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Research Collection
Doctoral Thesis
Influence of smear and compaction zones on the performance ofstone columns in lacustrine clay
Author(s): Gautray, Jean N.F.
Publication Date: 2014
Permanent Link: https://doi.org/10.3929/ethz-a-010247610
Rights / License: In Copyright - Non-Commercial Use Permitted
This page was generated automatically upon download from the ETH Zurich Research Collection. For moreinformation please consult the Terms of use.
ETH Library
DISS. ETH NO. 22107
INFLUENCE OF SMEAR AND COMPACTION ZONES ON THE PERFORMANCE
OF STONE COLUMNS IN LACUSTRINE CLAY
A thesis submitted to attain the degree of
DOCTOR OF SCIENCES of ETH ZURICH
(Dr. sc. ETH Zürich)
presented by
Jean Nicolas François-Xavier Gautray
MSc ETH Civil Eng.
born on 29.08.1987
citizen of France
accepted on the recommendation of
Prof. Dr. Sarah M. Springman
Dr. Jan Laue
Prof. Dr. Helmut Schweiger
2014
À et grâce à mon père
Acknowledgements
I
Acknowledgements
Many people in my personal and professional environment have helped me over numerous
years to become the person I am today, to reach my goals and to achieve this work.
A very special thanks goes of course to my deceased father François Gautray, who was my
dad, my best friend and my confident for twenty six years and whose heart decided to let go
last fall. He always put me back on track when troubles appeared, gave me everything he
could so I got to achieve my goals and not a single word of this thesis would have been
written without his help. His inspiration, his advice and his personality shall accompany and
help me forever.
My step-mother Catherine Rodier could find the words to teach me how to write properly, to
help me learn foreign languages and to bear me when I was a rather complicated kid. She
helped me to become a grown-up and has been a true mother at my side at any time for the
past seventeen years.
Julia Selberherr has been bringing me sunshine and warmth, this even in the darkest and
coldest moments. She has helped me to find new perspectives, has enlightened my world
with her smile, and will hopefully continue to do so. My thanks also go to her family, who
welcomed me with open arms and offered me a second home.
A requirement for the conduction of a PhD thesis is of course not only a favourable personal
situation but also an adequate professional environment.
This is why I would like to thank my supervisor Prof. Dr. Sarah Springman for taking the
decision to give me a position in her group and for giving me the opportunity to conduct this
research. The benefits from this position over the past four years will surely be helpful in the
future.
Dr. Jan Laue was supportive in his role as a co-supervisor by being available and excited
about news ideas and suggestions.
The assistance of my second co-supervisor, Prof. Dr. Helmut Schweiger, was decisive in the
conduction of the numerical modelling. His availableness during my stays in Graz, Austria
was remarkable and his advice of great value. The help from his assistants Dr. Franz
Tschuchnigg and Dr. Bert Schädlich was also very deeply appreciated.
Dr. Michael Plötze was always very helpful and his expertise in the domain of clay
mineralogy and of Mercury Intrusion Porosimetry was much esteemed. The expertise of
Gabriela Peschke was also of great help in order to obtain high quality Environmental
Scanning Electron Microscopy results.
Acknowledgements
II
A central part of this work is the modelling of boundary value problems under enhanced
gravity using the geotechnical drum centrifuge at the ETH Zürich. Such modelling activities
are unthinkable without a highly competent technical staff.
Markus Iten provided his expertise for the management of the geotechnical centrifuge and
his good mood, even when having to pop up at 4 o’clock in the morning. None of the tests
would have been possible without him.
The help Heinz Buschor and Andreas Kieper was also absolutely essential for the production
of the new centrifuge tools coming out of Dr Jan Laue’s and/or of my imagination. Their
ability to deliver very high quality products within short time periods was crucial and always
deeply appreciated.
Modern techniques always involve more complicated technologies which make the help of an
electrical engineering technician of immense value. Ernst Bleiker was always available to
help me out with the numerous electrical issues I had during the conduction of my thesis and
his ability to think “out of the box” to find imaginative and effective solutions in order to solve
complicated problems within very tight time periods was of invaluable support.
My thanks also go to Dr. Pierre Mayor for his valuable support and good advice.
Eventually, I would like to thank Ralf Herzog, Dr. André Arnold, Frank Fischli, and Dr. Ferney
Morales for their good company and nice conversations over the past years, and the other
members of the Institute for Geotechnical Engineering, whom I worked with, for their help
with professional matters.
Contents
III
Contents
Acknowledgements ................................................................................................................ I
Contents ................................................................................................................................III
List of figures ........................................................................................................................ XI
List of tables .................................................................................................................... XXXI
Abstract .......................................................................................................................... XXXV
Kurzfassung ................................................................................................................. XXXVII
1 Introduction .................................................................................................................... 1
1.1 Motivation .............................................................................................................. 1
1.2 Thesis layout .......................................................................................................... 2
2 State of the art of ground improvement with stone columns ........................................... 5
2.1 General considerations about ground improvement ............................................... 5
2.2 Objectives of ground improvement of soft soils with stone columns ....................... 7
2.3 Construction techniques ......................................................................................... 8
2.4 Bearing behaviour of stone columns submitted to vertical loading ........................10
2.4.1 Bearing behaviour ...........................................................................................10
2.4.2 Stress concentration on stone columns ...........................................................12
2.4.3 Ultimate Limit State response to vertical load ..................................................18
2.5 Design of stone columns .......................................................................................22
2.5.1 Bearing capacity ..............................................................................................22
2.5.1.1 Bulging failure ..........................................................................................22
2.5.1.2 Shear failure .............................................................................................24
2.5.1.3 Penetration of short columns ....................................................................29
2.5.2 Settlement calculation .....................................................................................30
2.5.2.1 Settlement calculations based on equilibrium considerations ...................31
2.5.2.2 Settlement calculations based on empirical methods ...............................35
2.5.3 Comparison of the design procedures .............................................................44
2.6 Load-transfer behaviour of stone columns ............................................................46
2.7 Load-transfer behaviour in inclusion-supported embankments ..............................53
2.8 Effect of stone columns on the consolidation time .................................................57
2.9 Analytical considerations about the installation of inclusions in soil .......................59
2.9.1 Cavity expansion theory ..................................................................................60
Contents
IV
2.9.2 Strain Path Method and Shallow Strain Path Method ......................................63
2.10 Observations concerning the installation effects of piles and stone columns on the
soil .............................................................................................................................66
2.10.1 Pile installation ................................................................................................66
2.10.2 Changes of host soil properties due to the installation of stone columns .........68
2.10.2.1 Effect on soil resistance and stress levels ................................................69
2.10.2.2 Smear and compaction zones: effect on permeability ..............................74
2.10.3 Radial drainage around stone columns ...........................................................85
2.11 Summary of the state of the art of ground improvement with stone columns .........88
3 Centrifuge modelling .....................................................................................................91
3.1 Historical background ...........................................................................................91
3.2 Principles of centrifuge modelling ..........................................................................92
3.2.1 Scaling factors.................................................................................................94
3.2.2 Advantages and disadvantages of physical modelling under enhanced gravity ..
........................................................................................................................94
3.3 Centrifuge modelling of ground improvement measures .......................................96
3.4 Techniques adopted ........................................................................................... 109
3.4.1 ETH Zürich geotechnical drum centrifuge and equipment ............................. 109
3.4.2 Pore pressure transducers (PPTs) ................................................................ 110
3.4.3 Load cells ...................................................................................................... 110
3.4.4 T-Bar penetrometer ....................................................................................... 111
3.4.5 Electrical impedance needle .......................................................................... 113
3.5 Model soils .......................................................................................................... 116
3.5.1 Birmensdorf clay ........................................................................................... 116
3.5.2 Quartz sand .................................................................................................. 117
3.5.3 Perth sand ..................................................................................................... 118
3.6 Soil model ........................................................................................................... 118
3.6.1 Preparation of Birmensdorf clay .................................................................... 119
3.6.2 Preparation of a soil model in a cylindrical strongbox .................................... 119
3.6.3 Preparation of a soil model in an oedometer container .................................. 121
3.6.4 Preparation of a soil model in an adapted oedometer container .................... 124
3.6.5 Installation of the PPTs ................................................................................. 124
3.6.5.1 Installation of the PPTs into a cylindrical strongbox and an adapted
oedometer container ............................................................................................... 125
Contents
V
3.6.5.2 Installation of the PPTs into a model consolidated in an oedometer
container and installed in the 400 mm diameter strongbox ..................................... 126
3.6.6 Identification of the locations of the stone columns ........................................ 127
3.7 Centrifuge tests ................................................................................................... 128
3.7.1 Overview ....................................................................................................... 128
3.7.2 Groundwater level ......................................................................................... 130
3.7.3 Tests conducted with specimens prepared in a cylindrical strongbox ............ 131
3.7.3.1 Loading of a single stone column (JG_v2, JG_v3 and JG_v6) ............... 131
3.7.3.2 Loading of a stone column group (JG_v8, JG_v10) ................................ 133
3.7.4 Tests conducted with specimens prepared in an oedometer container (JG_v1,
JG_v4, JG_v5) ............................................................................................................ 137
3.7.5 Tests conducted with specimens prepared in an adapted oedometer container
(JG_v7, JG_v9) .......................................................................................................... 141
4 Results from the centrifuge tests................................................................................. 143
4.1 Undrained shear strength .................................................................................... 143
4.1.1 Theoretical prediction .................................................................................... 143
4.1.2 Shear strength profile for pre-consolidation up to σ’v = 100 kPa .................... 145
4.1.3 Shear strength profile for pre-consolidation up to σ’v = 200 kPa .................... 147
4.1.4 Summary of the back-calculated values of the shear strength parameters a and
b ...................................................................................................................... 148
4.2 Pore pressure measurements conducted during the installation of stone columns ...
........................................................................................................................... 150
4.2.1 Measurements conducted during the installation of a single stone column .... 150
4.2.2 Measurements conducted during the installation of a stone column group .... 154
4.3 Measurements conducted during the footing loading of a single stone column
installed in a specimen prepared in a full cylindrical strongbox ....................................... 156
4.4 Measurements conducted during the footing loading of a single stone column
installed in a specimen prepared in an (adapted) oedometer container .......................... 159
4.4.1 Measurements conducted in a specimen consolidated up to σ’v = 100 kPa .. 159
4.4.2 Measurements conducted in a specimen consolidated up to σ’v = 200 kPa .. 163
4.4.3 Comparison of the results ............................................................................. 166
4.4.4 Load transfer around a single stone column .................................................. 168
4.5 Measurements conducted during the footing load of a stone column group ........ 176
4.5.1 Test JG_v8 (a / dsc = 2 [-]) ............................................................................. 176
4.5.2 Test JG_v10 (a / dsc = 2 [-]) ........................................................................... 179
Contents
VI
4.6 Comparison of the measurements around a single stone column and inside a stone
column group .................................................................................................................. 185
4.7 Electrical impedance measurements ................................................................... 188
4.7.1 Measurements around a single stone column ............................................... 188
4.7.1.1 Measurements conducted in specimens consolidated up to σ’v = 100 kPa
(test JG_v5) ............................................................................................................ 189
4.7.1.2 Measurements conducted in a specimen consolidated up to σ’v = 200 kPa
(test JG_v5) ............................................................................................................ 191
4.7.2 Measurements around a stone column group (test JG_v8) ........................... 193
4.8 Summary of the conducted modelling under enhanced gravity ........................... 196
5 Complementary investigations .................................................................................... 197
5.1 Oedometer tests conducted on samples extracted from the soil model used for the
centrifuge test JG_v9 ...................................................................................................... 197
5.2 Oedometer tests conducted on samples extracted from soil models after
consolidation .................................................................................................................. 203
5.3 Electrical impedance measurement under 1 g .................................................... 207
5.4 Microscopic investigations .................................................................................. 210
5.4.1 Description of the Scanning Electron Microscope .......................................... 210
5.4.2 Description of the Environmental Scanning Electron Microscope .................. 212
5.4.3 Results obtained ........................................................................................... 213
5.5 Mercury Intrusion Porosimetry (MIP) ................................................................... 215
5.5.1 General principle ........................................................................................... 215
5.5.2 Sample preparation ....................................................................................... 216
5.5.3 Apparatus used ............................................................................................. 216
5.5.4 Results obtained ........................................................................................... 217
6 Numerical modelling ................................................................................................... 219
6.1 Principles of numerical modelling of ground improvement ................................... 219
6.1.1 Improvement through compaction (embankment loading with installation of
vertical drains) ............................................................................................................ 219
6.1.2 Discrete modelling of improvement through material addition with displacement
...................................................................................................................... 221
6.2 Literature review of numerical modelling of ground improvement through stone
columns and prefabricated vertical drains ....................................................................... 224
6.2.1 Numerical modelling of ground improvement with stone columns and
prefabricated vertical drains ........................................................................................ 224
Contents
VII
6.2.2 Analogy to installation of rigid inclusions ....................................................... 234
6.3 Constitutive models ............................................................................................. 237
6.3.1 Mohr-Coulomb model .................................................................................... 237
6.3.1.1 Description ............................................................................................. 237
6.3.1.2 Limitations of the Mohr-Coulomb model ................................................. 239
6.3.1.3 Input parameters of the Mohr-Coulomb model ....................................... 239
6.3.2 Hardening Soil Model .................................................................................... 239
6.3.2.1 Stiffness moduli ...................................................................................... 240
6.3.2.2 Yield surfaces ........................................................................................ 241
6.3.2.3 Shear strain hardening ........................................................................... 242
6.3.2.4 Volumetric hardening ............................................................................. 243
6.3.2.5 Limitations of the Hardening Soil Model ................................................. 244
6.3.2.6 Input parameters of the Hardening Soil Model ....................................... 245
6.4 Axisymmetric numerical modelling ...................................................................... 246
6.4.1 Options discarded ......................................................................................... 246
6.4.2 Model ............................................................................................................ 248
6.4.3 Results .......................................................................................................... 251
6.5 3D numerical modelling ....................................................................................... 266
6.5.1 Model ............................................................................................................ 266
6.5.2 Results .......................................................................................................... 269
6.6 Summary of numerical modelling ........................................................................ 276
7 Summary .................................................................................................................... 279
7.1 General considerations ....................................................................................... 279
7.2 Findings from centrifuge modelling and complementary investigations ............... 279
7.3 Numerical modelling ........................................................................................... 282
7.4 Outlook ............................................................................................................... 287
8 Appendices ................................................................................................................. 289
8.1 Pore pressure and load measurements conducted during loading with a footing on
a single stone column installed in a specimen prepared in an oedometer container (test
JG_v1) ........................................................................................................................... 290
8.2 Pore pressure and load measurements conducted during loading a single stone
column installed in a specimen prepared in an oedometer container (test JG_v5) with a
circular footing ................................................................................................................ 292
Contents
VIII
8.3 Pore pressure and load measurements conducted during loading a single stone
column installed in a specimen prepared in a full cylindrical stongbox (test JG_v6) with a
circular footing ................................................................................................................ 293
8.4 Values of the J4 factor according to Grasshoff (1978) ......................................... 294
8.5 Comparison of the analytical and measured excess pore water pressure around a
single stone column when the maximum load is applied (P = 80 kPa, test JG_v1) ......... 294
8.6 Electrical impedance measurements conducted during the test JG_v9 ............... 295
8.7 Electrical impedance measurements conducted under 1 g ................................. 298
8.8 Vertical strain increments computed numerically for test JG_v7 .......................... 301
8.9 Shear strain increments computed numerically for test JG_v7 ............................ 303
8.10 Development of plastic points (test JG_v7) ......................................................... 306
8.11 Deformed mesh (test JG_v9) .............................................................................. 309
8.12 Total stress distribution computed numerically for test JG_v9 ............................. 310
8.13 Vertical strain increments computed numerically for test JG_v9 .......................... 311
8.14 Shear strain increments computed numerically for test JG_v9 ............................ 314
8.15 Excess pore water pressures computed numerically for test JG_v9 .................... 317
8.16 Development of plastic points (test JG_v9) ......................................................... 318
8.17 Total vertical stress distribution as a function of the radial distance at depths of 0
m, 2 m, 4 m and 6 m (test JG_v7) .................................................................................. 321
8.18 Total vertical stress distribution as a function of the radial distance at depths of 0
m, 2 m, 4 m and 6 m (test JG_v9) .................................................................................. 322
8.19 Comparison of the measured and modelled excess pore water pressures for the
test JG_v7 ...................................................................................................................... 323
8.20 Comparison of the measured and modelled excess pore water pressures for the
test JG_v9 ...................................................................................................................... 324
8.21 Distribution of the total vertical stresses for footing settlements of 100 mm and of
400 mm (test JG_v10) .................................................................................................... 326
8.22 Total vertical stress distribution below the footing for a settlement of 100 mm (test
JG_v10) .......................................................................................................................... 329
8.23 Total vertical stress distribution below the footing for a settlement of 400 mm (test
JG_v10) .......................................................................................................................... 332
8.24 Total vertical stress distribution below the footing for a settlement of 850 mm (test
JG_v10) .......................................................................................................................... 335
8.25 Comparison of the measured and modelled excess pore water pressures for the
test JG_v10 .................................................................................................................... 337
8.26 Development of plastic points (test JG_v10) ....................................................... 338
Contents
IX
9 List of subscripts and symbols .................................................................................... 343
10 References ................................................................................................................. 353
Contents
X
List of figures
XI
List of figures
Figure 1.1: Installation effects around a stone column at a model depth of 40 mm @ 50 g
(Weber, 2008). ...................................................................................................................... 2
Figure 2.1: Dry replacement technique: (a) filling the supply hopper, (b) penetration, (c)
compaction by step-wise withdrawal and reinsertion (d) finishing (Keller Grundbau, 2013). . 9
Figure 2.2: Wet top feed technique: (a) penetration, (b) filling, (c) compacting, (d) finishing
(International Construction Equipment Holland, 2013). .......................................................... 9
Figure 2.3: Ramming installation technique: (a) inserting granular plug, (b) driving up to the
desired depth, (c) filling with granular soil, (d) compacting and withdrawing casing, (e)
finishing (Van Impe et al., 1997b). ........................................................................................10
Figure 2.4: Interactions at stake under a footing (after Kirsch, 2004). ...................................11
Figure 2.5: Loading situations of stone columns (Kirsch, 2004). ...........................................12
Figure 2.6: Total vertical stress distribution of a uniform vertical stress σ (a) plan view
showing respective areas of stone columns (Asc) and soft soil (As), (b) cross-section showing
stress distribution onto the column (σsc) and the host soil (σs) (after Aboshi et al., 1991). .....13
Figure 2.7: Measured stress concentration factors at (a) St. Helens and (b) Canvey Island
(Greenwood, 1991). .............................................................................................................15
Figure 2.8: Cross-section of the test site at Humber Bridge (after Greenwood, 1991). .........16
Figure 2.9: Measured stress concentration factors at Humber Bridge (Greenwood, 1991). ..16
Figure 2.10: Stress concentration factors in 1 g small-scale model and field tests (Muir Wood
et al., 2000). .........................................................................................................................17
Figure 2.11: Failure mechanisms for a single stone column (a) bulging (b) bearing failure (c)
shear failure (d) penetration of short columns (e) shortening of long columns (f) deflection of
slender columns (Muir Wood et al., 2000) based on Waterton & Foulsham (1984). ..............18
Figure 2.12: Failure mechanisms for groups of stone columns (a) bulging failure and loss of
horizontal support (b) shearing failure (c) block failure and column penetration (Kirsch, 2004).
.............................................................................................................................................19
Figure 2.13: Deformed sand columns at the end of the footing penetration (Muir Wood et al.,
2000). ...................................................................................................................................20
Figure 2.14: Zone of influence of a footing on the underlying soil (a) “rigid” cone beneath
footing (b) variation of angle β with area replacement ratio (Muir Wood et al., 2000). ..........21
Figure 2.15: Deformed stone columns at the end of the footing penetration (McKelvey et al.,
2004). ...................................................................................................................................21
Figure 2.16: Shear failure of a stone column (after Muir Wood et al., 2000). ........................24
Figure 2.17: Truncated conical failure mechanism according to Brauns (1978a) (a) cross-
section, (b) plan view and (c) forces acting on volume A. .....................................................25
Figure 2.18: Stone column group analysis – firm to stiff fine-grained soil (Barksdale &
Bachus, 1983). .....................................................................................................................27
Figure 2.19: Clay and columns represented (a) discretely and (b) as an equivalent plane wall
(Springman et al., 2014). ......................................................................................................28
List of figures
XII
Figure 2.20: Stability considerations on a slip circle passing through soft soil and the
equivalent plane walls (numbers 1 to 11 show the sequence of the slices) (Springman et al.,
2014). ...................................................................................................................................29
Figure 2.21: Various stone column arrangements with the domain of influence of each
column (Balaam & Poulos, 1983). ........................................................................................30
Figure 2.22: Stress distribution on a rigid footing. .................................................................31
Figure 2.23: Evaluation of the stress distribution parameter depending on the different kinds
of mixture (Omine & Ohno, 1997). ........................................................................................33
Figure 2.24: Settlement diagram for stone columns installed in uniform soft clay (Greenwood,
1970). ...................................................................................................................................36
Figure 2.25: Values of the ground improvement factor n0 depending on the area replacement
ratio, for a Poisson’s ratio of 1/3 (after Priebe, 1995). ...........................................................37
Figure 2.26: Values of an additional component of the area replacement ratio to account for
column compressibility, for a Poisson’s ratio of 1/3 (after Priebe, 1995). ..............................38
Figure 2.27: Determination of an influence factor y for the calculation of a depth coefficient fd
for a Poisson’s ratio of 1/3 (γs: unit weight of the host soil; d: improvement depth; p: footing
load) (after Priebe, 1995). .....................................................................................................39
Figure 2.28: Priebe method best-fit line, with data sorted based on the site soil conditions
(Douglas & Schaefer, 2012). ................................................................................................40
Figure 2.29: Static system for the settlement calculation of groups of floating stone columns,
according to Priebe (2003) (Kirsch, 2004). ...........................................................................41
Figure 2.30: (a) Rheological modelling of the behaviour of stone columns, (b) Calculation
approach in plane-strain (Van Impe et al., 1997b). ...............................................................42
Figure 2.31: Graphical determination of the settlement reduction factor β (Van Impe & De
Beer, 1983). .........................................................................................................................43
Figure 2.32: Comparison of the ultimate bearing capacities as a function of the angle of
friction calculated using different procedures (after Greenwood & Kirsch, 1983). .................45
Figure 2.33: Comparison of results obtained from empirical models and elastic theories with
field observations (after Greenwood & Kirsch, 1983). ...........................................................46
Figure 2.34: Experimental setup for a single stone column loaded vertically through a rigid
footing (Sivakumar et al., 2011). ...........................................................................................47
Figure 2.35: Pressure distribution with depth during footing loading of a 60 mm diameter
stone column for different settlements (Sivakumar et al., 2011). ...........................................48
Figure 2.36: Representative borehole and selected soil properties from Red River research
site in Winnipeg, Canada. w: natural water content (horizontal bars display Atterberg limits);
γwet: unit weight of saturated soil; σ: stress; σ’pc: preconsolidation pressure; σ’v0: initial vertical
effective stress; u0: initial pore water pressure (Thiessen et al., 2011). .................................49
Figure 2.37: Red River test site in Winnipeg, Canada: stabilisation of river bank using a
combination of void and rockfill columns (a) cross-section and (b) plan view of the research
site (Thiessen et al., 2011). Elevations and distances in metres. ..........................................50
Figure 2.38: Red River test site in Winnipeg, Canada: pore water response to loading
(Thiessen et al., 2011). .........................................................................................................51
List of figures
XIII
Figure 2.39: Red River test site in Winnipeg, Canada: instrumentation layout (Thiessen et al.,
2011). ...................................................................................................................................51
Figure 2.40: Measured deformations along A axis (in downslope direction) at Red River test
site in Winnipeg, Canada: (a) SI-1 at crest of slope; (b) SI-4 in between columns along upper
row; (c) SI-7 in a column in upper row; (d) SI-10 downslope of columns (Thiessen et al.,
2011). ...................................................................................................................................53
Figure 2.41: Soil arching in stone column-supported embankment (after Deb, 2010). ..........54
Figure 2.42: Proposed foundation model for soft soil reinforced with stiffer inclusions (after
Deb, 2010). ..........................................................................................................................55
Figure 2.43: Effect of (a) ultimate bearing capacity of the soft soil and (b) the shear modulus
of the embankment soil on the arching ratio (Deb, 2010). .....................................................56
Figure 2.44: Arching effect in the embankment (Indraratna et al., 2013). ..............................57
Figure 2.45: Consolidation process for (a) a single stone column and (b) a group of stone
columns (after Black et al., 2007). ........................................................................................58
Figure 2.46: Comparison of the excess pore water pressure dissipation for displacement
piles and stone columns (McCabe et al., 2009). ...................................................................59
Figure 2.47: Geometric representation of cylindrical cavity expansion in either two or three
(spherical) dimensions (Vesic, 1972). ...................................................................................60
Figure 2.48: Deformation paths during penetration of a cone into clay calculated using the
SPM (Baligh, 1985). .............................................................................................................64
Figure 2.49: (a) Radial and (b) vertical deformation profiles after the installation of a simple
pile obtained with the SSPM analysis (Sagaseta & Whittle, 2001). .......................................65
Figure 2.50: Deformation and density changes during the penetration of a pile in dense sand
(after Linder, 1977). ..............................................................................................................67
Figure 2.51: (a) Half-cone inserted in sand (b) test set up (Davidson et al., 1981). ...............68
Figure 2.52: Displacements (in mm) and volumetric strains (in %) for jacking a half-CPT cone
into (a) loose sand (relative density = 25 %) (b) dense sand (relative density = 115 %)
(Davidson et al., 1981). ........................................................................................................68
Figure 2.53: Evolution of the undrained shear strength ratio (normalised to pre-installation
values) over time after the installation of stone columns (Aboshi et al., 1979). .....................69
Figure 2.54: Evolution of the unconfined compressive strength of clay over time (Asaoka et
al., 1994). .............................................................................................................................70
Figure 2.55: Profile of the host soil treated by SCP installation at the Bothkennar test site, as
investigated by Watts et al. (2000). ......................................................................................71
Figure 2.56: Lateral stress changes measured by earth pressure cells following poker
penetration and retraction during stone column compaction at the Bothkennar test site (Watts
et al., 2000). .........................................................................................................................72
Figure 2.57: Dynamic probing of the radial densification of the fill around a stone column at
the Bothkennar test site (Watts et al., 2000). ........................................................................72
Figure 2.58: Illustration of the different stress zones around the pier (rf = 1.9 m) in the
Memphis, USA case history (Handy et al., 2002). .................................................................73
Figure 2.59: Response of pore pressure transducers installed 2 m, resp. 4 m, below the
ground surface to column loading at the Raploch test site (Egan et al., 2009). .....................74
List of figures
XIV
Figure 2.60: Suggested variation of horizontal permeability with radius according to Onoue et
al. (1991) (after Saye, 2001). ................................................................................................75
Figure 2.61: Section of the test setup showing the smear zone (after Indraratna & Redana,
1998). ...................................................................................................................................75
Figure 2.62: Ratio of horizontal to vertical coefficient of permeability against the radial
distance from the axis of the SCP (denoted as drain) (Indraratna & Redana, 1998). ...........76
Figure 2.63: Excess pore water pressures during the insertion of the installation mandrel
(Sharma & Xiao, 2000). ........................................................................................................77
Figure 2.64: Variation of the horizontal permeability with radial distance to the drain for an
installation that causes a smear zone (Sharma & Xiao, 2000). .............................................77
Figure 2.65: Back-calculated sets of coefficients of relative horizontal permeability in the
undisturbed host soil (kh) and in the smear zone (k’h) and horizontal coefficient of
consolidation ch values, assuming ds = 2 dm (Bergado et al., 1991). .....................................79
Figure 2.66: Directions of the horizontal penetration tests (Shin et al., 2009). ......................80
Figure 2.67: Electrical resistivity and estimated outer boundary of the smear zone (Shin et
al., 2009). .............................................................................................................................81
Figure 2.68: Dimensions of the smear zone derived from the electrical resistance probe. All
dimensions in millimetres (Shin et al., 2009). ........................................................................81
Figure 2.69: Variation of the porosity as a function of the distance from the stone column axis
(Weber et al., 2010). .............................................................................................................83
Figure 2.70: Variation of the dry bulk density as a function of the distance from the stone
column axis (Weber et al., 2010). .........................................................................................83
Figure 2.71: Compression and smear zone around sand compaction piles (Juneja et al.,
2013). ...................................................................................................................................84
Figure 2.72: Scanning Electron Microscopy images of kaolin clay specimen adjacent to the
stone column installed and sheared (CIU) at 50 kPa (a) without smear and (b) with smear
(Juneja et al., 2013). .............................................................................................................84
Figure 2.73: Radial drainage within a unit cell (after Barron, 1948). ......................................85
Figure 3.1: Acceleration acting on a body rotating with angular velocity ω (Springman, 2004).
