane strain cells converted by the proposed method, in each of the two clay layers, the vertical permeability of the equivalent zone, kvp, is the same as that of the undisturbed zone in the axisymmetric cell It should be noted here that kvsp is not needed in the proposed plane strain cell

In summary, all computed cases for this two-clay-layer case are illustrated in Fig 4.2., and all input

parameters used for these plane strain cells are shown in Tables 4.2 and 4.3

In this two-clay-layer case, when Indraratna et al (2005) conversion expression was used for the lower clay layer, then a question arose as to whether we should choose the length of drainage path of the

drain (l) of the axisymmetric unit cell in the lower clay layer equal to the total thickness of both two clay layers (l = 10 m) or equal to the thickness of the lower clay layer only (l = 5.05 m) All

permeability for the lower clay layer of plane strain cells converted by Indraratna et al (2005)

conversion expression, in accordance with both l = 10 m and l = 5.05 m, are listed in Table 4.3

In Table 4.3, in the case that the plane strain cell has B = 0.75 m together with qwp = 0.0043 m3/day (i.e corresponding to Fd = 0.1 of the axisymmetric cell), the converted permeability of the lower clay

layer based on Indraratna et al (2005) method becomes negative, even though using either l = 10 m or

l = 5.05 m Therefore, the author could not model this case by Indraratna et al (2005) method For this

reason, the comparison between the proposed method and Indraratna et al (2005) method was not conducted for this case

Also in Table 4.3, when the plane strain cell has B = 0.45 m together with qwp = 0.0043 m3/day, the converted permeability of the lower clay layer based on Indraratna et al (2005) method also becomes

negative if using l = 10 m (i.e equal to the total thickness of both two clay layers) Hence, for the lower clay layer, the author did not choose l = 10 m, but choosing l = 5.05 m (i.e equal to the

thickness of this layer) to input into the conversion expression of Indraratna et al (2005)

4.4 Detailed description of the case of clay-sand-clay sandwich

It is observed that the subsoil usually has many soil layers; especially in Japan, the subsoil that has sand layers sandwiched between clay layers is usually found in many places Based on this fact, the author conducted a series of FE analyses to check Indraratna et al (2005) method as well as the proposed method in the case of the sandwich subsoil in this study

In this subsoil, the author assumed that the upper clay layer, the sand layer, and the lower clay layer has a thickness of 10 m, 1 m, and 5 m, respectively as shown in Fig.4.1c And the geometric parameters of the axisymmetric unit cell in this subsoil, which are R, rs, and rw were chosen the same

Chapter 4: Verification of the proposed method via Finite Element Method

as those in the case of one clay layer listed in Table 4.1 In general, the boundary conditions and the mesh of the cells are shown in Fig 4.6

Regarding determination of the length of drainage path of the drain (l) in each soil layer, it was

assumed as follows

In Indraratna et al (2005) method, the length of drainage path of the drain (l) in each soil layer has to

be determined, but, in this sandwich case, its determination is difficult as mentioned previously Therefore, in this sandwich case, the most acceptable assumption is to assume that the length of

drainage path of the drain (l) in each soil layer is equal to the thickness of the corresponding soil layer

as shown in Table 4.5 On the contrary, in the proposed conversion method, the determination of the

length of drainage path of the drain (l) in each soil layer is not needed

The following are the soil properties and material parameters used for the axisymmetric unit cell in this sandwich case:

For the vertical drain, all material parameters were assumed to be the same as those in the case of one homogeneous clay layer

For clay layers, all soil properties were also assumed to be the same as those on the one homogeneous clay layer analyzed in the previous case, except that the horizontal and vertical permeability coefficients of the lower clay layer were assumed to be 1/5 of the corresponding permeability coefficients of the upper clay layer

Regarding the sand layer of 1 m thickness, which is sandwiched between two clay layers, the permeability coefficient was assumed to be 4.00E-02 m/day; this value is obtained from the laboratory test on the subsoil located in the suburb of Kushiro city, Hokkaido, Japan However, when the author inputs this value to the conversion expression of permeability of Indraratna et al (2005), it yielded a negative permeability coefficient for the equivalent plane strain unit cell

In contrast to this, such a problem did not occur in the case of the proposed conversion expression For these reasons, the author was unable to use this value of permeability in FE analyses for the purpose of comparing results of Indraratna et al (2005) method and the proposed method

In other words, this means that the conversion expression of permeability proposed by Indraratna et al (2005) is inapplicable to the case of a sand layer having high permeability and a thin thickness Therefore, the author had to assume another permeability coefficient being 1.00E-02 (m/day), which is smaller than the value 4.00E-02 m/day, for this sand layer In addition, because this is a sand layer, the author assumed that there is no smear zone in this layer and that its horizontal permeability is equal to its vertical permeability

