SoilImprovementUsingStoneColumns-MahmoudS Hammad2014

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Soil Improvement Using Stone Columns
Thesis · June 2014
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AIN SHAMS UNIVERSITY
FACULTY OF ENGINEERING
STRUCTURAL ENGINEERING DEPARTMENT
SOIL IMPROVEMENT USING STONE COLUMNS
Thesis
Submitted in Partial Fulfillment of the
Requirements for the Degree of
MASTER OF SCIENCE
In
CIVIL ENGINEERING (STRUCTURES)
By
MAHMOUD EL SHAWAF ABDUL AZIM ALY HAMMAD
Supervised by
Prof. Dr. Ali Abdul Fattah Prof. Dr. Yasser Moghazy

Ali Ahmed El-Mossallamy
Professor of Geotechnical Professor of Geotechnical
Engineering Engineering
Structural Department Structural Department
Faculty of Engineering Faculty of Engineering
Ain Shams University Ain Shams University
Cairo – 2014
Name : Mahmoud El Shawaf Abdul Azim Aly Hammad

Thesis : Soil improvement using stone columns
Degree : Master of science in civil engineering (Structural)
EXAMINERS COMITEE
Name and Affiliation Signature

Prof. Dr. Ashraf Abd El-Hay El-Ashaal
Professor at the Construction Research
Institute – National Water Research Center
Prof. Dr. Mona Mohamed Mostafa Eid

Professor of Geotechnical Engineering
Faculty of Engineering
Prof. Dr. Ali Abdul Fattah Ali Ahmed

Prof. Dr. Yasser Moghazy El Mossallamy

Date: 23 / 6 / 2014
Name : Mahmoud El Shawaf Abdul Azim Aly Hammad

Thesis : Soil improvement using stone columns
Degree : Master of science in civil engineering
SUPERVISORS COMITEE
Name and Affiliation Signature
Prof. Dr. Ali Abdul Fattah Ali Ahmed

Prof. Dr. Yasser El Moghazy El

Mossallamy
Date: 23 / 6 / 2014
Postgraduate Studies
Authorization stamp: The thesis is authorized at / / 2014
College Board approval University Board approval

/ / 2014 / / 2014
CURRICULUM VITAE
Name Mahmoud El Shawaf Abdul Azim Aly
Date of Birth 19, April, 1986
Place of Birth Cairo, Egypt
Nationality Egyptian
Scientific BSc. of Structural Engineering, Faculty of Engineering,

degree Ain Shams University, 2008
Current Job Demonstrator of Geotechnical Engineering and

Foundations, Structural Engineering Department,
Faculty of Engineering, Ain Shams University
STATEMENT
This thesis is submitted to Ain Shams University for the degree of

M.Sc. in Civil Engineering.
The work included in this thesis was carried out by the author at the
Department of Structural Engineering, Faculty of Engineering, Ain
Shams University, Cairo, Egypt.
No part of this thesis has been submitted for a degree or a

qualification at any other University or Institution.
Name: Mahmoud El Shawaf Abdul Azim Aly
Signature:
Date: 23 / 6 / 2014
ACKNOWLEDGMENT
First and foremost thanks to GOD for his many graces and blessings.
I wish to express my deepest gratitude and appreciation to Prof. Dr. Ali

Abdul Fattah Aly, Professor of Geotechnical engineering, Structural
Department, Faculty of Engineering, Ain Shams University for his kind
supervision, fruitful comments and valuable advice.
My grateful appreciation also extends to Prof. Dr. Yasser Moghazy El

Mossallamy, Professor of Geotechnical engineering, Structural
Department, Faculty of Engineering, Ain Shams University for his
patience, help, guidance, useful suggestions, dedication and encouragement
throughout this research till its completion which is gratefully
acknowledged and sincerely appreciated.
An appreciation and thanks to Prof. Dr. Mona Mustafa Aid and Prof. Dr.
Ashraf Abdul Hay El Ashaal for their kind remarks and comments upon
reviewing the thesis.
Most important, my deepest thanks and love for my father, mother and
sister. Your constant and everlasting support is the reason I was able to
finish this research.
Last but certainly not least, I would like to thank my gorgeous wife and my
great kids for their support and patience through the ups and downs, the
sleepless nights and hard times throughout this long but rewarding journey.
ABSTRACT
The special nature of soft soil deposits makes it one of the most
complicated soil types to work with from a geotechnical engineering point
of view. There are two main problems encountered when undertaking
construction projects on soft soil deposits, excessive settlement and low
shear strength. Stone columns are considered one of the most effective
techniques used to improve soft soil deposits. Using stone columns to
reinforce soft soil deposits increases the bearing capacity, reduces the
settlement and accelerates the consolidation and construction time.
Different analytical methods used to calculate and study the improvement

of the soft soil deposits when reinforced with stone columns represent the
final behavior of the reinforced soil, i.e. after consolidation process takes
place. Finite element analysis programs can be used to study the behavior
of the soft soil deposits reinforced with stone columns both during and after
the consolidation process. In order to represent the problem accurately a
three dimensional finite element analysis is required. However, due to the
fact that two dimensional finite element analysis programs are common in
practical application, a need to convert the three dimensional problem to a
two dimensional configuration is needed.
One of the main objectives of this research is to investigate the different

possible two dimensional configurations that can be used to model the
problem of soft soil deposits reinforced with stone columns, and to
recommend the configuration whose behavior is nearest to the actual
behavior of the reinforced soft soil.
A case study of an embankment construction for Penchala Toll Plaza

project at New Pantai Expressway, Malaysia is undertaken (Tan et al.,
2008). A three dimensional finite element analysis of this case history using
PLAXIS 3D is performed. A comparison between the numerically
predicted behavior and the documented actual behavior of the embankment
showed a good agreement between them.
A Comparison is then done between the results of the three dimensional

numerical model and different configurations for a two dimensional
numerical model of the same case study. An Axisymmetric configuration, a
Plane strain configuration using an equivalent width for the column and a
Plane strain configuration using equivalent parameters for the column are
used to numerically model the case. The results show that the axisymmetric
configuration shows the best agreement with the three dimensional
analysis. Also, the plane strain with equivalent parameters configuration
shows more reliability by giving more accurate results than the plane strain
configuration using equivalent column parameters.
A study for the performance of the stone columns is conducted. The effect
of various parameters such as the embankment height, column spacing and
modulus of elasticity of the column on the stress concentration factor and
column load share ratio as well as settlement and time reduction factors is
studied through a parametric study. The findings of this parametric study
are then compared to some of the other analytical approaches used to
estimate the behavior of the reinforced soft soil and a recommendation is
done for the appropriate analytical methods to be used.
Keywords: Stone columns, soil improvement, embankments, soft soil

SUMMARY
Stone columns are considered one of the most effective techniques used to
improve soft soil deposits. Using stone columns as reinforcement for soft
soil deposits provides the advantages of reduced settlement as well as
reduction in the consolidation and hence construction time. The main
objective of this research is to investigate the behavior of the soft soil
reinforced using stone columns and the interaction between the soft soil
and the stone columns. The research presents a numerical study using finite
element analysis method using three dimensional modelling and different
problem of soft soil deposits reinforced with stone columns. A case study
reported by Tan et al. (2008) of an embankment construction for Penchala
Toll Plaza project at New Pantai Expressway, Malaysia, in 2003 is
undertaken, and the behavior of the different models is compared to
readings taken from field during and after construction. Finally, a
parametric study is performed to study the effect of different factors
affecting the behavior of the stone columns and its efficiency in the
reduction of the settlement of the soft soil deposits and improving the soft
soil behavior.
The thesis consists of seven chapters
Chapter (1) is the introduction to this research; it discusses the importance,

the scope and the main objectives of the research.
Chapter (2) is a literature review which briefly discusses through the past
researches the behavior of soft soil deposits reinforced with stone columns.
Chapter (3) presents a brief discussion about finite element method

including analysis sequence and different types of elements that may be
used in the analysis. Also, the finite element analysis program which is
used during this research is briefly discussed.
Chapter (4) presents the case study description. Also, it illustrates all the
three dimensional and two dimensional numerical models conducted to
investigate the behavior of soft soil reinforced with stone columns under an
embankment. Finally, a comparison between the numerically predicted
response of different models and the monitored response is presented.
Chapter (5) is a parametric study performed to investigate the effect of

different parameters on the behavior and performance of stone columns.
Chapter (6) presents a comparison between the findings of the numerically

performed parametric study with some of the analytical methods used to
estimate the behavior of the soft soil reinforced via stone columns
Chapter (7) presents the summary and the conclusions of the research. It
ends up with the suggestions for future studies and research topics relevant
to the subject.
TABLE OF CONTENTS
TABLE OF CONTENTS .................................................................................... i
LIST OF FIGURES ..........................................................................................vii
LIST OF TABLES .......................................................................................... xiv
NOMECLATURE ............................................................................................ xv
Chapter (1).......................................................................................................... 1
1.1 Introduction............................................................................................ 1
1.2 Research scope and objectives .............................................................. 2
1.3 Thesis outline ......................................................................................... 3
Chapter (2).......................................................................................................... 5
2.1 Introduction............................................................................................ 5
2.2 Stone columns ........................................................................................ 6
2.2.1 Stone column installation ................................................................ 7
2.3 Engineering behavior of the composite ground .................................... 9
2.3.1 Unit Cell .......................................................................................... 9
2.3.2 Load sharing and stress concentration .......................................... 10
2.4 Settlement and bearing capacity of stone column reinforced soil ....... 18
2.4.1 Experimental Studies ..................................................................... 18
i
2.5 Consolidation rate of soft soil reinforced with stone columns ............ 28
2.6 Stone columns-soft soil reinforcement system under embankment .... 34
2.7 Discussion ............................................................................................ 36
Chapter (3)........................................................................................................ 38
3.1 Introduction.......................................................................................... 38
3.2 Finite element method ......................................................................... 38
3.2.1 General .......................................................................................... 38
3.2.2 Analysis procedure of finite element method ............................... 39
3.2.3 Elements shapes ............................................................................ 41
3.2.4 Two dimensional simulation of special three dimensional

problems ..................................................................................................... 43
3.2.4.1 Plane strain .................................................................................... 43
3.2.4.2 Plane stress .................................................................................... 44
3.2.4.3 Axisymmetric problems ................................................................ 44
3.3 Material Modeling Basics .................................................................... 46
3.3.1 General.............................................................................................. 46
3.3.2 Stresses ............................................................................................. 46
3.3.3 Strains ............................................................................................... 47
3.4 Constitutive Material Models .............................................................. 48
ii
3.4.1 Linear elastic constitutive law .......................................................... 49
3.4.2 Nonlinear elastic constitutive laws ................................................... 49
3.4.3 Elastoplastic constitutive laws.......................................................... 50
3.4.4 Elasto-visco plasticity constitutive laws........................................... 55
3.5 Finite element modeling program used in this research ...................... 57
3.5.1 General.............................................................................................. 57
3.5.2 Input program ................................................................................... 58
3.5.2.1 Soil elements..................................................................................... 59
3.5.2.2 Types of soil behavior ...................................................................... 60
3.5.2.3 Boundary conditions ......................................................................... 62
3.5.2.4 Mesh Generation .............................................................................. 62
3.5.2.5 Initial conditions ............................................................................... 62
3.5.3 Calculation ........................................................................................ 63
3.5.3.1 Types of calculations ........................................................................ 63
3.5.3.2 Loading types ................................................................................... 64
3.5.4 Output ............................................................................................... 65
3.6 The Mohr Coulomb model .................................................................. 65
3.6.1 Young’s Modulus ............................................................................. 66
3.6.2 Poisson’s ratio................................................................................... 67
iii
3.6.3 Shear strength parameters................................................................. 68
Chapter (4)........................................................................................................ 70
4.1 Introduction.......................................................................................... 70
4.2 Case study description ......................................................................... 71
4.3 Back analysis of the case study ........................................................... 74
4.3.1 Constitutive law and soil parameters ............................................ 74
4.3.2 Model geometry and boundary conditions .................................... 74
4.3.3 Comparison between back analysis results and the field

measurements ............................................................................................. 77
4.4 Numerical modeling using 2D FE analyses ........................................ 80
4.4.1 Axisymmetric model ..................................................................... 83
4.4.2 Plane strain model using equivalent width.................................... 86
4.4.3 Plane strain model using equivalent parameters ........................... 90
4.5 Comparison between the 3D and 2D FE analyses .............................. 93
4.5.1 Settlement ...................................................................................... 94
4.5.2 Excess pore water pressure ........................................................... 96
4.5.3 Development of soil shear strength ............................................... 99
4.5.4 Stability of the embankment slope .............................................. 102
Chapter (5)...................................................................................................... 105
iv
5.1 Introduction........................................................................................ 105
5.2 Definitions ......................................................................................... 106
5.2.1 Modular ratio .................................................................................. 106
5.2.2 Stress concentration factor ............................................................. 106
5.2.3 Column load share ratio.................................................................. 106
5.2.4 Settlement reduction factor............................................................. 107
5.2.5 Time reduction factor ..................................................................... 107
5.3 The effect of column spacing ............................................................ 107
5.4 The effect of Stress level ................................................................... 111
5.5 The effect of modular ratio ................................................................ 114
Chapter (6)...................................................................................................... 118
6.1 Introduction........................................................................................ 118
6.2 The stress concentration factor (SCF) ............................................... 118
6.3 The column load share ratio .............................................................. 122
6.4 The settlement reduction factor ......................................................... 126
6.5 The time reduction factor .................................................................. 129
Chapter (7)...................................................................................................... 133
7.1 Summary ............................................................................................ 133
7.2 Conclusions........................................................................................ 134
v
7.3 Recommendations for future studies ................................................. 136
REFERENCES ............................................................................................... 137
vi
LIST OF FIGURES
Figure 2-1 – Installation techniques of stone columns (After Keller, 2002) ..... 8
Figure 2-2 – Installation patterns of stone columns and influence area of the
unit cell (After Weber, 2008) ............................................................................. 9
Figure 2-3 – Diagram of composite ground (After Bergado et al., 1996) ....... 10
Figure 2-4 – Effect of area ratio on stress concentration factor of non-

homogeneous granular pile (After Shahu et al., 2000) .................................... 13
Figure 2-5 – Influence of radial deformation and plastic strains on stress

concentration factor (After Castro, 2008) ........................................................ 14
Figure 2-6 – Variation in the stress concentration ratio as the applied

foundation load increased (After Mckelvey et al., 2004) ................................ 17
Figure 2-7 – Failure modes of stone columns (After Wood et al., 2000) ........ 20
Figure 2-8 – Photos of sand column beneath circular footing at beginning,

middle and end of foundation loading process: (a) 150 mm length; (b) 250 mm
length (After Mckelvey et al., 2004) ................................................................ 22
Figure 2-9 – Stress-settlement behavior under entire area loading (After

Ambily and Gandhi, 2007) ............................................................................... 24
Figure 2-10 – Comparison of group column test and single column test (After
Ambily and Gandhi, 2007) ............................................................................... 24
Figure 2-11 – Modified Greenwood curves (After Greenwood and Kirsch,