.............................................................................................................................................92
Figure 3.2: Principle of centrifuge modelling (after Schofield, 1980). ....................................93
Figure 3.3: Comparison of the stress profiles (a) in a prototype, (b) in a small-scale model
and (c) in a centrifuge model (after Laue, 1996). ..................................................................93
Figure 3.4: Distribution of the vertical stress with depth in a prototype situation and in the
centrifuge (zs denotes the depth of the sample) (after Taylor, 1995). ....................................95
Figure 3.5: (a) Cross-section of the centrifuge model of a clay sample reinforced by wick
drains and basal reinforcement loaded by an embankment and (b) influence of the drains on
the dissipation of excess pore water pressures during and after embankment construction
(Sharma & Bolton, 2001). .....................................................................................................96
Figure 3.6: Comparison between settlement improvement ratios obtained with the solution of
Priebe (1995) solution and from centrifuge tests (Al-Khafaji & Craig, 2000). .........................97
Figure 3.7: Pile lateral pressure as a function of the lateral displacement y normalised by the
pile radius d (Dyson & Randolph, 1998). ..............................................................................98
List of figures
XV
Figure 3.8: Sand compaction pile installation tool used at the National University of
Singapore. All dimensions are in mm (Ng et al., 1998). ........................................................99
Figure 3.9: Embankment constructed on soft clay (U2) and when improved by SCPs installed
at 1g (R1_20) or at 50 g (D50_20) (a) deformation grid lines in clay improved with SCPs built
in-flight (b) maximum lateral displacement (in mm) of the grid line L2 with g-level (Lee et al.,
2001). ................................................................................................................................. 100
Figure 3.10: Layout of SCPs and transducers for the installation of SCPs in test T7, D 20 mm
(D: SCP diameter) (Lee et al., 2004). ................................................................................. 100
Figure 3.11: Layout of SCPs and transducers for tests: (a) T1, D 18 mm; (b) T2, D 20 mm;
(c) T3, D 16 mm; (d) T4, D 17 mm; (e) T5, D 20 mm; (f) T6, D 20 mm (D SCP diameter) (Lee
et al., 2004). ....................................................................................................................... 101
Figure 3.12: (a) Total horizontal stress at 60 mm depth and (b) pore pressures at 80 mm
depth during SCP installation in clay. Line 1: time at which the casing tip reaches the depth
of the transducers. Line 2: time at which the casing tip reaches the full penetration and
withdrawal starts. Line 3: time at which the casing tip reaches the depth of the transducers
during withdrawal. Line 4: end of the SCP installation (Lee et al., 2004). ............................ 102
Figure 3.13: Ratios of measured to calculated horizontal stresses and pore pressures plotted
against (a) dt / D and (b) rt / D (Lee et al., 2004). ................................................................ 103
Figure 3.14: Ratios of measured to calculated horizontal stresses and pore pressures plotted
against the ratio of the depth of the transducers dt to the radial distance of the transducers rt
(Lee et al., 2004). ............................................................................................................... 103
Figure 3.15: Layout of sand compaction piles (P1 to P4) and location of the T-Bar test
(denoted as s) for pile group tests featuring either a) 2 piles or b) 4 piles (all dimensions in
mm) (Yi et al., 2013). .......................................................................................................... 104
Figure 3.16: Undrained shear strengths measured in the centrifuge for different tests (Yi et
al., 2013). ........................................................................................................................... 105
Figure 3.17: Experimental setup for the in-flight installation of stone columns (Weber et al.,
2005). ................................................................................................................................. 106
Figure 3.18: Detailed view of the stone column installation tool developed by Weber (2004).
........................................................................................................................................... 107
Figure 3.19: Settlements measured with and without stone columns at the toe of the
embankment (1), and on top of the embankment (2) (after Weber 2008). .......................... 107
Figure 3.20: Evolution of the pore water pressure after embankment construction in the
improved ground within the sand pile grid (---) in comparison with unimproved ground (––) at
three depths in the model with a groundwater table located at the surface of the model in the
middle of the container: P1 = 120 mm, P2 = 70 mm, P3 = 25 mm equivalent to prototype
depths of 6 m, 3.5 m and 1.25 m respectively (after Weber, 2008). .................................... 108
Figure 3.21: Installation effects around a stone column at a model depth of 40 mm @ 50 g
(Weber, 2008). ................................................................................................................... 108
Figure 3.22: Cross-section of the ETH Zürich geotechnical drum centrifuge (Springman et al.,
2001). ................................................................................................................................. 109
Figure 3.23: Cross-section of the transducer DRUCK PDCR 81 (König et al., 1994). ......... 110
Figure 3.24: Load cell produced by Hottinger Baldwin Messtechnik GmbH (Arnold, 2011). 111
List of figures
XVI
Figure 3.25: T-Bar penetrometer (a) front view and (b) side view. ...................................... 112
Figure 3.26: T-Bar penetrometer (after Weber, 2008). ........................................................ 112
Figure 3.27: T-Bar penetrometer mounted on the working arm of the tool platform in the
centrifuge (after Weber, 2008). ........................................................................................... 112
Figure 3.28: T-Bar calibration setup (after Weber, 2008). ................................................... 113
Figure 3.29: Electrical impedance needle (a) side view and (b) tilted view of the tip (outer
diameter 1 mm). ................................................................................................................. 114
Figure 3.30: Schematic views of the electrical impedance needle (a) covered, (b) with the
cover retracted and (c) cross-section A-A (Gautray et al., 2014). ....................................... 115
Figure 3.31: Ultrasonic bath Emmi 4, produced by EMAG AG (Gautray et al., 2014). ........ 115
Figure 3.32: Vacuum mixer. ............................................................................................... 119
Figure 3.33: General view (a) cylindrical strongbox used for the consolidation of Birmensdorf
clay under the hydraulic press, (b) view of the channels filled with Perth sand and (c) filling
with clay suspension. ......................................................................................................... 120
Figure 3.34: Hydraulic press used for the consolidation of clay. ......................................... 121
Figure 3.35: Preparation of the clay model (a) slurry inside the oedometer container (b) under
consolidation in the oedometer container. .......................................................................... 122
Figure 3.36: Schematic representation (a) of the possible cylindrical rupture zones when
extracting the clay sample from the container and (b) of the use of the plastic sheet in order
to prevent the adhesion between clay and oedometer container. ....................................... 122
Figure 3.37: (a) Removal of the oedometer container from the sample (b) view of the model
with clay sample surrounded by Perth sand. ...................................................................... 123
Figure 3.38: Ports for the installation of PPTs into the soil model prepared in 250 mm
diameter containers ............................................................................................................ 124
Figure 3.39: PPT installation tool. ....................................................................................... 125
Figure 3.40: Installation of the PPTs in the cylindrical strongbox (a) introduction of the PPTs
through the dedicated ports into the pre-drilled hole (b) filling of the pre-drilled hole with slurry
(Weber, 2008). ................................................................................................................... 125
Figure 3.41: PPT installation setup for a specimen consolidated in an oedometer container.
........................................................................................................................................... 126
Figure 3.42: Insertion of the PPT installation tool into the clay specimen using the installation
device. ................................................................................................................................ 127
Figure 3.43: Pin used to mark the positions of the stone columns to be installed (a) plan view
and (b) side view. ............................................................................................................... 128
Figure 3.44: Tilted view of the pin used to mark the positions of the stone columns to be
constructed with the stone column installation tool. ............................................................ 128
Figure 3.45: Vertical cross-section of the experimental setup in the centrifuge for the
specimens prepared in a cylindrical strongbox (Section 3.6.2), in an oedometer container and
in an adapted oedometer container (Section 3.6.4) (after Weber, 2008). ............................ 129
Figure 3.46: Position of the water level in the soil and in the standpipe for specimens
prepared in a cylindrical strongbox (tests JG_v2, JG_v3, JG_v6, JG_v8 and JG_v10). ...... 130
Figure 3.47: Position of the water level in the soil and in the standpipe for specimens
prepared in an oedometer container (tests JG_v1 and JG_v5). .......................................... 131
List of figures
XVII
Figure 3.48: Position of the water level in the soil and in the standpipe for specimens
prepared in an adapted oedometer container (tests JG_v7 and JG_v9). ............................ 131
Figure 3.49: Specimens prepared in a cylindrical strongbox: (a) plan view and (b) cross-
section of the soil model with positions of the PPTs and of the stone columns. .................. 133
Figure 3.50: Specimens prepared in a cylindrical strongbox: cross-section of the soil model,
with positions of the PPTs and of the stone columns (a / dsc = 2 [-]). .................................. 134
Figure 3.51: Specimens prepared in a cylindrical strongbox: plan view of the soil model with
positions of the PPTs and of the stone columns (a / dsc = 2 [-]). .......................................... 135
Figure 3.52: Specimens prepared in a cylindrical strongbox: insertion points of the electrical
impedance needle: positions of the reference points RP1 and RP2 and the points A2 to J2 (a
/ dsc = 2 [-]). ........................................................................................................................ 136
Figure 3.53: Specimens prepared in an oedometer container and surrounded by Perth sand:
(a) plan view and (b) cross-section of the soil model with positions of the PPTs and of the
stone column. ..................................................................................................................... 138
Figure 3.54: Comparison of the lateral stresses acting on the clay sample for specimens
prepared in an oedometer container and surrounded by Perth sand (σ’h Perth sand, calculated
based on the silo theory) and for specimens prepared in a rigid container (cylindrical
strongbox or adapted oedometer) with a pre-consolidation of σ’v = 100 kPa (σ’h clay, 100 kPa) or
of σ’v = 200 kPa (σ’h clay, 200 kPa). ........................................................................................... 140
Figure 3.55: Specimens prepared in an oedometer container and surrounded by Perth sand:
insertion points of the electrical impedance needle and positions of the reference points RP1
and RP2 and the points A to F. ........................................................................................... 141
Figure 3.56: Specimens prepared in an adapted oedometer: (a) plan view and (b) cross-
section of the soil model with positions of the PPTs and of the stone column. .................... 142
Figure 4.1: Profile of the vertical effective stress in the centrifuge (σ’v,centrifuge) and under the
press (σ’v,press) for a pre-consolidation of 100 kPa. .............................................................. 144
Figure 4.2: Profile of the vertical effective stress in the centrifuge (σ’v,centrifuge) and under the
press (σ’v,press) for a pre-consolidation of 200 kPa. .............................................................. 144
Figure 4.3: Profiles of the over-consolidation ratio for pre-consolidation stresses of 100 kPa
and 200 kPa. ...................................................................................................................... 145
Figure 4.4: Profiles of the undrained shear strength obtained with the T-Bar during tests
JG_v2 (su,JG_v2), JG_v8 (su,JG,v8) and JG_v10 (su,JGv10,A and su,JG,v10,B) compared with
theoretical predications based on Trausch-Giudici (2003, su,TG) and Küng (2003, su,K) and
with the back-calculated values of the parameters a and b (su,JG). ...................................... 146
Figure 4.5: Profiles of the undrained shear strength obtained with the T-Bar during test
JG_v9 (su,JG_v9,A and su,JG_v9_B) compared with theoretical predictions based on Trausch-
Giudici (2003, su,TG) and on Küng (2003, su,K), and with the back-calculated values of the
parameters a and b (su,JG). ................................................................................................. 146
Figure 4.6: Profiles of the undrained shear strength obtained with the T-Bar during tests
JG_v1 (su,JG_v1) and JG_v5 in the specimen consolidated up to 200 kPa (su,JG_v5) compared
with the profile obtained with back-calculated values of the parameters of a and b (su,JG). .. 147
Figure 4.7: Profiles of the undrained shear strength obtained with the T-Bar during test
JG_v7 (su, JG_v7,A and su, JG_v7_B) compared with theoretical predictions based on Trausch-
List of figures
XVIII
Giudici (2003, su,TG) and on Küng (2003, su,K) and with the back-calculated values of the
parameters a and b (su,JG). ................................................................................................. 148
Figure 4.8: Profiles of the back-calculated undrained shear strength for a specimen prepared
in a full cylindrical strongbox (su,JG,1) and in adapted oedometers (su,JG,2 and su,JG,3). .......... 149
Figure 4.9: Installation of a compacted column in a specimen pre-consolidated up to 200 kPa
(test JG_v7) (a) pore water pressures (b) depth of the tip of the installation tool with time. . 151
Figure 4.10: Insertion of the stone column installation tool in a specimen pre-consolidated up
to 200 kPa (tests JG_v7) (a) excess pore water pressures (b) depth of the tip of the
installation tool with time. .................................................................................................... 152
Figure 4.11: Insertion of the stone column installation tool in a specimen pre-consolidated up
to 200 kPa (test JG_v7): excess pore water pressures. ...................................................... 153
Figure 4.12: Insertion of the stone column installation tool in a specimen pre-consolidated up
to 100 kPa (test JG_v9) (a) excess pore water pressures (b) depth of the tip of the
installation tool with time. .................................................................................................... 154
Figure 4.13: Insertion of the stone column installation tool in a specimen consolidated up to
100 kPa (test JG_v10) (a) excess pore water pressures (b) location of the stone columns
installed. ............................................................................................................................. 155
Figure 4.14: Insertion of the stone column installation tool in a specimen consolidated up to
100 kPa (test JG_v10): excess pore water pressures. ........................................................ 156
Figure 4.15: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v2) (a) excess pore water pressures (b) evolution of the footing load (c)
deformation controlled footing settlement. .......................................................................... 158
Figure 4.16: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v9) (a) excess pore water pressures (b) evolution of the footing load (c) deformation
controlled footing settlement. .............................................................................................. 160
Figure 4.17: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v9), dissipation with time of the excess pore water pressures at a depth of 48 mm
around the stone column (a) from 0 s to 2000 s and (b) from 3000 s to 9000 s after reaching
the peak footing load (which corresponds to t = 0 s). .......................................................... 161
Figure 4.18: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v9): dissipation of the excess pore water pressures at a depth of 96 mm with time
around the stone column (a) from 0 s to 2000 s and (b) from 3000 s to 9000 s after reaching
the peak footing load (which corresponds to t = 0 s). ......................................................... 162
Figure 4.19: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v9): rate of dissipation of excess pore water pressures with time after reaching the
peak footing load (which corresponds to t = 0 s). ................................................................ 163
Figure 4.20: Loading of a single stone column in a specimen pre-consolidated up to 200 kPa
(JG_v7) (a) excess pore water pressures (b) evolution of the footing load (c) deformation
controlled footing settlement. .............................................................................................. 165
Figure 4.21: Distribution of the excess pore water pressure with increasing radial distance to
the axis of the stone column at a depth of 48 mm as a percentage of the applied footing load
P. ....................................................................................................................................... 167
List of figures
XIX
Figure 4.22: Distribution of the excess pore water pressure with increasing radial distance to
the axis of the stone column at a depth of 96 mm as a percentage of the applied footing load
P. ....................................................................................................................................... 168
Figure 4.23: Isobars of vertical stress increments under a vertically loaded quadratic plate
(Lang et al., 2007). ............................................................................................................. 169
Figure 4.24: Isobars of peak values of excess pore pressures measured in the centrifuge
under a vertically loaded circular footing resting on top of a stone column. ......................... 174
Figure 4.25: Isobars of vertical stress increments under a vertically loaded circular footing
(after Grasshoff, 1978). ...................................................................................................... 174
Figure 4.26: Distribution of the total vertical stress increase as a function of the radial
distance from the stone column at 96 mm depth as a percentage of the applied footing load
P, and in comparison with the depth factor J4 according to Grasshoff (1978). .................... 175
Figure 4.27: Excess sand shown on top of columns B, D and E within the footprint of the
footing on the surface of the clay model after test JG_v8. .................................................. 177
Figure 4.28: Loading of a stone column group in a specimen pre-consolidated up to 100 kPa
(JG_v8) (a) excess pore water pressures (b) evolution of the footing load (c) deformation
controlled footing settlement. .............................................................................................. 178
Figure 4.29: Position of the footing used for the loading phase during test JG_v10 (a = 24
mm). ................................................................................................................................... 180
Figure 4.30: Test JG_v10: (a) excess pore water pressures (b) evolution of the footing load
(c) footing settlement during the loading phase of a stone column group. .......................... 181
Figure 4.31: Distribution of the excess pore water pressure with increasing radial distance to
the axis of the centre stone column at depths of 30 mm and of 80 mm as a percentage of the
applied footing load P (test JG_v10). .................................................................................. 182
Figure 4.32: Loading of a stone column group in a specimen pre-consolidated up to 100 kPa
(test JG_v10), dissipation with time of the excess pore water pressures at a depth of 30 mm
around the stone column (a) from 0 s to 2000 s and (b) from 3000 s to 7000 s after reaching
the peak footing load (which corresponds to t = 0 s). ......................................................... 183
Figure 4.33: Loading of a stone column group in a specimen pre-consolidated up to 100 kPa
(test JG_v10), dissipation with time of the excess pore water pressures at a depth of 80 mm
around the stone column (a) from 0 s to 2000 s and (b) from 3000 s to 7000 s after reaching
the peak footing load (which corresponds to t = 0 s). ......................................................... 184
Figure 4.34: Loading of a stone column group in a specimen pre-consolidated up to 100 kPa
(test JG_v10): rate of dissipation of excess pore water pressures with time after reaching the
peak footing load (which corresponds to t = 0 s). ................................................................ 185
Figure 4.35: Excess pore water pressures during the footing load test on a single stone
column (test JG_v9). .......................................................................................................... 186
Figure 4.36: Excess pore water pressures during the footing load test on a stone column
group (test JG_v10). The maximum load was reached at 1000 s. ...................................... 186
Figure 4.37: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v9): rate of dissipation of excess pore water pressures with time after reaching the
peak footing load (which corresponds to t = 0 s). ................................................................ 187
List of figures
XX
Figure 4.38: Loading of a stone column group in a specimen pre-consolidated up to 100 kPa
(test JG_v10): rate of dissipation of excess pore water pressures with time after reaching the
peak footing load (which corresponds to t = 0 s). ................................................................ 187
Figure 4.39: Installation effects around a stone column at a model depth of 40 mm @ 50 g
(Weber, 2008). ................................................................................................................... 188
Figure 4.40: Positions of the needle insertion points around a single stone column, and
extent of the zones 2 and 3 according to Weber (2008)...................................................... 188
Figure 4.41: Impedance recorded at reference points RP1 and RP2 during test JG_v5. .... 189
Figure 4.42: Impedance recorded at the points A, B and C during test JG_v5. ................... 190
Figure 4.43: Impedance recorded at the points D, E and F during the test JG_v5. ............. 190
Figure 4.44: Impedance recorded at the reference points RP1 and RP2 during test JG_v5.
........................................................................................................................................... 191
Figure 4.45: Impedance recorded at the points A, B and C during test JG_v5. ................... 192
Figure 4.46: Impedance recorded at the points D, E and F during test JG_v5. ................... 193
Figure 4.47: Positions of the needle insertion points around a stone column group (test
JG_v8, a / dsc = 2 [-]. .......................................................................................................... 194
Figure 4.48: Impedance recorded at the reference points RP1 and RP2 during test JG_v8.
........................................................................................................................................... 194
Figure 4.49: Impedance recorded at the points A2, B2 and C2 during test JG_v8. ............. 195
Figure 4.50: Impedance recorded at the points D2, E2 and F2 during test JG_v8. ............. 195
Figure 4.51: Impedance recorded at the points G2, H2, I2 and J2 during test JG_v8. ........ 196
Figure 5.1: Plan view of the extraction positions of the specimens for oedometer tests (test
JG_v9). .............................................................................................................................. 197
Figure 5.2: Cross-section of the extraction positions of the specimens for oedometer tests
(test JG_v9). ....................................................................................................................... 198
Figure 5.3: Distribution of the over-consolidation ratio of the specimens used for the
oedometer tests during the centrifuge test. ......................................................................... 198
Figure 5.4: Evolution of the void ratio with one dimensional loading in an oedometer. ........ 199
Figure 5.5: Distribution of the confined stiffness moduli as a function of the vertical effective
stress. ................................................................................................................................ 200
Figure 5.6: Distribution of the mean vertical (ME, v, average) and horizontal (ME, h, average) confined
stiffness moduli as a function of the vertical effective stress. .............................................. 202
Figure 5.7: Distribution of the mean settlements for the samples extracted in the vertical and
horizontal directions with one-dimensional loading in an oedometer. .................................. 202
Figure 5.8: Definition of Eoedref from oedometer test results (after Brinkgreve & Broere, 2008).
........................................................................................................................................... 203
Figure 5.9: Evolution of the permeability with one dimensional loading in an oedometer. ... 203
Figure 5.10: Extraction positions of the samples for oedometer tests (test JG_v10). .......... 204
Figure 5.11: Distribution of the over-consolidation ratio in the horizontal direction of the
specimens used for the oedometer tests. ........................................................................... 204
Figure 5.12: Evolution of the void ratio with one dimensional loading in an oedometer. ...... 205
Figure 5.13: Distribution of the horizontal confined stiffness moduli as a function of the
vertical stress. .................................................................................................................... 206
List of figures
XXI
Figure 5.14: Evolution of the coefficient of permeability with one dimensional loading in an
oedometer. ......................................................................................................................... 207
Figure 5.15: Setup for the insertion of the electrical impedance needle under 1 g in the
laboratory (a) schematic view (b) picture. ........................................................................... 208
Figure 5.16: Positions of the insertion points of the electrical impedance needle under 1 g. All
dimensions in mm. ............................................................................................................. 209
Figure 5.17: Impedance recorded under 1 g after completion of the first consolidation stage.
........................................................................................................................................... 209
Figure 5.18: Impedance recorded under 1 g after completion of the fifth consolidation stage.
........................................................................................................................................... 210
Figure 5.19: Illustration of the contact between the electron beam and the surface of the
sample (Peschke, 2013). .................................................................................................... 211
Figure 5.20: Electron interaction volume within a sample (after Science Education Resource
Center, 2013). .................................................................................................................... 211
Figure 5.21: Types of interaction between electrons and a sample (Science Education
Resource Center, 2013). .................................................................................................... 212
Figure 5.22: Schematic of an ESEM illustrating the different pressures zones (Donald, 2003).
........................................................................................................................................... 213
Figure 5.23: ESEM picture of zone 2, located a radial distance of 1 mm from the edge of the
column and at a depth of 40 mm below the surface, with the radial axis horizontal (Weber,
2008). ................................................................................................................................. 214
Figure 5.24: ESEM picture of the zone 3 located at a radial distance of 5 mm from the edge
of the column and at a depth of 20 mm below the surface, with the radial axis horizontal. .. 214
Figure 5.25: ESEM pictures of the zone 3 at a radial distance of 5 mm from the edge of the
column and at (a) 60 mm depth and (b) 100 mm depth, with the radial axis horizontal. ...... 215
Figure 5.26: Vacuum pump. ............................................................................................... 216
Figure 5.27: Dilatometer (a) containing the soil specimen before and (b) containing mercury
after the investigation using the macro pore unit Pascal 140. ............................................. 217
Figure 5.28: Porosity as a function of the radial distance from the axis of the stone column at
a depth of (a) 20 mm (b) 60 mm (c) 100 mm. ..................................................................... 218
Figure 6.1: Conversion of axisymmetric unit cell into plane-strain for drains (a) axisymmetric
radial flow (b) plane-strain (Indraratna & Redana, 1997). ................................................... 220
Figure 6.2: Cross-sections of the stone column (a) unit-cell; and plane-strain conversions
according to (b) method 1 and (c) method 2 (Tan et al., 2008). .......................................... 222
Figure 6.3: Plan view of 2D stone columns strips (a) width of an equivalent strip (b) strip
spacing (Chan & Poon, 2012). ............................................................................................ 224
Figure 6.4: Soil profile and properties at Tianjin Port in Beijing, China (Rujikiatkamjorn et al.,
2007). ................................................................................................................................. 225
Figure 6.5: Case study at Tianjin Port in Beijing, China: embankment and vacuum loading on
soft soil stabilised by drains (a) loading history and (b) comparison of the predicted (FEM)
and measured (Field) consolidation settlements (Rujikiatkamjorn et al., 2007). .................. 226
List of figures
XXII
Figure 6.6: Embankment pre-loading at Tianjin Port in Beijing, China (a) loading history (b)
comparison of the results obtained via 2D and 3D modelling with field observations
(Indraratna et al., 2009). ..................................................................................................... 227
Figure 6.7: Development of settlement at the crest of an embankment constructed in-flight on
remoulded Birmensdorf clay reinforced with stone columns – comparison between numerical
model and centrifuge results (Weber et al., 2009). ............................................................. 228
Figure 6.8: Numerical and experimental study of PVDs installed in clay (a) plan view with
dimensions of the smear and transition zones in terms of mandrel size (b) comparison of
settlement obtained with results by Indraratna & Redana (1998) (Basu et al., 2010). ......... 229
Figure 6.9: Model geometry and axisymmetric finite element mesh with applied radial
deformation of the stone column wall (Castro & Karstunen, 2010). .................................... 230
Figure 6.10: Normalised excess pore pressures generated by the stone column installation
(Castro & Karstunen, 2010). ............................................................................................... 230
Figure 6.11: Decrease of the undrained shear strength after column or pile installation
(Castro & Karstunen, 2010). ............................................................................................... 231
Figure 6.12: Unit cell (a) typical stone column–reinforced soft clay deposit supporting an
embankment; (b) unit cell idealisation; (c) cross-section (Indraratna et al., 2013). .............. 233
Figure 6.13: Influence of clogging on the normalised average excess pore water pressure
and on the normalised average ground settlement (Indraratna et al., 2013). ...................... 234
Figure 6.14: Boundary conditions for (a) the fixed pile approach (b) the moving pile approach
(Dijkstra et al., 2011). ......................................................................................................... 235
Figure 6.15: Comparison of calculated and measured stress response at the tip of the pile
during installation for the fixed pile approach (after Dijkstra et al., 2011). ........................... 236
Figure 6.16: Comparison of calculated and measured stress response at the tip of the pile
during installation for the moving pile approach (after Dijkstra et al., 2011). ....................... 236
Figure 6.17: Modelling technique for the simulation of the pile insertion (after Grabe &
Pucker, 2012). .................................................................................................................... 237
Figure 6.18: Elastic perfectly plastic model. ........................................................................ 238
Figure 6.19: Impact of the effective cohesion on the failure line in a – σ’ diagram. ........... 238
Figure 6.20: Effective stress paths followed real soil and FEM prediction using the Mohr-
Coulomb model. ................................................................................................................. 239
Figure 6.21: Hyperbolic stress-strain relation in primary loading and unloading-reloading in a
CDC triaxial test (Schanz et al., 1999). ............................................................................... 240
Figure 6.22: Successive yield loci for shear strain hardening for various values of the plastic
shear strain γp and failure surface for m = 0.5 [-] (Brinkgreve & Broere, 2008).................... 241
Figure 6.23: Cap yield surface of the HSM for volumetric hardening in the - plane (after
Brinkgreve & Broere, 2008). ............................................................................................... 242
Figure 6.24: Representation of yield contours of the HSM in effective principal stress space
(Schanz et al., 1999). ......................................................................................................... 244
Figure 6.25: Representation of the associated flow rule in triaxial space. ........................... 245
Figure 6.26: Modelling of the insertion of the stone column installation tool by means of
application of prescribed displacements on the wall of an initial cavity (Weber, 2008). ....... 246
List of figures
XXIII
Figure 6.27: 2D axisymmetric numerical model for a unit cell including a single stone column.