In general, the permeability of each soil layer assumed in the clay-sand-clay sandwich is illustrated in Fig.4.1c

Lastly, the elastic modulus and the Poisson's ratio of each soil layers were assumed to be 1000 KN/m2

and zero, respectively

As previously mentioned, in the conversion method of Indraratna et al (2005), the plane-strain smear zone exists; therefore, the author had to choose a ratio of the smear zone permeability to the undisturbed zone permeability, ksp/khp. And, in this study, this ratio is chosen to be 1/5, i.e the same as that of the axisymmetric unit cell On the other hand, the plane strain cell of the proposed method has

no plane strain smear zone; therefore, the author needs not to choose such kind of ratio for the proposed plane strain cell

And, in the case of VTD-ES, of all the three layers, the vertical permeability coefficients in the smear and undisturbed zones of both the axisymmetric cell and Indraratna et al (2005) plane strain cell are

assumed to be zero Similarly, the vertical permeability coefficient in the equivalent zone of the proposed plane strain cell is also assumed to be zero in this VTD-ES case

In the case of VS-FS, the vertical permeability of the plane strain cells was assumed as follows: (1) For the plane strain cells based on Indraratna et al (2005) method, for the upper and lower clay layers, both the vertical permeability of the smear zone, kvsp, and of the undisturbed zone, kvp, were chosen equal to the vertical permeability of the undisturbed zone, kva, of the corresponding layer in the axisymmetric cell This means kvsp = kvp = kva for the upper and lower clay layers For the sand layer, its vertical permeability was set to be equal to the vertical permeability of the sand layer of the axisymmetric cell, i.e kvp = kva

(2) For the plane strain cells based on the proposed conversion method, the same way as that used in Indraratna et al (2005) method was employed except kvsp, since it is not needed in the proposed plane strain cell

In summary, all computed cases for this sandwich case are illustrated in Fig.4.2, and all input parameters used for these plane strain cells are shown in Tables 4.4 and 4.5

4.5 Results and discussion

4.5.1 In the case of one homogeneous clay layer

FE results of the degree of consolidation of the axisymmetric cell, and of the proposed plane strain cell are shown in Figs 4.7 and 4.8, in which all curves of degree of consolidation were calculated based on the surface settlement of the soil layer

The proposed conversion method was validated in the case of VTD-ES with k1=1 As shown in Fig 4.7a, it can be seen that a good agreement in the degree of consolidation, between the proposed plane strain cell and the axisymmetric cell, was obtained not only under no-well-resistance condition, but also under high well resistance condition After that, the proposed method was also examined in the case of VS-FS with k1=1; as shown in Fig 4.7b, the same good agreement as that in Fig 4.7a was obtained under both no-well resistance and high well resistance conditions

Further, the author tested the proposed method in the case of VTD-ES with k1=0.5 The results in Fig 4.8a indicated that, under both no well resistance and high well resistance conditions, the proposed method produced good matching results Finally, the author tested the proposed method in the case of VS-FS with k1=0.5 As can be seen in Fig 4.8b, the same good matching results as those in the case of VTD-ES (k1=0.5) were also obtained

4.5.2 In the case of two clay layers

The FE results of the degree of consolidation of the axisymmetric cell, of the Indraratna et al (2005) plane strain cell, and of the proposed plane strain cell are presented in Fig 4.9, in which all curves of degree of consolidation were calculated based on the surface settlement of soil layers

Besides, the difference in the degree of consolidation (Ua - Up) between the axisymmetric unit cell (Ua) and the plane train unit cell (Up) of Indraratna et al (2005) method, and the plane strain cell (Up) of the proposed method is illustrated in Fig 4.10

It is observed that the computed results of the case k1 = 1 and of the k1 = 0.5 are very similar each other; therefore only the results of k1 = 0.5 are showed here, and the results of k1 = 1 are presented in Appendix C

Chapter 4: Verification of the proposed method via Finite Element Method

As shown in Fig 4.9a and 4.9b, the results of degree of consolidation revealed that both methods are very good under no well resistance condition (Fd = 20)

Fig 4.10a shows that, under the condition of Fd =20, the maximum difference in the degree of consolidation (Ua - Up), of the whole two clay layers is 2% for Indraratna et al (2005) method, and 4% for the proposed method This means that the proposed method produced matching results which are almost as good as those of Indraratna et al (2005) method (under condition of Fd=20) Besides, also under condition of Fd = 20, Fig 4.10b showed that the maximum difference in the degree of consolidation of the lower clay layer is 1% for Indraratna et al (2005) method, and 2.5 % for the proposed method