1983) ................................................................................................................. 26
vii
Figure 2-12 – Modified Priebe design curves (After Priebe, 1995) ................ 27
Figure 2-13 – Definition of terms for modelling (After Han and Ye, 2001) ... 30
Figure 2-14 – Vertical stress on soil and columns with time (After Han and
Ye, 2001) .......................................................................................................... 32
Figure 2-15 – Stress concentration ration with time (After Han and Ye, 2001)
.......................................................................................................................... 32
Figure 2-16 – Dissipation of excess pore water pressure (After Han and Ye,
2001) ................................................................................................................. 33
Figure 2-17 – Total settlement-time relationship of reinforced soft clay by (a)

granular piles; (b) Vertical drain (After Bergado and Long, 1994) ................. 34
Figure 3-1 – One-dimensional element (After Rao, 2005) .............................. 41
Figure 3-2 – Two-dimensional elements (After Rao, 2005) ............................ 42
Figure 3-3 – Axisymmetric elements (After Rao, 2005) ................................. 42
Figure 3-4 – Three-dimensional elements (After Rao, 2005) .......................... 43
Figure 3-5 – Examples for Plane strain problems ............................................ 44
Figure 3-6 – Example for Plane stress problems ............................................. 45
Figure 3-7 – Examples for axisymmetric problems ......................................... 45
Figure 3-8 – Geometrical representation of the yield criterion in the principal

stress space (After Hill, 1950) .......................................................................... 51
Figure 3-9 – Different yield criterion (After Chen and Mccarron, 1986; After
Chen and Mizuno, 1990) .................................................................................. 52
viii
Figure 3-10 – Associated flow rule (After Atkinson, 1993) ............................ 53
Figure 3-11 – Schematic representation of the yield surface and the flow rule
(After Kempfert and Gebreselassie, 2006)....................................................... 53
Figure 3-12 – Perfect plasticity model (After Brinkgreve, 2002) .................... 56
Figure 3-13 – Isotropic hardening rule (After Chakrabarty. 2006) ................. 56
Figure 3-14 – Kinematic hardening rule (After Chakrabarty, 2006) ............... 57
Figure 3-15 – Distribution of nodes and stress points in Plaxis finite elements
(After Plaxis version 8 manuals) ...................................................................... 59
Figure 3-16 – Mohr’s circle of stress used to drive relation between undrained
shear strength and drained shear parameters (After Brinkgreve, 2002) .......... 61
Figure 3-17 – Mohr-Coulomb yield criterion .................................................. 66
Figure 3-18 – Definition of E0 and E50 for standard drained triaxial test results
(After Brinkgreve, 2002) .................................................................................. 67
Figure 3-19 – Mohr-coulomb failure envelope with one Mohr failure circle
(After Brinkgreve, 2002) .................................................................................. 68
Figure 4-1 – Case study profile (modified after Tan et al., 2008) ................... 72
Figure 4-2 – Stress and settlement at (SP1) and (SP2) (modified after Tan et
al., 2008) ........................................................................................................... 73
Figure 4-3 – Geometry of the 3D finite element model ................................... 75
Figure 4-4 – Generated Finite element mesh ................................................... 76
ix
Figure 4-5 – Settlement at (SP1) and (SP2) for field measurements and FEM
.......................................................................................................................... 77
Figure 4-6 - Excess pore water pressure at points (A) and (B) using the 3D
model ................................................................................................................ 79
Figure 4-7 - Soil shear strength development at points (A) and (B) using the
3D model .......................................................................................................... 80
Figure 4-8 – 2D finite element analyses configurations .................................. 81
Figure 4-9 – Embankment profile .................................................................... 82
Figure 4-10 – Geometry and boundary conditions for the Axisymmetric model
.......................................................................................................................... 84
Figure 4-11 – Generated Finite element Mesh for the axisymmetric model ... 84
Figure 4-12 – Settlement at SP1 using Axisymmetric model .......................... 85
Figure 4-13 – Excess pore water pressure at point (A) using axisymmetric
model ................................................................................................................ 85
Figure 4-14 – Soil shear strength development at point (A) using axisymmetric
model ................................................................................................................ 86
Figure 4-15 – Geometry and boundary conditions for Plane strain with
equivalent width Finite element model ............................................................ 87
Figure 4-16 – Generated finite element mesh for Plane strain with equivalent
width Finite element model .............................................................................. 87
Figure 4-17 – Settlements at (SP1) and (SP2) for Plane strain with equivalent
width Finite element model .............................................................................. 88
x
Figure 4-18 – Excess pore water pressure at points (A) and (B) for Plane strain
with equivalent width Finite element model .................................................... 89
Figure 4-19 – Soil shear strength development at points (A) and (B) for Plane
strain with equivalent width Finite element model .......................................... 89
equivalent parameters Finite element model.................................................... 91
parameters Finite element model ..................................................................... 91
Parameters Finite element model ..................................................................... 92
with equivalent parameters Finite element model ........................................... 92
strain with equivalent parameters Finite element model ................................. 93
Figure 4-25 – Comparison of settlements at (SP1) .......................................... 95
Figure 4-26 – Comparison of settlements at (SP2) .......................................... 95
Figure 4-27 – Comparison of Excess pore water pressure at point (A) ........... 97
Figure 4-28 – Comparison of Excess pore water pressure at point (B) ........... 97
Figure 4-29 – Comparison of shear strength development at point (A) ........ 100
Figure 4-30 – Comparison of shear strength development at point (B) ........ 100
Figure 4-31 – Stability of embankment slope for different finite element

models ............................................................................................................ 103
xi
Figure 4-32 –- Failure mode of embankment slope (H=2m) ......................... 104
Figure 5-1 – Effect of columns spacing on the stress concentration factor ... 109
Figure 5-2 – Effect of columns spacing on the pile load share ratio ............. 109
Figure 5-3 – Effect of columns spacing on the settlement reduction factor .. 110
Figure 5-4 – Effect of columns spacing on the time reduction factor ........... 110
Figure 5-5 – Effect of stress level on the stress concentration factor ............ 112
Figure 5-6 – Effect of stress level on the Column load share ratio................ 112
Figure 5-7 – Effect of stress level on the settlement reduction factor ........... 113
Figure 5-8 – Effect of stress level on the time reduction factor..................... 114
Figure 5-9 – Effect of Modular ratio on the stress concentration factor ........ 115
Figure 5-10 – Effect of Modular ratio on the Column load share ratio ......... 116
Figure 5-11 – Effect of Modular ratio on the settlement reduction factor ..... 117
Figure 5-12 – Effect of Modular ratio on the time reduction factor .............. 117
Figure 6-1 – The effect of embankment height on the stress concentration

factor using numerical modeling and theoretical approaches ........................ 121
Figure 6-2 – The effect of column spacing on the stress concentration factor
using numerical modeling and theoretical approaches .................................. 121
Figure 6-3 – The effect of Modular ratio on the stress concentration factor
xii
Figure 6-4 – The effect of embankment height on the column load share ratio
Figure 6-5 – The effect of column spacing on the column load share ratio
Figure 6-6 – The effect of the modular ratio on the column load share ratio
Figure 6-7 – The effect of embankment height on the settlement reduction

factor using numerical modeling and theoretical approaches ........................ 128
Figure 6-8 – The effect of column spacing on the settlement reduction factor
Figure 6-9 – The effect of modular ratio on the settlement reduction factor
Figure 6-10 – The effect of embankment height on the time reduction factor
Figure 6-11 – The effect of column spacing on the time reduction factor using
numerical modeling and theoretical approaches ............................................ 130
Figure 6-12 – The effect of modular ratio on the time reduction factor using
numerical modeling and theoretical approaches ............................................ 131
xiii
LIST OF TABLES
Table 4-1 – Material Parameters for Case study .............................................. 73
Table 4-2 – Stone columns parameters (equivalent parameters plane strain

model) ............................................................................................................... 90
Table 4-3 – Comparison of settlement ............................................................. 96
Table 4-4 – Comparison of excess pore water pressure .................................. 99
Table 4-5 – Comparison of Shear strength development for soft clay .......... 102
xiv
NOMECLATURE
as Area replacement ratio
Ac Area of stone column
As Area of soil surrounding the stone column within the unit

cell
cu Undrained shear strength of soil
c’, φ’ Effective soil shear parameters
c, φ Total soil shear parameters
Cr, Cv Coefficients of consolidation in radial and vertical directions

respectively
Cr, Cv Modified Coefficients of consolidation in radial and vertical

directions respectively
d Stone column diameter
de Influence diameter of unit cell
dw Diameter of drain well
[De] Elastic material stiffness matrix
{dεp} Incremental plastic strain vector
{dεe} Incremental elastic strain vector
{dεvp} Visco-plastic strain rate
xv
Ec Elasticity modulus of stone column material
Es Elasticity modulus of soft soil deposits
Egpi Elasticity modulus granular pile at depth (Zi)
Egp Elasticity modulus non homogeneous granular pile
E0 Initial elasticity modulus
E50 Elasticity modulus corresponding to 50% of strength
Eur Un-loading re-loading elasticity modulus
{F} Global force vector
g Plastic potential function
H Thickness of soft soil layer
Kh, Kv Coefficient of permeability in horizontal and vertical

direction respectively
[k] Stiffness matrix
[K] Global stiffness matrix
Mr Modular ratio
N Diameter ratio
ns Steady stress concentration ratio
P Isotropic mean
q Deviatoric stress
xvi
{Q} Nodal force vector
r Stone column radius
re Influence radius of unit cell
S Spacing between stone columns
SCF Stress concentration factor
Tr, Tv Consolidation time factors for radial and vertical flow

T’r, T’v Modified Consolidation time factors for radial and vertical
flow directions respectively
Ur, Uv Average degree of consolidation for radial and vertical flow

Urv Combined degree of consolidation for both radial and

vertical flow directions
Ux, Uy, Uz Displacements in X, Y and Z directions respectively
{u} Nodal displacement vector
{U} Global displacement vector
w Strain energy function
Xeq. Equivalent stone column parameter
Xc Original parameter for the stone columns
Xs Original parameter for the soil surrounding the stone column
Zi Depth of stone column
xvii
α Rate of increase of elasticity modulus of stone column with
depth
αL Stone column load share ratio
αs Soft soil settlement reduction factor
αt Consolidation time reduction factor

γb Bulk unit weight
γsat Saturated unit weight
γxy, γyz, γzx Shear strain components
εx, εy, εz Normal strain component in X, Y and Z directions

respectively
ε1, ε2, ε3 Principle strain component in corresponding to principle

stresses
εv Volumetric strain
εq Deviatoric strain
λ Plastic multiplier
μ Pore water pressure
σ Total stresses
σ' Effective stresses
σc Stress transferred to the stone column
σs Stress transferred to the soil
σcs Steady stress in the stone column after the end of

consolidation
xviii
σss Steady stress in the soil after the end of consolidation
σx, σy, σz Total stresses in X, Y and Z directions respectively
σ'x, σ'y, σ'z Effective stresses in X, Y and Z directions respectively
σ'1, σ'2, σ'3 Principle effective stresses

{σ}T Transpose of the stress tensor
τxy, τyz, τzx Shear stress components
υ Poison’s ratio
ψ Dilatancy angel
xix
Chapter )1(
Introduction
1.1 Introduction
The special nature of soft soil deposits makes it one of the most complicated
soil types to work with from a geotechnical engineering point of view. There
are two main problems encountered when undertaking construction projects
on soft soil deposits, excessive settlement and low shear strength. Due to large
void ratio and inherent compressibility of such deposits, consolidation and
displacement can be noticeable during and continue for long time after
construction. Low shear strength is particularly hazardous when constructing
large embankments on soft clay base, facilitating potential circular or sliding
failure planes.
Different techniques can be used to improve the performance of soft soil

deposits such as, Geosynthetic or fiber reinforcement of the soft soil deposits,
preloading with or without vertical drains, deep compaction, and stone
columns. Ground improvement by stone columns is considered to be a very
effective and reliable method to improve soil properties as it solves the soft
soil problems by providing the advantage of reducing settlement and
accelerating consolidation process. Another advantage of this method is the
simplicity of its construction.
1
Chapter (1) Introduction
Different analytical methods used to calculate and study the improvement of

the soft soil deposits when reinforced with stone columns represent the final
behavior of the reinforced soil, i.e. after consolidation process takes place.
Numerical modeling programs can be used to study the behavior of the soft
soil deposits reinforced with stone columns both during and after the
consolidation process. In order to represent the problem accurately three
dimensional finite element modeling can be used. However, due to the fact
that two dimensional finite element analysis programs are more common in
practice, a need to convert the three dimensional problem to a two dimensional
configuration is needed.
1.2 Research scope and objectives
One of the main objectives of this research is to investigate the different

problem of soft soil deposits reinforced with stone columns, and to
recommend the configuration whose behavior is nearest to the actual behavior
of the reinforced soft soil.
A case study of an embankment construction for Penchala Toll Plaza project

at New Pantai Expressway, Malaysia, in 2003 is undertaken, full description
of the case study was given by Tan et al., 2008. Three dimensional finite
element analyses of this case history using PLAXIS 3D is performed. A
comparison between the numerically predicted behavior and the documented
actual behavior of the embankment showed a good agreement between them.
A Comparison is then done between the results of the three dimensional

numerical model and different configurations for a two dimensional numerical
model of the same case study. An Axisymmetric configuration, a Plane strain
configuration using an equivalent width for the column and a Plane strain
configuration using equivalent parameters for the column are used to
2
numerically model the case. The results show that the axisymmetric
configuration shows the best agreement with the three dimensional analysis.
Also, the plane strain with equivalent parameters configuration shows more
reliability by giving more accurate results compared to the plane strain
configuration using equivalent column parameters.
A study for the performance of the stone columns is conducted. The effect of
various parameters such as the stress level, columns spacing and modular ratio
on the stress concentration factor and pile load share ratio as well as settlement
and time reduction factors is studied through a parametric study. The findings
of the parametric study are then compared with different analytical approaches
used to predict the behavior of soft soil reinforced via stone columns.
1.3 Thesis outline
The thesis is divided into the following chapters:
Chapter (1) is the introduction to this research; it discusses the importance, the
scope and the main objectives of the research.
Chapter (2) is a literature review which briefly discusses through the past
researches the behavior of soft soil deposits reinforced with stone columns, the
effect of the stone columns on the reduction of settlement and acceleration of
consolidation of the soft soil. The load share between the stone columns and
the soft soil is also briefly studied. In addition, the behavior of the
embankments built on soft soil deposits reinforced with stone columns is
reviewed.
Chapter (3) presents a brief discussion about finite element method including
analysis sequence and different types of elements that may be used in the
analysis. Also, different constitutive laws in geomechanics are highlighted. In
addition, the utilized material models formulations are provided. Finally, the
3
finite element analysis program PLAXIS which is used during this research is
briefly discussed.
Chapter (4) presents a case study description including its location, the soil
profile, and the soil parameters. In addition, deformation values monitored
during the construction of the embankment are introduced. A set of three
dimensional and two dimensional numerical models are conducted to
investigate the behavior of soft soil reinforced with stone columns under an
embankment. A comparison between the numerically predicted response of
the three dimensional model and the monitored response is presented to assess
the adequacy of the adopted numerical modeling. Finally, A comparison
between the response of the three dimensional numerical model and the
different two dimensional models is performed to predict which two
dimensional configuration is more adequate to model this problem.
Chapter (5) presents a parametric study performed to investigate the effect of

different parameters on the load carrying capacity of the stone column. The
effect of stress level, column spacing and modular ratio on the stress
concentration factor and the pile load share ratio as well as settlement and time
reduction factors is investigated.
Chapter (6) presents a comparison between the findings of the numerically

performed parametric study with some of the analytical methods used to
estimate the behavior of the soft soil reinforced via stone columns
Chapter (7) presents the summary and the conclusions of the research. It ends
up with the suggestions for future studies and research topics relevant to the
subject.
4
Chapter )2(
Literature Review
2.1 Introduction
Various geotechnical engineering challenges are encountered when dealing

with soft soil deposits. One of the main characteristics of soft soil is having
large void ratio which contributes in the excessive settlement and low shear
strength of these soil deposits. Embankment slope failure is a common
problem associated to the low shear strength of the soft soil deposits.
Consolidation is also a major issue, as it has high value and it takes a long
time for the process to end.
Soft soils are usually located near most river estuaries and coastal areas all
over the world. In Egypt, the soft soil deposits are most commonly found near
the northern and north eastern coasts where future development planes are
undertaken. Most structures built on these soft soil deposits are incompatible
with the weak foundation soil conditions. However, some techniques could be
used to overcome these problems, such as:
1. Excavating and replacing the soft soils with suitable soils.

2. Using deep foundations
3. Stabilizing the soft soils with injected additives such as lime.
5
Chapter (2) Literature Review
Additionally, there are some soil improvement techniques such as:
1. Geosynthetic or fiber reinforcement of the soft soil deposits.

2. Preloading with or without vertical drains.
3. Vacuum preloading method.
4. Deep dynamic compaction.
5. Ordinary or encased stone columns
Ground improvement by stone columns is considered to be a very effective

and reliable method to improve soil properties as it solves the soft soil
problems by providing the advantage of reduced settlement and accelerated
consolidation process. Another advantage of this method is the simplicity of
its construction. Within this research, the construction on soft soil will be
treated using stone columns as a soil improvement technique.
A lot of researches on stone columns as a soft soil improvement technique

were previously carried out by many researchers all over the world. These
researches discussed the stone column effect on the enhancement of the
properties of the soft soil. During this chapter some of these researches are
discussed.
2.2 Stone columns
Stone columns are an extension of the vibro-compaction method, conceived in

Germany in the mid-1930s, which was used to treat cohesive soils. Within the
past 30 years, the value of stone columns as a technique for reinforcing
cohesive soils has been recognized. The stone columns technique, developed
mainly in Europe, is now increasingly used in many regions all over the world
6
primarily for embankment and road works (Slocombe and Moseley, 1991).
Among the various methods for improving soft ground conditions, stone
columns are considered one of the most versatile and cost-effective ground
improvement technique. Stone columns have been used extensively in week
deposits to increase the load carrying capacity, reduce settlement of structural
foundations and accelerate consolidation settlements due to the reduction in
flow path lengths.
The grain size of the stone column material is one of the main controlling
parameters in the design of the stone columns. Dipty and Girish (2009) studied
the influence of column material on the performance of the stone columns
through laboratory experiments on model stone columns installed in clay. Five
reinforcement materials were studied: stone, gravel, river sand, sea sand and
quarry dust. It was found that stone then gravel are more effective than other
materials.
2.2.1 Stone column installation
A stone column consists of crushed rock with particle size less than one-
seventh of the column diameter. Stone columns are normally installed in a
triangular, squared or hexagonal pattern, as shown in Figure 2-2.
7
As shown in Figure 2-1, the following methods are commonly used to install
stone columns:
1. Replacement method involves replacing in-situ soil with stone column

materials. A vibratory probe (vibroflot), accompanied by a water jet, is
used to create the holes for the columns. This technique is suitable when
the ground water level is high and the in-situ soil is relatively soft.
2. Displacement method is utilized when the water table is low and the in-
situ soil is firm. It involves using a vibratory probe, which uses
compressed air, to displace the natural soil laterally.
3. Case-borehole or rammed columns method is also used. In this method,

the piles are constructed by ramming the granular materials in the pre-
bored holes in stages using heavy falling weight.
(a) Replacement method (b) Displacement method