........................................................................................................................................... 251
Figure 6.28: Comparison of the experimental and numerical load-settlement curves for tests
JG_v7 and JG_v9. .............................................................................................................. 252
Figure 6.29: Deformed mesh obtained for test JG_v7 for a settlement of 850 mm and a
footing load of 145.44 kPa. ................................................................................................. 253
Figure 6.30: Vertical strain increment computed numerically for test JG_v7 for a settlement of
850 mm and a footing load of 145.44 kPa. ......................................................................... 253
Figure 6.31: Total vertical stress distribution computed numerically for test JG_v7 for a
settlement of 850 mm and a footing load of 145.44 kPa. .................................................... 254
Figure 6.32: Development of plastic points during the loading phase for test JG_v7 (a) for a
settlement of 100 mm (P = 85 kPa), (b) for a settlement of 400 mm (P = 115.2 kPa) and (c)
for a settlement of 850 mm (P = 145.44 kPa). .................................................................... 255
Figure 6.33: Vertical stress distribution from a line load below a rigid strip footing (a) for the
self-weight of the footing and for a global safety factor equal to (b) 3.0, (c) 2.0, (d) 1.5 and (e)
1.0 (Jessberger, 1995). ...................................................................................................... 256
Figure 6.34: Pressure distribution under the footing for a settlement of 100 mm at prototype
scale (2 mm under 50 g) for (a) test JG_v7 (P = 55.6 kPa) and (b) test JG_v9 (P = 46.1
kPa). ................................................................................................................................... 257
Figure 6.35: Pressure distribution under the footing for a settlement of 400 mm at prototype
scale (8 mm under 50 g) for (a) test JG_v7 (P = 110.1 kPa) and (b) test JG_v9 (P = 90.1
kPa). ................................................................................................................................... 257
Figure 6.36: Pressure distribution under the footing for a settlement of 800 mm at prototype
scale (16 mm under 50 g) for test JG_v9 (P = 118.1 kPa). ................................................. 258
Figure 6.37: Total vertical stress distribution under the footing, as a function of the radial
distance, for settlements of 100 mm, 400 mm and 850 mm for test JG_v7 (σc = 200 kPa).
........................................................................................................................................... 259
Figure 6.38: Total vertical stress distribution under the footing, as a function of the radial
distance, for settlements of 100 mm, 400 mm and 850 mm for test JG_v9 (σc = 100 kPa).
........................................................................................................................................... 259
Figure 6.39: Total vertical stress distribution, as a function of the radial distance, under the
footing (z = 100 mm) and at depths of 2 m, 4 m and 6 m for a footing settlement of 100 mm
during the footing loading for test JG_v7 (P = 85 kPa). ....................................................... 260
Figure 6.40: Total vertical stress distribution, as a function of the radial distance, under the
footing (z = 100 mm) and at depths of 2 m, 4 m and 6 m for a footing settlement of 100 mm
during the footing loading for test JG_v9 (P = 70.8 kPa). .................................................... 260
Figure 6.41: Distribution of the stress concentration factor m over depth for footing
settlements of 100 mm, 400 mm and 850 mm for test JG_v7 (σc = 200 kPa). .................... 261
Figure 6.42: Distribution of the stress concentration factor m over depth for footing
settlements of 100 mm, 400 mm and 850 mm for test JG_v9 (σc = 100 kPa). .................... 261
Figure 6.43: Distribution of the normalised stress concentration factor m over depth for
footing settlements of 100 mm, 400 mm and 850 mm for test JG_v7 (σc = 200 kPa). ........ 262
List of figures
XXIV
Figure 6.44: Distribution of the normalised stress concentration factor m over depth for
footing settlements of 100 mm, 400 mm and 850 mm for test JG_v9 (σc = 100 kPa). ......... 263
Figure 6.45: Direction of the total principal stress at the end of the loading phase for test
JG_v7 (P = 141.90 kPa). .................................................................................................... 263
Figure 6.46: Distribution of the excess pore water pressures computed numerically for test
JG_v7 for a footing settlement of 850 mm and a footing load of 141.90 kPa. ..................... 264
Figure 6.47: Comparison of the values of the excess pore water pressures measured during
test JG_v7 with the values obtained numerically with Plaxis 2D (P1 till P3). ....................... 265
Figure 6.48: Comparison of the experimental load-settlement curves for test JG_v7 (σc = 200
kPa) with the numerical simulations, with and without installation effects. .......................... 266
Figure 6.49: Comparison of the experimental load-settlement curves for test JG_v9 (σc = 100
kPa) with the numerical simulations, with and without installation effects. .......................... 266
Figure 6.50: General view of the 3D mesh (groundwater table 0.5 m below the surface). ... 267
Figure 6.51: Plan on the stone column group. .................................................................... 267
Figure 6.52: Side view of the stone columns and zones created to represent the installation
effects (the base of the box is located 1 m below the toe of the compaction zone), in which
clay and compaction zone (left hand side), respectively clay, compaction zone and smear
zone (right hand side), were hidden in order to expose the smear zone, respectively the
stone column. ..................................................................................................................... 268
Figure 6.53: Plan on the stone column group with position of the square footing, as applied in
the centrifuge test JG_v10. ................................................................................................. 268
Figure 6.54: Comparison of the experimental and numerical load-settlement curves for the
test JG_v10. ....................................................................................................................... 269
Figure 6.55: Pressure distribution under the square footing for a settlement of 100 mm at
prototype scale (2 mm under 50g) for test JG_v10. ............................................................ 270
Figure 6.56: Distribution of the total vertical stresses for test JG_v10 (σc = 100 kPa) for a
settlement of 850 mm and a footing load of 138.66 kPa (section 1-1, Figure 6.51). The
dimensions are given in Figure 6.51 and Figure 6.52. ........................................................ 270
Figure 6.57: Distribution of the total vertical stresses for test JG_v10 (σc = 100 kPa) for a
settlement of 850 mm and a footing load of 138.66 kPa (section 2-2, Figure 6.51). The
dimensions are given in Figure 6.51 and Figure 6.52. ........................................................ 271
Figure 6.58: Total vertical stress distribution under the footing for a settlement of 850 mm for
test JG_v10 (σc = 100 kPa, P = 138.66 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 272
Figure 6.59: Total vertical stress distribution at a depth of 6 m for a settlement of 850 mm for
test JG_v10 (σc = 100 kPa, P = 138.66 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 273
Figure 6.60: Comparison of the values of the excess pore water pressures measured during
test JG_v10, with the values obtained numerically with Plaxis 3D (P1 till P3). .................... 273
Figure 6.61: Distribution of the excess pore water pressures computed numerically for test
JG_v10 (σc = 100 kPa) for a settlement of 850 mm and a footing load of 138.66 kPa (section
1-1, Figure 6.51). ................................................................................................................ 274
List of figures
XXV
Figure 6.62: Distribution of the excess pore water pressures computed numerically for test
JG_v10 (σc = 100 kPa) for a settlement of 850 mm and a footing load of 138.66 kPa (section
2-2, Figure 6.51). ................................................................................................................ 274
Figure 6.63: Deformed columns A, C and D for test JG_v10, for a settlement of 850 mm and
a footing load of 138.66 kPa, in which clay, compaction zone and smear zone were hidden in
order to expose the deformation of the stone columns. ...................................................... 275
Figure 6.64: Comparison of the experimental load-settlement curves for test JG_v10 with the
numerical simulations, with and without, installation effects. ............................................... 276
Figure 7.1: Distribution of the vertical stress increase as a function of the radial distance from
the stone column at 96 mm depth as a percentage of the applied footing load P and
comparison with the depth factor J4 according to Grasshoff (1978). ................................... 280
Figure 7.2: Electrical impedance needle (a) side view and (b) tilted view of the tip (outer
diameter 1 mm). ................................................................................................................. 280
Figure 7.3: ESEM picture of zone 3, located at a radial distance of 5 mm from the edge of the
column and at a depth of 20 mm below the surface, with the radial axis horizontal. ........... 281
Figure 7.4: Porosity as a function of the radial distance from the axis of the stone column at a
depth of (a) 20 mm (b) 60 mm (c) 100 mm. ........................................................................ 282
Figure 7.5: Total vertical stress distribution under the footing for a settlement of 850 mm for
test JG_v10 (σc = 100 kPa, P = 138.66 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 284
Figure 7.6: Total vertical stress distribution at a depth of 6 m for a settlement of 850 mm for
test JG_v10 (σc = 100 kPa, P = 138.66 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 284
Figure 7.7: Total vertical stress distribution computed numerically for test JG_v7 for a
settlement of 850 mm and a footing load of 145.44 kPa. .................................................... 285
Figure 7.8: Distribution of the total vertical stresses for test JG_v10 (σc = 100 kPa) for a
settlement of 850 mm and a footing load of 138.66 kPa (section 1-1, Figure 6.51). The
dimensions are given in Figure 6.51 and Figure 6.52. ........................................................ 285
Figure 7.9: Distribution of the excess pore water pressures computed numerically for test
JG_v7 for a footing settlement of 850 mm and a footing load of 141.90 kPa. ..................... 286
Figure 7.10: Isobars of peak values of excess pore pressures measured in the centrifuge
under a vertically loaded circular footing resting on top of a stone column. ......................... 286
Figure 8.1: Loading of a single stone column in a specimen pre-consolidated up to 200 kPa
(test JG_v1): (a) excess pore water pressures (b) evolution of the footing load (c) footing
settlement. .......................................................................................................................... 291
Figure 8.2: Loading of a single stone column in a specimen pre-consolidated up to 200 kPa
(test JG_v5): (a) excess pore water pressures (b) evolution of the footing load (c) footing
settlement. .......................................................................................................................... 292
Figure 8.3: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v6): (a) Excess pore water pressures (b) evolution of the footing load (c) footing
settlement. .......................................................................................................................... 293
Figure 8.4: Impedance recorded at the reference points RP1 and RP2 during test JG_v9
(Container A). ..................................................................................................................... 295
List of figures
XXVI
Figure 8.5: Impedance recorded at the reference points RP1 and RP2 during test JG_v9
(Container B). ..................................................................................................................... 295
Figure 8.6: Impedance recorded at the points A, B and C during test JG_v9 (Container A).
........................................................................................................................................... 296
Figure 8.7: Impedance recorded at the points D, E and F during test JG_v9 (Container A).
........................................................................................................................................... 296
Figure 8.8: Impedance recorded at the points A, B and C during test JG_v9 (Container B).
........................................................................................................................................... 297
Figure 8.9: Impedance recorded at the points D, E and F during test JG_v9 (Container B).
........................................................................................................................................... 297
Figure 8.10: Positions of the insertion points of the electrical impedance needle under 1 g. All
dimensions in mm. ............................................................................................................. 298
Figure 8.11: Impedance recorded under 1 g after completion of the second consolidation
stage. ................................................................................................................................. 299
Figure 8.12: Impedance recorded under 1 g after completion of the third consolidation stage.
........................................................................................................................................... 299
Figure 8.13: Impedance recorded under 1 g after completion of the fourth consolidation
stage. ................................................................................................................................. 300
Figure 8.14: Vertical strain increments computed numerically for test JG_v7 for a settlement
of 100 mm and a footing load of 85 kPa. ............................................................................ 301
Figure 8.15: Vertical strain increments computed numerically for test JG_v7 for a settlement
of 400 mm and a footing load of 115.2 kPa ........................................................................ 302
Figure 8.16: Shear strain increments computed numerically for test JG_v7 for a settlement of
100 mm and a footing load of 85 kPa. ................................................................................ 303
Figure 8.17: Shear strain increments computed numerically for test JG_v7 for a settlement of
400 mm and a footing load of 115.2 kPa. ........................................................................... 304
Figure 8.18: Shear strain increments computed numerically for test JG_v7 for a settlement of
850 mm and a footing load of 145 kPa. .............................................................................. 305
Figure 8.19: Development of plastic points during the loading phase for test JG_v7 for a
settlement of 100 mm (P = 85 kPa). ................................................................................... 306
Figure 8.20: Development of plastic points during the loading phase for test JG_v7 for a
settlement of 400 mm (P = 115.2 kPa). .............................................................................. 307
Figure 8.21: Development of plastic points during the loading phase for test JG_v7 for a
settlement of 850 mm (P = 145.44 kPa). ............................................................................ 308
Figure 8.22: Deformed mesh obtained for test JG_v9 for a settlement of 850 mm and a
footing load of 115.11 kPa. ................................................................................................. 309
Figure 8.23: Total vertical stress distribution computed numerically for test JG_v9 for a
settlement of 850 mm and a footing load of 119.67 kPa. .................................................... 310
Figure 8.24: Vertical strain increment computed numerically for test JG_v9 for a settlement of
100 mm and a footing load of 70.8 kPa. ............................................................................. 311
Figure 8.25: Vertical strain increment computed numerically for test JG_v9 for a settlement of
400 mm and a footing load of 94.3 kPa. ............................................................................. 312
List of figures
XXVII
Figure 8.26: Vertical strain increment computed numerically for test JG_v9 for a settlement of
850 mm and a footing load of 119.67 kPa. ......................................................................... 313
Figure 8.27: Shear strain increment computed numerically for test JG_v9 for a settlement of
100 mm and a footing load of 70.8 kPa. ............................................................................. 314
Figure 8.28: Shear strain increment computed numerically for test JG_v9 for a settlement of
400 mm and a footing load of 94.3 kPa. ............................................................................. 315
Figure 8.29: Shear strain increment computed numerically for test JG_v9 for a settlement of
850 mm and a footing load of 119.67 kPa. ......................................................................... 316
Figure 8.30: Distribution of the excess pore water pressures computed numerically for test
JG_v9 for a footing settlement of 850 mm and a footing load of 115.11 kPa ...................... 317
Figure 8.31: Development of plastic points during the loading phase for test JG_v9 for a
settlement of 100 mm (P = 70.8 kPa). ................................................................................ 318
Figure 8.32: Development of plastic points during the loading phase for test JG_v9 for a
settlement of 400 mm (P = 94.3 kPa). ................................................................................ 319
Figure 8.33: Development of plastic points during the loading phase for test JG_v9 for a
settlement of 850 mm (P = 119.67 kPa). ............................................................................ 320
Figure 8.34: Total vertical stress distribution as a function of the radial distance under the
footing (z = 400 mm) and at depths of 2 m, 4 m and 6 m for a footing settlement of 400 mm
during the footing loading for test JG_v7 (P = 115.2 kPa). .................................................. 321
Figure 8.35: Total vertical stress distribution as a function of the radial distance under the
footing (z = 850 m) and at depths of 2 m, 4 m and 6 m for a footing settlement of 850 mm
during the footing loading for test JG_v7 (P = 145.44 kPa). ................................................ 321
Figure 8.36: Total vertical stress distribution as a function of the radial distance under the
footing (z = 400 m) and at depths of 2 m, 4 m and 6 m for a footing settlement of 400 mm
during the footing loading for test JG_v9 (P = 94.3 kPa). .................................................... 322
Figure 8.37: Total vertical stress distribution as a function of the radial distance under the
footing (z = 850 m) and at depths of 2 m, 4 m and 6 m for a footing settlement of 850 mm
during the footing loading for test JG_v9 (P = 119.67 kPa). ................................................ 322
Figure 8.38: Comparison of the values of the excess pore water pressures measured during
test JG_v7 with the values obtained numerically with Plaxis 2D (P4 till P6). ....................... 323
Figure 8.39: Comparison of the values of the excess pore water pressures measured during
test JG_v7 with the values obtained numerically with Plaxis 2D (P7). ................................. 323
Figure 8.40: Comparison of the values of the excess pore water pressures measured during
test JG_v9 with the values obtained numerically with Plaxis 2D (P1 till P3). ....................... 324
Figure 8.41: Comparison of the values of the excess pore water pressures measured during
test JG_v9 with the values obtained numerically with Plaxis 2D (P4 till P6). ....................... 324
Figure 8.42: Comparison of the values of the excess pore water pressures measured during
test JG_v9 with the values obtained numerically with Plaxis 2D (P7). ................................. 325
Figure 8.43: Distribution of the total vertical stresses for test JG_v10 for a settlement of 100
mm and a footing load of 76 kPa (section 1-1, Figure 6.51). The dimensions are given in
Figure 6.51 and Figure 6.52. .............................................................................................. 326
List of figures
XXVIII
Figure 8.44: Distribution of the total vertical stresses for test JG_v10 for a settlement of 100
mm and a footing load of 76 kPa (section 2-2, Figure 6.51). The dimensions are given in
Figure 6.51 and Figure 6.52. .............................................................................................. 327
Figure 8.45: Distribution of the total vertical stresses for test JG_v10 for a settlement of 400
mm and a footing load of 108.60 kPa (section 1-1, Figure 6.51). The dimensions are given in
Figure 6.51 and Figure 6.52. .............................................................................................. 327
Figure 8.46: Distribution of the total vertical stresses for test JG_v10 for a settlement of 400
mm and a footing load of 108.60 kPa (section 2-2, Figure 6.51). The dimensions are given in
Figure 6.51 and Figure 6.52. .............................................................................................. 328
Figure 8.47: Total vertical stress distribution under the footing for a settlement of 100 mm for
test JG_v10 (σc = 100 kPa, P = 76.00 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 329
Figure 8.48: Total vertical stress distribution at a depth of 2 m for a settlement of 100 mm for
test JG_v10 (σc = 100 kPa, P = 76.00 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 330
Figure 8.49: Total vertical stress distribution at a depth of 4 m for a settlement of 100 mm for
test JG_v10 (σc = 100 kPa, P = 76.00 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 330
Figure 8.50: Total vertical stress distribution at a depth of 6 m for a settlement of 100 mm for
test JG_v10 (σc = 100 kPa, P = 76.00 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 331
Figure 8.51: Total vertical stress distribution under the footing for a settlement of 400 mm for
test JG_v10 (σc = 100 kPa, P = 108.60 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 332
Figure 8.52: Total vertical stress distribution at a depth of 2 m for a settlement of 400 mm for
test JG_v10 (σc = 100 kPa, P = 108.60 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 333
Figure 8.53: Total vertical stress distribution at a depth of 4 m for a settlement of 400 mm for
test JG_v10 (σc = 100 kPa, P = 108.60 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 333
Figure 8.54: Total vertical stress distribution at a depth of 6 m for a settlement of 400 mm for
test JG_v10 (σc = 100 kPa, P = 108.60 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 334
Figure 8.55: Total vertical stress distribution at a depth of 2 m for a settlement of 850 mm for
test JG_v7 (σc = 100 kPa, P = 138.66 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 335
Figure 8.56: Total vertical stress distribution at a depth of 4 m for a settlement of 850 mm for
test JG_v7 (σc = 100 kPa, P = 138.66 kPa). Negative stresses denote compression. The
dimensions are shown in Figure 6.51 and Figure 6.53........................................................ 336
Figure 8.57: Comparison of the values of the excess pore water pressures measured during
test JG_v10 with the values obtained numerically with Plaxis 3D (P4 till P6). ..................... 337
Figure 8.58: Comparison of the values of the excess pore water pressures measured during
test JG_v10 with the values obtained numerically with Plaxis 3D (P7). ............................... 337
List of figures
XXIX
Figure 8.59: Development of plastic points during the loading phase for test JG_v10 (section
1-1, Figure 6.51) for a settlement of 100 mm (P = 76 kPa). ................................................ 338
Figure 8.60: Development of plastic points during the loading phase for test JG_v10 (section
1-1, Figure 6.51) for a settlement of 400 mm (P = 108.60 kPa). ......................................... 339
Figure 8.61: Development of plastic points during the loading phase for test JG_v10 (section
1-1, Figure 6.51) for a settlement of 850 mm (P = 138.66 kPa). ......................................... 339
Figure 8.62: Development of plastic points during the loading phase for test JG_v10 (section
2-2, Figure 6.51) for a settlement of 100 mm (P = 76 kPa). ................................................ 340
Figure 8.63: Development of plastic points during the loading phase for test JG_v10 (section
2-2, Figure 6.51) for a settlement of 400 mm (P = 108.60 kPa). ......................................... 340
Figure 8.64: Development of plastic points during the loading phase for test JG_v10 (section
2-2, Figure 6.51) for a settlement of 850 mm (P = 138.66 kPa). ......................................... 341
List of figures
XXX
List of tables
XXXI
List of tables
Table 2.1: Classification of ground improvement methods (after Chu et al., 2009). ............... 5
Table 2.2: Classification of ground improvement methods (after Chu et al., 2009). ............... 6
Table 2.3: Suggested values of the stress concentration ratio m for specific combinations of
area replacement ratio as and effective friction angle of the stone column material for
compacted sand columns after Ichimoto & Suematsu (1981). ..............................................14
Table 2.4: Summary of 1 g small-scale model tests (σc: pre-consolidation stress; su:
undrained shear strength; rsc: radius of the stone columns; L: length of the stone columns; a:
spacing of the stone columns; as: area replacement ratio) (after Muir Wood et al., 2000). ....18
Table 2.5 Summary of inclinometer and piezometer installations at Red River test site in
Winnipeg, Canada: (SI: slope inclinometer; PZ: piezometer; VW: vibrating wire) (after
Thiessen et al., 2011). ..........................................................................................................52
Table 2.6: Geotechnical properties of Busan clay (Shin et al., 2009). ...................................80
Table 3.1: Summary of the main scaling factors (after Schofield, 1980). ..............................94
Table 3.2: Classification and selected mechanical properties of Birmensdorf clay (after
Weber, 2008). .................................................................................................................... 116
Table 3.3: Selected properties of Birmensdorf clay, determined from oedometer tests. ...... 117
Table 3.4: Parameters for the quartz sand used as stone column material (Weber, 2008). 117
Table 3.5: Selected properties of Perth Sand (Nater, 2005). .............................................. 118
Table 3.6: Summary of the containers used for the preparation of soil models for centrifuge
tests. .................................................................................................................................. 118
Table 3.7: Summary of the main system parameters. ......................................................... 129
Table 3.8: Overview of the experimental setup used for the different tests. ........................ 130
Table 3.9: Testing procedure for tests conducted using the full cylindrical strongbox (loading
of a single stone column, tests JG_v2, JG_v3 and JG_v6). ................................................ 132
Table 3.10: Overview of the experimental setups used for the different tests. .................... 134
Table 3.11: Testing procedure for tests conducted using the full cylindrical strongbox (loading
of a stone column group, tests JG_v8 and JG_v10). .......................................................... 137
Table 3.12: Testing procedure for tests conducted using the specimens prepared in an
oedometer or in an adapted oedometer (tests JG_v1, JG_v4, JG_v5, JG_v7 and JG_v9). 140
Table 4.1: Values of a and b obtained by Trausch-Giudici (2003) and Küng (2003). .......... 144
Table 4.2: Comparison of the back-calculated values of the parameters a and b with those
proposed by Trausch-Giudici (2003) and Küng (2003). ...................................................... 149
Table 4.3: Response of the PPT to the applied footing load on a single stone column during
test JG_v2. ......................................................................................................................... 157
Table 4.4: Response of the PPT to the applied footing load on a single stone column during
the test JG_v6. ................................................................................................................... 157
Table 4.5: Response of the PPT to the applied footing load on a single stone column during
the test JG_v9. ................................................................................................................... 159
Table 4.6: Response of the PPT to the applied footing load on a single stone column during
test JG_v7 (adapted oedometer). ....................................................................................... 164
List of tables
XXXII
Table 4.7: Response of the PPT to the applied footing load on a single stone column during
test JG_v1 (cylindrical strongbox with clay surrounded by sand). ....................................... 164
Table 4.8: Response of the PPT to the applied footing load on a single stone column during
test JG_v5 (cylindrical strongbox with clay surrounded by sand). ....................................... 166
Table 4.9: Calculation of the over-consolidation ratio at the installation depths of the PPTs
depending on the pre-consolidation stress σc. .................................................................... 170
Table 4.10: Coefficients of earth pressure at rest of the over-consolidated Birmensdorf clay
with depth. .......................................................................................................................... 170
Table 4.11: Values of the pore pressure parameter A depending on the type of clay
(Skempton, 1954). .............................................................................................................. 171
Table 4.12: Comparison of the peak analytical and measured excess pore water pressure at
end of the loading phase of a single stone column (test JG_v9, σc = 100 kPa, P = 119.67
kPa). ................................................................................................................................... 172
Table 4.13: Comparison of the peak analytical and measured excess pore water pressure at
end of the loading phase of a single stone column (test JG_v7, σc = 200 kPa, P = 145.44
kPa). ................................................................................................................................... 172
Table 4.14: Comparison of the peak analytical and measured excess pore water pressure at
end of the loading phase of a single stone column (test JG_v5, σc = 200 kPa, P = 120.14
kPa). ................................................................................................................................... 172
Table 4.15: Back-calculated values of the vertical stress increases at the locations of the
PPTs P4, P5, P6 and P7 for tests JG_v1, JG_v5, JG_v7 and JG_v9. ................................ 175
Table 4.16: Response of the PPT to the applied footing load on a stone column group during
test JG_v8. ......................................................................................................................... 179
Table 4.17: Response of the PPT to the applied footing load on a stone column group during
test JG_v10. ....................................................................................................................... 180
Table 5.1: Compression indexes CC obtained from the oedometer tests. ........................... 199
Table 5.2: Compression indexes CS obtained from the oedometer tests. ............................ 200
Table 5.3: Vertical (ME,v) and horizontal (ME,h) confined stiffness moduli obtained from the
oedometer tests and values of the over-consolidation ratios for samples extracted vertically
(OCRv) and horizontally (OCRh). ........................................................................................ 201
Table 5.4: Compression indexes CC obtained from the oedometer tests. ........................... 205
Table 5.5: Compression indexes CS obtained from the oedometer tests. ............................ 206
Table 5.6: Overview of the consolidation stages for the implementation of the electrical
impedance needle under 1 g. ............................................................................................. 208
Table 6.1: Input parameters for the Mohr-Coulomb model. ................................................. 239
Table 6.2: Stiffness moduli used in the HSM. ..................................................................... 240
Table 6.3: Input parameters for the soil stiffness in the HSM. ............................................. 245
Table 6.4: Description of the calculation phases used in the axisymmetric model. ............. 247
Table 6.5: Summary of the Hardening Soil parameters for the clay. ................................... 248
Table 6.6: Mohr-Coulomb parameters for the stone column material. ................................. 249
Table 6.7: Summary of the Hardening Soil parameters for the smear zone. ....................... 249
Table 6.8: Summary of the Hardening Soil parameters for the compaction zone. ............... 250
List of tables
XXXIII
Table 6.9: Comparison of the experimental and numerical values of the maximum footing
loads. ................................................................................................................................. 251
Table 6.10: Comparison of the values of the maximum footing loads obtained numerically
with (Pmax, Plaxis) and without (Pmax, Plaxis, no smear) installation effects for a settlement of 850 mm.