Under high well resistance condition, Figs 4.9a and 4.10a revealed that, of the whole two clay layers, the maximum difference (Ua - Up) of Indraratna et al (2005) method is 9%, which is considered to be fairly high, whereas, the maximum difference of the proposed method is 2% Figs 4.9b and 4.10b show that, of the lower clay layer, the maximum difference (Ua - Up) under high well resistance condition of Indraratna et al (2005) method even reach 14%, whereas the difference of the proposed method is less than 1%

Further, the author examined both methods in the case of VS-FS with k1 = 0.5 as shown in Figs 4.9c, 4.9d, 4.10c, and 4.10d As can be seen in these figures, almost the same results as those in the case of VTD-ES (k1 = 0.5) are obtained However, in this VS-FS case, the maximum difference in the degree

of consolidation in the lower clay layer of Indraratna et al (2005) method is 12%, i.e little smaller than the value 14% in the case of VTD-ES, but this value 12% is still be considered to be significant

4.5.3 In the case of clay-sand-clay sandwich

In the case of clay-sand-clay sandwich, the FE results of the degree of consolidation of the axisymmetric cell, of the Indraratna et al (2005) plane strain cell, and of the proposed plane strain cell are shown in Fig 4.11

And, the difference in the degree of consolidation between the axisymmetric unit cell (Ua) and the plane train unit cell (Up) of Indraratna et al (2005) method and of the proposed method is illustrated in Fig 4.12

The author observed that the computed results of the case k1 = 1 and of the k1 = 0.5 are almost the same; therefore only the results of k1 = 0.5 are presented here, and the results of k1 = 1 are given in Appendix C

It is shown in Figs 4.11a, 4.11c, 4.12a, and 4.12c that, in the case of no well resistance (Fd = 20), under both free strain and equal strain condition, of the whole three layers, the difference in the degree

of consolidation between the axisymmetric cell and the plane strain cell converted by Indraratna et al (2005) method (Ua - Up) is less than 1% And, of the proposed method, (Ua - Up) is less than 3 %

As far as the case of no well resistance (Fd = 20) is concerned, the author can see in Figs 4.11b, 4.11d, 4.12b, and 4.12d that, of the lower clay layer, the difference (Ua - Up) of Indraratna et al (2005) method is less than 3 % Similarly, of the proposed method, the difference (Ua - Up) is less than 5 %, i.e still small and acceptable Hence, the author can say that, the proposed method can yield matching results, which is as good as that of Indraratna et al (2005) method in the case of no well resistance

In Figs 4.11a, 4.11c, 4.12a, and 4.12c, under both free strain and equal strain, it can be seen that, in the case of Fd = 0.1 (high well resistance), of the whole three layers, the difference (Ua - Up) of Indraratna et al (2005) method can reach 8%, whereas, the difference of the proposed method is less than 1%

Besides, in Figs 4.11b, 4.11d, 4.12b, and 4.12d, also in the case of Fd = 0.1, the results revealed that, of the lower clay layer, the difference (Ua - Up) of Indraratna et al (2005) method can even reach 12%

On the contrary, the difference of the proposed method is only less than 3%

4.6 Conclusions of this chapter

In this study, a new conversion method has been proposed to convert the axisymmetric unit cell to an equivalent plane strain unit cell under vacuum-surcharge preloading condition

In the proposed method, the half-width B of the plane strain cell can be chosen in a way so that it is convenient in making a FE mesh for a full-scale 2D simulation In addition, the widths and the permeability of the smear and undisturbed zones in the axisymmetric cell are converted theoretically into an equivalent plane-strain permeability of the equivalent zone in the proposed plane strain cell Therefore, in contrast to the method by Indraratna et al (2005), the proposed plane strain unit cell does not need the inclusion of the plane-strain smear zone For this reason, by using the proposed method, a finite element plane-strain simulation of soft ground improved by vertical drains under embankment combined with vacuum preloading can be conveniently made, since the total number of elements and material parameters required in the analysis are extremely reduced

Besides, the proposed conversion method is simpler and more convenient than the conversion method

of Indraratna et al (2005), because, in the proposed method, the determination of the length of

drainage path of the drain (l) in each soil layer of a multi-layer subsoil is not required