Figure 2-1 – Installation techniques of stone columns (After Keller, 2002)
8
Bulging and subsequent failure of granular piles mainly occur due to high
stress concentration near top of the granular pile. Hence, after installing stone
columns, a blanket of sand or gravel with a thickness of 0.3 m or more is
usually placed over the top. This blanket works as both a drainage layer and
also to distribute the stresses under the structure uniformly (Ali and Abolfazle,
2005 and Shahu, 2006).
Figure 2-2 – Installation patterns of stone columns and influence area of the
unit cell (After Weber, 2008)
2.3 Engineering behavior of the composite ground
2.3.1 Unit Cell
The unit cell is defined as a cylinder with an influence diameter (d e) enclosing

one stone column and the surrounding soil influenced by the presence of the
stone column. The influence diameter of the unit cell depends on the stone
column installation pattern. The triangular, squared or hexagonal patterns in
installation of the stone columns experiences different influence diameters as
indicated previously in Figure 2-2.
9
2.3.2 Load sharing and stress concentration
Beregado et al. (1996) presented a theory to determine the stability of ground

reinforced with stone columns. The theory is based on the stress concentration
in the granular pile. Figure 2-3 illustrates a diagram of the composite ground.
Each stone column is separated into its own unit cell, as shown in figure 2-3-a.
The area replacement ratio (as) is the ratio of the granular pile area (Ac) over
the whole area of the equivalent cylindrical unit cell.
Equation 2.1
Where (As) is the soft soil area within the unit cell.
This ratio can also be expressed in terms of the stone column diameter (d) and
spacing between stone columns (S). The area replacement ratio for stone
columns installed in square and equilateral triangular patterns is respectively.
Equation 2.2.a
√ Equation 2.2.b
Figure 2-3 – Diagram of composite ground (After Bergado et al., 1996)
10
The distribution of the stress in the column (σ c), surrounding soil (σs) and
average stress (σ) is illustrated in figure 2-3-b. when ground reinforced with
stone columns is loaded; stress concentrations develop in the column
accompanied by reduction in stress in the surrounding soil. This can be
explained by the fact that, when loaded, the vertical settlement of the stone
column and the surrounding soil is approximately the same, causing
generation of stress concentration in the stiffer columns. A stress
concentration factor (SCF) is used to express the distribution of the vertical
stress within the unit cell
Equation 2.3
The relative stiffness of the stone column and the surrounding soil influences
the magnitude of the stress concentration. The average stress (σ) over the unit
cell area is given by:
Equation 2.4
The stress in the stone column (σc) and the stress in the surrounding soil (σ s)
are then given as follows:
Equation 2.5.a
Equation 2.5.b
As mentioned above, the applied load is divided between the stone column
and the surrounding soil relative to their stiffness values. Therefore, the stone
column carries the most applied load due to its higher stiffness. The modular
11
ratio is defined as the ratio between the elasticity modulus of the stone column
and the elasticity modulus of the soil, (E c/Es), which according to Beregado et
al. (1996) represents the stiffness ratio. Depending on that, the final stress
concentration ratio for a laterally confined stone column having a full elastic
behavior is equal to the constrained modular ratio. However, this is in
contradiction with experience as the modular ratio is usually in the range 10-
50, while the stress concentration ratio measured in actual cases is in the range
2-10 which is a much lower value. This is due to the fact that the column is
not confined and it has a lateral displacement. Additionally, the behavior of
the column is elastoplastic because the lateral bulging of the column is due to
its yielding.
2.3.2.1 Analytical studies
Shahu et al. (2000) presented a simple theoretical approach to analyze soft

ground reinforced by non-homogeneous granular piles with granular mat on
top based on the unit cell concept. He suggested that the modulus of elasticity
of the non-homogeneous granular pile (Egp) increases with depth while the
modulus of elasticity of the homogeneous granular pile is constant with depth.
Equation 2.6
Where (α) is defined as the rate of the increase of the modulus of elasticity of
the pile (Egp) with depth, (Egpi) is the modulus of elasticity of granular pile at
depth (zi), and (H) is the thickness of the soil.
A parametric study was also carried out by Shahu et al. (2000) to evaluate the
relative influence of various parameters on the effect of non-homogeneity of
12
the granular pile on treated ground process. The results showed that the
variation of stress concentration factor with depth tends to become more
uniform as the rate of variation of the granular pile stiffness with depth
increases, as shown in Figure 2-4. The higher the area replacement ratio is, the
higher the increase in the stress concentration factor is. In the theoretical
approach of Shahu et al. (2000) the values of the stress concentration factor
are high because the stone column and the soil behave linearly and the lateral
displacement of the stone column is prevented.
Figure 2-4 – Effect of area ratio on stress concentration factor of non-

homogeneous granular pile (After Shahu et al., 2000)
Deb (2007) developed a simple generalized mechanical foundation model for

soft soil improved by stone columns incorporating the nonlinear behavior of
the granular fill and soft soil. Parametric studies for a uniformly loaded strip
footing have been carried out to show the effects of various parameters on the
total settlement, the differential settlements and the stress concentration ratio.
It has been observed that the presence of the granular bed on top of the stone
13
columns helps to transfer stress from soil to stone columns and reduces both
total and differential settlement. The results also indicated that when the
modular ratio increases the stress concentration ratio increases. As the spacing
between the columns increases, the stress concentration ratio also increases
Castro (2008) studied analytically the influence of the horizontal deformation

and plastic behavior of the column on the distribution of stress between soil
and column, which is pointed out in Figure 2-5. With lateral confinement the
stress concentration factor SCF starts from zero and reaches a final value 40
which is equal the confined modular ratio which is not realistic as commented
before. The consideration of radial deformations, keeping elastic behavior,
reduces the final value to 25. Plastic strains in the column further reduce the
final value of SCF to about 5 which is more realistic.
Figure 2-5 – Influence of radial deformation and plastic strains on stress

concentration factor (After Castro, 2008)
14
2.3.2.2 Experimental studies
Stewart and Fahey (1994) conducted a number of centrifuge tests to study the
load share between the stone columns and the surrounding soft clay due to
construction of an iron ore stockpiles imposing a surface load up to 500 kPa.
The results of the tests proved that the loading on the surrounding soft soil is
partially carried by the stiffer columns. Thus, resulting in less stress being
transferred to the soft clay, and therefor leads to smaller settlement taking
place. Stress concentration factors of about 4 were estimated from the
centrifuge tests. In addition, a confined compression test was performed on a
stone column and surrounding clay. The loads were applied through a stiff
steel plate. The results showed that stress transfer from the clay to the column
depends mainly on consolidation around the column. The stress concentration
factor in the confined test was found to be increasing with the increased
loading from about 2 to about 3.5.
Kirsch and Sondermann (2003) concluded that the stress concentration is

depending on various parameters such as the loading type (Flexible or rigid),
the surcharge, the material parameters of column and soil, the geometrical
dimensions. They performed numerical and analytical analyses for an
embankment resting on soft soil reinforced with stone columns and compared
their results with the measurements carried out on another embankment site in
Kuala Lumpur. The columns were installed using a square pattern at 2.2 m.
The stress concentration factor was measured at 2.6 which agree well with the
field measurements of 2.8 under the same surcharge value of 120 kPa.
15
Mckelvey et al. (2004) examined the load sharing mechanism of the

composite clay/granular fill material by analyzing pressures measured beneath
a square rigid footing. Variations in the stress concentration ratio as the
applied footing pressure increased are presented in Figure 2-6. Initially in TS-
11 and TS-13 (two identical tests on long columns) the columns seemed to
take large proportions of the applied load with SCF>4. In contrast, the stress
concentration ratio was less than 2 for the shorter column in TS-14
immediately after the loads were applied. Short columns appear to provide
less resistance to loading compared with long columns, which shows some
resistance to punching.
In practice, columns are not loaded to failure, and working loads are
considerably smaller than the ultimate bearing capacity. Examining a possible
operational region, shown by the shaded area in Figure 2-6, it appears that the
stress distribution underneath the footing supported by stone columns is
significantly different between short and long columns. At higher loading,
beyond the possible working range, the stress ratio appeared to reach a
constant value of approximately 3, regardless to the column length. This
observation agrees well with previously stored data, particularly from field
studies. Barksdale and Bachus (1983) suggested that typical values of SCF
usually fall within range 2.5-5. Based on a case study of a footing supported
on long columns 10 m long and 0.75 m in diameter, Greenwood (1991) and
Han and Ye (1991) reported that the stone columns carried a high percentage
of the load as the loading progressed; In this case, the stress concentration
ratio was initially as high as 25 and reduced to 5 at higher loadings. In
contrast, Bell et al. (1986) examined the stress concentration ratio underneath
16
a footing supported on short columns, less than 4 m long, the value of SCF
increased from as low as 1.24 to 3 as the loading progressed.
Figure 2-6 – Variation in the stress concentration ratio as the applied

foundation load increased (After Mckelvey et al., 2004)
Weber (2008) carried out centrifuge tests for an embankment resting on soft
soil reinforced with stone columns. The results emphasized that the stress
concentration in the stone column is not only load dependent but also depends
on column depth. Small loads give stress concentration up to 8, which
decreases with depth, while high loads result in a constant distribution of
stress concentration over depth with values between 2.5 and 4.
Castro and Sagaseta (2009) also found that the stress concentration factor is in
the range of 3-6 from the field measurements for an embankment with 10 m
height constructed on soft soil reinforced with stone columns.
17
2.4 Settlement and bearing capacity of stone column reinforced soil
The presence of the stone column creates a composite material of lower

overall compressibility and higher shear strength than the original insitu soil.
Confinement, and thus stiffness of the stone column is provided by the lateral
stress within the weak soil. When an axial load is applied at the top of a single
stone column, an extension of the column diameter is produced beneath the
surface. This extension in turn, increases the lateral stress within the clay,
which provides an additional confinement for the stone column. An
equilibrium state is eventually reached, resulting in a reduction in the vertical
displacement, when compared with the untreated ground.
2.4.1 Experimental Studies
2.4.1.1 Field Tests
Mitchell & Huber (1985) performed 28 field load tests on individual stone
columns constructed in soft estuarine deposits during the installation of 6500
stone columns. The stone columns reinforced soft soil is used to support a
large waste water treatment plant. All stone columns extended completely
through the soft soil layer which ranged from 9 m to 15 m. the diameter of the
stone columns ranged from 0.5 m to 0.75 m. the column spacing ranged from
a 1.2 m x 1.5 m pattern to a 2.1 m x 2.1 m pattern. The results of the load test
showed that the existence of the stone column led to a reduction in settlements
to about 30% - 40% compared to the settlement of the untreated ground.
18
Han and Ye (1991) presented the results of full scale load tests on stone
columns reinforced soft soil in coastal areas. A total of 16 stone columns were
used in soft soil having a length of 14 m and an average diameter of 0.85 m
arranged in triangular pattern. The treated and untreated grounds were loaded.
It was found that the stone columns increase the bearing capacity of the treated
ground to two times the untreated ground. Also, it was established that using
stone columns to reinforce soft soil is very effective in decreasing the initial
excess pore water pressure and to keep the foundation stable.
2.4.1.2 Laboratory tests
Wood et al. (2000) performed model tests to determine the mechanisms of

response for beds of clay reinforced with stone columns subjected to surface
footing loads. An exhumation technique was used to discover the deformed
shapes of the stone columns. The laboratory model tests showed that there was
significant interaction between the footing and the individual stone columns
within a group. As a consequence, the load-settlement relationship for
neighboring columns in different locations would be different. Thus, it will be
more accurate in design of the stone column reinforced foundation to consider
increasing stiffness towards center of the group. The kinematic constrains that
the rough base of the footings imposes, push the load to greater depths toward
the center of the footing. Based on the study of wood et al. (2000), the
following failure modes of stone columns have been proposed:
1. The bulging failure of a stone column takes place when it is not

prevented from expanding radially by adjacent columns as show in
Figure 2-7-a.
19
2. The bearing capacity failure plan occurs in the head of the column,
Figure 2-7-b.
3. Failing by a diagonal shear plane if the stone column has a little lateral
restrains and is subjected to high loads, Figure 2-7-c.
4. Failing by penetration through an underlying soft clay layer if the stone
column is short column, Figure 2-7-d.
5. The compression failure happens when the stone column is long, Figure
2-7-e.
6. A slender stone column can fail by bending if it is laterally loaded,
Figure 2-7-f.
Figure 2-7 – Failure modes of stone columns (After Wood et al., 2000)
Bae et al (2002) investigated also the failure mechanism and various

parameters of the behavior of end-bearing single and group stone columns by
laboratory loading tests. Results of the laboratory tests were verified by finite
element model (FEM) analyses. The laboratory tests and the FEM analyses
results showed that the bulging failure mode appeared in the depth of 1.6 to
2.8 times the column diameter. The major failure mode of stone columns
20
group is conical failure, the conical failure angle in short columns is smaller
than that in long columns. The results also showed that the bearing capacity of
the stone column is affected by the undrained shear strength of the
surrounding soil, the spacing distance between columns and the installation of
granular mat at the top of the columns.
Mckelvey et al. (2004) carried out a series of laboratory model tests on a

consolidated clay bed using two different materials:
1. Transparent material with clay like properties prepared by mixing

fumed silica in an oil blend of mineral spirits and crystal light liquid
paraffin.
2. Speswhite Kaolin clay.
The tests on the transparent material permitted visual examination for the
deforming of the stone column during loading. For the Foundation loading on
transparent material samples, three sand columns, 25 mm in diameter, were
installed in a triangular pattern beneath a circular footing, 100 mm in
diameter, and also in a row arrangement beneath a strip footing to depths of
150 mm and 250 mm. For the foundation loading of Kaolin clay tests, four
sand columns, 25 mm in diameter, were installed in a square pattern beneath a
model pad footing, 90 mm x 90 mm, to depths of 150 mm and 250 mm.
The presence of the granular columns significantly improved the load-carrying

capacity of the soft clay. However, columns with length to diameter ratio (L/d)
more than 6 seem to show further increase in the load capacity. The results of
the tests showed that columns can fail in 3 different ways: bulging, punching
and pending. Punching is more prevalent in short columns whilst bending
21
failure is predominant in perimeter columns located beyond the center of the

footing. Bulging was more generally common in long columns, as shown in
figure 2-7. Beneath the rigid footing, the central column in the stone columns
group deformed or bulged uniformly, while the edge columns bulged away
from the neighboring columns, as shown in Figure 2-8.
Figure 2-8 – Photos of sand column beneath circular footing at beginning,

middle and end of foundation loading process: (a) 150 mm length; (b) 250 mm
length (After Mckelvey et al., 2004)
Ambily and Gandhi (2007) carried out a detailed experimental study on the
behavior of single column and group of seven columns by varying parameters
like spacing between the columns, undrained shear strength of the clay, angle
of internal friction of the stones and the loading type. Laboratory tests were
22
carried out on a column of 100 mm diameter surrounded by soft clay of

different consistencies. The tests were performed with both loading the entire
equivalent area and loading the stone column only. During the experimental
tests, the actual stress on the column and the clay were measured by fixing
pressure cells in the loading plate. FEM analyses were then performed by
using axisymmetric analyses. The numerical results were compared with the
experimental results which showed good agreement. The following
conclusions were drawn based on this study:
1. When the column area alone was loaded, the failure was by bulging
with maximum bulging at depth of about 0.5 times the diameter of
the stone column.
2. As spacing increases, the axial capacity of the column decreases and

the settlement increases up to spacing ratio (S/d) of 3, beyond which
the change is negligible.
3. The load-settlement behavior of a unit cell with an entire area loaded

is almost linear and it is possible to find the stiffness of the improved
ground as shown in Figure 2-9.
4. Single column tests with an entire unit cell area loaded compared
well with the group test results. Hence, the single column behavior
with unit cell concept can simulate the field behavior for an interior
column when a large number of columns are simultaneously loaded
as shown in Figure 2-10.
23
5. Stiffness improvement is found to be independent on the shear

strength of clay and depends mainly on column spacing and the
friction angle of the stone column material.
Figure 2-9 – Stress-settlement behavior under entire area loading (After

Ambily and Gandhi, 2007)
Figure 2-10 – Comparison of group column test and single column test (After
Ambily and Gandhi, 2007)
24
Black et al. (2007) conducted an experimental study in which samples of soft

kaolin clay, 100 mm in diameter and 200 mm in height, were reinforced with
vertical columns of sand, 32 mm in diameter with 120 mm and 200 mm
depths, were tested under triaxial conditions. It was found that the undrained
shear strength of samples containing full-depth columns was greatly improved
compared with that of the unreinforced samples.
Andreou et al. (2008) performed triaxial compression tests on specimens of

non-reinforced and reinforced soft kaolin clay with granular columns. The
specimens were 200 mm high and 100 mm diameter while the column
diameter was 20 mm. The results showed that the response of a soft
foundation soil reinforced by granular columns to vertical loading is highly
dependent upon the drainage conditions, the material of the stone column and
the loading rate of the soil.
2.4.1.3 Theoretical studies
Laboratory research, testing and field studies undertaken over the last years
have contributed to the understanding of the conventional stone column
behavior. This has led to the development of empirical, analytical and
numerical techniques used to assess column capacity and load-settlement
behavior. In the following section, a brief description of the design methods
used for assessment of settlement reduction is introduced.
Greenwood (1970) introduced design curves to asses settlement reduction

associated with the use of conventional stone columns. The empirical curves
were derived from column groups placed under widespread loads on uniform
soft soil. They represent settlement reduction as a function of column spacing
25
and the undrained shear strength of the natural soil (For c u = 20 kPa and 40
kPa). Later, Greenwood and Kirsch (1983) presented updated curves as a
function of area ratio, as illustrated in Figure 2-11.
Figure 2-11 – Modified Greenwood curves (After Greenwood and Kirsch,