........................................................................................................................................... 265
Table 6.11: Comparison of the experimental and numerical values of the maximum footing
loads for a footing settlement of 850 mm. ........................................................................... 269
Table 6.12: Comparison of the values of the maximum footing loads obtained numerically
with (Pmax, Plaxis) and without (Pmax, Plaxis, no smear) installation effects. ....................................... 275
Table 8.1: Values of the factor J4 in ‰ (Lang et al., 2007) (z: depth of the point considered;
R: radius of the circular footing; a: radial distance of the point considered from the centre of
the footing). ........................................................................................................................ 294
Table 8.2: Comparison of the analytical and measured excess pore water pressure during
the loading phase of a single stone column (test JG_v1, σc = 200 kPa, P = 80 kPa). ......... 294
Table 8.3: Overview of the consolidation stages conducted for the insertion of the electrical
impedance needle under 1 g. ............................................................................................. 298
List of tables
XXXIV
Abstract
XXXV
Abstract
Switzerland’s main infrastructures and industrial installations are located in the so-called
Mittelland, the plain between Jura and the Alps, where the flat areas are mostly formed of
soft lacustrine deposits which are the residual products of the last glaciations. Such soil
conditions cause challenges for constructions in terms of the Ultimate Limit State (ULS) as
their bearing capacity may be insufficient. Moreover, the Serviceability Limit State (SLS) may
become problematic, as the low permeability of lacustrine deposits cause consolidation times
of several years, which leads to continuous settlements of the structures over this period.
Stone columns have proven to be an efficient ground improvement method in soft soils, as
they increase the stiffness and strength of the subsoil, as well as reducing the consolidation
time through shorter (radial) drainage paths. This allows higher loads to be carried, with
lower post-construction settlements.
However, installation effects, identified e.g. by Weber (2008) impair the performance of stone
columns in terms of drainage capacity. The spatial distribution of these effects long remained
unclear, which represented a gap of knowledge that this research aimed to fill.
The load-settlement behaviour of composite foundations with stone columns was
investigated in the geotechnical drum centrifuge at the ETH Zürich. The use of a centrifuge
enables a reproduction of the in-situ stress states and thus an accurate reproduction of the
mechanisms developing during the installation and subsequent loading of granular
inclusions. Stone columns were installed in-flight in a geotechnical drum centrifuge and
subsequently loaded with rigid footings while monitoring the footing load and the pore water
pressures in the clay specimen. The findings of the physical modelling under enhanced
gravity gave an insight into the load-transfer behaviour with depth around a stone column.
The insertion of an electrical needle measuring the impedance in the host soil surrounding
the inclusions also gave valuable information about the extent of the micromechanical
reorganisation of the clay particles caused by the installation and loading of a stone column.
Samples were extracted from the soft soil bed around the piles after the centrifuge tests in
order to conduct complementary investigations, namely oedometer tests, Mercury Intrusion
Porosimetry (MIP) and Environmental Scanning Electron Microscopy (ESEM). The MIP
investigations showed that the extent of the macromechanical installation effects remains
constant with depth and reaches 2.5 times the radius of the stone column from the centre of
the inclusions. The ESEM observations confirmed the interpretation of the measurements
made with the electrical needle, according to which a progressive reorganisation of the clay
particles occurs up to a distance corresponding to 5 times the radius of the inclusions from
the axis of the stone column.
The outcomes of the physical modelling under enhanced gravity as well as those of the
subsequent micromechanical investigations were used for the construction of a numerical
model aiming to reproduce the centrifuge tests.
Abstract
XXXVI
The numerical model developed in the course of this research allows the installation effects
to be considered in a simplified form as it does not require modelling the installation phase of
the inclusions. A good prediction of the load-settlement behaviour of composite foundations
using stone columns in soft clays has been reached. Interesting insights into the variation of
the stress concentration with depth were obtained, as it was shown that the stress
concentration factor reaches values ranging from 1.0 to 1.5 in the lower third of the stone
column. This could open up the way for a more economical design of granular inclusions, as
the diameter and thus the quantity of material needed, could be reduced in zones where the
stress concentration is low. This should be confirmed by further research, taking the
influence of varying soil characteristics and stress history as well as of different types of
loading, into account.
Kurzfassung
XXXVII
Kurzfassung
Die wichtigsten Infrastrukturen der Schweiz befinden sich in dem sogenannten Mittelland,
der Ebene zwischen Jura und den Alpen, mit flachen Zonen aus weichem Seebodenlehm,
die Restprodukte von den letzten Vereisungen darstellen. Derartige Bodenverhältnisse
stellen eine Herausforderung für bauliche Anlagen in Bezug auf den Grenzzustand der
Tragsicherheit (ULS) dar, da deren Tragfähigkeit unzureichend sein kann. Auch der
Grenzzustand der Gebrauchstauglichkeit (SLS) kann sich problematisch erweisen, da die
niedrige Durchlässigkeit von Seebodenlehmen zu Konsolidationszeiten von mehreren Jahren
bis hin zu Jahrzehnten führt, was kontinuierliche Setzungen der Anlagen über diese
Zeitperiode zur Folge hat.
Kiessäulen stellen eine effiziente Methode der Baugrundverbesserung dar. Auf der einen
Seite steigern sie durch die Verdichtung des umgebenden Bodens und die Zuführung von
tragfähigem Material die Steifigkeit des Baugrundes. Auf der anderen Seite werden die
Drainagewege verkürzt, was eine Reduktion der Konsolidierungszeit bewirkt. Allerdings
beeinträchtigen Effekte, die sich aus der Herstellung der Säulen ergeben, vor allem die
Wirkung der Drainage durch das Entstehen von Schmierzonen wie auch durch die
Verdichtung, die eine Reduktion der Durchlässigkeit im weichen Boden bewirkt. Die
Verteilung dieser Zonen in radialer wie auch vertikaler Richtung ist dabei nicht abschliessend
geklärt.
Das Last-Setzungsverhalten von starren Fundamenten auf Kiessäulen wurde in der
geotechnischen Trommelzentrifuge der ETH Zürich untersucht. Mittels einer Zentrifuge
können die in-situ Spannungszustände im Boden simuliert werden. Damit stellen sich
realitätsnahe Mechanismen während der Installation und der darauf folgenden Belastung von
Kiessäulen ein. Dabei wurden Kiessäulen im Flug in einer geotechnischen
Trommelzentrifuge installiert und anschliessend mit starren Fundamenten unter
Überwachung der Fundamentbelastung und der Porenwasserdrücke im Tonmodell belastet.
Die Ergebnisse der physikalisen Modellierung unter den Bedingungen eines erhöhten
Schwerfeldes gaben einen Einblick in das Lastabtragungsverhalten einer Kiessäule über die
Tiefe. Die Einbringung einer elektrischen Nadel zur Messung der Impedanz des Bodens um
die Inklusionen lieferte wertvolle Informationen über den Umfang der durch die
Kiessäuleninstallation und -belastung verursachten mikromechanischen Reorganisation der
Tonpartikel.
Im Anschluss an die Zentrifugenversuche wurden Proben aus dem weichen Bodenmodell
um die Säulen entnommen, um ergänzende Untersuchungen, nämlich Oedometerversuche,
Quecksilberporosimetrie und Environmental Scanning Electron Microscopy (ESEM)
durchzuführen. Die Quecksilberporosimetrie-Untersuchungen haben gezeigt, dass der
Umfang der makromechanischen Installationseffekte über die Tiefe konstant bleibt und sich,
gemessen ab der Säulenachse, über einen Bereich bis zum 2.5-fachen Radius der Kiessäule
erstreckt. Die ESEM-Beobachtungen haben die Interpretation der Messungen mit der
elektrischen Nadel bestätigt, der zufolge eine progressive Reorganisation der Tonpartikel in
einem Bereich mit dem fünffachen Radius der Kiessäulen, gemessen ab deren Achse,
stattfindet.
Kurzfassung
XXXVIII
Basierend auf den Ergebnissen der physikalischen Modellierungen unter den Bedingungen
eines erhöhten Schwerefeldes sowie der nachfolgenden mikromechanischen
Untersuchungen wurde ein numerisches Modell zur Reproduktion der Zentrifugentests
entwickelt.
Dieses numerische Modell ermöglicht eine vereinfachte Berücksichtigung der
Installationseffekte, da es auf die Modellierung der Installationsphase im numerischen Modell
verzichtet. Die gefundenen Zonen wurden mit veränderten Parametern als Makromodell
vorgegeben. Es wurde eine gute Prognose des Last-Setzungsverhaltens von
Verbundfundationen mit Kiessäulen erreicht. Dies lieferte interessante Einblicke in die
Variation der Spannungskonzentration über die Tiefe, da gezeigt wurde, dass der
Spannungskonzentrationsfaktor Werte zwischen 1,0 und 1,5 in dem tieferen Drittel der
Kiessäulen erreicht. Dies könnte den Weg zu einer wirtschaftlicheren Bemessung von
Kiessäulen weisen, da der Durchmesser und damit die benötigten Rohstoffmengen in den
Zonen mit niedriger Spannungskonzentration reduziert werden könnten. Dies sollte durch
weitere Forschung bestätigt werden, indem der Einfluss von wechselnden
Bodeneigenschaften, der Spannungsgeschichte sowie verschiedener Belastungstypen,
berücksichtigt wird.
1 Introduction
1
1 Introduction
1.1 Motivation
The Swiss Mittelland is a zone of vital importance for the country, where key infrastructure,
and lifelines industrial complexes and numerous buildings of national relevance are to be
found and where the demand for land suitable for construction is still growing. However, the
ground in the flat areas is mostly constituted of soft normally or slightly over-consolidated
lacustrine deposits that were formed after the last glaciations.
Building on soft soils presents challenges concerning design in respect of both the Ultimate
Limit State (ULS) and the Serviceability Limit State (SLS). Normally or slightly over-
consolidated lacustrine soils may not provide sufficient bearing capacity to support the
planned infrastructure. In addition, the low permeability will lead to consolidation times that
may last years or decades, and which will lead to on-going settlements during this period.
Numerous methods are available in foundation engineering to reduce settlements or
circumvent low bearing capacities by increasing the load transfer capacity, such as
preloading, grouting, deep mixing or support piling (Fleming et al., 1992). Drains, sometimes
combined with preloading or vacuum techniques, are used to accelerate the consolidation
process. An amalgamation of these two effects can be achieved by implementing stone
columns as a ground improvement measure. These elements are usually less costly and
more environmentally friendly than deep foundations, as no manmade substances are
introduced into the soil. The host soil will also react in a stiffer way as the ratio of area
improved by stone columns is increased, while the main direction of dissipation of excess
pore water pressure will change from vertical to radial in the horizontal plane. The soil
structure in varved lacustrine soils aids this process since the horizontal permeability is often
one to two orders of magnitude higher than the vertical permeability, leading to acceleration
of the consolidation process by several orders of magnitude.
However, the installation of stone columns disturbs the soil microstructure around the
inclusion, causing the appearance of smear and compaction zones, as shown in Figure 1.1
(Weber, 2008). The mixed zone (zone 1) can be considered to be part of the stone column
and contains a mix of host soil and column material. The smear zone (zone 2), located
directly at the boundary of the stone column, exhibits an alignment of the clay minerals along
a residual shear surface, which will counteract the improving effect of the inclusions both in
load transfer and in blocking seepage flow. The compaction zone (zone 3) is located in an
annulus around the smear zone and is associated with increased density and reduced pore
space, which likewise impedes radial drainage. The fourth zone can be considered as a free-
field zone, where no significant installation effects can be identified.
1.2 Thesis layout
2
Figure 1.1: Installation effects around a stone column at a model depth of 40 mm @ 50 g
(Weber, 2008).
Although numerous scientific projects have been conducted on this topic, the spatial
distribution of these installation effects, namely zones 2 and 3, remained unclear. This
represents a gap in knowledge, which has significant relevance for optimising design.
In order to fill this gap, tests under enhanced gravity were conducted in the ETH Zürich
geotechnical drum centrifuge (Springman et al., 2001), during which stone columns were
installed into a soft soil specimen in-flight. Some of the inclusions were loaded using a stiff
foundation and the distribution of the density of the host soil around the stone columns was
investigated using an electrical impedance needle. Some samples were extracted from the
specimen used for the centrifuge investigations for further investigations in the laboratory.
These included oedometer tests, microscopic investigations and Mercury Intrusion
Porosimetry to determine the spatial distribution of the installation effects with depth and
radial distance to the inclusions.
The findings from the centrifuge tests and subsequent complementary investigations were
validated by a numerical model.
1.2 Thesis layout
This thesis is divided into six chapters:
- Chapter 1 introduces the motivation of the research and the methodology adopted,
- Chapter 2 consists of a literature review of research on stone columns in terms of
objectives, construction techniques, bearing behaviour, design, load-transfer
behaviour, effect on the consolidation time and impact onto the host soil,
- Chapter 3 presents the basic physics of centrifuge modelling before a literature
review of how centrifuge modelling has been used to develop process understanding
of various forms of ground improvement. The methods adopted in this thesis, as well
as the materials used, the preparation techniques for the soil specimens and the
centrifuge tests conducted are described subsequently,
- Chapter 4 shows the results of the centrifuge tests in terms of undrained shear
strength, pore pressure measurements during the installation of the stone columns,
1 Introduction
3
measurements conducted while loading the stone columns and electrical impedance
measurements,
- Chapter 5 displays the complementary investigations conducted using specimens
extracted from soft soil specimens used for the centrifuge tests. These investigations
included oedometer tests, electrical impedance measurements under Earth’s gravity
and microscopic and Mercury Intrusion Porosimetry investigations,
- Chapter 6 presents the basic principles and a literature review of numerical modelling
of ground improvement methods, focussing particularly on columnar inclusions and
drains. Subsequently, the numerical models used in this thesis and the results
obtained in the axisymmetric and three-dimensional numerical modelling are
presented.
- Chapter 7 synthesises the outcomes of the thesis.
1.2 Thesis layout
4
2 State of the art of ground improvement with stone columns
5
2 State of the art of ground improvement with stone columns
2.1 General considerations about ground improvement
Ground improvement aims to enhance the engineering properties of a soil in terms of bearing
capacity and / or stiffness in order to make it suitable for construction. Chu et al. (2009)
present an overview of the different ground improvement methods and suggest a subdivision
into five categories: ground improvement without admixtures in coarse-grained soils, ground
improvement without admixtures in fine-grained soils, ground improvement with admixtures,
ground improvement with grouting type admixtures, and earth reinforcement (Table 2.1 and
Table 2.2).
Table 2.1: Classification of ground improvement methods (after Chu et al., 2009).
A. Ground
improvement
without
admixtures in
coarse-grained
soils
A1. Dynamic compaction Densification of granular soil by dropping a heavy
weight from air onto ground
A2. Vibro compaction Densification of granular soil using a vibratory probe
inserted into ground
A3. Explosive compaction Shock waves and vibrations are generated by blasting
to cause granular soil to settle through liquefaction or
compaction
A4. Electric pulse compaction Densification of granular soil using the shock waves and
energy generated by electric pulse under high voltage
A5. Surface compaction (including
rapid impact compaction)
Compaction of fill or ground at the surface or shallow
depth using a variety of compaction machines
B. Ground
improvement
without
admixtures in
fine-grained
soils
B1. Replacement / displacement
(including load reduction using
lightweight materials)
Remove poor quality soil by excavation or displacement
and replace it by good quality soil or rocks. Some
lightweight materials may be used as backfill to reduce
the load or earth pressure
B2. Preloading using fill (including
the use of vertical drains)
Fill is applied to pre-consolidate compressible soil so
that its compressibility will be considerably reduced
when future loads are applied, and removed
subsequently
B3. Preloading using vacuum
(including fill and vacuum)
Vacuum pressure of up to 90 kPa is used to pre-
consolidate compressible soil so that its compressibility
will be much reduced when future loads are applied
B4. Consolidation with enhanced
drainage (including the use of
vacuum)
Similar to dynamic compaction except vertical or
horizontal drains (or possibly together with a vacuum)
are used to dissipate pore pressures generated in soil
during compaction
B5. Electro-osmosis or electro-
kinetic consolidation
DC current causes water in soil or solutions to flow from
anodes to cathodes which are installed in soils
B6. Thermal stabilisation using
heating or freezing
Change the physical or mechanical properties of soil
permanently or temporarily by heating or freezing the
soil
B7. Hydro-blasting compaction Collapsible soil (loess) is compacted by a combined
action of wetting and deep explosion along a borehole
2.1 General considerations about ground improvement
6
Table 2.2: Classification of ground improvement methods (after Chu et al., 2009).
C. Ground
improvement
with
admixtures or
inclusions
C1. Vibro replacement or stone
columns
Hole jetted into soft, fine-grained soil and back filled with
densely compacted gravel or sand to form columns
C2. Dynamic replacement Aggregates are driven into soil by high energy dynamic
impact to form columns. The backfill can be either sand,
gravel, stone or demolition debris
C3. Sand compaction piles Sand is fed into ground through a casing pipe and
compacted by either vibration, dynamic impact, or static
excitation to form columns when the casing has been
removed
C4. Geotextile confined columns Sand is fed into a closed bottom geotextile lined
cylindrical hole to form a column
C5. Rigid inclusions (or composite
foundation)
Use of piles, rigid or semi-rigid bodies or columns which
are either premade or formed in-situ to strengthen soft
ground
C6. Geosynthetic reinforced
column or pile supported
embankment
Use of piles, rigid or semi-rigid columns / inclusions and
geosynthetic grids to enhance the stability and reduce
the settlement of embankments
C7. Microbial methods Use of microbial materials to modify soil to increase its
strength and stiffness or reduce its permeability
C8. Other methods Unconventional methods, such as formation of sand
piles using blasting and the use of bamboo, timber and
other natural products
D. Ground
improvement
with grouting
admixtures
D1. Particulate grouting Grouting granular soil or cavities or fissures in soil or
rock by injecting cement or other particulate grouts
either to increase the strength or reduce the
permeability of soil or ground
D2. Chemical grouting Solutions of two or more chemicals react in soil pores to
form a gel or a solid precipitate either to increase the
strength or reduce the permeability of soil or ground
D3. Mixing methods (including
premixing or deep mixing)
Treating the weak soil by mixing it with cement, lime, or
other binders in-situ using a mixing machine or before
placement
D4. Jet grouting High speed jets at depth erode the soil and inject grout
to form columns or panels
D5. Compaction grouting Very stiff, mortar-like grout is injected into discrete soil
zones and remains in a homogeneous mass so as to
densify loose soil or lift settled ground
D6. Compensation grouting Medium to high viscosity particulate suspensions are
injected into the ground between a subsurface
excavation and a structure in order to negate or reduce
settlement of the structure due to ongoing excavation
E. Earth
reinforcement
E1. Geosynthetics or
mechanically stabilised earth
Use of the tensile strength of various steel or
geosynthetic material to enhance the shear strength of
soil and stability of roads, foundations, embankments,
slopes, or retaining walls
E2. Ground anchors or soil nails Use of tensile strength of embedded nails or anchors to
enhance the stability of slopes or retaining walls
E3. Biological methods using
vegetation
Reinforcing and stabilising slopes using roots of
vegetation
2 State of the art of ground improvement with stone columns
7
Table 2.1 and Table 2.2 show the wide range of possibilities available in the field of ground
improvement. The aim of this dissertation is the investigation of the behaviour of stone
columns installed in clay, which is why the categories C1 and C3 (Table 2.2) will be
considered into more detail, while the other possibilities will not be considered further.
Stone columns are installed using a vibrating poker, creating a cavity in the ground, which is
subsequently filled with coarse-grained material (gravel or sand). The advantage of this
replacement technique is the flexibility of its application, as it can be used over a wide range
of possible host soil grain sizes. This work is focused on fine-grained lacustrine soils for
which this technique offers a very cost effective solution to improve the relevant properties.
2.2 Objectives of ground improvement of soft soils with stone
columns
The objectives of ground improvement in soft soils by means of granular inclusions can be
divided into two categories:
- the increase of stiffness and shear strength of the host soil, which can prove to be
very valuable e.g. underneath an embankment, to prevent the formation of a slip
circle or under foundations to reduce settlements and
- the amelioration of the drainage conditions through a reduction of the length of the
drainage path. This usually also takes advantage of the structural anisotropy, which is
often to be found in soft soils and which results in a significantly higher permeability in
the horizontal than in the vertical direction. The improved drainage conditions lead to
a reduction of the consolidation times and a quicker increase in stiffness and
strength.
Detailed considerations concerning ground improvement can be found, among others, in
Greenwood & Kirsch (1983), Barksdale & Bachus (1983), Van Impe (1989), Bergado et al.
(1994), Ou & Woo (1995), Van Impe et al. (1997b, 1997a, 1997c) and Chu el al. (2009).
Granular inclusions, also called stone columns, stone compaction piles, sand columns or
sand compaction piles, refer to the same type of construction. The differences in
denomination are due to variations of the material fed in, or to the construction technique.
None of the columns contain binding material (such as cement or lime). As a consequence,
they need to be supported by the surrounding material, which makes them flexible. This is a
major difference to rigid inclusions such as concrete piles, which rely on tip bearing and shaft
friction.
A limiting factor for the implementation of granular inclusions in soft soils is the capacity of
the host soil to provide lateral support to the inclusions. A range of values can be found in the
literature for the minimal acceptable undrained shear strength. Older studies suggested
values ranging from su = 7.5 kPa (Greenwood, 1970) to su = 15 kPa (Greenwood & Kirsch,
1983), while the German authorities (Forschungsgesellschaft für das Strassenwesen, 1979)
advised values of su from 15 kPa to 25 kPa. However, further research (Raju, 1997; Priebe,
2.3 Construction techniques
8
2003; Wehr & Maihold, 2012) showed that these values were significantly too conservative
and that the construction of granular inclusions was possible in soils exhibiting undrained
shear strengths as low as 5 kPa.
A geotextile encasement can also be installed if the undrained shear strength of the soil is
too low to provide the necessary horizontal support (Raithel et al., 2005). A geotextile “sock”
with a tensile strength of 200 kPa to 400 kPa is therefore inserted into the installation
mandrel and filled with sand or gravel. The mandrel is subsequently extracted from the host
soil and the geotextile “sock” provides lateral support to the inclusion. The bearing
mechanisms of encased stone columns remain the same as for non-encased inclusions.
Typical dimensions of granular inclusions in soft soils range from 6 to 20 m in length and,
with a diameter between 0.6 and 1.2 m. It is possible to achieve depths up to 30 m,
depending on the construction technique and host soil (Keller Grundbau, 2013).
2.3 Construction techniques
There are many different techniques that are applied to construct granular inclusions,
although all exhibit two steps. Firstly, the coarse soil (gravel or sand) is introduced into the
subsoil without contamination from the surrounding weak soil. Subsequently, the granular
soil is compacted, either by means of vibration or tamping.
The installation of stone columns is mainly achieved by using the dry bottom feed process
(Figure 2.1). After the vibrator has been filled with gravel featuring grain sizes ranging from
10 to 40 mm (Greenwood & Kirsch, 1983), it is introduced into the subsoil aided by
compressed air. Once the desired depth is reached (Figure 2.1 b), the vibrator is pulled up
(typically 0.3 m to 1.0 m), which causes the gravel to fill the cavity created. The compaction
is then achieved through re-penetration of the vibrator (typically 0.2 m to 0.8 m, Figure 2.1 c).
The compaction steps are repeated until the surface is reached. Surface compaction is
subsequently necessary to achieve a flat plane. This technique is most appropriate for
penetration depths of less than 20 m (Kirsch & Kirsch, 2010).
The wet top feed technique is an alternative to the dry bottom feed technique (Figure 2.2), in
which the penetration of the vibrator is assisted by water under high pressure, which loosens
the soft soil and supports the cavity created. The gravel is then fed from the surface through
the cavity, while the vibrator is pulled up (typically 0.5 m). Compaction is achieved by the re-
penetration of the vibrator (typically 0.4 m). An advantage of this technique, compared to the
dry bottom feed technique, is that it is possible to reach greater depths (in the range of 30
m). However, the negative influence of the added water on the soil behaviour has to be taken
into account. Moreover, flushing water charged with fine particles is produced, which has to
be recycled or cleaned before being returned to nature.
2 State of the art of ground improvement with stone columns
9
(a) (b) (c) (d)
Figure 2.1: Dry replacement technique: (a) filling the supply hopper, (b) penetration,
(c) compaction by step-wise withdrawal and reinsertion (d) finishing
(Keller Grundbau, 2013).
(a) (b) (c) (d)
Figure 2.2: Wet top feed technique: (a) penetration, (b) filling, (c) compacting, (d) finishing
(International Construction Equipment Holland, 2013).
It is also possible to install the column by ramming rather than using vibration techniques.
Figure 2.3 illustrates this process, which was originally proposed by Franki (Franke, 1997).
Granular material is fed into a casing, which is then driven into the soil up to the desired
depth, with the help of an internal pile hammer, to form a stone column. The casing is then
Water
Flushing
waterFlushing
water
2.4 Bearing behaviour of stone columns submitted to vertical loading
10
filled with granular material through the top of the installation tube and the compaction is
carried out with the internal pile hammer, while the installation tube is withdrawn gradually
while ensuring that no gaps are able to form between the casing, the ground and the
granular pile. Information about the host soil can be gained based on the number of blows of
the pile hammer (Van Impe et al., 1997b) to reach a certain depth.
Figure 2.3: Ramming installation technique: (a) inserting granular plug, (b) driving up to the
desired depth, (c) filling with granular soil, (d) compacting and withdrawing
casing, (e) finishing (Van Impe et al., 1997b).
2.4 Bearing behaviour of stone columns submitted to vertical
loading
2.4.1 Bearing behaviour
The basic response of stone columns to vertical loading consists of:
- internal deformations (shear and volumetric),
- the mobilisation of a shaft friction at the cylindrical interface between columns and
host soil, and
- the mobilisation of a tip resistance.
2 State of the art of ground improvement with stone columns
11
The mechanisms of the mobilisation of shaft friction and tip resistance are similar to the case
of long, slender piles (Fleming et al., 1992), particularly in respect of the load transfer
between the shaft and the pile base as a function of the load.
The bearing behaviour of stone columns is governed by a complicated set of interaction
mechanisms between stone column and the ground, column and column in a group, column
and any form of footing and finally the footing with the ground (Figure 2.4, Kirsch, 2004).