The proposed method was validated via analyzing consolidation of the axisymmetric and plane-strain unit cells in three cases, the first is a homogeneous clay layer, the second is a two-clay-layer case, and the third is a three-layer subsoil being clay-sand-clay sandwich The vertical drains analyzed in these cases are under two conditions, no well resistance (Fd = 20), and high well resistance (Fd = 0.1) In contrast to the results of Indraratna et al (2005) method in the two-clay-layer case, in which the difference in the degree of consolidation between their plane strain cell and the axisymmetric cell can reach more than 10% (and in some cases, the converted permeability by their method even becomes negative), the analyzed results by the proposed method showed excellent agreements in all three cases

In addition, the results also indicated that the proposed method can be used well, not only under equal strain condition, but also under free strain condition

Regarding Indraratna et al (2005) method in the case of clay-sand-clay sandwich with the vertical drain having high well resistance, this study pointed out that if the sand layer has a thin thickness and a high permeability, their conversion method was found to be incorrect, and therefore it cannot be used

Therefore, it can be said that the proposed method in this study is simple, but showing excellent results; besides, this study indicated that the proposed method is more convenient and better than Indraratna et

al (2005) method

Chapter 4: Verification of the proposed method via Finite Element Method

REFERENCES OF CHAPTER 4

[1] Biot MA (1941): General theory of three dimensional consolidation Journal of Applied Physics;

12: 155-164

[2] Hansbo S (1987) Design aspects of vertical drains and lime column installations In:

Proceedings of the 9th Southeast Asian Geotechnical Conference, Bangkok, Thailand; Vol 2 p 8-12

[3] Indraratna B, Rujikiatkamjorn C, and Sathananthan I (2005) Analytical and numerical

solutions for a single vertical drain including the effects of vacuum preloading Canadian

Geotechnical Journal; 42: 994-1014

[4] Jamiolkowski M and Lancellotta R (1981): Consolidation by vertical drains-uncertainties

involved in prediction of settlement rates, Panel Discussion, In: Proceedings of the 10thInternational Conference on Soil Mechanics and Foundation Engineering, Stockholm, Sweden

[5] Kobayashi M, Minami J, and Tsuchida T (1990): Determination method of horizontal

consolidation coefficient of clay, Technical Report of Research Center of Harbor Engineering of Transport Ministry; 29(2): 63-83

[6] Mesri G and Lo DOK (1991): Field performance of prefabricated vertical drains In:

Proceedings of the International Conference on Geotechnical Engineering for Coastal Development - Theory and Practice on Soft Ground, Yokohama, Coastal Development Institute

of Technology, Japan; Vol 1 p 231-236

[7] Rixner JJ, Kraemer SR, and Smith AD (1986): Prefabricated vertical drains, Engineering

Guidelines (Vol 1), Federal Highway Administration, Report No FHWA-RD-86/168,

Washington D C

[8] SAGE Engineering, Ltd SAGE CRISP user’s manual (1999) SAGE Engineering, Ltd., Bath,

United Kingdom

[9] Tran TA, Mitachi T, and Yamazoe N (2004a): 2D finite element analysis of soft ground

improvement by vacuum-embankment preloading In: Proceedings of the 44th Annual Conference on Geotechnical Engineering of Hokkaido Branch of Japanese Geotechnical Society, Sapporo City, Japan; p 127-132

effectiveness of improvement of peaty soft ground by vacuum preconsolidation In: Proceedings

of the 39th Annual National Conference on Geotechnical Engineering of Japanese Geotechnical

Society, Niigata City, Japan; p 963-964

The case of one clay layer

Lower clay layer

The case of two clay layers

k

= 5 (k ) 1ha

Upper clay layer

Embankment Embankment

k = k = k = 4.30E-4 (m/day)

1ha 1vsa 1va

1va 1vsa 1ha= k = k = 4.30E-4 (m/day)k

3sa= (1/5)k

3ha

3ha 3va 3vsa

k = k = k

1ha

= (1/5)k1sak

LC

The case of clay-sand-clay sandwich Lower clay layer

Sand layer Thickness H = 1m

Chapter 4: Verification of the proposed method via Finite Element Method

Fig 4.2 Illustration of computed cases The case of two clay layers

The case of one homogeneous clay layer

o

o

pp

o o

VS-FS VTD-ES

VS-FS VTD-ES

VS-FS VTD-ES

o

o

pp

o o

The case of clay-sand-clay sandwich

(a) Axisymmetric unit cell, R = 0.45 m

w Impermeable boundary

Vertical roller boundary Vertical roller boundary

Centreline of the drain

Impermeable boundary

Fixed, impermeable boundary

Horizontal roller boundary Half drain, r = 2.5 cm Smear zone, 15.5 cm Fixed, impermeable boundary Undisturbed zone, 27 cm