1983)
Priebe (1976) proposed a method for assessing settlement reduction based on

the unit cell, elastic theory and Rankine earth pressure theory. In this model,
the stone column was assumed to be incompressible and surrounded by an
elastic material. Soil settlement occurred when lateral pressure in the column
exceeded the confining pressure in the surrounding soil. Priebe generated a
series of design curves where the basic settlement improvement factor was
plotted against the area ratio for a range of granular materials. The
improvement factor is the ratio between the settlement of the untreated and
treated soil. Later Priebe (1995) presented a revised version which included
compressibility, modular ratio of column and soil, confinement from
overburden pressure and solutions for single and strip footings. An example of
these modified design curves is presented in Figure 2-12.
26
Figure 2-12 – Modified Priebe design curves (After Priebe, 1995)
Balaam and Booker (1985) studied the settlement of a rigid foundation

supported by a layer of clay stabilized with stone columns. The results of an
analytic solution for the settlement, assuming no yield occurs in the clay or the
columns, were presented. Later, this solution was used to develop an
interaction analysis, which considered yielding within the stone columns. The
solutions were obtained from the analysis of a unit cell. In order to check the
validity of these assumptions elastoplastic FEM analyses were performed and
the agreement between the two methods was very good. The results were
plotted as a settlement correction factor, which is the ratio between actual
settlement and elastic settlement of the foundation. The results showed that the
most significant reduction in settlement occur when the columns are closely
spaced and the column-soil modular ratio is high.
Pulko and Majes (2006) introduced a simple analytical method for the analysis
of stone-column reinforced foundations. The stone-column and the
surrounding soil are modelled as a unit cell, consisting of elastic soil and rigid
plastic column material according to the Mohr-Coulomb failure law. The
dilation of the column material according to the Rowe stress-dilatancy theory
27
is directly incorporated into the method. The closed-form solution method is

used for the prediction of the effects of stone-columns on settlement reduction
and stresses in the soil and column. A parametric study is also presented to
study the influence of area replacement ratio and material properties of the
granular material on settlement reduction factor. The study showed the
significant effect of the dilatancy of granular material on the settlement
reduction and stress concentration.
2.5 Consolidation rate of soft soil reinforced with stone columns
Field observations, according to a number of publications, showed that stone

columns accelerate the consolidation rate in the soft soil. The acceleration in
the consolidation rate is accredited to the stone column for providing a
drainage path and relieving excess pore water pressure by transferring load
from the surrounding soft soil to the stone column.
Baron (1947) proposed a solution which dealt with the consolidation of fine
grained soil by vertical drain. The average degree of consolidation in the radial
direction is
[ ]
Equation 2.7
Where [ ] Equation 2.8
(N) Is defined as the diameter ratio which is the ratio between the diameter of
a drain well (dw) and the diameter of its influence zone (d e); (Tr) is the
consolidation time factor for radial flow, respectively as shown in Figure 2-13.
28
This solution dealt with the consolidation of fine-grained soils by vertical

drains. Stone columns and vertical drains have two major differences. First,
stone columns have larger drainage ability. Baron’s solution ignored the effect
of the stiffness difference between the vertical drain and the surrounding soil
on the consolidation rate. However, the stone columns are much stiffer than
vertical drains and carry a substantial part of the applied load. Second, the
stone columns have a smaller diameter ratio than vertical drain wells. Typical
diameter ratio for stone columns ranges from 1.5 to 5. However, the values for
well diameter ratios used in this method ranged from 5 to 100.
Han and Ye (2001) presented a simplified method for computing the rate of
consolidation of the soft soil around stone columns considering stiffness ratio,
although stone columns and surrounding soil were assumed linearly elastic in
their study while in reality they have a nonlinear behavior. Stone columns act
as drain wells where vertical and radial flows are similar to those of Terzhagi
one dimensional solution and the Barron solution for drain wells in fine
grained soils, respectively. The following relationship is still applicable to
calculate time rate and settlement of the stone column improved ground:
Equation 2.9
Where,
Urv is the combined degree of consolidation for both radial and vertical
drainage.
Ur is the degree of consolidation for radial flow only.
Uv is the degree of consolidation for vertical flow only.
29
Figure 2-13 – Definition of terms for modelling (After Han and Ye, 2001)
Thus, an approximate solution can be obtained as follows:
[ ] [ ]
Equation 2.10
Where,
, a modified time factor for the radial flow.
, a modified time factor for the vertical flow; (H) is the vertical
drainage path.
And the modified coefficients of consolidation are as follows:
( ) Equation 2.11.a
( ) Equation 2.11.b
30
Where (N) is the diameter ratio; steady stress concentration ratio (n s) is the
ratio between steady stress in column (σcs) and steady stress in soil (σss) at the
end of consolidation.
The new solution demonstrates stress transfer from the soil to the stone
columns and dissipation of excess pore water pressure due to the drainage and
vertical stress reduction during consolidation. Ignoring consolidation due to
vertical flow, the calculated average total stress on the soil and columns for
case N = 3 and ns = 5 are plotted in Figure 2-14. This figure, demonstrates that
the stress on columns increase with time, while the stress in soil decreases.
This stress transfer from the soil to the columns is called “stress
concentration”.
The stress concentration can also be presented in terms of the stress

concentration ratio, as shown in Figure 2-15. The stress concentration ratio
increases with time and approaches the steady-stress concentration ratio (ns =
5 in this case). This proposal method indicated the general trend that the
steady-stress concentration ratio increased with the applied loads .at larger
loads than the yield load of the stone columns, the steady-stress concentration
started to decrease.
31
Figure 2-14 – Vertical stress on soil and columns with time (After Han and
Ye, 2001)
Figure 2-15 – Stress concentration ration with time (After Han and Ye, 2001)
In Han and Ye (2001) study however, no lateral displacement was assumed in

the theoretical development. Therefore, the dissipation of the excess pore
water pressure depends mainly on two factors, drainage and reduction of
vertical stresses. The dissipation of excess pore water pressure, due to the
32
vertical stress reduction, is about 40% of the total dissipation for this special
case, as shown in Figure 2-16. The contribution of vertical stress reduction to
the dissipation of excess pore water pressure explains why stone columns are
more effective than drain wells in accelerating consolidation rate of the soft
clays.
Since the coupled effect of deformation and dissipation of pore water pressure
has been incorporated in the FEM analyses such as Plaxis program which uses
Biot’s (1941) system of differential equations solved by integration over time,
the stress concentration effect is automatically taken in consideration in the
Plaxis analysis of the unit cell of the stone column-soft soil. (Malarvizhi and
Ilamparuthi, 2007).
Figure 2-16 – Dissipation of excess pore water pressure (After Han and Ye,
2001)
Bergado and Long (1994) presented the use of the FEM for embankment
simulations. Based on revised Cam clay model for 2-D consolidation analysis,
33
two test embankments where constructed on soft Bangkok clay improved by

granular piles and vertical drains. The embankments have 4m height. The soft
clay which has 8 m depth is over-lained by a medium stiff clay layer. In 2-D
plane strain model, the vertical drains and granular piles were converted into
continuous walls. The analysis results showed the granular piles imply more
acceleration of consolidation and more reductions in the total settlement of the
soft clay than vertical drains, as shown in Figure 2-17.
Figure 2-17 – Total settlement-time relationship of reinforced soft clay by (a)

granular piles; (b) Vertical drain (After Bergado and Long, 1994)
2.6 Stone columns-soft soil reinforcement system under embankment
Terzaghi and Peck (1967) stated that the instability of embankment

constructed on soft soil foundations is mainly of two types: a) where the
embankment sinks into the foundation soils and b) failure by spreading.
Hence, stone columns reinforced soils as embankment foundation have been
used as a more effective method to prevent sinking and spreading failure.
Therefore, this technique improves the performance of the embankment over it
34
by increasing shear strength and bearing capacity as well as decreasing

consolidation settlement and lateral displacement of the soft foundation soil.
Cheung and Wagner (1998) investigated the construction of the South Eastern
Arterial Road project in Auckland, a 4 m high geogrid reinforced embankment
with steep side slope (1H:4V). The embankment is underlain by soft ground
and is in close proximity to three adjacent structures. The site is underlain by a
22 m thick layer of alluvium clay. Stone columns were installed along the
edges of the embankment to strengthen its foundations and reduce the
influence of ground displacement on the adjacent buildings. A layer of woven
geotextiles was placed at the base of the embankment. Stone columns of 90
cm diameter were arranged in four rows distanced 2 m center to center around
the perimeter of the embankment. Wick drains were used in the central part of
the embankment where stone columns were not installed. The embankment
was constructed in two stages and the embankment settlement responses were
monitored.
Measurements of settlement gauges installed in the non-reinforced zone of the

embankment indicated ground settlement of 100 mm under a 2 m high light
weight fill and the embankment settlement rate became very small within 6
month after construction. Within the stone columns reinforced zone, ground
settlements of 40 mm to 70 mm were measured under the 4m high
embankment load. At a distance of about 2 m outside the embankment area,
no detectible ground deformation was recorded.
Saroglou et al. (2008) presented the ground improvement using stone columns
for the construction of a new high way road from Keratea to Lavrio, in Attika
35
peninsula, Greece. An embankment of maximum height of 3 m was

constructed on subsoil comprises of soft clays of low plasticity with
intercalations of silty to clayey sands of medium density with gravel. Using
stone columns with a depth of 14 m reduced the total settlement from 14 cm to
7 cm and accelerated consolidation time from 16 months to a period of 4
months.
Borges et al. (2009) conducted a parametric study to investigate the influence

of several factors on the behavior of soft soils reinforced with stone columns
under embankment loads. Parameters such as replacement ratio, deformability
of the column, thickness of the soft soil, deformability of the fill and friction
angle of the column material were examined. The confined axisymmetric
cylindrical unit cell was used. The analysis was performed by a finite element
program that incorporates the Biot consolidation theory. The results confirmed
that increasing replacement area ratio or stiffness of the column material
significantly reduces settlements and horizontal displacement and accelerates
the consolidation.
2.7 Discussion
Previous researches especially in the last two decades discussed the

improvement mechanism of soft soil with stone columns. Although the studies
were performed by using experimental small and somewhat large models, the
full scale reflects the real behavior of the studied cases. The previous
researches also contained FEM analyses and analytical solutions of the stone
columns reinforced soft soil. Although most of the used FEM analyses and
analytical solutions included elastic behavior for the soft soil and the column
36
material, the behavior of these materials are elastoplastic. Most of the past
studies were performed in undrained conditions in spite that the long term
stability is more important when dealing with soft soil. The experimental and
theoretical studies indicated that stone columns are most often used in soft
clay soil to reduce and accelerate ground settlement and to increase the
bearing capacity of the soft ground.
The bearing capacity of the improved site is governed by the degree of the
lateral bulging of the stone column that occurs during loading. Some
researchers studied the stone columns bulging but the variation of bulging
with column diameter and spacing between columns especially in long term
was not explained. The previous researchers discussed that the stress
concentration and load transferring within stone columns increase with
increasing stone column stiffness, but the stress concentration in the stone
columns and reducing stress in the surrounding soil correlation with the
applied loads along consolidation process need to be illustrated. The role of
the stress concentration on acceleration of the consolidation needs also to be
outlined.
37
Chapter )3(
Numerical Modeling
3.1 Introduction
Implementation of various constitutive relationships into finite element

method facilitated the study of numerous problems in geotechnical
Engineering. Finite element procedure is a powerful tool that can analyze
problems incorporating non uniform geometry and complicated boundary
conditions. In this thesis, 3D, plane strain and axisymmetric finite element
models were used to analyze the behavior of soft soil reinforced with stone
columns using PLAXIS finite element code.
In this chapter, a brief description of the finite element method, the different
constitutive laws in geomechanics and the utilized material models
formulations are illustrated. Also, the numerical models used in this study are
discussed in detail.
3.2 Finite element method
3.2.1 General
An analytical solution is a mathematical expression that gives the value of the

unknown quantity at any location in a body. Analytical solutions are valid for
infinite locations in the body, but it can be used only for simplified problems.
38
Chapter (3) Numerical Modeling
For complicated problems involving heterogeneous, anisotropic, nonlinear

stress-strain relationship, non-uniform geometrical shapes, and complicated
boundary conditions; the numerical methods are convenient. The numerical
methods provide approximate values of the unknown quantities at a certain
number of discrete points in the body. Several numerical methods have been
developed, such as boundary element method, finite difference method, and
finite element method.
The finite element method is applicable to a wide range of boundary value

problems. The development of this method began in the late 1950s, where the
key concepts of the stiffness matrix and element assembly existed in the form
used now. Three approaches are usually used to solve finite element problems
in the soil mechanics field:
1. Displacement method, where the displacements are assumed as primary

unknown quantities.
2. Equilibrium method, where the stresses are assumed as primary

unknown quantities.
3. Mixed method, where some displacements and some stresses are

assumed as primary unknown quantities.
3.2.2 Analysis procedure of finite element method
The finite element analysis can be summarized in the following steps:
1. Continuum discretization. The continuum is divided into an equivalent

system of finite elements.
39
2. Selection of the displacement functions. The displacement functions are

used to approximate the variation of the actual displacements over each
finite element. Displacement functions may be polynomials, or
trigonometric.
3. Derivation of the element stiffness matrix. The stiffness matrix contains

the material and geometric properties of elements. The stiffness matrix
[k] relates the nodal displacements to the applied nodal forces, as
follows:
{ } [ ]{ } Equation 3.1
Where {Q} is the nodal force vector and {u} is the nodal displacement
vector.
4. Assembly of the equations for the entire discretized continuum. This

step includes the assembly of the global stiffness matrix [K] for the
entire continuum from the individual element stiffness matrices, and
global force vector {F} from the element nodal force vectors.
{ } [ ]{ } Equation 3.2
Where {U} is the nodal displacement vector for the entire body.
5. Calculation of the unknown displacements. The equations assembled in

step 4 are solved for the unknown displacements.
40
3.2.3 Elements shapes
A finite element has a one, two, or three dimensional configurations. In each

configuration elements have different shapes.
1. One dimensional element: one dimensional element is used for

structures that can be idealized by a line, as shown in Figure 3-1. The
line ends are called the nodal points.
2. Two dimensional elements: Figure 3-2 shows the different shapes of

two dimensional elements, such as triangular, rectangular, and
quadrilateral elements.
3. Axisymmetric elements: Three dimensional problem can be represented

by a two dimensional axisymmetric configuration in certain cases. The
finite element used for these problems are circular rings, called toroidal
elements. The cross section of the toroidal element is triangular, or
quadrilateral, as presented in Figure 3-3.
4. Three dimensional elements: Figure 3-4 illustrates the different shapes

of three dimensional elements. A tetrahedron corresponds to the
triangular element in the two dimensional configuration. Also the
rectangular prism, and hexahedron correspond to the rectangle, and
quadrilateral elements used in two dimensional configuration.
Figure 3-1 – One-dimensional element (After Rao, 2005)
41
Triangle Rectangle Quadrilateral Parallelogram
Figure 3-2 – Two-dimensional elements (After Rao, 2005)
One dimensional axisymmetric Two dimensional axisymmetric

(Shell) element (Toroidal) element
Figure 3-3 – Axisymmetric elements (After Rao, 2005)
42
Tetrahedron Rectangular Prism Hexahedron
Figure 3-4 – Three-dimensional elements (After Rao, 2005)
3.2.4 Two dimensional simulation of special three dimensional problems
In some situations, the geometry and loading scheme of a three dimensional

problem can be reduced to an equivalent two dimensional configuration.
3.2.4.1 Plane strain
A plane strain problem, as shown in Figure 3-5, involves a long body with
uniform geometry and loading in the longitudinal direction (z). Practical
examples of these configurations are strip footings, tunnels, retaining walls.
Displacements and strains in the longitudinal direction are zero. The stress in
the longitudinal direction (σz) can be expressed as in terms of (σx) and (σy) as:
Equation 3.3
Where (σx) and (σy) are the normal stresses in x and y directions respectively;
(υ) is the soil’s poisons ratio.
43
3.2.4.2 Plane stress
A plane stress problem, as presented in Figure 3-6, involves a body with very
small dimensions in z direction. The shear stress components τ yz, τzx vanish on
the surface, and σz is zero. The shear strain components γyz, γzx vanish on the
surface. The normal strain component in longitudinal direction (ε z) is
expressed in terms of εx, εy as:
Equation 3.4
Where (εx) and (εy) are the normal strains in x and y directions respectively.
3.2.4.3 Axisymmetric problems
An axisymmetric problem, as shown in Figure 3-7, involves circular

structures, or circular loading. Deformations and stresses are assumed to be
identical in any radial direction. In this case a cylindrical coordinate system (r,
z, and ϴ) is used. In this configuration the shear strains and stresses in rϴ, ϴz
planes are zero.
Figure 3-5 – Examples for Plane strain problems
44
Figure 3-6 – Example for Plane stress problems
Figure 3-7 – Examples for axisymmetric problems
45
3.3 Material Modeling Basics
3.3.1 General
Material model is a set of mathematical equations that describe the

relationship between stress and strain called Constitutive relationships. They
have multiple forms depending on the studied material behavior, and loading
technique. Material models can be expressed in the form of a relationship
between stresses and strains, or a relationship between stress rates and strain
rates.
3.3.2 Stresses
The state of stress in any element of a loaded continuum can be expressed in

terms of six independent components of stress in the Cartesian coordinate
system.
{ } [ ] Equation 3.5
Where {σ}T is the transpose of the stress tensor, σx, σy, σz are the components
of the normal stress and τxy, τyz, τzx are the components of shear stress.
According to Terzaghi’s principle, stresses are divided into effective stresses