Figure 2.4: Interactions at stake under a footing (after Kirsch, 2004).
Some typical loading situations of stone columns are summarised in Figure 2.5, which
represent a first insight into the behaviour of stone columns under normally vertical loading.
The first difference is of course whether the ground improvement concerns a single column
or a column group. The type of loading also plays a role, in terms of the general behaviour of
stone columns:
- a conventional vertical load applied directly to the top of a stone column (Figure
2.5 a) leads to settlement at the surface and lateral squeeze into the host soil, due
to lateral deformation of the column,
- a vertical load acting on a rigid footing built on top of a stone column (Figure 2.5
b) applies a constant settlement over the length of the footing and hence volume
loss occurs in the soft ground. The load causes a lateral deformation of the
column,
- loads applied onto stone column groups (Figure 2.5 c and d) trigger a similar
response to the case of single stone columns (Figure 2.5 a and b), with the
addition of the interaction between the columns. Figure 2.5 (d) shows the
response of a group of stone columns to an inclined load (in this case
embankment loading). This specific case shows similar mechanisms to those
observed with a rigid footing, however in a different geometric arrangement.
Loading
Flexural rigidity EI
Interaction:
Footing – column
Footing – soil
Column – soil
Column – column
2.4 Bearing behaviour of stone columns submitted to vertical loading
12
A fact common to all loading situations is that the undrained response of the host soil causes
the column to deform laterally or “bulge”, which in turn loads the subsoil laterally. This leads
to further settlements of the footing. However, these settlements increase the vertical loading
of the host soil, which cause a rise of the horizontal stresses supporting the stone column.
(a) (b) (c) (d)
Single
column
Single column loading by
a rigid footing
Column group loaded
by a rigid footing
Column group with
flexible loading, e.g.
dam
Figure 2.5: Loading situations of stone columns (Kirsch, 2004).
2.4.2 Stress concentration on stone columns
The load applied is transmitted to both host soil and compacted granular inclusions.
However, a stress concentration will occur above the stiff stone columns and the stress
repartition between subsoil and stone column needs to be accounted for. Equilibrium has to
be satisfied, which Aboshi et al. (1991) formulated for a unit cell in a quadratic organisation
(Figure 2.6) as follows:
( ) 2.1
2.2
with a distance between the axis of the stone columns in a quadratic grid
Asc stone column cross-section
As soft soil cross-section in the unit cell (As = a2 – Asc)
σ total stress acting on the unit cell
σsc total stress acting on the stone column
σs total stress acting on the soft soil surface
2 State of the art of ground improvement with stone columns
13
(a) (b)
Figure 2.6: Total vertical stress distribution of a uniform vertical stress σ (a) plan view
showing respective areas of stone columns (Asc) and soft soil (As), (b) cross-
section showing stress distribution onto the column (σsc) and the host soil (σs)
(after Aboshi et al., 1991).
The determining geometrical factor is the area replacement ratio as, which is the ratio
between the column cross-section and the total cross-section of the unit cell:
( ) 2.3
The total vertical stress distribution can be described by means of a stress concentration
factor m, defined as the ratio between the stress acting on the stone column and the stress
acting on the host soil:
2.4
These two values then enable the vertical stresses acting on the stone column and on the
subsoil to be calculated:
( ) 2.5
( ) 2.6
The total vertical stress acting on the stone column depends strongly on the geometric
boundary conditions, namely the diameter and distance from axis to axis of the columns.
Barksdale & Takefumi (1991) show that the stress concentration factor m decreases with an
increasing replacement ratio as, while Ichimoto & Suematsu (1981) suggest values of m for
design depending on the area replacement ration as and on the angle of friction of the
column material φ’sc (Table 2.3).
Asc
As
a
a
a
a
sstσs
σsc
σ
a
2.4 Bearing behaviour of stone columns submitted to vertical loading
14
Table 2.3: Suggested values of the stress concentration ratio m for specific combinations of
area replacement ratio as and effective friction angle of the stone column
material for compacted sand columns after Ichimoto & Suematsu (1981).
as [%] φ’sc m [-]
0 – 30 30 3
30 – 70 30 2
> 70 35 1
The values after Ichimoto & Suematsu (1981) should only be considered for a first evaluation
of the stress concentration on top of sand columns, as they do not account either for group
effects or variations of the properties of the host soil and only consider a certain range of
possible effective angles of friction of the stone column material. Although these limitations
are present, the values proposed by Ichimoto & Suematsu (1981) have been partially verified
by various field measurements. Gruber (1995) measured the load carried by some of the
stone columns constructed in soft ground under an embankment by means of load cells.
Stress concentrations of between 2.5 and 3.5 could be measured at the surface for area
replacement ratios between 4% and 11%. Kirsch (2004) conducted diverse tests in which he
installed load cells under footings built on groups of stone columns with area replacement
ratios varying from 10 % to 70 %, and recorded stress concentrations of 1.5 to 2.5. The
column located in the middle of the group was less loaded than those on the borders, which
is also known from investigations into the response of pile groups to axial load.
Greenwood (1991) shows that, besides the geometric parameters, the type and magnitude of
loading also plays a role in the distribution and magnitude of the stress concentration.
Greenwood (1991) conducted three different loading tests at three different locations in the
United Kingdom and in three different sets of ground conditions. In the first case (located in
St Helens), a stiff footing was used to load stone columns (φ’sc = 42°) constructed in sandy
silt (φ’s = 30°). The area replacement ratio was of about 45 %.The stress concentration was
approximately equal to 3.5 for a load of 40 kPa and decreased to about 2.0 for the failure
load of 200 kPa (Figure 2.7 a). The stress concentration factor decreased slightly with
increasing cycles of loading.
2 State of the art of ground improvement with stone columns
15
(a) (b)
Figure 2.7: Measured stress concentration factors at (a) St. Helens and (b) Canvey Island
(Greenwood, 1991).
The second test was conducted on Canvey Island, where a group of stone columns
(featuring a diameter of 750 mm and reaching a depth of 10 m) was constructed in silty clay
and loaded with a 36 m diameter oil storage tank, which is assumed to be a flexible loading
scenario (Figure 2.7 b). The columns were installed in a triangular pattern with a spacing of
1.5 m (as 45 %) and the pressure cells were placed close to the centre of the tank. The
tank was subsequently filled slowly with water over 100 days up to a failure load of 130 kPa.
Again, the stress concentration was found to decrease with increasing loads. However, the
values were significantly higher, as they started from 25 to reach 5 at high loads. Greenwood
(1991) explains that this is due to the low strength of the subsoil on Canvey Island (su = 20
kPa), which supported very little charge at the beginning of the loading, before starting to
consolidate, which led to a redistribution of the stress at the surface and to a decrease in the
stress concentration factor. This shows that the stress concentration is not only controlled by
geometrical parameters, as assumed by Ichimoto & Suematsu (1981) as the stress
concentrations factors measured by Greenwood (1991) on Canvey Island show significant
higher values of up to 25 in comparison with the maximum value of 3 suggested by Ichimoto
& Suematsu (1981) (Table 2.3).
The third and last test was performed at Humber Bridge, where stone columns installed in
silty clays (su ranging from 11 to 121 kPa, denoted as cu in Figure 2.8) were loaded by an 8
m high embankment built with compacted chalk fill to a unit weight of 20.8 kN/m3. The stone
columns had a diameter of 775 mm, reached a depth of 9 m and were installed in a triangular
pattern with a spacing of 2.25 m.
Average ground pressure [kN/m2]
040 80 120 160 200S
tre
ss c
on
ce
ntr
atio
nfa
cto
rm
[-]
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
1st cycle loading
2nd cycle loading
3rd cycle loading
Surcharge [kN/m2]0 20 40 60 80 100 120
0
5
10
15
20
25
Str
ess c
once
ntr
ation
facto
rm
[-]
2.4 Bearing behaviour of stone columns submitted to vertical loading
16
Figure 2.8: Cross-section of the test site at Humber Bridge (after Greenwood, 1991).
The trend of the measured stress concentrations was found to be contrary to the two other
sites in this case. Greenwood (1991) explains why it rose with increasing loading (Figure 2.9)
by the significant initial compaction of the host soil: the initial stiffness of the subsoil was
comparable to that of the granular inclusions at low stresses, thus leading to a stress
concentration factor lower than one for the initial phase of the embankment loading (up to
about 1.5 m embankment height).
Figure 2.9: Measured stress concentration factors at Humber Bridge (Greenwood, 1991).
The results presented by Greenwood (1991) show the difficulty in assessing the stress
concentration factor m. This factor does not only depend on geometrical parameters, as
suggested by Ichimoto & Suematsu (1981), but also on the parameters of the host soil, as
Applied stress [kN/m2]0 40 80 120 160 200
Applied stress [kN/m2]
Str
ess c
once
ntr
ation
facto
rm
[-]
1
2
3
4
5
6
2 State of the art of ground improvement with stone columns
17
demonstrated by the results of the test conducted on Canvey Island (Figure 2.7 b). The
measured stress concentrations at Humber Bridge (Figure 2.9) also show that the difference
between the properties of the host soil and those of the stone column material can play a
decisive role in the development of stress concentrations. As a consequence, reliable
predictions of the stress concentration factor m can only be achieved by considering the
geometrical parameters of the boundary value problem, the parameters of the host soil as
well as by comparing these to those of the stone column material.
Muir Wood et al. (2000) conducted small scale loading tests on a rigid footing placed on
groups of stone columns with varying lengths and spacing (Table 2.4) and installed in kaolin
which was consolidated up to 120 kPa and unloaded to 30 kPa. The unscaled data (denoted
as Model test in Figure 2.10) from tests with an area replacement ratio of 24 % were then
compared with the field data obtained by Greenwood (1991) at the Humber Bridge (denoted
as Field data in Figure 2.10). The trends observed are very similar, although the replacement
ratios (denoted as As in Figure 2.10) were not equal.
Although a good agreement between test data and field observations was reached in terms
of stress concentration in this case, questions can be raised regarding the limitations of
small-scale tests in terms of reproduction of the in-situ stress states within the host soil in a
full-scale boundary value problem. This leads to a higher influence of dilatancy on the
loading behaviour of the composite foundation, thus affecting the determination of the
Serviceability Limit State (SLS).
Figure 2.10: Stress concentration factors in 1 g small-scale model and field tests (Muir Wood
et al., 2000).
Str
ess c
on
ce
ntr
atio
n fa
cto
r m
[-]
Ratio of applied pressure to initial undrained shear
strength of the host soil [-]
Field data: As = 21% (Humber Bridge, afterModel test: As = 24%
Field data: As = 21% (Humber Bridge,
after Greenwood, 1991)
2.4 Bearing behaviour of stone columns submitted to vertical loading
18
Table 2.4: Summary of 1 g small-scale model tests (σc: pre-consolidation stress; su:
undrained shear strength; rsc: radius of the stone columns; L: length of the stone
columns; a: spacing of the stone columns; as: area replacement ratio) (after Muir
Wood et al., 2000).
Test σc
[kPa]
su
[kPa]
rsc
[mm]
L
[mm]
a
[mm] L / rsc
as
[%]
TS02 180 23 5.5 100 25.3 18.2 15
TS03 110 5 5.5 100 30.8 18.2 10
TS04 160 16.5 5.5 150 19.8 27.2 24
TS05 120 10.5 5.5 100 17.6 18.2 30
TS07 120 8 5.5 150 30.8 27.2 10
TS08 120 15 5.5 100 19.8 18.2 24
TS09 120 11.5 8.75 160 31.5 18.2 24
TS10 120 11.5 8.75 100 28 11.4 30
TS16 120 11.5 5.5 100 30.8 18.2 10
TS17 120 14 5.5 160 19.8 29 24
TS19 120 10 5.5 160 19.8 18.2 24
TS21 120 10 5.5 100 19.8 18.2 24
TL02 120 17 8.75 160 31.5 18.2 24
TS20 120 14 5.5 100 19.8 18.2 24
TS11 120 14 - - - - -
TS20/2 120 14 - - - - -
2.4.3 Ultimate Limit State response to vertical load
Stone columns exhibit different Ultimate Limit State (ULS) responses to vertical loading,
depending on the boundary conditions. Figure 2.11 shows a summary of the different
possible failure mechanisms that were presented in Muir Wood et al. (2000): bulging, surficial
bearing failure, shear failure, penetration of short columns, shortening of long columns and
deflection of slender columns.
(a) (b) (c) (d) (e) (f)
Figure 2.11: Failure mechanisms for a single stone column (a) bulging (b) bearing failure (c)
shear failure (d) penetration of short columns (e) shortening of long columns (f)
deflection of slender columns (Muir Wood et al., 2000) based on Waterton &
Foulsham (1984).
σ’v,max
3-4 d
d
σ’vmax
2 State of the art of ground improvement with stone columns
19
Hughes & Withers (1974) suggest that the most likely failure mechanism is bulging failure
(Figure 2.11 a), even though it is relatively difficult to make a clear difference between
deformation and failure. Lateral deformation is necessary to activate the support of the
surrounding host soil and thus the resistance. Shear failure can occur either near to the
surface in the form of a bearing failure (Figure 2.11 b) or deeper as a shear surface develops
in the column (Figure 2.11 c). Datye (1982) describes the penetration of short columns
(Figure 2.11 d), which is due to inadequate dimensions (diameter and length of the column)
with respect to the load and the undrained shear strength in the ground. Muir Wood & Hu
(1997) and Muir Wood et al. (2000) describe the last two possible failure mechanisms
illustrated in Figure 2.11 (e & f), namely the shortening of long columns and the deflection of
slender columns. The last case is caused by uneven application or inclined of load and
insufficient lateral support of the host soil.
Groups of stone columns encounter similar failure mechanisms to those faced by single
columns. Kirsch (2004) illustrates some of the different possible failures for a group of stone
columns (Figure 2.12), which are governed by complex interactions (Figure 2.4): bulging
failure and loss of horizontal support, shearing failure, block failure and column penetration.
(a) (b) (c)
Figure 2.12: Failure mechanisms for groups of stone columns (a) bulging failure and loss of
horizontal support (b) shearing failure (c) block failure and column penetration
(Kirsch, 2004).
These theoretical considerations could also be observed in small scale model tests
conducted using clay specimens (Muir Wood et al., 2000). The sand used to model the
inclusions had an effective angle of friction of 30°. The penetration of short columns can
clearly be observed in Figure 2.13 (a). The shape of the deformed columns was obtained by
removing the sand from the inclusions after the footing loading and by pouring in a plaster in
order to represent the deformed shape. The excavation of the clay then allowed for a clear
identification to be obtained of the features of the response of stone columns to loading. The
horizontal arrows at the sides of the pictures show the original depth of the columns. Bulging
(marked with the letter A in Figure 2.13 a and b) occurs when the radial expansion of the
column is not prevented by adjacent soil. Shear planes (marked with the letter B in Figure
2.13 a and b) may appear through the column when the inclusion is subjected to high shear
stresses with limited lateral restraint.
2.4 Bearing behaviour of stone columns submitted to vertical loading
20
(a) (b)
Figure 2.13: Deformed sand columns at the end of the footing penetration (Muir Wood et al.,
2000).
Muir Wood et al. (2000) suggest using the deformation observed in the plastered small-scale
model stone columns to estimate the zone of influence of the footing on the underlying
improved soil. This zone of influence may be assumed to be conical, as shown in Figure
2.14, while the angle β increases with an increasing area replacement ratio (denoted as Area
ratio in Figure 2.14). The angle β can be used to determine the average mobilised angle of
friction of the improved ground with the following formula (Muir Wood et al., 2000):
2.7
The experimental results show that the average mobilised angle of friction of the improved
ground is about 41° for an area replacement ratio as of 30%, which is close to the angle of
friction for pure sand at the stress levels in a small scale model during the test. The value of
falls to approximately 23° for area replacement ratios of 24% and 10%, which is close to
the value of the drained angle of friction of pure clay (Figure 2.14 a).
However, the tests were conducted at small-scale and hence low stress states within the
host soil. Moreover, Muir Wood et al. (2000) noted that there was dilatancy of the sand,
although the column material was medium dense. These issues raise some concern about
the possibility to extrapolate the mechanical results correctly in terms of effective angle of
friction to a full-scale boundary value problem. This is especially the case for high area
replacement ratios (30 %), for which is close to the effective angle of friction of sand.
However, this should only have a limited impact for area replacement ratios in the range of
10 %, as is then close to the effective angle of friction of clay.
2 State of the art of ground improvement with stone columns
21
(a) (b)
Figure 2.14: Zone of influence of a footing on the underlying soil (a) “rigid” cone beneath
footing (b) variation of angle β with area replacement ratio (Muir Wood et al.,
2000).
McKelvey et al. (2004) conducted loading tests of groups of stone columns installed in a
saturated transparent soil, which allowed the deformation mechanisms to be observed
directly. The rigid footing was circular with a diameter of 100 mm while the stone columns
featured a diameter of 25 mm and a length of 150 mm (area replacement ratio of 23 %).
Sand with an effective angle of friction of 34 ° was used to build the columns. McKelvey et al.
(2004) noted that the columns deformed into a barrel shape (Figure 2.15), the depth of which
was shown to depend on the length of the columns. The vertical oval lighter zones in Figure
2.15 however tend to indicate that boundary effects occurred in this case as the columns
were installed near to the edges of the strongbox. Although this does not impact strongly on
qualitative aspects of the deformations mechanisms, quantitative conclusions should be
handled with care.
Figure 2.15: Deformed stone columns at the end of the footing penetration (McKelvey et al.,
2004).
β
2.5 Design of stone columns
22
2.5 Design of stone columns
A number of different methods are available to design stone columns, as well as to anticipate
their behaviour, both for Serviceability Limit State (SLS) and Ultimate Limit State (ULS).
The ULS embraces situations in which safety is involved, i.e. collapse of a structure or where
a risk for human lives, or severe economic loss, is present (Orr & Farrell, 1999). In the case
of stone columns, the verification of the ULS is based on the calculation of the bearing
capacity of single inclusions, as well as that of the whole group of stone columns. The
bearing capacity of the inclusions is dependent on the properties of the host soil and on the
boundary conditions. The host soil provides lateral support to the column, while complex
interactions (Figure 2.4) will develop in case of groups of stone columns depending on
relative effects of geometry and strength, which in turn will also have an influence on the
bearing capacity of the system.
The SLS corresponds to situations in which the requirements of a structure are no longer met
to guarantee serviceability (Orr & Farrell, 1999). In the case of stone columns, the verification
of the SLS is based on the comparison of the calculated value of the expected settlements
with the acceptable value.
Summaries of the most common design methods can be found e.g. in Soyez (1987),
Bergado et al. (1994), Gruber (1995), Daramalinggam (2003) and Kirsch (2004).
2.5.1 Bearing capacity
The three most common failure mechanisms: bulging failure, shear failure and penetration of
short columns (Figure 2.11 a, c and d) are investigated here. The calculation of the bearing
capacity of a single column is usually conducted based on the solutions for a single pile.
However, models developed in order to calculate the bearing capacity of groups of stone
columns for bulging failure (Madhav & Viktar, 1978), as well as for shear failure when loaded
by a footing (Barksdale & Bachus, 1983) and by an embankment (Springman et al., 2014)
are also presented.
2.5.1.1 Bulging failure
Bergado et al. (1994) propose a design procedure based on Greenwood (1970) to determine
the bulging failure load of a stone column. The fundamental idea is that the bulging failure
load is reached when the horizontal load of the column exceeds the passive resistance of the
host soil, mixing drained and undrained behaviour. The bulging failure load can be expressed
as follows:
( √ )
2.8
with qsc, bulging bulging failure load of a single stone column
unit weight of the host soil
z depth from surface of composite foundation
2 State of the art of ground improvement with stone columns
23
Kp, s coefficient of passive earth pressure of the host soil
su undrained shear strength of the host soil
φ’sc effective angle of friction of the stone column material
Vesic (1972) and Datye (1982) propose a procedure based on the cavity expansion theory.
The ultimate bearing capacity for bulging is then formulated as:
(
)
2.9
with F’c cavity expansion parameter
F’q cavity expansion parameter
q0 overburden pressure at the depth where the bulging occurs
The two cavity expansion parameters can be determined, assuming fully undrained
behaviour so that the volumetric strain in the plastic region is equal to zero:
(
) ( )
2.10
2.11
with Ir stiffness index
φ’s effective angle of friction of the host soil
The stiffness index Ir is defined as:
( )
2.12
with Es Young’s modulus of the host soil
u undrained Poisson’s ratio ( )
G shear modulus
Hughes & Withers (1974) also suggest a design procedure based on cavity expansion theory
and formulate the ultimate bearing capacity as:
( )
2.13
Stuedlein & Holtz (2012) proposed the following modification of Equation 2.13, based on the
analysis of load tests:
( ( ) )
2.14
with σr0 ultimate total in-situ lateral stress at the depth where the bulging occurs
2.5 Design of stone columns
24
Madhav & Viktar (1978) extend the solution recommended by Hughes & Withers (1974) to
the plane-strain case in order to be able to consider groups of stone columns and propose
the following formulation:
( ) (
)
[ (
)
] 2.15
with φ’sc effective angle of friction of the stone column material
su undrained shear strength of the host soil
γs unit weight of the host soil
z depth from surface of composite foundation
K0 coefficient of earth pressure at rest
qs bearing capacity of the host soil expressed as ( ⁄ )
Nc dimensionless bearing capacity parameter according to Terzaghi (1943)
W width of equivalent granular pile strip
B width of loaded area
2.5.1.2 Shear failure
Bergado et al. (1994) present an overview of different possible design procedures in order to
assess the ultimate load acting on a stone column that would cause a shear failure (Figure
2.16), which, in reality, would only occur after significant bulging would have taken place.
Figure 2.16: Shear failure of a stone column (after Muir Wood et al., 2000).
Barksdale & Bachus (1983) point out that Greenwood (1970) and Wong (1975) assume that
the lateral resistance developed by the surrounding soil is equal to the passive resistance
mobilized behind a retaining wall. This is a synonym of plane-strain loading and does not
take the three-dimensional aspects of a stone column into account. Barksdale & Bachus
(1983) however admit that the design procedure proposed by Wong (1975) appears to
2 State of the art of ground improvement with stone columns
25
correlate well with field measurements. Bergado et al. (1994) formulate the procedure
suggested by Wong (1975) in the following manner:
( √ )
[ (
)] 2.16
with qsc, shear shear failure load of a single stone column
As soft soil cross-section in the unit cell (Equation 2.2)
Kp,s coefficient of passive earth pressure of the host soil
q0 over-burden pressure at the depth where the shear failure occurs
su undrained shear strength of the host soil
Ka, sc coefficient of active earth pressure of the column material
dsc stone column diameter
L length of stone column
Brauns (1978a, 1980) suggests a design procedure based on a truncated conical failure
mechanism, illustrated in Figure 2.17. The assumptions made in this case are that the
volume of the stone column remains constant and that the shear stress around the columns
(shaft) as well as the tangential stress along the failure mechanism may be neglected. The
resulting bearing capacity is formulated as:
(
( )) (
) ( )
2.17
2.18
with q surcharge at the surface
δ inclination of the failure mechanism within the host soil
δsc inclination of the failure surface within the stone column
(a) (b) (c)
Figure 2.17: Truncated conical failure mechanism according to Brauns (1978a) (a) cross-
section, (b) plan view and (c) forces acting on volume A.
Barksdale & Bachus (1983) propose a formulation of the ultimate shear failure load of stone
column groups. They therefore consider the situation of a square or an infinitely long footing
suc
c
su fksu fk
suc
c
su fksu fk
suc
c
su fksu fk
A A
2.5 Design of stone columns
26
located at the surface of a soft soil reinforced with stone columns (Figure 2.18). Several
assumptions are made in this model:
- the speed of loading is high enough in order to ensure that undrained behaviour may
be assumed in the soft soil,
- the full shear strength of the soft soil and of the stone column is mobilised,
- the failure surface can be approximated by two straight lines,
- the soil immediately beneath the foundation reaches failure along a straight rupture
surface, triggering the shear resistance of a triangular block (Figure 2.18).
According to Barksdale & Bachus (1983), the shear failure load can be calculated from the
following set of equations:
2.19
2.20
2.21
(
) 2.22
( ) 2.23
( ) 2.24
with σ3 average lateral confining pressure
β inclination of the failure surface
su, avg composite undrained shear strength
B foundation width
γs unit weight of the host soil
su undrained shear strength of the host soil
φ’avg composite angle of friction between host soil and stone columns
as area replacement ratio
μsc stress concentration factor for the stone column
μs stress concentration factor for the host soil
2 State of the art of ground improvement with stone columns
27
Figure 2.18: Stone column group analysis – firm to stiff fine-grained soil (Barksdale &
Bachus, 1983).
Springman et al. (2014) present solution taking the influence of the stone columns on the
external bearing capacity of an embankment with base reinforcement into account. They
therefore first consider the series of stone columns and host soil (Figure 2.19 a) as being
replaced by an equivalent plane wall (Figure 2.19 b).
The shear strength of the i-th element in an equivalent plane wall can be formulated as:
⁄
⁄
2.25
( ) 2.26
(
⁄
)
⁄
2.27
2.5 Design of stone columns
28
with qsc,shear,PW,i shear strength of the i-th equivalent plane wall
su undrained shear strength
dsc stone column diameter (denoted as d in Figure 2.19 and Figure 2.20)
a spacing between stone columns
σ’n,PW,i normal effective stress acting on the shear plane in the i-th equivalent
plane wall element (Figure 2.20)
φ’sc effective angle of friction of the stone column material
γs unit weight of the host soil
γeq equivalent unit weight of the composite plane wall
zi depth of the i-th slice below the surface
σ’sc,i normal effective stress applied to the shear plane in the i-th stone
column
αi inclination angle of the shear plane in the i-th slice
ui pore water pressure acting on the slip surface for the i-th slice
γsc unit weight of the stone column material
(a) (b)
Figure 2.19: Clay and columns represented (a) discretely and (b) as an equivalent plane wall
(Springman et al., 2014).
The external ultimate resistance should be checked based on a slip-circle analysis according
to Bishop (1955) and taking the influence of the base reinforcement (denoted as ZRd in Figure
2.20) into account (Figure 2.20), in which the system is divided into slices in order to
calculate the stability. σ’n,PW,d and σ’c,d denote the design values of the effective normal stress
acting on the slip surface in the plane walls and in the clay slices, respectively. The
difference between the stresses acting on top of the stone columns may be ignored, which is
on the safe side, or the suggestions made by Aboshi et al. (1991) may be used to assess the
stress distribution between granular inclusions and soft soil. The ratio between the resisting
moment and the moment triggered by the actions is the safety factor for a slip surface.
Although this approach is widely used in practice, it has limitations, as it assumes that the
geometry of the boundary value problem may be modelled by plane wall, which is not always
the case.
2 State of the art of ground improvement with stone columns
29
Figure 2.20: Stability considerations on a slip circle passing through soft soil and the
equivalent plane walls (numbers 1 to 11 show the sequence of the slices)
(Springman et al., 2014).