Impermeable boundary Periphery of the cell

50 kPa

L C

Undrained rigid plate Top of the cell Periphery of the drain

Vertical roller boundary

(b) Plane strain unit cell, B = 0.45 m

Equivalent zone, 42.5 cm

Half drain wall, b = 2.5 cm Horizontal roller boundary Fixed, impermeable boundary

Half drain wall, b = 2.5 cm Horizontal roller boundary Impermeable boundaryw

Fixed, impermeable boundary Equivalent zone, 72.5 cm

(c) Plane strain unit cell, B = 0.75 m

Top of the cell Undrained rigid plate

CL

50 kPa

Periphery of the cell Impermeable boundary Vertical roller boundary Vertical roller boundary

Periphery of the drain

Impermeable boundary Centreline of the drain Vertical roller boundary

Vertical roller boundary Centreline of the drain Impermeable boundary

Vertical roller boundary Periphery of the drain

Top of the drain Negative 50 kPa excess pwp Top of the drain

Negative 50 kPa excess pwp

Fig 4.3 Boundary conditions and meshes of the axisymmetric and plane strain cells used

in FE analyses for the case of one homogeneous clay layer

Fig 4.4 The element type used for FE analyses

Integration pointPore pressure unknownDisplacement unknown

Centreline of the drain

Vertical roller boundary

Impermeable boundary

Periphery of the drain

Vertical roller boundary

Top of the drain

Negative 50 kPa excess pwp

w

Impermeable boundary Half drain wall, b = 2.5 cm Horizontal roller boundary

Fixed, impermeable boundary Equivalent zone, 42.5 cm

(b) Plane strain unit cell, B = 0.45 m

w

Half drain, r = 2.5 cm Impermeable boundary Horizontal roller boundary

Fixed, impermeable boundary

Fixed, impermeable boundary Smear zone, 15.5 cm Undisturbed zone, 27 cm

(a) Axisymmetric unit cell, R = 0.45 m

Negative 50 kPa excess pwp

Top of the drain

50 kPa L

C

Vertical roller boundary Periphery of the drain

Centreline of the drain

Vertical roller boundary Impermeable boundary

Undrained rigid plate Top of the cell

Periphery of the cell Impermeable boundary Vertical roller boundary

Fig 4.5 Boundary conditions and meshes of the axisymmetric and plane strain cells used

in FE analyses for the case of two clay layers

Chapter 4: Verification of the proposed method via Finite Element Method

Vertical roller boundary Vertical roller boundary

Centreline of the drain

Impermeable boundary

Impermeable boundary Periphery of the cell

50 kPa

L C

Undrained rigid plate Top of the cell

Periphery of the drain

Vertical roller boundary

Top of the drain

Negative 50 kPa excess pwp

Fixed, impermeable boundary

Horizontal roller boundary Half drain, r = 2.5 cm Smear zone, 15.5 cm Fixed, impermeable boundary Undisturbed zone, 27 cm

Impermeable boundaryw

(b) Plane strain unit cell, B = 0.45 m

50 kPa

L C

Undrained rigid plate Top of the cell Top of the drain wall

Negative 50 kPa excess pwp

Periphery of the drain wall Vertical roller boundary

Periphery of the cell Impermeable boundary

Impermeable boundary Centreline of the drain wall Vertical roller boundary

Vertical roller boundary

Fig 4.6 Boundary conditions and meshes of the axisymmetric and plane strain cells used in

FE analyses for the case of clay-sand-clay sandwich

<The case: Fd = 0.1>

B0.45: Proposed R0.45: Axisymmetric B0.75: Proposed

<The case: Fd = 0.1>

B0.45: Proposed R0.45: Axisymmetric B0.75: Proposed

<The case: Fd = 0.1>

B0.45: Proposed R0.45: Axisymmetric B0.75: Proposed

Fig 4.7 Comparison of FEM results of the axisymmetric unit cell having R = 0.45 m (R0.45: Axisymmetric) with that of the plane strain unit cell having B = 0.45 m (B0.45: Proposed) and with that of the plane strain unit cell having B = 0.75 m (B0.75: Proposed); these graphs corresponding to the case of one clay layer and k1=1

<The case: Fd = 0.1>

B0.45: Proposed R0.45: Axisymmetric B0.75: Proposed

<The case: Fd = 0.1>

B0.45: Proposed R0.45: Axisymmetric B0.75: Proposed

<The case: Fd = 0.1>

B0.45: Proposed R0.45: Axisymmetric B0.75: Proposed