(σ'), and pore water pressures (μ).
Equation 3.6
Water has no shear resistance, and therefore effective shear stresses of soil are
equal to total shear stresses. In soil material models, stresses are usually
expressed in the form of incremental effective stresses.
46
For any stressed element, there are three mutually orthogonal planes on which
the shear stress components are zero. These planes are called the principal
planes. The stresses acting on these planes are called the principal stresses.
Principal stresses are the eigenvalues of the stress tensor.
Stress invariants are stress measures which are independent of the direction of
the coordinate system. Two forms of stress invariants are the isotropic mean
stress (p) and the deviatoric stress (q).
Equation 3.9.a
Equation 3.9.b
√( ) ( ) ( )
Equation 3.10.a
√( ) ( ) ( )
Equation 3.10.b
3.3.3 Strains
For every stress component, there is a corresponding strain component. The

state of strain at a point can be expressed in terms of six components in the
Cartesian coordinate system.
{ } [ ] Equation 3.11
47
Where (εx, εy and εz) are the normal strain components, while (γxy, γyz and γzx)
are the shear strain components.
In the soil material models, strains are usually expressed in the form of strain
rates.
{ } [ ] Equation 3.12
Principal strains corresponding to the principal stresses can be expressed by

the following strain vector
{ } [ ] Equation 3.13
Strain invariants analogous to the pre mentioned stress invariants are the
volumetric strain (εv), and the deviatoric strain (εq).
Equation 3.14.a
Equation 3.14.b
√ Equation 3.15.a
√ Equation 3.15.b
3.4 Constitutive Material Models
Soil is a natural complicated material. It is an elastoplastic, nonlinear,

anisotropic, and time dependent material. Soil stiffness is variable and affected
by different factors, such as the stress level, strain level, loading rate, loading
48
direction, and stress path. Soil has a preloading memory which influences its
response to external loads.
Numerous constitutive laws have been formulated to simulate the soil material
behavior. There is no available soil constitutive model capable of describing
the intricate soil behavior under all conditions. The ideal soil model is the one
which can match the real soil behavior for a large extent for a certain studied
soil problem. In the following, the different constitutive soil models will be
illustrated.
3.4.1 Linear elastic constitutive law
The simplest constitutive law, it is based on Hook’s law of linear elastic

stress-strain relationship.
{ } [ ]{ } Equation 3.16
Where {dεe} is the incremental elastic strain vector and [D e] is the elastic
material stiffness matrix.
This constitutive relationship is inappropriate to model the soil behavior. It

could be used to model stiff materials, such as concrete or steel structural
elements, where plasticity has no effect.
3.4.2 Nonlinear elastic constitutive laws
Nonlinear elastic constitutive law is considered as an extension of linear

elastic relationship using infinitesimal linear intervals. Many nonlinear elastic
soil models have been developed, like hyper-elastic and hypo-elastic models.
49
The hyper-elastic model states that the current stress is a function of current
strain only and does not depend on the material deformation history. The
hyper-elastic constitutive relationship general form is:
{ } Equation 3.17
{ }
Where (w) is the strain energy function which is independent of the load path.
In the hypo-elastic models, the incremental stress and strain vectors are
linearly related by a variable tangent material stiffness matrix whose moduli
are function of the stress or the strain state. Hypo-elastic models are stress
path dependent.
3.4.3 Elastoplastic constitutive laws
In elasto-plasticity, the strain and strain rates are divided into an elastic part
(εe) and a plastic part (εp). Elastic strains are calculated based on Hook’s law.
{ } { } { } Equation 3.18.a
{ } { } { } Equation 3.18.b
Three functions are necessary for full description of elastoplastic constitutive

laws: a) yield function, b) flow rule, c) hardening law.
Hill (1950) stated that, a yield criterion is a law which defines the limit of
elasticity under any possible combination of stresses. It shows if the soil
undergoes plastic deformations or not. The yield function (f) is represented by
a curve in 2D stress space, and surface in 3D stress space. The intersection of
50
the yield surface with the π plane, whose equation is σ 1 = σ2 = σ3 = 0, is called

the yield locus as shown in Figure 3-8.
Figure 3-8 – Geometrical representation of the yield criterion in the principal

stress space (After Hill, 1950)
Any stress point inside the yield surface represents an elastic stress state. A
stress point that lays on the yield surface represents a yielding state and plastic
irreversible deformations occur. A stress point outside the yield surface is an
impossible stress state according to perfect plasticity behavior. Different
common yield surfaces are presented in Figure 3-9.
Flow rule is a relationship between the yield surface and the direction of the
vector of plastic strain. A plastic potential surface is an envelope orthogonal to
all the plastic strain vectors. An associated flow rule (normality condition)
means that the plastic strain vector is normal to the yield surface, in other
words the plastic potential surface coincides the yield surface, as illustrated in
Figure 3-10.
51
(a) Von-Mises (b) Tresca
(c) Mohr-Coulomb (d) Drucker-Prager
(e) Lade-Duncan
Figure 3-9 – Different yield criterion (After Chen and Mccarron, 1986; After
Chen and Mizuno, 1990)
52
Figure 3-10 – Associated flow rule (After Atkinson, 1993)
Figure 3-11 – Schematic representation of the yield surface and the flow rule
(After Kempfert and Gebreselassie, 2006)
53
Brinkgreve (2002) showed that incremental plastic strains (dε p) can be

calculated using the following equation
{ } Equation 3.19
{ }
Where (g) is the plastic potential function, (λ) is the plastic multiplier; λ is
zero for pure elastic state as given by Equation (3.20.a) and positive in the
case of plastic behavior as expressed by Equation (3.20.b).
[ ]{ } Equation 3.20.a
{ }
[ ]{ } Equation 3.20.b
{ }
The following relationship between the effective stress rates and strain rates
for elastoplastic behavior can be obtained using equations (3.20.a, b)
according to Smith and Griffith (1982), and Vermeer and De Borst (1984).
{ } ([ ] [ ] [ ]) { } Equation 3.21
{ } { }
[ ] Equation 3.22
{ } { }
Where the parameter (α) is zero for elastic behavior, and equals one for
plasticity.
Elastoplastic models are classified into two forms: a) Perfect plasticity, b)

hardening plasticity. A perfect plasticity model has a fixed yield surface which
is not affected by plastic straining. Perfect plasticity models formulations are
based on normality condition. Tresca, Von-Mises, Mohr-Coloumb, and
54
Drucker-Prager are perfect plasticity models. Figure 3-12 shows the stress-
strain relationship for a perfect plasticity model.
In a hardening plasticity model, the yield surface is not fixed. A hardening law
is defined as the relationship between the increase in the yield stress and the
plastic strain. Lade-Duncan, double hardening soil models are hardening
plasticity models. Strain hardening could be formulated by three methods:
isotropic hardening, kinematic hardening, and mixed hardening. Isotropic
hardening rule means that the yield surface expands uniformly without any
change in its shape, as presented in Figure 3-13. In the kinematic hardening
rule, the yield surface translates as a rigid body without change in its size,
shape, and orientation, as shown in Figure 3-14. Mixed hardening rule is a
combination of the isotropic and kinematic hardening.
3.4.4 Elasto-visco plasticity constitutive laws
Elasto-visco plasticity constitutive relationships are used to model time

dependent behavior of the soil due to the viscous properties of the soil such as,
creep and relaxation. The total strain could be written as:
{ } { } { } Equation 3.23
Where {dεvp} is the visco-plastic strain rate
55
Figure 3-12 – Perfect plasticity model (After Brinkgreve, 2002)
Figure 3-13 – Isotropic hardening rule (After Chakrabarty. 2006)
56
Figure 3-14 – Kinematic hardening rule (After Chakrabarty, 2006)
3.5 Finite element modeling program used in this research
3.5.1 General
The finite element method has been applied to geotechnical engineering

problems, since 1960’s, having been developed a decade earlier for
applications in structural engineering and continuum mechanics. The name
finite element was, however, first mentioned in a paper by Clough (1960), in
which the technique was presented for plane stress analysis. Since then, a
large amount of research has been devoted to this technique and many
research papers and text books have been published on this subject. The
method is now firmly established as an engineering tool of wide applicability.
The main advantage of the method is that it can be applied to the materials
exhibiting non-linear stress-strain behavior. In the current research the
behavior of soft soil reinforced with ordinary stone columns under
57
embankment loads was studied using the finite element program Plaxis
Version 8.2.
Development of Plaxis began in the nineteen seventies at the Technical

University of Delft as an initiative of the Dutch Department of Public Works
and Water Management. The initial brief was to develop and easy-to-use finite
element code for the analysis of a river embankment on soft soil of low lands
in Holland. In subsequent years, Plaxis was extended to cover most other areas
of geotechnical engineering.
Plaxis is a finite element package specially intended for the analysis of

deformation and stability in geotechnical engineering projects as it provides
the advanced constitutive models required to simulate the nonlinear time-
dependent behavior of soil. The program can be used in plan-strain and
axisymmetric modeling as well as three dimensional modeling. In the
following sections a brief description of the different parts of Plaxis program
while highlighting the different options used in this study.
3.5.2 Input program
In the Input program of the Plaxis the geometry of the problem is given by
entering different soil layers, structural parts, external loads, etc. a choice
between various available material models, such as Linear Elastic Model,
Mohr-Coulomb, Hardening soil, etc., is made at the input for each material.
The material is given relative material properties, such as stiffness and density,
which differs according to the used material model. Appropriate boundary
conditions are then assigned to the whole model. When the model is complete,
58
a mesh is automatically generated and initial stress and pore water pressure are
initiated before moving to the Calculation program.
3.5.2.1 Soil elements
During generation of the mesh, soil clusters are divided into triangular
elements. Plaxis provides two types of triangular elements, 6-nodes elements
and 15-nodes elements, as shown in Figure 3-15. During the finite element
calculations, displacements are calculated at those nodes. On other hand, stress
is calculated at individual points called stress points rather than at the nodes. A
15-nodes triangular element contains 12 stress points while a 6-nodes
triangular element contains 3 stress points. In this research a 15-nodes
triangular element was used.
Figure 3-15 – Distribution of nodes and stress points in Plaxis finite elements
(After Plaxis version 8 manuals)
59
3.5.2.2 Types of soil behavior
An important feature of the soil is the presence of pore water. Pore water
pressure significantly influences the soil response. To enable incorporation of
the pore water influence in the soil response Plaxis offers for each model a
choice of three types of behavior:
1. Drained behavior where no excess pore water pressure is generated.

This behavior is used for dry soils and also for soil types providing full
drainage due to high permeability, as in sand, or due to low rate of
loading. This option can also be used to model the long-term behavior
of soil without the need to model the precise history of the undrained
loading and consolidation.
2. Undrained behavior in which a full development of excess pore water

pressure is present. This occurs when a soil has low permeability, as in
clay, or under a high rate of loading. The undrained behavior is usually
followed by consolidation in loading phases.
3. Non-porous behavior is used in modeling of rock or structural elements

such as concrete. In this behavior neither initial nor excess pore water
pressure is taken into account.
During this research the undrained soil behavior was used in simulating both
the soft and stiff clay layers as they had very low permeability also to be able
to study both the short-term and the long-term behavior of the soft soil when
reinforced via stone columns. Drained soil behavior was used in simulating
60
the stone columns, the embankment and the crust as they have high
permeability.
In Plaxis, it is possible to specify undrained behavior in an effective stress

analysis using effective soil parameters, as Plaxis distinguishes between
effective stresses and excess pore pressures. The advantage of using effective
strength parameters in undrained conditions is that the increase of shear
strength with consolidation is automatically calculated. Thus, the soft clay and
the stiff clay layers are modeled using the undrained behavior while imputing
the effective strength parameters. The undrained shear strength of soil is
calculated in according to equation 4.1. This equation can be deduced from
Figure 3-16
Equation 3.24
Figure 3-16 – Mohr’s circle of stress used to drive relation between undrained
shear strength and drained shear parameters (After Brinkgreve, 2002)
61
3.5.2.3 Boundary conditions
Boundary conditions are used to describe the fixities for the boundaries of the
problem geometry. In this research the boundary conditions were set by fixing
the vertical boundaries in the horizontal direction only (U x = 0) while allowing
displacement to take place in the vertical direction; while a fixation of both
vertical and horizontal displacement (Ux = Uy = 0) for the lower horizontal
boundary of the problem.
3.5.2.4 Mesh Generation
Plaxis uses unstructured mesh, which is generated automatically with options

for global and local mesh refinement. Plaxis provides several options of mesh
density ranged from very coarse to very fine mesh. In this research medium
mesh size was chosen. Mesh was then refined in zones which stresses and
strains are expected to be high i.e. the soil area surrounding the stone columns,
the crust layer and the embankment body.
3.5.2.5 Initial conditions
Once the geometry of the model has been created and the finite element mesh
has been generated, the initial conditions of the problem are specified. Plaxis
provides two modes required to specify the initial conditions:
1. Water condition mode is used to generate the initial pore water pressure
where the phreatic line is specified and the boundary conditions for the
ground water flow are presented.
62
2. Initial geometry configuration mode is used to generate the initial

effective stresses in the soil body by specifying the initial geometry of
the problem and the initial properties of the soil clusters.
3.5.3 Calculation
After generation of a finite element model, calculation can be executed. Both

calculation type and loading type has to be specified in this step.
3.5.3.1 Types of calculations
Choices between different ways of analysis of the actual problem are mode in
the calculation program. Distinction is made between three basic types of
calculations:
1. Plastic calculation should be selected to carry out an elastic-plastic

deformation analysis in which it is not necessary to take excess pore
water pressure with time into account.
2. Consolidation analysis should be selected when it is necessary to

analyze the development or the dissipation of excess pore water
pressure in water-saturated clay-type soils as a function in time. Plaxis
allows for true elastic-plastic consolidation analyses.
3. Phi-c reduction (safety analysis) can be executed by reducing shear

parameters. A safety analysis can be performed after each individual
calculation phase and thus for each construction stage to calculate the
factor of safety.
63
In the current research consolidation analysis was used to be able to detect the
behavior of the stone column-soft soil settlement with time. Also, a safety
analysis was performed after all stages to study the effect of the stone column
reinforcement on the stability of slope for the embankment.
3.5.3.2 Loading types
In order to simulate the problem accurately, the method in which the external
load is applied to the soil mass must be specified, the following types of
loading can be applied in most finite element softwares:
1. Staged construction is the most important type of loading. In this Plaxis

feature, it is possible to change the geometry and load configuration by
deactivating or reactivating different loads, volume clusters and
structural objects as created in geometry input. Staged construction
enables an accurate and realistic simulation for various loading,
construction and excavation processes.
2. Total multipliers is used to specify the ultimate values of external loads.

When the total multipliers loading is selected, the ultimate values of
external loads will be applied exactly at the end of calculation.
3. Incremental multipliers is selected when the external load is applied

incrementally. Before entering a load increment, an increment of time
can be entered. Increments of time are not relevant when using plastic
calculation except when time-dependent models are used. The input of
time increments is essential when using consolidation analysis.
64
In this research staged construction type was used to simulate the construction
of the embankment over the reinforced soft clay. Also, the incremental
multipliers is used with different time increments to simulate the post
construction consolidation process.
3.5.4 Output
When the calculations are completed the results can be viewed in the output
program. A large amount of data can be obtained from finite element
calculation such as stresses, pore water pressure and displacements.
3.6 The Mohr Coulomb model
The Mohr Coulomb model is an elastic perfectly plastic model. As explained

before in this chapter, the yield surface of the elastoplastic model with perfect
plasticity is a fixed surface. The yield surface of the Mohr Coulomb model, as
shown in Figure 3-17, is fully defined by model parameters and not affected
by plastic straining. The Mohr Coulomb model requires a total of five
parameters, which are generally familiar to most geotechnical engineers and
which can be obtained from basic tests on soil samples. These parameters are
briefly explained in the following part.
65
Figure 3-17 – Mohr-Coulomb yield criterion
3.6.1 Young’s Modulus
Plaxis uses the young’s modulus (E) as the basic stiffness in the elastic model
and the Mohr Coulomb model. A stiffness modulus has the dimension of
stress. The values of the stiffness parameter adopted in a calculation require a
special attention as many materials show nonlinear behavior from the
beginning of loading. For soils, the initial slope is usually indicated as (E 0),
and secant modulus at 50 % strength is donated (E 50), as shown in Figure 3-
18. For materials with a large linear elastic range it is realistic to use E 0, but
for loading of soils E50 is generally used. Considering case of unloading
problems, as in tunnels and excavation, the unloading reloading elasticity
66
modulus (Eur) is used instead of E50. Plaxis also offers a special option of
layers in which the stiffness increases with depth.
Figure 3-18 – Definition of E0 and E50 for standard drained triaxial test results
(After Brinkgreve, 2002)
3.6.2 Poisson’s ratio
The selection of a Poisson’s ratio (υ) is particularly simple when the elastic
model or the Mohr Coulomb model is used for gravity loading. For this type
of loading Plaxis should give realistic ratios of as both models
will give the well-known ratio . For one-dimensional
compression it is easy to select Poisson’s ratio that gives a realistic value of
K0; hence, υ is evaluated by matching K0. In many cases the value of
Poisson’s ratio is ranged between 0.3 and 0.4, however, it is in the range of 0.5
in the case of undrained behaviour.
67
3.6.3 Shear strength parameters
The cohesive strength of soil (c) has the dimension of stress. Plaxis can handle
cohesion-less soils (c = 0), but some options will not perform well. Plaxis also
offers a special option of layers in which cohesion can increase with depth.
The friction angle of soil (φ) is entered in degrees. The friction angle largely
determines the shear strength by means of Mohr’s stress circle, as shown in
Figure 3-19. The Mohr Coulomb failure criterion proves to be better for
describing soil behavior than the Druker-Prager approximation, as the latter
failure surface tends to be highly inaccurate for axisymmetric configurations.
Figure 3-19 – Mohr-coulomb failure envelope with one Mohr failure circle
(After Brinkgreve, 2002)
68
The dilatancy angle (ψ) is specified in degrees. A part from heavily over-
consolidated layers, clay soils tend to show little dilatancy (ψ = 0). The
dilatancy of sand depends on both the density and the friction angle.
69
Chapter )4(
Case Study
4.1 Introduction
As established in past researches, using stone columns to reinforce soft soil