2.5.1.3 Penetration of short columns
The failure mechanism of penetration is limited to short columns, which are assumed to
behave like piles and to be subjected to tip resistance and friction, under the assumption that
the lateral support of the host soil is high enough to mobilise the frictional resistance qf. The
failure load can then be assessed as:
2.28
with qt tip resistance
qf frictional resistance around the shaft
As this mechanism develops only when there are inadequate geometrical dimensions
(diameter and length) of the stone column, Brauns (1978a) proposes an upper and a lower
bound for the length of the column, under the assumption that the self-weight of the column
may be ignored and that the tip resistance is equal to (Scott, 1963). The upper
bound, above which a verification of the penetration may be regarded as superfluous, is:
2.29
with rsc radius of the stone column
σsc total stress acting on the stone column
su undrained shear strength of the host soil at the base of the stone column
2.5 Design of stone columns
30
The minimal length of the stone column in order to prevent a punch-through failure can be
estimated as:
(
) 2.30
with rsc radius of the stone column
σsc total stress acting on the stone column
su undrained shear strength of the host soil at the base of the stone column
These proofs are simplistic in the sense that they assume a constant value of the undrained
shear strength throughout the depth of the soft soil. However, as long as the failure
mechanism of penetration only occurs for short inclusions, this simplification can be
considered to be valid for practical cases.
2.5.2 Settlement calculation
The settlement calculation is normally conducted under the assumption that the situation at
hand can be modelled in plane-strain. Each column and its surrounding area will respond in
the same manner and that a unit cell approach can be applied. Three possible arrangements
of the columns are shown in Figure 2.21: a hexagonal, a square or a triangular pattern
(Balaam & Poulos, 1983).
(a) (b) (c)
Figure 2.21: Various stone column arrangements with the domain of influence of each
column (Balaam & Poulos, 1983).
Depending on the pattern, the domain of influence D of the column can be estimated as:
- (
)
for a hexagonal pattern,
- (
)
for a square pattern, and
- (
)
for a triangular pattern,
with a spacing between the axis of the stone columns.
A ground improvement factor n0 is used to assess the settlement of the improved ground,
whereby the factor n0 is commonly defined as the ratio between the settlements of the virgin
2 State of the art of ground improvement with stone columns
31
host soil over the settlements of the improved ground. The settlement reduction factor
represents the inverse of n0.
2.31
with n0 ground improvement factor
s0 settlement of the host soil
si settlement of the improved layer
β settlement reduction factor
Three main types of settlement calculation may be identified, namely based on equilibrium
considerations (Section 2.5.2.1), empirical methods (Section 2.5.2.2) and numerical
modelling. A detailed description of the development of numerical modelling of stone
columns using finite elements can be found in Chapter 6 of this thesis.
2.5.2.1 Settlement calculations based on equilibrium considerations
Models proposed by Baumann & Bauer (1974), Aboshi et al. (1979), Omine & Ohno (1997)
and Goughnour (1983) for the calculation of settlements are presented in this section.
Baumann & Bauer (1974) consider the situation of a stone column loaded by a rigid footing,
assuming a constant Young’s Modulus of the stone column material with regard to depth and
horizontal extent. Similar to Equation 2.1, the equilibrium condition can be defined as:
2.32
with σ0 average load intensity on the footing
A footing area
σs total stress acting on the host soil
As soft soil cross-section in the unit cell
σsc total stress acting on the stone column
Asc stone column cross-section
Figure 2.22: Stress distribution on a rigid footing.
sstσs
σsc
σ0
dsc
zsc
ssc ss
a
2.5 Design of stone columns
32
Due to the rigidity of the footing, the settlements of the stone column (ssc) and of the subsoil
(ss) have to be equal and can be calculated as:
( )
(
)
2.33
2.34
with ssc settlement of the stone column
Esc Young’s modulus of the stone column material
zc depth to which the column has been compacted
a √ ⁄
A footing area
rsc radius of the stone column
σsc total stress acting on the stone column
σs total stress acting on the soft soil surface
ss settlement of the host soil
Es Young’s modulus of the host soil
The Young’s moduli of the stone column material and of the host soil should be determined
for the equivalent effective stress levels, based on laboratory or field tests. The radial
deformation of the column can be assessed as:
(
)
2.35
Equations 2.33, 2.34 and 2.35 can be rearranged and expressed as:
[
(
)]
[
(
)] 2.36
with Ks coefficient of earth pressure of the host soil
Ksc coefficient of earth pressure of the column material
The value of Ks lies between the at-rest and the passive coefficient in order to take the lateral
loading of the host soil by the stone column triggered by the bulging deformation of the
inclusion into account. Ksc varies between the at-rest and the active coefficient, so that the
influence of the deformation caused by the loading is considered.
Hughes et al. (1975) proposed to divide the subsoil into different layers, based on the
assumption of constant volume, triggering a radial expansion of the column, while settlement
occurred. This approach enables to avoid assuming that the stiffness parameters are
constant over the whole depth of the inclusion and thus enables a more precise assumption
2 State of the art of ground improvement with stone columns
33
of the settlements. The following formulation for the evaluation of the settlement of a stone
column is used:
∑
2.37
with Hi thickness of the i-th layer
δri / r radial strain of the i-th layer, obtained from pressuremeter tests
Aboshi et al. (1979) and Omino & Ohno (1997) propose simple solutions, which can be
implemented as a first approximation. Aboshi et al. (1979) formulate the settlement sv of a
composite foundation under a stiff footing load as follows:
( )
2.38
( ) 2.39
with sv settlement of the composite foundation
P footing load
H thickness of the layer
Es Young’s modulus of the host soil
m stress concentration ratio
as area replacement ratio
Omine & Ohno (1997) also present a simple procedure based on the two-phase mixture
model (Omine et al., 1993). A two-phase mixture consists of a basic material (the matrix) and
another material (the inclusion). A summary of the different possible situations, which can be
encountered for the determination of the stress distribution parameter, is given in Figure
2.23.
Horizontal laminate Vertical laminate
Mixture with
inclusions distributed
at random
Mixture with rod
shaped inclusions
Constant stress Constant strain Constant strain
energy
Approximation based
on Finite Element
Method
Figure 2.23: Evaluation of the stress distribution parameter depending on the different kinds
of mixture (Omine & Ohno, 1997).
2.5 Design of stone columns
34
Due to the geometrical organisation of granular inclusions, the case of a group of stone
column may be assumed to be represented by a “Mixture with rod shaped inclusions” (Figure
2.23), for which the parameter evaluating the stress distribution can be assessed as:
2.40
The homogenised Young’s modulus for the two-phase mixture, which can be implemented in
a settlement calculation, is expressed as:
( )
( )
2.41
with fsc volume content of the stone column
Goughnour (1983) conducted a one-dimensional elastic-plastic analysis of the settlement of
a vertically loaded stone column for stiff loads (thus assuming that the stone column and the
surrounding soil have deformed equally at the top). The stone column response was taken
as rigid-plastic and incompressible in the plastic phase. As in the solution proposed by
Hughes et al. (1975), the subsoil is subdivided into layers. In this case however, this
calculation is used to assess the stress increments at different depths. The set of equations
for the evaluation of the settlement is:
( )
[
( )
( ) ] 2.42
( )
[ (
)] 2.43
( )
( ) ( ) ( ) (
⁄ )
(
⁄ )
( )
2.44
(√
)
√
√
2.45
(
) ( )
[(
)(
)
]
2.46
2 State of the art of ground improvement with stone columns
35
(
) ( )
[
(
)(
⁄
)
]
2.47
with εv vertical strain (same for stone column and host soil)
as replacement ratio
Cc compression index of the host soil
e0 initial void ratio of the host soil
(P0)V, s initial effective vertical stress in the host soil
ΔP footing load increment
K0 coefficient of earth pressure at rest
(ΔP)*V, s effective vertical stress increase in the host soil averaged over the
horizontal projected area of host soil
(ΔP)*V effective vertical stress increase averaged over horizontal projected
area of the unit cell
(P0)V, sc initial effective vertical stress in the stone column
φ’sc effective angle of friction of the stone column material
K coefficient of earth pressure
F parameter depending on K0 and as
This is an iterative solution, which can be used to carry out parametric studies to investigate
the interaction between various factors.
2.5.2.2 Settlement calculations based on empirical methods
Empirical models introduced by Greenwood (1970), Priebe (1976), Priebe (1995), Priebe
(2003) and Van Impe & De Beer (1983) are presented in this section.
Greenwood (1970) proposes the design diagram shown in Figure 2.24, based on field
measurements, in order to assess the settlement of a composite foundation. Two important
assumptions are made in this case:
- the curves do not take the immediate settlement and shear displacement into
consideration,
- the stone columns rest on firm clay, sand or harder ground.
Although this approach presents an opportunity to determine the settlement of a composite
foundation based on the undrained shear strength of the host soil and on the spacing
between the columns, the degree of uncertainty leads Greenwood (1970) to point out that
these curves are to be used with caution.
2.5 Design of stone columns
36
Figure 2.24: Settlement diagram for stone columns installed in uniform soft clay (Greenwood,
1970).
Priebe (1976) presented his first design procedure in order to calculate the settlements of
composite foundations, which he then developed over subsequent years (Priebe, 1988,
1995). The basic approach adopted by Priebe (1976) is based on that of the cavity
expansion theory presented in Gibson (1961). This application leads to the calculation of the
ground improvement factor n0 (graphically illustrated in Figure 2.25 for different values of the
angle of friction of the column material, denoted in this formula as ), formulated as follows:
[
⁄ ( )
( )
] 2.48
( ) ( ) ( )
2.49
(
⁄ )
2.50
with n0 ground improvement factor
as area replacement ratio
Ka, sc coefficient of active earth pressure of the stone column material
’ Poisson’s ratio of the host soil
φ’sc effective angle of friction of the stone column material
In his original approach, Priebe (1976) assumes that the stone columns rest on a hard soil
layer, that the host soil is isotropic, that the self-weight of the stone columns and of the soil
may be neglected and that the stone columns are incompressible and reach internal shear
failure. Neglecting the self-weight means that the pressure difference between the stone
columns and the surrounding soil only depends on the loading applied on top of the inclusion
and its repartition over depth. This assumption was found to be very conservative when
comparing the predictions obtained with field measurements (Ulrichs & Wernick, 1986).
2 State of the art of ground improvement with stone columns
37
Figure 2.25: Values of the ground improvement factor n0 depending on the area replacement
ratio, for a Poisson’s ratio of 1/3 (after Priebe, 1995).
Two of the assumptions made earlier in the settlement calculations proposed by Priebe
(1976) can be avoided, as shown in Priebe (1995).
Firstly, the influence of the compressibility of the stone columns can be taken into account
using an additional ground improvement factor n1, which is also shown in Figure 2.26, where
denotes the angle of friction of the stone column material:
[
⁄ ( )
( )
] 2.51
⁄ ( ⁄ )
2.52
( ⁄ )
( ) 2.53
( ) ( )
( )
√[
( )
]
( )
2.54
2.55
with n1 ground improvement factor for column compressibility
as area replacement ratio
Ka, sc coefficient of active earth pressure of the stone column material
n0 ground improvement factor (Equation 2.48)
ME, s confined stiffness modulus of the host soil
ME, sc confined stiffness modulus of the stone column material
Area ratio 1 / as [-]
Impro
vem
ent
facto
rn
0[-
]
⁄
2.5 Design of stone columns
38
Figure 2.26: Values of an additional component of the area replacement ratio to account for
column compressibility, for a Poisson’s ratio of 1/3 (after Priebe, 1995).
Secondly, the effect of the overburden pressure and of the self-weight of the stone column
and soft soil can be considered using a depth coefficient fd (Figure 2.27) and the resulting
improvement factor n2. Due to the fact that n2 is dependent on the depth, Priebe (1995)
suggests that the subsoil should be subdivided into layers of thickness Δt, so that the
settlement calculations can be carried out:
with n2 depth-dependent ground improvement factor
n1 ground improvement factor for compressibility (Equation 2.51)
fd depth coefficient
K0,sc coefficient of earth pressure at rest of the stone column material
Δt layer thickness
γs unit weight of the host soil
σsc total stress acting on the stone column
Figure 2.27 shows curves allowing for a graphical determination of the factor fd. It assumed
that the host soil and the stone column material exhibit the same unit weight. This is not
conservative, and so it is recommended that Equations 2.56 and 2.57 are used for the
determination of n2, in order to be on the safe side.
Confined stiffness modulus ratio ME,sc / ME,s [-]
Additio
n t
oth
eA
rea R
atio
Δ(1
/ a
s)
[-]
⁄
2.56
∑( )
2.57
2 State of the art of ground improvement with stone columns
39
Figure 2.27: Determination of an influence factor y for the calculation of a depth coefficient fd
for a Poisson’s ratio of 1/3 (γs: unit weight of the host soil; d: improvement depth;
p: footing load) (after Priebe, 1995).
The total settlement for homogeneous conditions can be estimated on the basis of these
parameters as:
2.58
with P vertical footing load
H thickness of the layer of soft soil
ME, s confined stiffness modulus of the host soil
n2 depth-dependent ground improvement factor (Equation 2.56)
The approaches proposed by Priebe (1976) and Priebe (1995) can now be compared. The
overburden pressure and the self-weight of the stone column reduce the improvement factor
when the compressibility of the stone column material is taken into account.
Priebe’s (1995) method is widely used in practice for the calculation of settlements for soft
layers reinforced with stone columns, although Ellouze et al. (2010) note some
inconsistencies in this process as Priebe (1995) uses the same unit cell model for two
different situations:
- firstly, it is assumed that no vertical deformation takes place during the cavity
expansion, so a solution based on plane-strain conditions can be found,
- secondly, the solution obtained is used to model the application of a uniform
vertical load, which would cause uniform settlement throughout the cell.
Area ratio 1 / as [-]
Influence
facto
ry
[-]
⁄
2.5 Design of stone columns
40
Douglas & Schaefer (2012) investigate the reliability of Priebe’s (1995) method based on
case studies with a maximum measurement of settlement of 8 cm. They conclude that the
settlements calculated have an 89 % probability of being larger than the measured values,
which means that the results obtained from Priebe (1995) are conservative (Figure 2.28).
They also note that the site conditions only have a minor influence on the results of the
calculations.
As a conclusion, it can be said that the simplicity of Priebe’s (1995) approach, as well as its
strong tendency to deliver an overestimation of the settlements, has assured its success
among practising engineers over the years, although it is bound by some restrictive
assumptions.
Figure 2.28: Priebe method best-fit line, with data sorted based on the site soil conditions
(Douglas & Schaefer, 2012).
Priebe (2003) also develops a design procedure in order to be able to consider the case of
floating stone columns (Figure 2.29). This avoids the necessity of assuming that the columns
rest on a hard soil layer. The total settlement can be calculated as:
2.59
with si calculated settlement of the improved layer
sc calculated settlement of the untreated layer
reduced settlement due to the stress concentration at the tip of the stone
column =
( )⁄
se settlement due to the stress concentration at the tip of the stone column
s0 settlement of the treated layer without ground improvement
2 State of the art of ground improvement with stone columns
41
Improved layer Untreated layer Stress concentration
Figure 2.29: Static system for the settlement calculation of groups of floating stone columns,
according to Priebe (2003) (Kirsch, 2004).
psc,0 and pc,0 denote the load acting on top of the stone column and on top of the host soil,
respectively, evaluated with the stress concentration factor m. psc,u and pc,u refer to the
stresses present at the depth of the tip of the column and in the host soil, respectively.
The settlements se can be calculated based on the stress concentration psc, u at the base of
the stone columns and on the resulting force Pe, illustrated in Figure 2.29.
( ) 2.60
with psc, u stress acting within the column at the depth of the tip of the column
P footing loading
A footing area
m stress concentration factor
as replacement ratio
The correction of the settlements se should allow for a reduction of the error due to the stress
concentration in the stone columns.
Tre
ate
dla
ye
rU
ntr
ea
ted
laye
r
sisi sc se
psc,0psc,0
psc,0
pc,0
psc,upsc,u
psc,u
pc,u
s = 0
teσc
σe
pepe
Pe
2 rsc
P
P
2.5 Design of stone columns
42
(a)
(b)
Figure 2.30: (a) Rheological modelling of the behaviour of stone columns, (b) Calculation
approach in plane-strain (Van Impe et al., 1997b).
Van Impe & De Beer (1983) propose an approach based on a model of a stone column
resting on an undeformable bearing layer in plane-strain (Figure 2.30 b). It is further
assumed that the stone column material mobilises its plastic peak resistance while the host
soil remains elastic, which is considered using the rheological model shown in Figure 2.30
(a). The calculation aims at the determination of the settlement reduction factor , as defined
in Equation 2.31 and calculated here resolving a system of non-linear equations. The
fundamental behaviour difference between host soil and columns (elastic versus plastic) may
however raise some questions concerning the validity of the model to solve practical
P
Soil
Column
H
L
L
sv
Rigid base
2 State of the art of ground improvement with stone columns
43
boundary value problems, given that the soft soil would be more likely than the columns to
reach a plastic state.
2.61
( ) (
)
2.62
(
) 2.63
with s total settlement
β settlement reduction factor
s0 settlement of the treated layer without ground improvement
H thickness of the treated layer
effective Poisson’s ratio
P footing load
Es Young’s modulus of the host soil
Figure 2.31: Graphical determination of the settlement reduction factor β (Van Impe & De
Beer, 1983).
L, sr and in Figure 2.30 (b) denote the spacing between the stone column, the radial
deformation of the inclusions and the radial stress acting on the columns, respectively.
0 0.2 0.4 0.6 0.8 1.0
0
20
40
60
80
100
as
2.5 Design of stone columns
44
The settlement reduction factor β can be determined graphically depending on the angle of
friction of the stone column material, denoted as ϕ1 in Figure 2.31. p0 denotes the footing
load and Es the Young’s modulus of the host soil.
2.5.3 Comparison of the design procedures
The overview of the design procedures gives a summary of the main calculations and an
indication of the possibilities available. It is not surprising to note a relatively large variation of
the results obtained due to the different assumptions, models, methods and simplifications
used for the determination of the bearing capacity and settlements of soft ground reinforced
with stone columns. This is shown, for example, in Greenwood & Kirsch (1983) (Figure 2.32,
Figure 2.33).
The results obtained for the ultimate bearing capacity with a solution based on the cavity
expansion theory (Vesic, 1972) are the highest (Figure 2.32), which can be assumed to be
due to the high compaction of the host soil triggered by the installation process. This is
confirmed by the proximity of the results obtained by the method proposed by Brauns
(1978a, 1978b), which assumes the mobilisation of a passive earth pressure in the host soil
surrounding the inclusion. The lowest values are obtained assuming a bulging failure of the
stone column (Hughes & Withers, 1974). The results obtained by Bell (1915) do not consider
granular inclusions, and are based on the horizontal pressures and can therefore be
considered to be a reference value in order to quantify the reinforcing effect of stone
columns.
The results of field experiments presented in Hughes et al. (1975) and Appendino &
Comastri (1970) all lie between the results obtained considering a bulging failure and based
on the cavity expansion theory, which can be considered to be lower and upper bounds,
respectively.
Figure 2.33 shows a comparison of the results obtained from different approaches with field
observations. The model proposed by Priebe (1976) represents a compromise between the
other approaches and enable reasonable predictions of the behaviour observed in the field.
The solution presented by Baumann & Bauer (1974), based on equilibrium considerations
(Section 2.5.2.1), overestimates the improvement effect of the granular inclusions regarding
the settlements compared to the great majority of the field observations. This is also the case
of the approach presented by Greenwood (1970), which is to be handled with care, as
already described in Section 2.5.2.2.
The results obtained using the approach presented by Balaam (1978) highlight the impact of
the ratio between the stiffness of the column (denoted as Ec in Figure 2.33) and of the host
soil (denoted as Es in Figure 2.33). This has a greater influence on the ground improvement
factor than the type of loading (flexible / rigid footing).
2 State of the art of ground improvement with stone columns
45
Figure 2.32: Comparison of the ultimate bearing capacities as a function of the angle of
friction calculated using different procedures (after Greenwood & Kirsch, 1983).
The significant variation in the results obtained from the different procedures, as well as the
large spread in the field data, makes it difficult to draw definitive conclusions. However, the
assumptions made by the different models have to be kept in mind when implementing them
in design calculations.
Terzaghi’s (1936) wise advice should not be forgotten either: “Whoever expects from soil
mechanics a set of simple, hard and fast rules for settlement computation will be deeply
disappointed… The nature of the problem strictly precludes such rules.”
’
2.6 Load-transfer behaviour of stone columns
46
Figure 2.33: Comparison of results obtained from empirical models and elastic theories with
field observations (after Greenwood & Kirsch, 1983).
2.6 Load-transfer behaviour of stone columns
Although a large amount of research on the ultimate limit state and serviceability limit state of
stone columns has been carried out, the actual mechanisms through which the load applied
on a stone column is transferred within the inclusion and to the surrounding host soil have
received little attention. Sivakumar et al. (2011) investigated the load transfer between a
stone column (the diameter of which varied from 40 mm to 60 mm) that expanded throughout
the depth of the sample and the surrounding clay during consolidation and subsequent
foundation loading means of small-scale experiments. Clay samples of 400 mm in height and
300 mm in diameter were used for the experimental setup (Figure 2.34), while the lateral
movements were blocked at the boundary by rigid walls. The stone columns were installed
by pouring basalt (particle sizes ranging from 2.5 mm to 3 mm) into a pre-bored hole and
compacting it manually. Pressure cells were installed in the stone column to record the
stresses measured in the inclusion. Once the installation of the stone column in the clay
specimen was complete, the sample was consolidated up to 300 kPa. A footing loading was
subsequently applied to the column only at a rate of 1 kPa / h.
1/as
2 State of the art of ground improvement with stone columns
47
Figure 2.34: Experimental setup for a single stone column loaded vertically through a rigid
footing (Sivakumar et al., 2011).
Figure 2.35 shows the vertical pressure distribution with depth during the footing for different
footing settlements. The minimum increase of vertical pressure is observed at a depth about
300 mm, corresponding to a distance of 5 times the diameter of the column from the surface.
Sivakumar et al. (2011) report that the minimal increase of vertical pressure during the
loading of a 40 mm diameter column was observed at 200 mm, there again corresponding to
a distance of 5 times the diameter of the inclusion. The rise of the recorded vertical
pressures below a depth of 300 mm is thought to be due to the fact that the compression of
the host soil at higher depths due to the bulging of the stone column is not as marked as
near the surface. This leads to the stiffness of the host soil being lower and thus a higher
stress concentration within the stone column. Although this is qualitatively plausible, the
reproduction of the in-situ stresses is impossible with small-scale laboratory experiments. As
a consequence, the validity of a quantitative extrapolation to a full-scale boundary value
problem is not granted. Moreover, the influence of the interaction between pressure cells and
2.6 Load-transfer behaviour of stone columns
48
stone column material is not discussed, which may raise some questions about the
quantitative results shown in Figure 2.35.
Figure 2.35: Pressure distribution with depth during footing loading of a 60 mm diameter
stone column for different settlements (Sivakumar et al., 2011).
Full-scale investigations are relatively rare, but Thiessen et al. (2011) considered the
behaviour of rockfill (maximum grain size of 150 mm) columns used to stabilise riverbanks.
The installation procedure used in this case differed from the commonly used vibro-
replacement or vibro-compaction techniques (Figure 2.1). The columns were pre-bored using
a rotary rig, filled from the surface and compacted with a vibro-lance.
Site characterisation was conducted prior to installing the columns. The results can be seen
in Figure 2.36. No significant difference between the mechanical properties of the clays and
silt were found. Values of c’ = 5 kPa for the effective cohesion and φ’ = 19° for the effective
angle of friction, both for large strains, were determined by means of consolidated undrained
triaxial tests with pore pressure measurement (CIU).
The columns were installed in a triangular pattern, with a diameter of 2.13 m, a centre-to-
centre spacing of 3.1 m and an average length of 6.9 m, reaching about 1 m into the till
(Figure 2.37 a). Void columns were excavated with a diameter of 0.48 m and to a depth just
above an elevation of 219 m (Figure 2.37 a and b). They were left empty during the duration
of the loading in order to provide a vertical plain around the edge of the research area to
concentrate shear strains and thereby to reduce undesired deformation outside the test
zone. This setup also allowed for plane-strain assumptions to be made for the analysis of the
experimental results.
Vertical pressure: kPa
Depth
: m
0 200 400 600 800 1000 1200
0
200
100
300
400
0
1 mm
2 mm
8 mm
16 mm
2 State of the art of ground improvement with stone columns
49
Figure 2.36: Representative borehole and selected soil properties from Red River research
site in Winnipeg, Canada. w: natural water content (horizontal bars display
Atterberg limits); γwet: unit weight of saturated soil; σ: stress; σ’pc:
preconsolidation pressure; σ’v0: initial vertical effective stress; u0: initial pore
water pressure (Thiessen et al., 2011).
An embankment was built at the test area over a period of 9 days on the ground surface with
fill material compacted a dry unit weight of 15.6 kN/m3 (Figure 2.37 a and Figure 2.38). The
load was maintained constant after completion of the embankment construction.
The response of the soil to the loading was monitored by means of slope inclinometers
(denoted as SI in Figure 2.37 b, Figure 2.38 and Figure 2.39) and piezometers (denoted as
VW in Figure 2.38 and Figure 2.39). Table 2.5 gives details above a summary of the
installation depths of the inclinometers (denoted as SI) and piezometers (denoted as PZ and
VW).
2.6 Load-transfer behaviour of stone columns
50
(a)
(b)
Figure 2.37: Red River test site in Winnipeg, Canada: stabilisation of river bank using a
combination of void and rockfill columns (a) cross-section and (b) plan view of
the research site (Thiessen et al., 2011). Elevations and distances in metres.
2 State of the art of ground improvement with stone columns
51
Figure 2.38: Red River test site in Winnipeg, Canada: pore water response to loading
(Thiessen et al., 2011).
Figure 2.39: Red River test site in Winnipeg, Canada: instrumentation layout (Thiessen et al.,
2011).
Time (days)
To
tal h
ea
d(m
)
Pla
ce
dfill
(t)
0 1 2 3 4 5 6 7 8 9
220
221
222
223
224
225
226
0
500
1000
1500
2000
1000
1500
VW-A (z = 11 m)
VW-B (z = 7.3 m)
VW-D (z = 5.8 m)
VW-E (z = 6.1 m)Loading
2.6 Load-transfer behaviour of stone columns
52
Table 2.5 Summary of inclinometer and piezometer installations at Red River test site in
Winnipeg, Canada: (SI: slope inclinometer; PZ: piezometer; VW: vibrating wire)
(after Thiessen et al., 2011).
Installation Ground [m] Bottom
reading [m] Installation Ground [m]
Bottom
reading [m]
SI-1 232.01 215.30 PZ-2 215.40 229.46
SI-2 229.46 215.27 VW-A 218.65 229.62
SI-4 225.65 211.95 VW-B 222.29 229.62
SI-5 231.86 212.16 VW-C 226.56 229.62
SI-6 - 218.30 PZ-7 Well 225.52
SI-7 225.52 216.69 VW-D 219.61 225.40
SI-8 225.40 214.28 VW-E 219.70 225.79
SI-9 - 217.61 VW-F 222.16 225.81
SI-10 223.47 213.87
SI-11 225.38 215.32
SI-12 225.57 212.86
It was observed that the pore water response to loading (Figure 2.38) changed with time and
as the footprint of the loading evolved. The response of the piezometers A and B, located
underneath the crest of the embankment (at depths of 11 m and 7.3 m below the ground
surface, respectively), increased with time as the loading was placed directly over the
piezometers. However, the response of the piezometer B to loading was stronger than that of
the piezometer A and the excess pore water pressures were dissipated faster as well. This
can be explained by the greater measurement depth of the piezometer A compared to that
of the piezometer B. The relatively low response of the piezometer D (located at a depth of
5.8 m below the ground surface) to loading was explained to be due to its vicinity to the void
columns. Inversely, the response of the piezometer E, located at mid-distance between the
crest and the toe of the embankment (at a depth of 6.1 m below the ground surface),
diminished with time. Thiessen et al. (2011) do not give any explanation for the slower
reaction to loading of the piezometer E. The rate of dissipation of the excess pore water
pressures is however similar to that measured by the piezometer B, which is consistent with
the similar installation depths.
The investigation of the deformations triggered by the loading of the test area delivered some
interesting information regarding the load transfer of the inclusions. Inclinometer SI-7 was
located at the toe of the embankment and shows a bending behaviour over the length of the
column, which would suggest that shear stresses developed in the column over the whole
depth of the clay and dissipated in the silt.
The measurements recorded by the inclinometers SI-4 and SI-10 cannot be regarded
as trustworthy due to some issues during the installation of the nearby rockfill columns.
2 State of the art of ground improvement with stone columns
53
SI-1, located under the crest of the embankment, did not record significant bending over the
length of the columns.
A local shear surface can be identified in each case at the top of the inclusions (SI-4, SI-7,
and SI-10) or at the crest of the embankment (SI-1).
Figure 2.40: Measured deformations along A axis (in downslope direction) at Red River test site in Winnipeg, Canada: (a) SI-1 at crest of slope; (b) SI-4 in between columns along upper row; (c) SI-7 in a column in upper row; (d) SI-10 downslope of columns (Thiessen et al., 2011).
2.7 Load-transfer behaviour in inclusion-supported embankments
The load-transfer behaviour of embankments supported by rigid inclusions has been
described in detail over the past decades (e.g. Hewlett & Randolph (1988); Fleming et al.
(1992); Han & Gabr (2002); Aslam & Ellis (2010); Baudouin et al. (2010)). These studies
highlighted the development of soil arching within the embankment between the inclusions
due to differential stiffness between the stiffer inclusions and the softer soil in-situ (Figure
2.41).
0
0(a) 0 0 0(b) (c) (d)
SI-1 SI-4 SI-7 SI-10
10 20 0 20 0 10 20A axis deflection (mm)
10 20 0
A axis deflection (mm)
10212
214
216
218
226
228
230
232
Ele
vation (
m)
220
222
224
Ele
vation (
m)
Clay
till
Top of
columns
ClayDay 3.3Day 6.5Day 11Day 18
Clay
till
Silt
till
2.7 Load-transfer behaviour in inclusion-supported embankments
54
Soil arching can be quantified using a soil arching ratio ρ. This ratio is equal to zero when the
formation of an arch between the inclusions is complete and defined as:
2.64
with ρ soil arching ratio (ρ = 0 represents complete soil arching and ρ = 1 represents
no soil arching)
σs stress acting on the soft soil midway between two inclusions
γe unit weight of fill
He embankment height
σ0 load acting on the top of the embankment
Recent studies (e.g. Deb, 2010 and Indraratna et al., 2013) showed that arching also occurs
when stone column-supported embankments are constructed with a geosynthetic base
reinforcement (σt in Figure 2.41 denotes the average pressure acting onto the base
reinforcement) as the soil between the stone columns settles more than the inclusions due to
the embankment load. This settlement (denoted as ΔS in Figure 2.41) is however reduced by
the shear resistance of the embankment soil (denoted as in Figure 2.41). As a
consequence, the load acting on top of the inclusions (denoted as σc in Figure 2.41)
increases whereas the loading of the soft soil (denoted as σs in Figure 2.41) decreases.
Figure 2.41: Soil arching in stone column-supported embankment (after Deb, 2010).
Top of embankment
Shear
planeGeosynthetic
reinforcement
Stone
columns
Soft soil
σs
σc
σt
σt
ΔΔS
Granular
Granular
layer
He
Top of embankment
σ0
2 State of the art of ground improvement with stone columns
55
The stiffness of the base reinforcement and the embankment height play a role in the
development of soil arching:
- as the stiffness of the base reinforcement increases, less differential settlements
occur and the arching is reduced,
- the shear strength of the embankment material is too low for arching to occur if
the embankment height is not large enough. An increase of the embankment
beyond the minimum height necessary to fully mobilise arching effects does not
have any consequence on arching.
Deb (2010) develops a mathematical model based on plane-strain considerations to
investigate the behaviour of stone column-supported embankments. The soft soil is modelled
using spring-dashpots and the columns are simulated using stiffer nonlinear springs (Figure
2.42). He highlights that soil arching is more marked with decreasing bearing capacity of the
soft soil (Figure 2.43 a), as well as with increasing shear modulus of the embankment soil
(Figure 2.43 b).
Figure 2.42: Proposed foundation model for soft soil reinforced with stiffer inclusions (after
Deb, 2010).
Although this model presents the advantage to enable a simulation of the soil with elements
featuring clear parameters, some drawbacks are present as well. First, the interaction
between granular inclusions and host soil as well as the bulging deformations cannot be
taken into account. Second, Deb (2010) does not give any indication either about the method
used to determine the spring constants implemented in this specific case or about the values
of the constants used, which may raise some questions about the validity of the results as
the spring constant for the spring with dashpot ks0 plays a major role in the outcomes
(Figure 2.43), many of the parameters used being normalised by ks0B2 (B being the half width
of the embankment).
2B
Rough elastic membrane
(geosynthetic layer)
Pasternak shear
layer (granular
layer)
Spring-dashpot
(soft soil)
Stiffer non-linear
springs (stone
column)
Firm soil or bedrock
Embankment modelled by
Pasternak shear layerTTT T
2.7 Load-transfer behaviour in inclusion-supported embankments
56
(a) (b)
ks0: spring constant per unit area for the spring with dashpot (Figure 2.42)
B: half width of the embankment
Ge*: shear modulus of the embankment soil normalised by ks0B2
Gr*: shear modulus of the granular layer (Figure 2.41) normalised by ks0B2
Ec: Young’s modulus of the stone column material
Es: Young’s modulus of the soft soil
U: degree of consolidation of the soft soil
: ultimate shear resistance of the embankment soil normalised by ks0B
2
: ultimate shear resistance of the granular layer normalised by ks0B
2
s: spacing between the columns
bw: width of the stone columns
qus*: bearing capacity of the host soil normalised by ks0B.
Figure 2.43: Effect of (a) ultimate bearing capacity of the soft soil and (b) the shear modulus
of the embankment soil on the arching ratio (Deb, 2010).
Indraratna et al. (2013) consider arching effects in a numerical analysis of a stone column-
supported embankment and formulate an expression for the normal stress within the
embankment along the radial R-direction σr (Figure 2.44):
( )
2.65
with σr normal stress in soil element along R-direction
R global radial coordinate
Kp, e coefficient of passive earth pressure of the embankment
γe unit weight of fill
qs in Figure 2.44 denotes the surcharge load intensity applied onto the surface of the
embankment, rc the radius of a stone column, re the radius of influence of a stone column, H
the thickness of the soft soil deposit and He the height of the embankment.
2 State of the art of ground improvement with stone columns
57
Figure 2.44: Arching effect in the embankment (Indraratna et al., 2013).
Soil arching causes an increase of the load applied onto the inclusions supporting the
embankment and a reduction of the load applied onto the soft soil. This can be an
explanation for the rising stress concentration observed by Greenwood (1991) at Humber
Bridge (Figure 2.9): once soil arching is complete, the additional load applied onto the
embankment is transmitted to the inclusions, triggering an increase of the stress
concentration factor.
2.8 Effect of stone columns on the consolidation time
Stone columns do not only increase the bearing capacity and improve the performance of the
subsoil by reducing deformations but they might also cause a significant reduction of the
consolidation time due to an inversion of the main drainage direction from vertical to
horizontal assuming one-dimensional consolidation conditions, through a reduction of the
drainage length. This often takes advantage of the natural anisotropy of most soft soils,
which exhibit a strikingly higher permeability in the horizontal direction than in the vertical
one. The reduction of the length of the drainage path d plays the most important role for the
reduction of the consolidation time, because the consolidation time is proportional to d2:
2.66
with t consolidation time
Tv dimensionless time factor
d length of the drainage path
γw unit weight of water
k coefficient of permeability
ME confined stiffness modulus
2.8 Effect of stone columns on the consolidation time
58
Sivakumar et al. (2004) conducted triaxial tests on sand columns installed in clay specimens
and show a significant reduction of the consolidation time for the specimens with stone
column.
Black et al. (2007) investigate the performance of single stone columns and of groups of
stone columns using small-scale laboratory experiments under 1 g (triaxial setup). They
show a considerable acceleration of the consolidation process with increasing length of the
inclusion (denoted as Hc in Figure 2.45) over the height of the layer to be treated (denoted as
Hs in Figure 2.45).
(a) (b)
Figure 2.45: Consolidation process for (a) a single stone column and (b) a group of stone
columns (after Black et al., 2007).
McCabe et al. (2009) present results from field tests enlightening the positive influence of
stone columns on the consolidation time compared to displacement piles (Figure 2.46). A
significant acceleration of the consolidation process can be noted when stone columns are
installed instead of piles. The dimensionless time factor for a radial flow Th is defined as:
2.67
2.68
with Th dimensionless time factor for radial flow
ch horizontal coefficient of consolidation (Equation 2.69)
t consolidation time
r radius of stone column or pile
req equivalent radius of a square pile
Square root time (min0.5) Square root time (min0.5)
20 40 60 800
20
40
Degre
eofconsolid
atio
n, U
(%
)
60
80
100
0
20
40
Degre
eofconsolid
atio
n, U
(%
)60
80
100
20 40 60 80
(a) (b)
100 mm
32 mm
2 State of the art of ground improvement with stone columns
59
The horizontal coefficient of consolidation ch can be formulated as:
2.69
with kh coefficient of horizontal permeability
ME, h horizontal confined stiffness modulus
γw unit weight of water
Figure 2.46: Comparison of the excess pore water pressure dissipation for displacement
piles and stone columns (McCabe et al., 2009).
However, although the beneficial effect of stone columns on the consolidation time is not to
be doubted, the insertion of the mandrel causes some installation effects that can be
detrimental to the drainage efficiency of stone columns. These effects have an influence
on the stress levels in the host soil surrounding the column and trigger the development of a
so-called smear zone.
2.9 Analytical considerations about the installation of inclusions in
soil
Due to the similarity of the physical mechanisms encountered during the construction of piles
and stone columns, the studies conducted on the installation effects of rigid inclusions are
considered here as well. The influence of the construction of piles on the pore pressures, and
on the total stress distribution in the surrounding soil, plays an important role in the
determination of the shaft resistance and in the tip bearing load.
Time factor, Th = cht/4r2 or cht/4r2eq
0.01 0.1 1 10 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
De
gre
eo
fco
nso
lida
tio
n, U
Driven five-pile group, r/R>5
(Belfast clay )40
Jacked single pile, r/R=1
(Bothkennar clay )51
Jacked single pile, r/R=1
(Belfast clay )52
Stone column, r/R=5
(Keller GE Contract B)
2.9 Analytical considerations about the installation of inclusions in soil
60
2.9.1 Cavity expansion theory
The installation of a driven pile or of a stone column can be modelled by a stress expansion
theory approach at the tip and by a cavity expansion theory approach along the shaft. This
section deals with the cavity expansion approach.
Vesic (1972) proposes a theoretical approach to calculate the total stress changes as well as
the excess pore water pressures generated by the expansion of a cavity in plane-strain
conditions.
The system considered by Vesic (1972) is that of a spherical cavity or of a cylindrical cavity
with an initial radius Ri expanded by a pressure p. The application of the pressure will trigger
the formation firstly of an elastic and then of a plastic zone around the cavity, the radius of
which will increase until the pressure reaches pu. The cavity will then exhibit a radius Ru and
the plastic zone will have expanded to Rp. The soil mass beyond Rp remains in an elastic
state.
Figure 2.47: Geometric representation of cylindrical cavity expansion in either two or three
(spherical) dimensions (Vesic, 1972).
Rp
Ru
Ripu
sq
sq
sr
sp
up
plastische Zone
elastische ZonePlastic zone
Elastic zone
2 State of the art of ground improvement with stone columns
61
A cylindrical cavity is considered here since it is more applicable to the installation of the
stone column, even more than for rigid piles, through the radial compaction applied. The
radius of the plastic zone can be estimated as:
√
(
⁄ )
√
2.70
with Rp radius of the plastic zone
Ru final cavity radius
Ir stiffness index of soil (Equation 2.12)
I’rr reduced stiffness index for cylindrical cavity
Δ volumetric strain in the plastic zone
φ’s effective angle of friction of the host soil
The theory developed by Vesic (1972) relies on a total stress approach. Undrained behaviour
( = 0 [-]) is simulated by setting the effective angle of friction of the host soil equal to 0,
which does not represent reality. Equation 2.70 can be simplified to:
√ 2.71
The ultimate cavity pressure pu can be computed as:
2.72
(
) 2.73
(
) (
)
(
)⁄
2.74
with su undrained shear strength of the host soil
φ’s effective angle of friction of the soil
F’c cavity expansion parameter
F’q cavity expansion parameter
For undrained behaviour, Fc’ can be simplified to:
2.75
The radial stress in the plastic zone can then be formulated as:
(
) 2.76
with pu ultimate cavity pressure (Equation 2.72)
r distance from centreline
2.9 Analytical considerations about the installation of inclusions in soil
62
When considering a cavity expansion under undrained conditions, the excess pore water
pressures generated are also to be determined. Within the plastic zone, the excess pore
water pressures can be formulated as:
[ ( )
] 2.77
2.78
with A pressure parameter according to Skempton (1954)
Rp radius of the plastic zone
q0 over-burden pressure
r distance from centreline
The formulation of the excess pore water pressures outside the plastic zone is:
(
)
2.79
with α Henkel’s (1959) pore pressure parameter for the particular stress level
( )
Henkel’s (1959) pore pressure parameter can be expressed as a function of the pressure
parameter A (Equation 2.78) according to Skempton (1954):
√ (
) 2.80
Randolph & Wroth (1979) consider the pile installation process as the expansion of a cavity
from zero radius to r0 (r0 being the radius of the pile) and formulate the generated excess
pore water pressure at a radius r within the plastic zone based on equations proposed by
Gibson & Anderson (1961) and Hill (1950). They assume that the mean effective stress
remains constant under undrained conditions:
( ) [ (
) (
)] 2.81
with δσr radial total stress change
δσθ circumferential total stress change
The radial and circumferential total stress changes are then estimated as:
[ (
) (
)] 2.82
[ (
) (
)] 2.83
2 State of the art of ground improvement with stone columns
63
This allows a determination of the undrained shear strength at failure:
( ) 2.84
2.9.2 Strain Path Method and Shallow Strain Path Method
Besides the stress changes induced by the installation of inclusions in the subsoil, the
deformation field generated is also of interest. Baligh (1985) first presented the Strain Path
Method (SPM) in order to predict the soil disturbances caused by the insertion of objects in
the subsoil. Whereas the SPM can be compared to the Stress Path Method, it replaces
stress-control with strain-control, which is relevant for many events that are dictated by rate
of displacement.
The main elements of the SPM are:
- the velocity fields.
The SPM assumes that the deformations of the subsoil during penetration can be
decoupled from the constitutive relationships of the soil, which simplifies the
problem considerably. It is recommended to integrate the deformations from the
velocity fields.
- the constitutive relationships.
The effective stress path can be determined along a strain path by using either an
effective stress model, or a total stress approach in which the deviatoric stresses
as well as the pore pressures are considered.
- equilibrium.
As long as strains are not completely decoupled from stresses, the solutions
based on approximate strain fields may not be considered totally exact and the
computed stress may not always satisfy all equilibrium requirements. Two
approaches can be used to solve this inconsistency. The first one is to solve a
series of Poisson equations to calculate the pore pressures until the requirements
are fulfilled. The second, more engineering-oriented approach is to apply
corrective total stresses on the element considered to satisfy equilibrium and
boundary conditions.
Baligh (1985) uses the SPM to assess the soil deformations during penetration of a cone into
the soil (Figure 2.48). Besides the expected outwards radial displacement of soil elements,
two interesting observations can be made. First, the elements retrieve their original elevation
once the penetration is achieved, with the exception of the point located on the surface of the
cone. Second, the circular shape of the deformation paths can be of interest when
investigating clayey soils, as it might indicate a reorganisation of the direction of the clay
platelets up to a distance of 5 times the radius of the installed cone. Although the final state
seems similar to that obtained with pure cavity expansion, the values of the strains of the
elements are different.
2.9 Analytical considerations about the installation of inclusions in soil
64
Figure 2.48: Deformation paths during penetration of a cone into clay calculated using the
SPM (Baligh, 1985).
The SPM is restricted to conditions of steady, deep penetrations. This is fine in order to
predict the soil movements near the tip but might lead to problems for the prediction of
movements in the far-field, where the surface may influence the deformations. Based on
Sagaseta (1987), Sagaseta & Whittle (2001) propose the Shallow Strain Path Method
(SSPM) in order to address these disadvantages of the SPM by integrating the boundary
conditions of the stress-free ground surface.
2 State of the art of ground improvement with stone columns
65
(a) (b)
Figure 2.49: (a) Radial and (b) vertical deformation profiles after the installation of a simple
pile obtained with the SSPM analysis (Sagaseta & Whittle, 2001).
The stress state of the ground surface is modelled by the incorporation of corrective shear
tractions. Large strains are taken into account by formulating the velocities of the soil
elements instead of their displacements. However, the consideration of large strains is only
partial, as long as the deformations induced by the corrective shear tractions are assessed
with small-strain elastic solutions.
The small-strain solutions (denoted here with the subscript ss) for surface displacements due
to the installation of a pile featuring a radius r0 and a length L at a distance r from the
centreline are:
( )
√ 2.85
( )
(
√ ) 2.86
2.10 Observations concerning the installation effects of piles and stone columns on the soil
66
with r distance from centreline
r0 pile radius
L pile length
Figure 2.49 shows the predicted vertical and radial displacements obtained using the SSPM
for a pile with an embedment depth of L/R = 10 which has been driven into the soil from the
surface (R denotes the radius of the pile). Only a tear-shaped zone, located in front of the tip,
undergoes downward displacements, while the rest of the mass exhibits upwards
movements. The size of this zone was found not to be related to the pile radius. The biggest
heave is observed in the immediate vicinity of the pile shaft. However, the extent of the
significant soil movements (5 times the pile radius) is comparable to those observed with
calculations made using the SPM (Figure 2.48). Although these results are very interesting,
the conditions applied at the interface between pile and soil are not specified, thus making
any generalisation of the outcomes difficult.
2.10 Observations concerning the installation effects of piles and
stone columns on the soil
2.10.1 Pile installation
Randolph et al. (1979) conducted numerical investigations based on the cavity expansion
theory presented by Ladanyi (1963) and Vesic (1972). The main emphasis was the
determination of the effective and total stresses around a pile during installation and
subsequent consolidation relative to the shaft resistance. A first finding was that the over-
consolidation ratio (OCR) does not have a major influence on the stress changes. Second, it
was observed that the sensitivity of clay does play an important role on the development of
shaft friction around driven piles in clay. The authors further show that the installation of a
pile causes excess pore pressure which can be evaluated within the plastic zone (that is for
, with r0 the radius of the pile and Rp the radius of the plastic zone) as:
2.87
√
2.88
with Δu excess pore pressure
su undrained shear strength of the host soil
Rp radius of the plastic zone
r distance from centreline
r0 pile radius
G shear modulus of the soil
The effect of pile installation on sand may also be regarded as interesting. Linder (1977)
conducted laboratory tests during which he installed piles in sand specimen by ramming the
inclusions into the host soil. He could identify different zones around a pile tip in sand,
2 State of the art of ground improvement with stone columns
67
depending on the depth of the pile, as illustrated in Figure 2.50. Densified and loosened
zones could be observed. A major difference can be noted in comparison with the analytical
predictions in clay (Figure 2.49) as no surface heave was detected in this case. This shows
that although tests conducted with sand specimens can deliver some interesting information
about the stress expansion at the tip of the pile, the consequences of the cavity expansion
along the shaft can be very different from those in clay.
Figure 2.50: Deformation and density changes during the penetration of a pile in dense sand
(after Linder, 1977).
The effect of the driving stiff inclusions into sand seems to depend strongly on the density of
the subsoil. Davidson et al. (1981) conduct model tests of the introduction of a CPT cone
adjacent to a glass wall fixed in the container with inside dimensions of length 1.0 m, width of
1.0 m and height 0.65 m (Figure 2.51 a) and follow the resulting displacements using a set of
two cameras (Figure 2.51 b). Several tests were conducted with sands prepared to different
densities.
A similar displacement pattern, as found by Linder (1977), could be observed in the case of a
loose sand (relative density of 25 %, Figure 2.52 a). No movements at the surface could be
observed beyond a distance of 3 times the cone radius from the cone centre. A similar
pattern to that predicted by Sagaseta & Whittle (2001) for clay was detected for a dense
sand (relative density of 115 %, Figure 2.52 b) as heave could be observed up to 8.5 times
the cone radius from the cone centre. This shows the impact of the soil density on the
Displacement
zone
Initial depth
Core
Compaction
zone
Shear and loosening zone
“Filling up” of the
loosening zone
(post-compaction)
~ 1.30 m
~ 3.30 m
2.10 Observations concerning the installation effects of piles and stone columns on the soil
68
measured results as Linder (1977) did not measure heave, although he described his sample
as being dense (no measure of the relative density was given).
(a) (b)
Figure 2.51: (a) Half-cone inserted in sand (b) test set up (Davidson et al., 1981).
(a) (b)
Figure 2.52: Displacements (in mm) and volumetric strains (in %) for jacking a half-CPT cone
into (a) loose sand (relative density = 25 %) (b) dense sand (relative density =
115 %) (Davidson et al., 1981).
2.10.2 Changes of host soil properties due to the installation of stone columns
The changes in subsoil caused by the installation of stone columns are theoretically
comparable to those observed during the penetration of piles. However, the bearing
mechanisms of rigid piles and stone columns are significantly different. The influence of
excess pore water pressures and of their dissipation on the bearing capacity of piles can be
of importance, as they affect the shaft as well as the tip resistances. In the case of stone
columns, the major issue is the lateral support of the granular material by the host soil and
not the shaft resistance.
Sand
Half-cone Half-cone
Sand
Glass wall
Camera
Jack
2 State of the art of ground improvement with stone columns
69
2.10.2.1 Effect on soil resistance and stress levels
Aboshi et al. (1979) examine data from field tests where stone columns were installed in
clay. They observe a short-term reduction of the undrained shear strength due to the
disturbance of the clay structure by the penetration of the installation tool. However, the ratio
of the measured undrained shear strength after the installation (denoted as c in Figure 2.53)
to the undrained shear strength measured before the construction process (denoted as c0 in
Figure 2.53) reaches unity at least after a month. The increase of the ratio c/c0 up to values
of 1.7 can be explained by an increase of the effective stresses in the host soil. These
observations show that the installation process of stone columns has only a short-term effect
in terms of the bearing capacity of a composite foundation. However, the authors do not
specify the method used to determine the undrained shear strength, do not give the
distances from the location of the measurements to the stone columns and do not state the
sort of clay and their stress history. This lack of information does not allow a detailed
interpretation of the results.
Figure 2.53: Evolution of the undrained shear strength ratio (normalised to pre-installation
values) over time after the installation of stone columns (Aboshi et al., 1979).
Asaoka et al. (1994) conducted in-situ loading tests with composite foundations on sand
compaction piles (SCP) with a relatively low replacement ratio (as = 25%) in Maizuru, Kyoto,
Japan. The soil at the test site is composed of clayey alluvial deposits. The SCP construction
phase lasted approximately 200 days. The loading was then conducted in two phases. First,
a sand mat and an empty concrete caisson were placed on the surface of the composite soil
after SCP installation. This phase was maintained for about 10 months. The caisson was
subsequently filled with sand and a steel tank was positioned on top of the caisson in order
to apply additional load by filling the tank with water. The second loading phase lasted for
9 days and the last 43 % of the load was applied within less than 2.5 hours. Figure 2.54
2.10 Observations concerning the installation effects of piles and stone columns on the soil
70
shows the profiles of the axial compressive strength before the installation of the SCP and
after the first loading phase. The measured unconfined compressive strength of the host soil
exhibits a significant increase at every depth after the SCP installation. Although the results
seem interesting, the authors do not give detailed information about the characteristics of the
host soil. They also do not specify how the unconfined compressive strength was determined
and do not mention the distances from the locations where the measurements were
conducted to the stone columns. As for Aboshi et al. (1991), this lack of information does not
allow any definitive conclusions to be drawn.
Figure 2.54: Evolution of the unconfined compressive strength of clay over time (Asaoka et
al., 1994).
The increase in horizontal stresses due to SCP installation was investigated based on field
tests, e.g. Gruber (1995) and Watts et al. (2000). Watts et al. (2000) conducted tests at the
Bothkennar test site in Scotland, where a soft clayey layer rests on a firm clayey or silty layer
(Figure 2.55).
Figure 2.56 shows the lateral stress changes during the penetration of the 0.3 m diameter
poker (- - -) and the subsequent compaction of the column, which generally exhibited a
radius of about 0.35 m. The continuous lines show the change in stress during the
compaction of the inclusion. The results obtained with the earth pressure cell G4 (installed at
a depth of 2.5 m from the surface at a distance of 0.9 m from the centre of the column) are
represented by black squares ( ) and those obtained with the earth pressure cell G1
(installed at a depth of 2.5 m and a distance of 1.5 m from the centre of the column) are
illustrated by black circles ( ). Compaction was achieved by extraction and reinsertion of the
poker.
An increase of the stress level could first be detected when the poker tip reached the depth
of the cells and the effect diminished with increasing distance from the column axis. A
significant increase of the lateral stresses could be observed during the penetration as well
as retraction and re-insertion of the poker during the compaction phases. The increases in
lateral earth pressures disappeared immediately after the extraction of the poker from the
2 State of the art of ground improvement with stone columns
71
host soil. This would indicate that the installation of stone columns does not change the
stress state of the host soil, although it causes an increase of the undrained shear strength
(e.g. Aboshi et al., 1991; and Asaoka et al., 1994) and of the density of the soft soil bed
around the inclusions.
The results of dynamic probing using a Standard Penetration Test (SPT) at increasing
distance from the column axis shown in Figure 2.57 indicate more significant compaction
close to the edge of the column than 0.6 m away from the centre of the inclusion. The
highest compaction is reached in the granular fill, but an increase in blow counts can also be
noted in the so-called “cohesive” fill.
Figure 2.55: Profile of the host soil treated by SCP installation at the Bothkennar test site,
as investigated by Watts et al. (2000).
2.10 Observations concerning the installation effects of piles and stone columns on the soil
72
Figure 2.56: Lateral stress changes measured by earth pressure cells following poker
penetration and retraction during stone column compaction at the Bothkennar
test site (Watts et al., 2000).
Figure 2.57: Dynamic probing of the radial densification of the fill around a stone column at
the Bothkennar test site (Watts et al., 2000).