deposits increases the bearing capacity, reduces the settlement and accelerates
the consolidation and construction time. During this research, the behavior of
embankments, built on soft soil deposits reinforced using stone columns, is
studied using numerical modeling. Numerical modeling needs to be verified in
order to make sure that the estimated soil parameters and the used constitutive
law represent the real behavior of soil. A case study with full monitored
settlement data is used to verify the numerical modeling. The verified
numerical model is then used to get better understanding of the performance
of soil deposits improved via stone columns.
In this chapter, the case study description is presented including its location
and geometry. The soil profile and parameters for the different soil layers, the
embankment and the used stone columns are also explained. In addition,
settlement values monitored during and after the construction of the
embankment are introduced. A comparison between the response of three
dimensional finite element analyses of the embankment and the insitu
measurements is presented. Two dimensional finite element analyses using
axisymmetric and plane strain configurations are also performed. Finally, a
comparison between the behavior of the reinforced soil deposits in the three
70
Chapter (4) Case Study
dimensional analyses and the different configurations for the two dimensional
analyses is studied.
4.2 Case study description
The finite element simulation has been applied for the modeling of an
embankment construction for Penchala Toll Plaza project at New Pantai
Expressway, Malaysia, in 2003. A brief description of the project was given
by Tan et al. (2008). The embankment geometry and the stone column
reinforced soil profile are shown in Figure 4-1 having a line of symmetry on
the left boundary. The 40 m wide and 1.8 m high embankment is filled by
sandy material. The embankment lies over a 6 m layer of soft clay underlain
by an extended layer of stiff clay. The stone columns, arranged in a square
grid 2.4 m x 2.4 m, extends through the entire soft clay layer. The upper crust
layer is a 1 m thick fill of hard soil, which was provided as a replacement of
soft clay surface to improve ground for a stable construction platform and to
distribute the load on the treated soil uniformly. The groundwater level is 1 m
below the ground surface. Table 4-1 presents the different parameters for all
soil layers, as well as, the embankment and the used stone columns. The
embankment was constructed over 3 stages, each stage involved a 3 days
construction of a 0.6 m height layer of the embankment, and thus the
embankment was built over entire construction duration of 9 days. As the
stone columns extends through the entire soft clay layer and ends at the top of
the stiff clay layer which has a very high elasticity modulus, the stone columns
in this case can be considered as end bearing columns. Also, the columns can
be considered as short columns as there entire length is 6 m only.
71
20.00 m
(+1.80)
SP1 SP2 2H
: 1V
Embankment Fill
(±0.00)
Crust (-1.00)
X Soft Clay X
(-6.00)
0.8 m Diameter Stone Columns at spacing 2.4 m (Square grid)
Stiff Clay
ELEVATION
2.40
2.40
0.80
300 m
SEC. (X-X)
Figure 4-1 – Case study profile (modified after Tan et al., 2008)
Two settlement plates, SP1 and SP2, were installed in situ to measure the
settlements at the center of the embankment and at 8 m from its edge during
and a long time after construction. The field measurements for the settlement
at SP1 and SP2 for a total time period of 90 days are shown in Figure 4-2.
72
Table 4-1 – Material Parameters for Case study

b / sat E’ c’ φ’
Material υ’ Kh (m/s) Kv (m/s)
(kN/m3) (kPa) (kPa) (deg)
Embankment 1.16 x 1.16 x
18 / 20 0.3 15000 3 33
Fill 10-5 10-5
3.47 x 1.16 x
Crust 17 / 18 0.3 15000 3 28
10-7 10-7
3.47 x 1.16 x
Soft Clay 15 / 15 0.3 1100 1 20
10-9 10-9
3.47 x 1.16 x
Stiff Clay 18 / 20 0.3 40000 3 30
10-9 10-9
Stone 1.16 x 1.16 x
19 / 20 0.3 30000 5 40
Column 10-4 10-4
Figure 4-2 – Stress and settlement at (SP1) and (SP2) (modified after Tan et
al., 2008)
73
4.3 Back analysis of the case study
The back analysis of the case study was performed using a three dimensional
finite element model. The model is built using the finite element code
PLAXIS 3D 2012.
4.3.1 Constitutive law and soil parameters
During the back analysis of the case, all the different soil layers including the
soft clay layer as well as the embankment and the stone columns are modeled
using Mohr-Coulomb model.
As mentioned previously, it is possible in Plaxis to specify undrained behavior

in an effective stress analysis using effective soil parameters. The advantage
of using effective strength parameters in undrained conditions is that the
increase of shear strength with consolidation is automatically calculated using
equation 3.24. Thus, the soft clay and the stiff clay layers are modeled using
the undrained behavior while imputing the effective strength parameters. The
embankment, stone columns and the crust layer are all modeled using the
drained behavior and their effective parameters. All the soil parameters used
to model the different soil layers as well as the embankments and the stone
columns are presented in table 4-1. All the parameters used are the same as the
parameters presented by Tan et al. (2008).
4.3.2 Model geometry and boundary conditions
Because the embankment extends for a long distance (over 300 m) in the
longitudinal direction, a slice can be used to present the whole embankment.
The used slice has a width of 2.4 m with the stone columns at its centerline.
74
The width of the modeled slice is based on the fact that the stone columns are
arranged in a square grid with 2.4 m spacing in both directions. The
longitudinal extents of the model, at Y=0 and Y=2.4 m, are considered to be
axes of symmetry, thus, representing the entire length of the embankment. For
the horizontal direction, the model extends to a distance of 20 m after the end
of the embankment to ensure that the boundary conditions at the extents of the
model does not affect the behavior of the reinforced soft soil beneath the
embankment. Although the stiff clay layer at the bottom extends for a large
depth, a 4 m depth of this layer was found to be enough to represent it without
affecting the accuracy of the model.
Figure 4-3 – Geometry of the 3D finite element model
The boundary conditions for all the vertical boundaries (X=0, X=40 m, Y=0
and Y=2.4 m) were set by allowing for the vertical displacement while
preventing the horizontal movement from taking place. Moreover, for the
lower horizontal boundary of the model (Z=0) both the vertical and horizontal
75
movements are not allowed. Finally, the flow of water during the
consolidation process is prevented through all the vertical and horizontal
boundaries of the model. The geometry and boundary conditions of the
numerical model as well as the generated finite element mesh are presented in
Figures 4-3 and 4-4 respectively.
Figure 4-4 – Generated Finite element mesh
76
4.3.3 Comparison between back analysis results and the field

measurements
A comparison between the results of the finite element model and the field
measurements for the case study is performed to verify the accuracy of the
used model. The comparison is performed through the settlement readings
taken at settlement plates (SP1) and (SP2) during and after the construction
process. The comparison between the model results and the field
measurements is done for a total time period of 90 days after which the field
settlement measurements are almost constant indicating that the consolidation
process is finished. Figure 4-5 show the settlement values for the numerical
model and the field measurements at both (SP1) and (SP2). The comparison
shows a very good agreement between the model and the field measurements.
Figure 4-5 – Settlement at (SP1) and (SP2) for field measurements and FEM
77
For settlement plate (SP1) the settlement values seem to coincide during the
first 9 days (construction period) as the settlement values at the end of this
period reached 30 mm and 27 mm for the model and field readings
respectively i.e. a 10% difference. The model readings seems to have higher
values for settlement than the field measurements for the time period from 10
days to 55 days, however, the difference did not exceed a 15% value. During
the period from 55 days till 90 days, the model gives settlement values less
than the field measurements. Settlement values of 75 mm and 77 mm are
recorded for the model and the field respectively with a difference of about
3%.
For settlement plate (SP2) the settlement values seem to coincide during the
first 9 days (construction period) as the settlement values at the end of this
period reached 38 mm and 36 mm for the model and field readings
respectively i.e. a 6% difference. The model readings seems to have higher
values for settlement than the field measurements for the time period from 10
days to 50 days, however, the difference did not exceed a 15% value. During
the period from 50 days till 90 days, the model gives settlement values less
than the field measurements. Settlement values of 77 mm and 79 mm are
recorded for the model and the field respectively with a difference of about
3%.
Generally, a good agreement is reached between the results of the three

dimensional model and the field measurements for settlement at both (SP1)
and (SP2) which in turn verifies both the numerical model and the soil
parameters used within the model.
78
The three dimensional finite element model is also used to study the
dissipation of the excess pore water pressure and the development of the shear
strength for the soft clay layer at points (A) and (B). Figures 4-6 and 4-7
display the excess pore water pressure and shear strength respectively.
Figure 4-6 - Excess pore water pressure at points (A) and (B) using the 3D
model
79
Figure 4-7 - Soil shear strength development at points (A) and (B) using the
3D model
4.4 Numerical modeling using 2D FE analyses
Due to the fact that the two dimensional finite element analyses programs are
more commonly used in practice than the three dimensional finite element
analyses program in solving different geotechnical engineering problems,
there is a need to convert the 3D problem to a 2D configuration.
Axisymmetric and plane strain 2D configurations are two options that can be
used to simulate a 3D problem using a 2D finite element model.
An axisymmetric problem involves circular structures or circular loading

where deformations and stresses are assumed to be identical in any radial
direction. While, a plane strain problem involves a long body with uniform
geometry and loading in the longitudinal direction.
80
The behavior of stone columns can be examined using the Unit cell concept
(i.e., a cylindrical cell with the stone column at the center and surrounded by
the soft soil within its effective diameter) which can be perfectly simulated
using the axisymmetric configuration. However, the unit cell concept is not
suitable to investigate the behavior of the embankment itself, especially when
studying the stability of the embankment slope; thus, the plan strain
configuration is more suitable in such cases. In the plane strain configuration,
the stone columns are modeled as continuous walls instead of discreet
columns, thus, affecting the results of the entire problem. Two approximations
can be used to overcome this problem:
1. Simulating the stone columns as walls having the same width as the
columns diameter, while changing the different parameters of the walls
with a set of equivalent parameters.
2. Keeping the parameters of the simulated walls the same as those of the
original stone columns but changing the width to an equivalent width.
2D FEM configuration
Axisymmetric
Plane strain model
model
Using equivalent
parameters
Using equivalent
width
Figure 4-8 – 2D finite element analyses configurations
81
During the following sections, three configurations are investigated.

Axisymmetric, plane strain with equivalent width and plane strain with
equivalent parameters configurations are used. Settlement at settlement plates
(SP1) and (SP2), excess pore water pressure dissipation and the shear strength
increase at points (A) and (B) as well as the embankment slope stability for
these three configurations are compared to the results of the three dimensional
finite element analysis. Figure 4-9 illustrates the position of (SP1), (SP2), (A)
and (B).
Figure 4-9 – Embankment profile
Mohr-Coulomb constitutive laws are used to model all different soil layers as
well as the embankment and the stone columns for all the two dimensional
finite element analyses used during the following part of the research. The soft
clay and stiff clay layers are modeled in the undrained conditions while using
their effective strength parameters, while, the stone columns, embankment and
the crust layer are modeled in a drained condition.
82
The boundary conditions for all two dimensional models are similar to those
of the three dimensional finite element model. The horizontal movement is
prevented while allowing the vertical movement for all the vertical
boundaries. For the bottom horizontal boundary both the vertical and
horizontal displacements are prevented. The water flow is prevented through
all the vertical and horizontal boundaries of the models. The stiff clay layer
was presented as a layer with 4 m depth only. The model extends to a distance
of 20 m beyond the embankment width for both configurations of the plane
strain models.
4.4.1 Axisymmetric model
The axisymmetric model is used to simulate the unit cell concept of the stone
column. Cross section of the model is shown in Figure 4-10. The radius of the
model is determined according to the influence diameter of the stone column.
The influence diameter depends on the columns arrangement pattern and the
distance between the columns. The influence diameter for a square pattern is
1.13 times the spacing between the columns. For this case the columns are
arranged with spacing of 2.4 m, thus, the radius of the model is 1.36 m. Figure
4-11, shows the generated finite element mesh of the model. The settlement at
(SP1) and the excess pore water pressure dissipation at point (A) are shown in
Figure 4-12 and 4-13, respectively. The increase in shear strength of the soft
clay layer is shown in Figure 4-14. The slope stability of the embankment is
not investigated due to the previously mentioned limitations in this model
type.
83
Figure 4-10 – Geometry and Figure 4-11 – Generated Finite

boundary conditions for the element Mesh for the axisymmetric
Axisymmetric model model
84
Figure 4-12 – Settlement at SP1 using Axisymmetric model
Figure 4-13 – Excess pore water pressure at point (A) using axisymmetric
model
85
Figure 4-14 – Soil shear strength development at point (A) using axisymmetric
model
4.4.2 Plane strain model using equivalent width
In this model, the columns are simulated as continuous walls having

equivalent width while maintaining the same parameters of original stone
column mentioned previously in table 4-1. The equivalent width of the walls is
calculated according to equation 4.1 reported by Tan et al. (2008) which is
based on retaining a constant ratio between the area of stone columns and the
area of the surrounding soil for both the Axisymmetric and the plane strain
configurations.
Equation 4.1
86
Where (bw) is the equivalent width of the wall, (r c) is the radius of the stone
columns, (re) is radius of the influence zone or the unit cell and (S) is the
spacing between the columns.
equivalent width Finite element model
width Finite element model
87
Figure 4-15 shows the geometry and boundary conditions for the used model,
while, the generated finite element mesh is shown in Figure 4-16. Results are
presented through Figure 4-17 showing the settlement occurring at (SP1) and
(SP2), and, Figure 4-18 and Figure 4-19 shows the excess pore water pressure
dissipation as well as the development of the soil shear strength at points (A)
and (B) respectively.
width Finite element model
88
with equivalent width Finite element model
strain with equivalent width Finite element model
89
4.4.3 Plane strain model using equivalent parameters
In this model, the columns are simulated as continuous walls having a width
of 0.8 m which is the same as original diameter of the stone columns. The
parameters of the wall used to simulate the stone column are replaced by a set
of equivalent parameters as shown in table 4-2. The equivalent parameters are
calculated according to equation 4.2 which in cooperates the parameters of the
stone column and the soil according to their areas relative to the area of the
wall.
Equation 4.2
Where Xeq., Xc and Xs are the equivalent parameter, column parameter and soil
parameter respectively; Ac and As are the column and the in between soil areas
respectively; dc is the stone column diameter and S is the spacing between
columns.
Table 4-2 – Stone columns parameters (equivalent parameters plane strain

model)
b / sat E c’ φ’
Material υ Kh (m/s) Kv (m/s)
(kN/m3) (kPa) (kPa) (deg)
Stone 3.04 x 3.04 x
16 / 16.3 0.3 8666 2 25
Column 10-5 10-5
90
equivalent parameters Finite element model
parameters Finite element model
Figures 4-20 and 4-21 shows the geometry and boundary conditions of the
used model and the generated finite element mesh, respectively. The
settlement at (SP1) and (SP2), also, the pore water pressure dissipation at
points (A) & (B) are shown in Figures 4-22 and 4-23, respectively. The
development of the soft soil shear strength at points (A) and (B) is illustrated
in Figure 4-24.
91
Parameters Finite element model
with equivalent parameters Finite element model
92
strain with equivalent parameters Finite element model
4.5 Comparison between the 3D and 2D FE analyses
A comparison between the behaviors of the three dimensional and two

dimensional analyses of the case study is introduced. The settlement, the
excess pore water pressure, the development of shear strength with time and
the stability of the embankment slope are evaluated for all the finite element
models. As the three dimensional finite element model is calibrated with the
field measurements through the back analyses of the case study, this model is
then used as a guide for the comparison between the different two dimensional
modeling configurations.
93
4.5.1 Settlement
Figure 4-25 shows the settlement at (SP1). The plain strain using equivalent
parameters analysis predicts higher values for settlement, during the
construction and early stages after, than all the other numerical simulations;
the settlement value reaches 39 mm at the end of the construction period of 9
days compared to values of 30 mm, 29 mm and 35 mm for the three
dimensional, Axisymmetric and plane strain with equivalent width analyses
respectively. However, the plane strain with equivalent parameters analysis
gives lower value for the final settlement (65 mm) when compared to the
predicted settlement of the 3D, Axisymmetric and equivalent width analyses
(75, 78 and 83 mm respectively); this is due to the fact that the diaphragm
stone walls in this case have a higher stiffness value ,relative to this of the soft
clay, when compared to the discreet columns in the 3D or the axisymmetric
analyses and to the diagram stone walls ,having a much smaller width, in the
plane strain with equivalent width analysis.
The settlement behavior at (SP2) is similar to that at (SP1) as shown in Figure

4-26. The equivalent parameters configuration reaches higher settlement
values during the earlier stages (48 mm at the end of construction) when
compared to the 3D and equivalent width analyses (38 and 47 mm at the end
of construction respectively). However, the settlement values at later stages
are lower for the equivalent parameters configuration (73 mm for the final
settlement) when compared to the 3D and equivalent width analyses (77 and
89 mm for the final settlement respectively).
94
Figure 4-25 – Comparison of settlements at (SP1)
Figure 4-26 – Comparison of settlements at (SP2)
95
Generally, as seen in table 4-3, the axisymmetric analysis gives a very good
agreement with both the 3D analysis and the field measurements throughout
the entire consolidation process. Moreover, the plane strain with equivalent
width analysis predicts higher final settlement values while the plane strain
with equivalent parameters analysis predicts lower final settlement values.
Table 4-3 – Comparison of settlement

Settlement at SP1 Settlement at SP2
(mm) (mm)
Case t=9 days t=90 days t=9 days t=90 days
(End of (After End on (End of (After End on
construction) consolidation) construction) consolidation)
Field
27 77 36 79
measurements
3D FEM 30 75 38 77
Axisymmetric 29 78 ------- -------
Plane strain
(equivalent 35 83 47 89
width)
Plane strain
(equivalent 39 65 48 73
Parameters)
4.5.2 Excess pore water pressure
The excess pore water pressure readings at points (A) and (B) through a total
period of 90 days are shown in Figures 4-27 and 4-28 respectively. The
axisymmetric finite element analysis could not be used to investigate point (B)
which is a downside for this configuration type as it cannot be used to study
the behavior of the soft clay layer beyond the embankment.
96
Figure 4-27 – Comparison of Excess pore water pressure at point (A)
Figure 4-28 – Comparison of Excess pore water pressure at point (B)
97
At point (A), the axisymmetric and the plane strain with equivalent width
simulations gives a higher value of excess pore water pressure during the
construction process and early stages after construction reaching values of 17
and 18 KPa (at the end of construction) when compared to the three
dimensional analysis which predicts a value of 15 KPa, However, at the later
stages the two configurations almost coincides with the 3D analysis. The
plane strain with equivalent parameters analysis predicts lower values for the
excess pore water pressure throughout the entire period reaching a maximum
value of 13 KPa at the end of construction.
Similarly, At point (B), the plane strain with equivalent width simulations
gives a higher value of excess pore water pressure during the construction
process and early stages after construction reaching a maximum value of 3.5
KPa when compared to the three dimensional analysis which predicts a value
of 3.3 KPa, Moreover, at the later stages this configurations almost coincides
with the 3D analysis. The plane strain with equivalent parameters analysis
predicts lower values for the excess pore water pressure throughout the entire
period reaching a maximum value of 3.1 KPa.
As shown in previous figures for the different finite element modeling

configurations, the dissipation of the excess pore water pressure takes place
during and after construction. This is noticeable as the maximum values for
the excess pore water pressure reached at the end of construction is less than
the loads applied to reinforced soil system by the embankment which is about
32.4 kPa.
98
Generally, the axisymmetric and the plane strain with equivalent width
analysis shows a relatively good agreement with the three dimensional
analysis. However, the plane strain with equivalent parameters analysis
predicts lower values for the excess pore water pressure. Another comment is
that the plane strain with equivalent parameters analysis shows a much higher
rate for the consolidation process, as mentioned before, this can be associated
to the fact that the radial flow paths is smaller in this configuration which in
turn speeds up the consolidation process. A summary for the values of excess
pore water pressure for different models is show in table 4-4.
Table 4-4 – Comparison of excess pore water pressure

Maximum Excess pore water Maximum Excess pore water
Case at Point (A) at Point (B)
(kPa) (kPa)
3D FEM 15 3.3
Axisymmetric 17 ------
Plane strain
(equivalent 18 3.5
width)
Plane strain
(equivalent 13 3.1
Parameters)
4.5.3 Development of soil shear strength
The development in the soft clay layer shear strength is evaluated at points (A)
and (B) over a period of 90 days as shown in Figures 4-29 and 4-30
respectively. The axisymmetric finite element analysis could not be used to
investigate point (B) as explained before.
99
Figure 4-29 – Comparison of shear strength development at point (A)
Figure 4-30 – Comparison of shear strength development at point (B)
100
At point (A), the shear strength before beginning of construction (t=0)

predicted by all the different 2D analyses almost coincide at a value of 23 kPa
while the 3D analyses gives a value of 24 kPa at the same time. However, at
the end of construction (t=9 days) both the 3D analyses and the plane strain
analyses using equivalent column parameters predict a higher value for the
shear strength (28 kPa) when compared to the axisymmetric and the plane
strain using equivalent column width analyses which predicts a value of about
26 kPa. Moreover, after the end of consolidation process both the 3D analyses
and the axisymmetric analyses gives a shear strength value of about 38 kPa,
which is less than the value of 39 kPa given by both of the plane strain
configurations.
The effect of the embankment construction on the shear strength development

at point (B) is almost insignificant .The shear strength of the soft clay layer
increase from a value of 22 kPa at t=0 days to a value of 23 kPa after the end
of consolidation process as given by the 3D analyses. The plane strain using
equivalent column width analyses and the plane strain using equivalent
column parameters analyses both show the shear strength increases from a
value of about 23 kPa before construction to a value of about 24 kPa after the
end of consolidation which is slightly higher than the value given by the 3D
analyses, however, the difference is about 4% between them.
Generally, the axisymmetric analyses show similar behavior to the 3D

analyses. However, the plane strain using equivalent parameters analyses
shows a higher rate of increase in the shear strength at early stages after
construction reaching a constant shear strength value before all the other
analyses types. Moreover, the plane strain using equivalent column width
101
analyses takes a longer period of time after construction to reach the final
shear strength when compared to the other analyses. A summary for the
development of shear strength for the different models is show in table 4-5
Table 4-5 – Comparison of Shear strength development for soft clay

Shear strength at point (A) Shear strength at point (B)
(kPa) (kPa)
consolidation
consolidation
construction
construction
Case
t=0
t=0
End
End
End
End
of
of
of
of
3D FEM 24 27.6 38.2 22.3 22 23
Axisymmetric 22.8 25.7 37.2 ----- ----- -----
Plane strain
(equivalent 22.9 25.5 39.3 22.5 22.5 23.7
width)
Plane strain
(equivalent 22.8 27.8 38.7 23 23 24.1
Parameters)
4.5.4 Stability of the embankment slope
The long term stability of the embankment is investigated using the three
dimensional analysis and the plane strain analyses using equivalent width as
well as equivalent parameters. The embankment height is varied using values
that ranged from 1.8 m to 5 m in order to investigate the difference between
the behaviors of the three models under different stress levels. A disadvantage
of the axisymmetric model is its inability to investigate the safety of the
embankment slope. A comparison of the factor of safety for the embankment
slope using the different analyses types is shown in Figure 4-31. The two
102
different configurations for the plane strain analyses predicts an almost equal
factor of safety under embankment heights of 3m or higher, while the
equivalent width configuration predicts a slightly higher safety factor of 2.4 at
an embankment height of 1.8 m when compared to the equivalent parameters
safety factor of 2.2. Generally, both two dimensional configurations seems to
give lower values than the three dimensional analysis, however, the difference
between them did not exceed 12% at 1.8 m embankment height and decreased
to 6% at embankment height of 5 m. Figure 4-32 shows the failure mode for
the 2 m height embankment using the plane strain with equivalent column
parameters configuration which illustrates that the failure surface pass through
the soft clay layer and intersects with the stone columns.
Figure 4-31 – Stability of embankment slope for different finite element

models
103
Figure 4-32 –- Failure mode of embankment slope (H=2m)
104
Chapter )5(
Parametric Study
5.1 Introduction
As mentioned previously during the review of past literatures concerning the

stone columns as a ground improvement technique, the stress concentration
and the load transferring within the stone column needs further study.
Studying the effect of various parameters on the stress concentration and the
load transferring within the stone column helps in having a better
understanding of the behavior of the soft soil deposits reinforced via stone
columns. During this chapter a parametric study is performed to monitor the
effect of various parameters on different aspects of the behavior of the soft soil
deposits reinforced using stone columns.
The performed parametric study is based on the case study previously

discussed in Chapter (4). It is performed using the finite element analyses
calibrated during the previous chapter. As established previously, the
axisymmetric model gives the best agreement with the behavior of the three
dimensional finite element analysis, thus, the axisymmetric model is used to
perform the parametric study. Three different parameters are studied during
this parametric study, column spacing, the stress level and the modular ratio.
The effect of these parameters on the behavior of the reinforced soft soil
deposits is monitored through four different aspects, the stress concentration
105
Chapter (5) Parametric Study
factor, the column load share ratio, the reduction of settlement and the
reduction in consolidation time.
5.2 Definitions
5.2.1 Modular ratio
The modular ratio (Mr) is defined as the ratio between Young’s modulus of
elasticity of the stone columns material and Young’s modulus of elasticity of
the surrounding soil. Equation 5.1 shows the Modular ratio definition.
Equation 5.1
5.2.2 Stress concentration factor
As shown in Equation 5.2, the stress concentration factor (SCF) is the ratio
between the stresses transferred to the stone columns and the stresses
transferred to the native soil.
Equation 5.2
5.2.3 Column load share ratio
The Column load share ratio (αL), as illustrated in equation 5.3, is defined as
the ratio between the load carried by the stone columns and the total applied
on the reinforced soil system.
Equation 5.3
106
5.2.4 Settlement reduction factor
It is defined as the ratio between the settlement of the soil deposits treated via
stone columns and the settlement of the untreated soil deposits. A definition of
the settlement reduction factor (αs) is shown in equation 5.4.
Equation 5.4
5.2.5 Time reduction factor
As illustrated in Equation 5.5, Time reduction factor (αt) is defined as the ratio
between the time required to reach 90% of the consolidation process for the
soil deposits reinforced using stone columns and the untreated soil deposits.
Equation 5.5
5.3 The effect of column spacing
The effect of column spacing is studied by varying the ratio between the
column diameter and the spacing between columns. The different spacing to
diameter ratios used ranges from a value of 2 to 6 based on the common
values used in various cases. During the study of the column spacing effect,
the stone column diameter is established at the value of 0.8 m which is the
original column diameter of the case study. The stress level on the reinforced
soil deposits is kept constant by preserving the embankment height at its
original value of 1.8 m. The value of elasticity moduli for the different soil
deposits, the embankment and the stone columns are the same as the case
study previously shown in table 4-1.
107
The effect of varying the columns spacing to diameter ratio on the stress
concentration factor is shown in Figure 5-1. It is found that the stress
concentration factor increases from a value of 1.7 at spacing to diameter ratio
of 2 to a value of 2.1 at spacing to diameter ratio of 4, however, this value
seems to be constant for spacing to diameter ratios from 4 to 6. This is because
the load transferred to the column due the arching effect that takes place
within the embankment increases as the spacing between column increases,
however, after a certain spacing (in this case at S/d=4) the load transferring to
the column reaches its highest value and does not seem to increase which in
turn makes the SCF almost constant. The effect of columns spacing to
diameter ratio on the column load share ratio is shown in Figure 5-2. The
column load share ratio decreases from a value of 24.5 % at spacing to
diameter ratio of 2 to a value of 4.8 % at spacing to diameter ratio of 6. This
can be contributed to the fact that although the stress transferred to the column
increase as the column spacing increase, the ratio between the column and the
soil areas increases radically which in turn increases the load ratio transferred
to the soil.
The effect of increasing the column spacing on the reduction in settlement and
the reduction of consolidation time are shown in Figures 5-3 and 5-4
respectively. The reduction of settlement for the reinforced soil increase from
39% to 93% when increasing the spacing to diameter ratio from 2 to 6, while
the reduction in time of the consolidation process increase from a value of 1%
to a value of 16% for the same increase in column spacing ratio. This shows
that the variation of the column spacing ratio has a much greater effect on the
108
reduction of settlement when compared to the reduction of the consolidation

time.
Figure 5-1 – Effect of columns spacing on the stress concentration factor
Figure 5-2 – Effect of columns spacing on the pile load share ratio
109
Figure 5-3 – Effect of columns spacing on the settlement reduction factor
Figure 5-4 – Effect of columns spacing on the time reduction factor
110
5.4 The effect of Stress level
As the stress acting on the reinforced soil system in the case study is due to an
embankment load, the variation of the stress level through the parametric
study is done by changing the embankment height. The embankment height
during the parametric study is taken with the values of 1 m, 2 m, 3 m, 4 m, 5
m and 6 m, these values correspond to total stress levels of 18 kPa, 36 kPa, 54
kPa, 72 kPa, 90 kPa and 108 kPa respectively. The diameter of the stone
columns and the spacing between them is 0.8 m and 2.4 m respectively, while
the elasticity moduli for all the soil layers, the embankment and the stone
columns are taken according to table 4-1. The effect of varying the
embankment height on the stress concentration factor, column load share ratio,
settlement reduction factor and consolidation time reduction factor are shown
through Figures 5-5 to 5-8 respectively.
The stress concentration factor (shown in Figure 5-5) was found to decrease
from a value of 2.1 at embankment height of 1 m (18 kPa) to a value of 1.6 at
embankment height of 6 m (108 kPa). This can return to the fact that the
increase of the stress level acting on the reinforced soft soil deposits increases
the confinement provided to the stone columns which in turn increases the
columns capacity and the stresses transferring to the columns, however, the
increase in the stresses transferred to the stone columns is a minimum when
compared to the increase of the total stresses transferring to the reinforced soil
system, thus, the ratio between the two stress levels decreases as the stress
level acting on the reinforced soil system increases. The same reason can be
applied to explain the decrease of the column load share ratio (shown in
111
Figure 5-6) from a value of 16.5% at an embankment height of 1 m to a value

of 11.8% at an embankment height of 6 m.
Figure 5-5 – Effect of stress level on the stress concentration factor
Figure 5-6 – Effect of stress level on the Column load share ratio
112
As shown in Figure 5-7, the reduction in the settlement of the reinforced soft
soil deposits increases from a value of 59% to a value of 80% when increasing
the embankment height from 1 m to 6 m. This can be associated with the
increase of the stress levels transferring to the soft soil, thus, increasing the
occurring settlement and in turn decreasing the reduction in settlement ratio.
However, the increase of the stress levels acting on the reinforced soft soil
deposits seems to be insignificant concerning the reduction in the
consolidation process duration as the time reduction ratio increased with a
value less than 1% when the stress level increased from 18 kPa to 108 kPa as
shown in Figure 5-8.
Figure 5-7 – Effect of stress level on the settlement reduction factor
113
Figure 5-8 – Effect of stress level on the time reduction factor
5.5 The effect of modular ratio
The modular ratio is defined as the ratio between Young’s elasticity moduli of
the stone columns and the soft soil deposits (E column/Esoil). During the
parametric study the diameter of the stone columns is taken at a value of 0.8
m, the spacing between the columns is kept at a value of 2.4 m and the
embankment height is at 1.8 m. The variation in the modular ratio during the
study is done by increasing the value of the elasticity modulus of the stone
columns while retaining the value of the elasticity modulus for the soft clay
layer at 1100 kPa. The values chosen for the elasticity modulus of the stone
columns varies from 11000 kPa to 77000 kPa which corresponds to a modular
ratio of 10 to 70 respectively. The effect of the modular ratio on the stress
concentration factor, column load share ratio, settlement reduction factor and
114
consolidation time reduction factor are shown through Figures 5-9 to 5-12
respectively.
Increasing the modular ratio was found to increase the stress concentration
factor and the column load share ratio, as shown in Figure 5-9 and 5-10
respectively. The increase of the modular ratio from 10 to 70 is accompanied
by an increase of the stress concentration factor and the column load share
ratio from values of 1.2 and 11.1% to values of 2.4 and 16.9% respectively.
This increase is explained by the fact that increasing the modular ratio leads to
an increase in the stone columns stiffness when compared to that of the soft
soil deposits, thus, higher stresses and more loads are transferred to the
column instead of the soft soil.
Figure 5-9 – Effect of Modular ratio on the stress concentration factor
115
Figure 5-10 – Effect of Modular ratio on the Column load share ratio
The effect of the modular ratio on the settlement reduction factor and the time
reduction factor are shown in Figures 5-11 and 5-12 respectively. As
explained previously, the increase in the modular ratio increases the relative
stiffness of the stone columns which leads to less loads and less stresses being
transferred to the soft soil. This can explain the decrease in the settlement
reduction factor from 71% to 68% as the modular ratio increased from 10 to
70. However, the effect of the modular ratio has proven to be insignificant
regarding the reduction in the consolidation process duration as the value of
the time reduction factor stayed almost constant at 3% despite the increase in
the modular ratio.
116
Figure 5-11 – Effect of Modular ratio on the settlement reduction factor
Figure 5-12 – Effect of Modular ratio on the time reduction factor
117
Chapter )6(
Comparison with analytical methods
6.1 Introduction
As mentioned previously in chapter (2), many researches presented simplified

analytical methods that can be used to predict the behavior of soil deposits
reinforced via stone columns. These methods can be used to estimate the
reduction in settlement of the soft soil deposits, the acceleration in
consolidation time and the stress and load share between the columns and the
surrounding soil. During this chapter, the results of the parametric study
previously performed using the calibrated finite element model are compared
to the behavior of the stone column improved soil estimated by some of the
theoretical methods discussed during the literature review to study the
compatibility of these theoretical approaches.
6.2 The stress concentration factor (SCF)
A comparison between the stress concentration factor given by the numerical

modeling and that calculated using different theoretical approaches is
performed. The analytical methods of Preibe (1995), Shahu et al. (2000) and
Pulko and Majes (2006) are used. A brief description of these approaches is
previously discussed in chapter (2). Figures (6-1) to (6-3) show the variation
of the stress concentration factor with embankment height, spacing between
columns and the modular ratio respectively.
118
Chapter (6) Comparison with analytical methods
It is noticed that the theoretical method of Preibe (1995) always estimates a

very high value for the SCF ranging between 70 and 80 which, according to
the previous researches mentioned in literature review as well as the numerical
study performed previously, is not logical as the stress concentration factor
usually lies within the range of 2 to 10. This can return to the large number of
inaccurate assumptions proposed by Preibe (1995) such as the assumption that
the stone column material is linear elastic and uncompressible which is not
true as the stone column has elastoplastic behavior, also, this is due to the fact
that the Preibe method gives the SCF at the mid height of the soft soil layer
while the SCF estimated by the numerical analyses is at the top of the column.
As a result, the shown Figures only show a comparison between the numerical
analyses and the theoretical approaches of Shahu et al. (2000) and Pulko and
Majes (2006) in order to be able to focus properly on the comparison between
them.
Although, the analytical method of Shahu et al. (2000) predicts a SCF higher
than that estimated through the numerical modeling, the values of the SCF
predicted by this method lies within the range of 1.5 and 9.5 which is a
reasonable value. The increase of the values of the SCF can return to the
assumption that no lateral bulging occurs in the column (column is
constrained) which in turn increases the capacity of the column, also a linear
elastic behavior is adopted for the stone column which is not accurate as
explained before. When examining the effect of the variation of spacing
between columns on the SCF, it is found that the behavior predicted by Shahu
et al. (2000) is similar to that of the numerical study as the SCF estimated by
this method increases from a value of 3.8 to a value of 4.1 for a spacing to
119
diameter ratios of 2 and 6 respectively, which corresponds to an increases

from a value of 1.7 to a value of 2.1 for the numerical analysis with a
difference of about 55% between the two approaches. However, when
increasing the embankment height (i.e. stress level increases) this approach
gives an increase in the SCF from a value of 3.5 to a value of 5.5 for
embankment heights of 1m and 6m respectively, which contradicts with the
decrease from a value of 2.1 to a value of 1.6 (at the same embankment
heights) estimated by the numerical analyses. Moreover, the relation between
the modular ratio and the SCF calculated using the approach of shahu et al.
(2000) seems to be linear as the SCF increases from a value of 1.5 to a value
of 9.5 when increasing the modular ratio from 10 to 70, this is not compatible
with the results of the numerical study as the SCF increases from a value of
1.2 to a value of 2.4 for modular ratios 10 and 70 respectively, which in
increases the difference between the two methods from 20% to 75% when
increasing the modular ratio from 10 to 70 respectively.
The analytical approach of Pulko and Majes (2006) gives higher values for the
SCF than the values estimated by the numerical analyses as the values of the
SCF lies in the range between 5 to 6, which is reasonable as explained before,
however, the difference between the two approaches is in the range of 40% to
70%. Moreover, the approach by Pulko and Majes (2006) ignores the effect of
the modular ratio and the embankment height (i.e. stress level) on the SCF as
it gives a value of 5.3 for the SCF regardless to the modular ratio and the
stresses applied to the reinforced soil system which is not accurate according
to the numerical study.
120
Figure 6-1 – The effect of embankment height on the stress concentration

factor using numerical modeling and theoretical approaches
Figure 6-2 – The effect of column spacing on the stress concentration factor
using numerical modeling and theoretical approaches
121
Figure 6-3 – The effect of Modular ratio on the stress concentration factor
6.3 The column load share ratio
The Load share ratio estimated by the theoretical approaches of Preibe (1995),
Shahu et al. (2000) and Pulko and Majes (2006) is compared to the results of
the performed numerical analyses. The results of the comparison are shown in
Figures (6-4) to (6-6). Due to the same reasons explained before, the column
load share ratio estimated by the theoretical approach of Preibe (1995) is much
higher than that estimated by the numerical analyses as well as all the other
numerical approaches. The values given by Preibe (1995) method ranges
between 61% and 95% while that estimated using the numerical analyses
range between 5% and 25%, however, the variation of the column load share
ratio with column spacing, embankment height and modular ratio is similar for
both analyses.
122
Figure 6-4 – The effect of embankment height on the column load share ratio
Figure 6-5 – The effect of column spacing on the column load share ratio
123
Figure 6-6 – The effect of the modular ratio on the column load share ratio
The performed study using Shahu et al. (2000) shows that the column load
share ratio increase from a value of 25% to a value of 35% when increasing
the embankment height from 1m to 6m. This contradicts with the findings of
the numerical analyses which show that the load share ratio decreases from
17% to 12% for the same embankment heights, thus, the difference between
the two methods increases from 32% at embankment height of 1m to 66% at
embankment height of 6m. However, the behavior of the column load share
ratio is similar for the two methods with the variation of the spacing between
columns, as the theoretical approach of Shahu et. al (2000) gives a decrease of
the load share ratio from 48% to 8% when increasing the spacing to diameter
ratio from 2 to 6, while the numerical analyses estimates a decrease from 25%
to 5% for the same increase in the spacing to diameter ratio. Moreover, the
124
effect of the modular ratio on the column load share ratio seems to be more
significant in the Shahu et al. approach than the numerical analyses. An
increase of the load share ratio from 13% to 47% is estimated by Shahu et al.
(2000) for the increase of modular ratio from 10 to 70 while for the same
increase in modular ratio the numerical analyses gives an increase in the load
share ratio from a value of 11% to a value of 17%.
Although the theoretical approach of Pulko and Majes (2006) does not take
the effect of the applies stress and the modular ratio on the column load share
ratio and gives a constant value of 17% in both cases, this approach shows a
good agreement with the results of the numerical study, as the load share ratio
given by the numerical analyses decreases from 17% to 12% when increasing
the embankment height from 1m to 6m and increases from 13% to 17% when
increasing the modular ratio from 10 to 70. Moreover, the effect of increasing
the spacing between columns on the column load share ratio is almost
identical for both methods especially at spacing to diameter ratios more than 4.
The column load share ratio estimated by the theoretical approach of Pulko
and Majes (2006) decreases from 30% to 5% when increasing the spacing to
diameter ratio from 2 to 6 which corresponds to a decrease from 25% to 5%
given by the numerical analyses for the same increase in column to diameter
ratio. This can return to the fact that this approach takes the column plasticity
as well as the lateral bulging of the column into consideration.
125
6.4 The settlement reduction factor
Figures (6-7) to (6-9) show the effect of the embankment height, column
spacing and modular ratio on the settlement reduction factor respectively. The
settlement reduction is estimated using numerical analyses as well as
theoretical approaches of Preibe (1995), Shahu et al. (2000) and Pulko and
Majes (2006). It is noticed that the approach of shahu et al. (2000) estimates a
much higher reduction in the settlement of the improved soil with a settlement
reduction factor ranging between 8% and 15% for all the cases compared to
the numerical analyses which ranges between 39% and 91%. However, the
behavior of the settlement reduction factor is similar for both cases with the
variation of embankment height and column spacing, while it estimates a more
significant effect for the modular ratio on the settlement reduction than the
numerical analyses.
Although the approach of Preibe (1995) predicts a higher value for the
settlement reduction factor than Shahu et al. (2000) (i.e. a lower reduction in
settlement), the improvement estimated by Preibe is still much higher than that
of the numerical analyses with a difference between them ranging from 24%
to 89%. The embankment height does not seem to have an effect on the
settlement reduction factor in the Preibe (1995) approach as the settlement
reduction factor has a constant value of 30% despite the increase in the
embankment height from 1m to 6m. However, the behavior of the settlement
reduction variation with the column spacing is similar to that of the numerical
analyses as it increases from 16% to 64% with the increase of the spacing to
diameter ratio from 2 to 6 which corresponds to an increase from 39% to 91%
estimated for the same increase in spacing to diameter ratio using the
126
numerical analyses. Moreover, this approach of Preibe (1995) estimates a

decrease in the settlement reduction factor from 56% to 14% when increasing
the modular ratio from 10 to 70 while the numerical analyses estimates an
almost constant value for the settlement reduction factor at 70% for the same
range of the modular ratios.
The theoretical approach of Pulko and Majes (2006) shows a very good
agreement with the results of the numerical analyses when used to estimate the
settlement reduction factor. The settlement reduction factors given by both the
Pulko and Majes (2006) approach and the numerical modeling almost coincide
with the variation of column spacing and the modular ratio. Although, this
theoretical approach does not depend on the applied stress in its calculations
and gives a constant value for the settlement reduction factor with the
variation of the embankment height, the value given by this approach (70%) is
almost the average value of the settlement reduction factor given by the
numerical analyses which gives a value of 59% at an embankment height of
1m and increases to 80% at an embankment height of 6m. as explained before,
this can return to the fact that this approach consider the column lateral
bulging as well as it’s plasticity.
127
Figure 6-7 – The effect of embankment height on the settlement reduction

factor using numerical modeling and theoretical approaches
Figure 6-8 – The effect of column spacing on the settlement reduction factor
128
Figure 6-9 – The effect of modular ratio on the settlement reduction factor
6.5 The time reduction factor
A comparison between the time reduction factor given by the numerical

analyses and that calculated using theoretical approaches is performed. The
analytical method of Han and Ye (2001) is used. A brief discussion of this
approach is previously stated in the literature review. Figures (6-10) to (6-11)
show the variation of the time reduction factor with embankment height,
spacing between columns and the modular ratio respectively.
129
Figure 6-10 – The effect of embankment height on the time reduction factor
Figure 6-11 – The effect of column spacing on the time reduction factor using
numerical modeling and theoretical approaches
130
Figure 6-12 – The effect of modular ratio on the time reduction factor using
numerical modeling and theoretical approaches
Generally, the theoretical approach of Han and Ye (2001) gives a lower values
for the time reduction factor than that estimated by the numerical analyses
throughout the entire parametric study, thus it estimates that the consolidation
process will end earlier than the numerical analyses. This approach agreed
with the numerical analyses in the fact that the embankment height (applied
stress level) and the modular ratio do not have a significant effect on the time
reduction factor as it gives a constant value of 0.37% for the time reduction
factor for embankment heights and modular ratios ranging from 1m to 6m and
10 to 70 respectively, however, the numerical analysis estimates an almost
constant value of 3% for the embankment heights from 1m to 6m and a value
of 2.8% for modular ratios from 10 to 70. The theoretical approach of Han and
Ye (2001) shows that the time reduction factor increases with increasing the
131
spacing between columns as it estimates an increase of the time reduction

factor from 0.1% to 3.1% when increasing the column to diameter ratio from 2
to 6. This agrees with the findings of the numerically performed parametric
study, however, the numerical analyses estimates a higher rate of increase
from 0.9% to 16.2% for the same range of column diameter to spacing ratios.
132
Chapter )7(
Summary, Conclusions and
Recommendations
7.1 Summary
A study of the behavior of soft soil deposits reinforced via stone columns
using numerical modeling is conducted. The numerical modeling was applied
to simulate a case study from Panchela Toll Plaza, Malaysia. The case study is
for an embankment overlaying layers of soft and stiff clay reinforced via stone
columns. Finite element analyses programs PLAXIS 3D and PLAXIS 2D
were used to simulate the case study. Three dimensional analyses as well as
two dimensional axisymmetric and plane strain analyses were conducted.
Mohr-Coulomb model as well as effective soil parameters were used during
the numerical modeling of the case. A comparison of the behavior of the
reinforced soft soil deposits between the recorded and the modeled behavior
for the case study was performed. A suitable 2D configuration to simulate the
behavior was stated.
A parametric study is conducted based on the same case study. The study was
performed using the calibrated 2D axisymmetric finite element model. The
study focused on the effect of various parameters on the stresses and loads
transferred to the stone columns from the embankment. The effect of the same
parameters on the reduction in both settlement and consolidation time was
illustrated.
133
Chapter (7) Summary, Conclusions and Recommendations
Finally, a comparison of the results of the numerically performed parametric

study with the results of different theoretical and analytical approaches used to
estimate the behavior of soil reinforced via stone columns.
7.2 Conclusions
The following conclusions were drown from this research
(1) Three dimensional finite element analyses is the most suitable method for
predicting the behavior of stone columns used to improve soft soil
deposits.
(2) Two dimensional axisymmetric analyses is an adequate tool that can be

used to study the behavior of soil deposits reinforced via stone columns as
it shows good agreement with both insitu monitored behavior and three
dimensional numerical modeling predicted behavior. However, a major
disadvantage is that it cannot be used to study the stability of the slope
embankment.
(3) Two dimensional plane strain numerical modeling is suitable to perform a

safety analyses for the slope of embankments overlaying soil deposits
reinforced using stone columns.
(4) Mohr-Coulomb model can be utilized to simulate the behavior of soft soil
deposits reinforced via stone columns, especially, when using the effective
shear parameters to simulate the soft soil deposits.
134
(5) Increasing the spacing between the stone columns leads to:
 Increase in the stresses transferring to the stone columns to a certain

extent then it becomes insignificant.
 Decrease in the column load share ratio.
 Decrease in the reduction of settlement.
 Decrease in the reduction of consolidation duration.
(6) Increasing the stress level on the reinforced soil system leads to:
 Decrease in the stress concentration factor.
 Decrease in the column load share ratio.
 Decrease in the reduction of settlement.
 Almost in significant in the reduction of consolidation duration.
(7) Increasing the Modular ratio leads to:
 Increase in the stress concentration factor.
 Increase in the column load share ratio.
 Slight increase in the settlement reduction.
 Almost in significant in the reduction of consolidation duration.
135
(8) The Pulko and Majes (2006) analytical approach is considered an adequate
tool that can be used for the preliminary prediction of the reduction in
settlement that takes place when using short end bearing stone columns to
reinforce soft soil deposits.
(9) The Han and Ye (2001) analytical approach can be used to estimate the
duration of the consolidation process of the reinforced soft soil. However,
for the case of short end bearing stone columns it estimates a shorter time
for the consolidation process to end than the numerical modeling.
7.3 Recommendations for future studies
(1) More case studies for soft soil reinforced with stone columns needs to be
analyzed using 3D and 2D models to ensure the findings of this research.
(2) Further study of the behavior of the stress concentration factor and the
column load share ratio with time especially in the cases of long columns
and floating columns.
(3) The application of geogrids or Geosynthetics as an encasement to increase

the confinement and thus the capacity of the columns.
(4) The effect of the cementation of the upper part of the column on its
capacity.
(5) The effect of using geogrids to reinforce the embankment on the stability
of slopes.
(6) The adequacy of different theoretical approaches for different cases

especially long or floating columns needs further study.
136
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Soil Improvement Using Stone Columns

Thesis · June 2014

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SOIL IMPROVEMENT USING STONE COLUMNS

MAHMOUD EL SHAWAF ABDUL AZIM ALY HAMMAD

Prof. Dr. Ali Abdul Fattah Prof. Dr. Yasser Moghazy

Name : Mahmoud El Shawaf Abdul Azim Aly Hammad

Name and Affiliation Signature

Prof. Dr. Mona Mohamed Mostafa Eid

Prof. Dr. Ali Abdul Fattah Ali Ahmed

Prof. Dr. Yasser Moghazy El Mossallamy

Name : Mahmoud El Shawaf Abdul Azim Aly Hammad

Name and Affiliation Signature

Prof. Dr. Ali Abdul Fattah Ali Ahmed

Prof. Dr. Yasser El Moghazy El

College Board approval University Board approval

Name Mahmoud El Shawaf Abdul Azim Aly

Date of Birth 19, April, 1986

Place of Birth Cairo, Egypt

Scientific BSc. of Structural Engineering, Faculty of Engineering,

Current Job Demonstrator of Geotechnical Engineering and

This thesis is submitted to Ain Shams University for the degree of

No part of this thesis has been submitted for a degree or a

Name: Mahmoud El Shawaf Abdul Azim Aly

I wish to express my deepest gratitude and appreciation to Prof. Dr. Ali

My grateful appreciation also extends to Prof. Dr. Yasser Moghazy El

Different analytical methods used to calculate and study the improvement

One of the main objectives of this research is to investigate the different

A case study of an embankment construction for Penchala Toll Plaza

A Comparison is then done between the results of the three dimensional

Keywords: Stone columns, soil improvement, embankments, soft soil

The thesis consists of seven chapters

Chapter (1) is the introduction to this research; it discusses the importance,

Chapter (3) presents a brief discussion about finite element method

Chapter (5) is a parametric study performed to investigate the effect of

Chapter (6) presents a comparison between the findings of the numerically

TABLE OF CONTENTS .................................................................................... i

LIST OF FIGURES ..........................................................................................vii

LIST OF TABLES .......................................................................................... xiv

1.2 Research scope and objectives .............................................................. 2

1.3 Thesis outline ......................................................................................... 3

2.2 Stone columns ........................................................................................ 6

2.2.1 Stone column installation ................................................................ 7

2.3 Engineering behavior of the composite ground .................................... 9

2.3.1 Unit Cell .......................................................................................... 9

2.3.2 Load sharing and stress concentration .......................................... 10

2.4.1 Experimental Studies ..................................................................... 18

2.6 Stone columns-soft soil reinforcement system under embankment .... 34

2.7 Discussion ............................................................................................ 36

3.2 Finite element method ......................................................................... 38

3.2.1 General .......................................................................................... 38

3.2.2 Analysis procedure of finite element method ............................... 39

3.2.3 Elements shapes ............................................................................ 41

3.2.4 Two dimensional simulation of special three dimensional

3.2.4.1 Plane strain .................................................................................... 43

3.2.4.2 Plane stress .................................................................................... 44

3.2.4.3 Axisymmetric problems ................................................................ 44