0 10 20 30 40 50 60
Increase in horizontal earth pressure at depth of cell: kN/m2
0
4.0
De
pth
ofp
oke
rtip
belo
wo
rig
ina
l g
rou
nd
leve
ld
urin
gtr
ea
tmen
t: m
0.5
1.0
1.5
2.0
2.5
3.0
3.5De
pth
ofp
oke
rtip
belo
wo
rig
ina
l g
rou
nd
leve
ld
urin
gtr
ea
tmen
t: m
Depth of cells
Cells G4 G1
0.9 1.5Distance from
column centre: m
Change in stress:
poker penetration
Change in stress:
column conpaction
2 State of the art of ground improvement with stone columns
73
Handy & White (2006) summarised field tests featuring the construction of rammed
aggregate piers at different sites. A site was located at Memphis, USA, where the subsoil
was over-consolidated low-plasticity clay featuring an effective angle of friction of 25° and a
unit weight of 19.6 kN/m3. The results obtained in Memphis, USA show different stress zones
appearing around the pier during the installation. A plastic zone forms in the vicinity of the
pier, while the subsoil further away from the inclusion remains in an elastic state. The
appearance of a passive zone, leading to the creation of radial cracks near the surface was
also noted (Figure 2.58).
Figure 2.58: Illustration of the different stress zones around the pier (rf = 1.9 m) in the
Memphis, USA case history (Handy et al., 2002).
Egan et al. (2009) conducted instrumented field tests in order to assess the response of pore
water pressure to loading (Figure 2.59). Three rows of five columns with dimensions of
550 mm diameter and 5.5 m length were therefore installed in normal-consolidated Carse
clay in Raploch, Scotland. Oedometer tests conducted on Carse clay indicated that the
vertical coefficient of consolidation cv was comprised between 0.6 and 1.7 m2 / year while the
horizontal coefficient of consolidation ch was determined in-situ to have values ranging from
1.3 to 7.0 m2 / year. The peak, denoted as Column construction in Figure 2.59, actually
corresponds to the behaviour just after completion of the installation of the inclusion, as the
pore pressure transducers (PPT) were removed during the construction phase, in order to
prevent any damage. Interestingly, the installation depth of the transducers does not seem to
play a major role in terms of the measured excess pore water pressures. It can, however, be
argued that the variation in installation depth of the PPT is minute as it varies from 2 m to
4 m underneath the ground surface, which does not allow for an analysis to be conducted of
the influence of this factor on the response. However, a good correlation between the
formulation of the excess pore pressures generated by the installation of a pile formulated by
Randolph et al. (1979, Equation 2.87) and the recorded values could be observed. Although
2.10 Observations concerning the installation effects of piles and stone columns on the soil
74
these results seem very interesting, the authors do not indicate the load applied on the
footing, which limits the interpretation of the results to a qualitative domain.
Figure 2.59: Response of pore pressure transducers installed 2 m, resp. 4 m, below the
ground surface to column loading at the Raploch test site (Egan et al., 2009).
2.10.2.2 Smear and compaction zones: effect on permeability
The installation of stone columns, or of vertical drains, not only influences the stress level but
causes a thin disturbed zone around the inclusion, which is dependent on the host soil
characteristics, and is usually described as smear zone (e.g. Onoue et al.,1991; Indraratna &
Redana, 1998; Sharma & Xiao, 2000; Bergado et al., 1991 and Shin et al., 2009). This zone
exhibits a reduced permeability and thus reduces the drainage performance of the inclusion.
Onoue et al. (1991) conducted small-scale loading tests on sand drains installed in Boston
Blue Clay, while recording the pore water pressures 10 mm under the surface of the clay
specimen. Based on their observations, they suggested dividing the soil surrounding the
drains into three zones:
- zone I or undisturbed zone, beginning at a distance of 6.5 times the radius of the
drain (rw) from the drain axis;
- zone II where the installation of the inclusion causes a decrease of the void ratio
and, as a consequence, a decrease of the permeability;
- zone III or remoulded zone where an additional decrease of the horizontal
coefficient of permeability kh is anticipated.
Figure 2.60 shows the evolution of the normalised horizontal coefficient of permeability (kho
denotes the undisturbed permeability) with the radial distance from the drain axis.
Date
Pre
ssur
e[k
Pa]
2 State of the art of ground improvement with stone columns
75
Figure 2.60: Suggested variation of horizontal permeability with radius according to Onoue et
al. (1991) (after Saye, 2001).
Indraratna & Redana (1998) and Indraratna et al. (2001) present the results of small-scale
tests modelling the installation of SCP in remoulded clay. Indraratna & Redana (1998)
evaluated the extent of the smear zone by determining the compressibility and permeability
parameters at different distances from the axis of the SCP. The main conclusions drawn from
these investigations are that the installation effect of the SCP on the soil structure is greatest
near the boundary of the SCP, while the radius of the smear zone (denoted as rs in Figure
2.61) can be taken to be equal to 100 mm or 4 to 5 times the radius of the column
(respectively equal in this case to 25 mm and denoted as rw in Figure 2.61).
Figure 2.61: Section of the test setup showing the smear zone (after Indraratna & Redana,
1998).
SandSand
SCP Sand Sand
Impermeable
l = 950 mm
Sand
D = 450 mm
Sand
Sand
Sa
nd
Sand
Sand
Smear zone
Sand
SandSa
nd
Sand Sand
rwrs
Remoulded
clay
Smear zone
Rigid
boundary
2.10 Observations concerning the installation effects of piles and stone columns on the soil
76
It could also be observed that the horizontal coefficient of permeability k’h in the smear zone
decreased in vicinity of the SCP, but that the vertical permeability k’v remained almost
identical to the original values in the host soil, even at the column interface (Figure 2.62).
However, this approach assumes that the smear zone remains homogeneous, which may
lead to some less accurate results than if a difference is made between smear zone (or
remoulded zone, Figure 2.60) and compaction zone (or disturbed zone, Figure 2.60).
Figure 2.62: Ratio of horizontal to vertical coefficient of permeability against the radial
distance from the axis of the SCP (denoted as drain)
(Indraratna & Redana, 1998).
Sharma & Xiao (2000) used a large-scale laboratory apparatus to install vertical sand drains
in kaolin samples with different pre-consolidation pressures. They measured the pore water
pressures at different distances from the drain axis during installation, using 6.4 mm diameter
miniature pore pressure transducers. The experimental setup allowed for an installation with,
and without, smear zone to be conducted. The mandrel consisted of an open-ended 54 mm
diameter outer tube with a thickness of 2 mm and of a 50 mm diameter inner tube with a
closed bottom end. In the first case, both tubes were fixed together and pushed into the clay,
thus reproducing the common installation process and causing a smear zone. In the second
case, only the outer tube was pushed into the host soil and subsequently the clay stuck in
the tube was removed carefully with an auger, so that the installation effects are limited in
such a way that they can be neglected in the analysis.
A comparative study of the response of the soil with and without smear zone is illustrated in
Figure 2.63. t0 corresponds to the start of the insertion of the installation mandrel, t1 to the
time when the tip of the mandrel reaches the depth of the transducers and t2 denotes the
time when the mandrel reaches the full penetration depth. The generated excess pore water
2 State of the art of ground improvement with stone columns
77
pressures are significantly higher in the case with smear, which is consistent with the
expected reduction of horizontal permeability in the smear zone.
Figure 2.63: Excess pore water pressures during the insertion of the installation mandrel
(Sharma & Xiao, 2000).
Figure 2.64: Variation of the horizontal permeability with radial distance to the drain for an
installation that causes a smear zone (Sharma & Xiao, 2000).
2.10 Observations concerning the installation effects of piles and stone columns on the soil
78
The horizontal permeability was determined from oedometer tests conducted on samples
extracted from the model after the installation of the sand drain (Figure 2.64). The results of
these investigations indicate that the coefficient of horizontal permeability decreased by a
factor of 1.3 in the smear zone compared to the intact zone. It was also observed that the
effect of the reconsolidation due the insertion of the mandrel was of higher significance than
the remoulding of the host soil. A comparable subdivision is proposed, as suggested by
Onoue et al. (1991), with a remoulded zone next to the drain surrounded by a reconsolidation
zone. The extents differ, as the remoulded zone is limited to a small extent in the immediate
vicinity to the drain, while the ratio of the radius of the reconsolidated zone rs to the radius of
the drain rw is set equal to s = rs / rw = 4 [-].
Bergado et al. (1991) conducted field tests as well as laboratory investigations, in order
to assess the extent and properties of the smear zone around vertical drains. Assuming that
the diameter of the smear zone (ds) is twice the diameter of the mandrel (dm),
Bergado et al. (1991) detected an influence of the size of the mandrel on the zone of
disturbance in field tests. The back-calculated value of the horizontal coefficient of
consolidation ch, by means of oedometer tests, is smaller for a large mandrel than for a small
mandrel (Figure 2.65), and the rate of increase of kh / kh’ was greater as a function of ch for
the large mandrel.
The laboratory experiments conducted could also confirm the suggestion made by
Hansbo (1987) that the horizontal coefficient of permeability within the smear zone k’h can be
considered equal to the vertical coefficient of permeability in the undisturbed zone kv. In
addition, the coefficient of horizontal permeability of the smear zone was found to be 1.75
times smaller (on average) than the coefficient of horizontal permeability in the undisturbed
zone.
2 State of the art of ground improvement with stone columns
79
Figure 2.65: Back-calculated sets of coefficients of relative horizontal permeability in the
undisturbed host soil (kh) and in the smear zone (k’h) and horizontal coefficient of
consolidation ch values, assuming ds = 2 dm (Bergado et al., 1991).
Chai & Miura (1999) also report that laboratory tests generally tend to underestimate the
values of the hydraulic conductivity of the natural deposits, as a consequence of the sample
disturbance and sample size effects. They therefore suggest a correction factor Cf:
(
)
2.78
with kh coefficient of horizontal permeability of the undisturbed host soil
k’h coefficient of horizontal permeability in the smear zone
Cf conversion factor between coefficients of permeability obtained in the
laboratory and in the field
l laboratory
Shin et al. (2009) conducted small-scale tests under 1 g using a micro-cone penetrometer
(denoted as MCP in Figure 2.66) with a diameter of 5 mm and an electrical resistance probe
(denoted as ERP in Figure 2.66) to identify the extent of the smear zone around rectangular-
shaped drains. Figure 2.66 shows the experimental setup used. The soil used was Busan
clay, the properties of which are given in Table 2.6. The specimen has a diameter of 700 mm
and a height of 1000 mm, while the mandrel has dimensions of 50 x 25 mm.
2.10 Observations concerning the installation effects of piles and stone columns on the soil
80
Table 2.6: Geotechnical properties of Busan clay (Shin et al., 2009).
Soil properties Values
Water content [%] 56
Liquid limit [%] 46.4
Plastic limit [%] 24.1
Plasticity index [%] 22.3
Specific density [g / cm3] 2.64
The electrical resistance probe consists of a needle with an outer diameter of 2.108 mm and
an inner diameter of 0.254 mm, in which a cable is installed and glued. The electrical
resistance R of the soil is measured at the tip of the tool, and the electrical resistivity can be
obtained as:
2.79
with α electrode shape factor, determined through calibration
ρ electrical resistivity
Figure 2.66: Directions of the horizontal penetration tests (Shin et al., 2009).
2 State of the art of ground improvement with stone columns
81
The smear zone was assumed to have been reached once the resistivity deviated from the
plateau reached within the undisturbed zone. The point of deviance is marked by the arrow
SR in Figure 2.67. Due to the rectangular form of the drain, the smear zone has an elliptical
form (Figure 2.68), the extent of which is about 4 times the equivalent mandrel radius in the
longer axis of the mandrel and 3.3 times the equivalent mandrel radius in the shorter axis of
the mandrel.
Figure 2.67: Electrical resistivity and estimated outer boundary of the smear zone (Shin et
al., 2009).
Figure 2.68: Dimensions of the smear zone derived from the electrical resistance probe. All
dimensions in millimetres (Shin et al., 2009).
aa
2.10 Observations concerning the installation effects of piles and stone columns on the soil
82
Weber et al. (2010) present the results of micro-mechanical investigations using Mercury
Intrusion Porosimetry (MIP) that were conducted in order to determine the extent of the
compacted zone around stone columns, using samples extracted after centrifuge model
tests. The MIP technique was first presented by Winslow & Shapiro (1959) and proposes to
determine the pore diameter by measuring the intrusion of mercury into a sample under a
specific pressure. The diameter is inversely related to the insertion pressure by the equation
proposed by Washburn (1921):
2.80
with σ surface tension
θ wetting angle for mercury, assumed to be equal to 130° for clay minerals at
room temperature (Diamond, 1970)
p mercury pressure
The measurements show an increase of the dry bulk density of the host soil as well as a
decrease of the porosity up to a distance of about three times the radius of the column from
the column axis (Figure 2.69 and Figure 2.70). A similar distribution of the porosity to that
suggested by Onoue et al. (1991) can be observed, which indicates that the assumption
made by Indraratna & Redana (1998) that the smear zone remains homogeneous may be
simplistic. However, the radial extent of the installation effects is less important in this case
than measured by Onoue et al. (1991) in small-scale laboratory experiments. This would
tend to indicate that this extent depends on the stress, which would speak for the physical
modelling under enhanced gravity, as it can reproduce the in-situ stress states, which small-
scale laboratory experiments cannot. However, the fact that Weber et al. (2010) do not
indicate the depth from which the samples used for the MIP investigations have been
extracted is a problem here, as it could be argued that the different radial extents measured
are due to different measurement depths.
2 State of the art of ground improvement with stone columns
83
Figure 2.69: Variation of the porosity as a function of the distance from the stone column axis
(Weber et al., 2010).
Figure 2.70: Variation of the dry bulk density as a function of the distance from the stone
column axis (Weber et al., 2010).
Juneja et al. (2013) make a distinction between a smear zone, where the clay is disturbed
and remoulded, and a compression zone, where the clay is laterally compressed due to
the installation of the inclusion (Figure 2.71), in a similar manner to Onoue et al. (1991),
Weber (2008) and Weber et al. (2010). The experimental process consisted of isotropic
consolidated undrained triaxial shear (CIU) tests conducted on clay samples in which sand
2.10 Observations concerning the installation effects of piles and stone columns on the soil
84
compaction piles (SCP) were installed. The surface of the cylindrical casing used to install
the SCP was varied from smooth (in order to limit the appearance of the smear zone to a
minimum) to gritty (in order to cause the appearance of a smear zone). Figure 2.72 shows
that a significant microstructural remoulding of clay occurs within the smear zone,
thus leading to a reduction of the pores and of the permeability. However, as in
Weber et al. (2010), no indication of the depth at which these measurements were conducted
is given, thus raising some questions about the effect of depth onto the development of
smear around stone columns.
Figure 2.71: Compression and smear zone around sand compaction piles (Juneja et al.,
2013).
(a) (b)
Figure 2.72: Scanning Electron Microscopy images of kaolin clay specimen adjacent to the
stone column installed and sheared (CIU) at 50 kPa (a) without smear and
(b) with smear (Juneja et al., 2013).
2 State of the art of ground improvement with stone columns
85
2.10.3 Radial drainage around stone columns
The analysis of the consolidation of a grid is usually conducted based on the drainage
conditions of a unit cell (Figure 2.73). The radius R of such a unit cell can be determined
based on the grid arrangement, as shown in Figure 2.21.
Figure 2.73: Radial drainage within a unit cell (after Barron, 1948).
The theory of radial drainage, considering the influence of a disturbed zone, was first
developed by Barron (1948), with the following assumptions:
- all vertical loads are initially carried by excess pore water pressure,
- all displacements within the soil mass occur in a vertical direction,
- a triangular pattern of drains is most economical,
- the zone of influence of a drain is circular,
- the load is uniformly distributed.
In case of equal vertical strain and of a radial flow to a central drain, the excess pore water
pressure due to radial flow can be formulated at a location r from the axis of the inclusion
as:
[ (
)
(
) ( )]
2.89
[
(
)
(
) ( )] 2.90
2.91
Störzone mit verminderterDurchlässigkeit
R
rw
rs
r
Drain
Disturbed zone with reduced
permeability
2.10 Observations concerning the installation effects of piles and stone columns on the soil
86
2.92
2.93
2.94
with Δur excess pore water pressure due to radial flow
r average excess pore water pressure due to radial flow
rs radius of the smear zone
r distance from centreline
kh coefficient of horizontal permeability in the undisturbed host soil
kh’ coefficient of horizontal permeability of the disturbed host soil
n radius ratio of the unit cell to the drain (Equation 2.93)
s radius ratio of the smear zone to the drain
Δu0 initial uniform excess pore water pressure
base of natural logarithms
Th dimensionless time factor for radial flow
R radius of the unit cell considered
rw radius of the drain
The time factor Th for a radial flow can be estimated at a certain time t as:
2.95
with ch horizontal coefficient of consolidation (Equation 2.69)
Alternatively, Th can also be determined based on the assumption of an average degree of
consolidation for a radial flow as:
(
) 2.96
Th in Equation 2.96 does not only depend on the average degree of consolidation but also on
the parameter (Equation 2.90), that is on the geometrical dimensions of the unit cell and on
the extent of the remoulded zone. Further research has been conducted based on the work
of Barron’s (1948), e.g. Hansbo et al. (1981), Han & Ye (2001) and Han & Ye (2002).
Hansbo et al. (1981) reformulated the factor (Equation 2.96), in order to take the influence
of depth on the radial flow, as:
(
)
( )
2.97
2 State of the art of ground improvement with stone columns
87
with z depth of soil
l half-length of drain
qw discharge capacity of the drain
The analytical solutions proposed by Barron (1948) and Hansbo et al. (1981) have to be
used with care when applied to stone columns, as they have been developed for vertical
drains and neglect the difference between the stiffnesses of the stone column and of the host
soil, as pointed out e.g. by Han & Ye (2002). Moreover, the radius of influence of stone
columns is usually smaller than the radius of influence of vertical drains. Han & Ye (2001)
suggest that a modified time factor is adopted for radial flow Thm (Equation 2.99), based on a
modified horizontal coefficient of consolidation chm (Equation 2.100), in order to assess the
average degree of consolidation for a radial flow taking these two aspects into account:
2.98
2.99
( )
⁄
⁄
( )
⁄
2.100
with average degree of consolidation for a radial flow
Thm modified dimensionless time factor for radial flow
factor defined in Equation 2.90
chm modified horizontal coefficient of consolidation
de diameter of the unit cell considered
ME, sc confined stiffness modulus of the stone column material
ME, s confined stiffness modulus of the undisturbed host soil
as replacement ratio
m stress concentration ratio
γw unit weight of water
Han & Ye (2002) propose to take the influence of the characteristics of composite
foundations reinforced by stone columns into account by formulating the average degree of
consolidation for a radial flow :
2.101
2.102
2.11 Summary of the state of the art of ground improvement with stone columns
88
(
)
2.103
( (
)
)
(
) (
)
(
)
2.104
with average degree of consolidation for a radial flow
t consolidation time
chm modified horizontal coefficient of consolidation
de diameter of the unit cell considered
ks coefficient of permeability of the undisturbed host soil
ks’ coefficient of permeability of the soil in the smear zone
H height of the unit cell considered
dsc diameter of the stone column
n radius ratio of the unit cell to the drain (Equation 2.93)
s radius ratio of the smear zone to the drain (Equation 2.94)
The solutions proposed by Barron (1948), Hansbo et al. (1981) and Han & Ye (2001, 2002)
all assume an instantaneous and uniform loading of the unit cell.
Wang (2009) formulates a solution using an expression for the degree of consolidation, and
thus taking the time-dependency of the loading into account, based on similar assumptions:
( )
∫ ( ( ) )
∫
2.105
with U average degree of consolidation
average final settlement
s(t) average settlement at time t
q(t) average applied loading at time t
average pore pressure throughout the soil-stone column cylinder
q0 ultimate loading
2.11 Summary of the state of the art of ground improvement with
stone columns
Ground improvement with stone columns has experienced massive development over the
past decades, which has largely been led by industry innovation and enabled by the
development of progressively computerised machinery. Stone columns increase the stiffness
and strength of the host soil and decrease the consolidation time by taking advantage of the
natural anisotropy of permeability in the ground and reducing the (radial) drainage paths. The
increase in stiffness and strength allows higher loads to be carried, with lower post-
construction settlements.
2 State of the art of ground improvement with stone columns
89
The bearing behaviour of stone columns is based on complex interactions between host soil,
inclusions and supported structure. These interactions are governed by the difference of the
characteristics of the host soil and of the stone column material and are further influenced by
installation effects.
The installation of stone columns changes the structure of the host soil, which influences
both the bearing behaviour and the drainage performance of the columns themselves. In
soils with low sensitivity, the insertion of the installation mandrel causes radial compaction of
the host soil around the column, through a cyclic increase in the horizontal stresses.
However, installation effects, mostly described as smear zone in the literature, also appear
around the inclusions, which have a negative influence on the reduction of the consolidation
time as they cause a reduction of the permeability. Some researchers (e.g. Indraratna &
Redana, 1998) assume that the installation of stone columns cause a so-called “smear
zone”, which they assume to be homogeneous over its whole extent. Others (e.g.
Onoue et al., 1991; Weber et al., 2010) however show that the permeability and the porosity
are not constant over the whole radial extent of the installation effects and suggest making a
distinction between smear and compaction zones. The vertical distribution of the installation
effects remains unknown, and is the main theme of this research.
In addition, geometrical aspects such as the spacing of the columns and their diameter play
a key role in the bearing behaviour of stone columns, which differs from that of rigid
inclusions such as piles. Inner deformation is necessary to mobilise the bearing mechanisms
of stone columns. Applying a load usually causes radial lateral spreading bulging of the
column in its upper part. The surrounding host soil provides the necessary lateral support to
the column material.
The vertical loading of a composite foundation provokes a re-distribution of the stress under
the foundation as stone columns exhibit a significantly higher stiffness than the host soil. The
ratio of the observed stresses at the top of stone columns to the stresses measured at the
surface of the host soil (stress concentration ratio m) usually ranges from 2 to 6 but can
reach peaks as high as 25. The evolution of this ratio shows different trends with increasing
loading of the composite foundation. A direct comparison is difficult, as the values of this ratio
depend on the type of loading (flexible / stiff), as well as on the stress history and
characteristics of the host soil. Deeper knowledge of the installation effects might help to
clarify the reasons for the differences in stress concentration factor m.
2.11 Summary of the state of the art of ground improvement with stone columns
90
3 Centrifuge modelling
91
3 Centrifuge modelling
3.1 Historical background
Craig (2002) gives an historical overview of centrifuge modelling over the last 150 years by
presenting seven scientists and engineers considered to have or have had an influence on
the development of this technique.
The first person known to have proposed physical modelling under enhanced gravity is
Edouard Phillips. He was born in 1821 and published on this topic from 1845 until his death
in 1889. Phillips used the rotation and related acceleration to conduct model tests on the
understanding of various forms of failure of railway bridges or parts thereof.
Bucky (1931) made use of centrifuge modelling in order to investigate mine roof stability at
Columbia University (USA). He was also reported to have been engaged in military
applications.
In the former USSR, two people (Davidenkov, 1933; Pokrovsky, 1933) seem to have worked
independently on the development of physical modelling using centrifugal acceleration. The
use of this modelling technique was restricted mainly to military purposes, and was aimed at
simulating the effect of explosives on various forms of buried infrastructure.
Karl Terzaghi is widely acknowledged as being the father of soil mechanics. Although he was
not keen on using physical models, he was aware of the activity of Bucky, having been
offered a position at Columbia University in the early 1930s and he corresponded with Peter
Rowe in the 1950s.
Peter Rowe conducted the first centrifuge studies using large models in the 1970s, which
was contemporary to the development of the oil and gas offshore industry in the UK. He
demonstrated the advantages of using physical modelling under enhanced gravity through
programmes aiming at simulating the effect of cyclic loading due to waves and wind on
offshore platforms and piles and jack-up structures on a range of natural and laboratory soils.
Andrew Schofield covered a wide range of subjects in his research, but most of interest here
are the summary and extension of scaling laws (Schofield, 1980), without which the correct
conduction and interpretation of centrifuge modelling is not possible. He expended a
considerable effort in the context of modelling soft ground behaviour, thus developing the
Critical State Soil Mechanics framework (Schofield & Wroth, 1968).
Eventually, Sarah Springman, former student of Andrew Schofield’s in Cambridge, can be
considered as one of the leading current researchers in the field of centrifuge modelling.
After bringing centrifuge technology to the ETH Zürich (Springman et al., 2001), she has lead
a number of studies aimed at resolving problems investigated elsewhere by means of small-
scale physical or numerical models, such as natural hazards (Chikatamarla, 2005; Imre,
2010), ground improvement in soft soils (Weber, 2008) or soil-structure interaction (Nater,
2005; Arnold, 2011).
3.2 Principles of centrifuge modelling
92
3.2 Principles of centrifuge modelling
A main challenge of physical modelling is the reproduction of the stress state present in a soil
mass. Due to the range of dimensions in civil engineering, full-scale tests are not often
conducted and small-scale tests under 1 g cannot reproduce the stress fields active in reality
(Figure 3.3). The use of centrifuge modelling allows the stress states acting in soil mass in
reality to be reproduced while using small-scale models, by taking advantage of the
acceleration acting on a rotating body. A body rotating around an axis is submitted to a radial
(centripetal) and tangential acceleration, as shown in Figure 3.1.
Figure 3.1: Acceleration acting on a body rotating with angular velocity ω (Springman, 2004).
Centrifuge modelling takes advantage of the action of the centripetal acceleration a, which
depends on the angular velocity ω and on the radius r. The centripetal acceleration can be
formulated as:
3.1
Equation 3.1 can, in case of a constant radius, be reduced to
3.2
The centripetal acceleration causes a field of increased gravity to act on the rotating body,
which allows the scaling factors to be derived:
3.3
with n factor of increase of the Earth’s gravity
g Earth’s gravity
3 Centrifuge modelling
93
Figure 3.2: Principle of centrifuge modelling (after Schofield, 1980).
Due to the field of increased gravity generated by the rotational movement, a centrifuge
model can be subjected to the same stress state and distribution as a prototype model
(Figure 3.3), although its dimensions are divided by the factor of increase of the Earth’s
gravity n (Laue, 1996, 2002).
Figure 3.3: Comparison of the stress profiles (a) in a prototype, (b) in a small-scale model
and (c) in a centrifuge model (after Laue, 1996).
The dimensions of the model built under 1 g can be reduced by the factor n, as they are then
exposed to an acceleration increased by the same factor under n times the Earth’s gravity.
The gradient of the increase of stress with depth is significantly higher in the centrifuge than
in reality, which can play an important role when modelling shallow boundary value
problems. It also should be noted that the increase in acceleration field is not linear (Taylor,
1995).
PrototypeSmall-scale model
Small-scale model
CentrifugeCentrifuge model
n – Factor of increase of Earth’s
gravity
σ’v – Effective vertical stress
γ – Specific unit weight
zm – Depth in model
zp – Depth in prototype
(a)
(b)
(c)
3.2 Principles of centrifuge modelling
94
3.2.1 Scaling factors
The lengths of a model built under 1 g are scaled with a factor n when submitted to n times
the Earth’s gravity. The scaling factors are derived from these and from the stress
equivalence between model and prototype. A summary of the main scaling factors is given in
Table 3.1.
Table 3.1: Summary of the main scaling factors (after Schofield, 1980).
Parameter Scale
(model / prototype)
Acceleration [m/s2] n
Linear dimension [m] 1/n
Stress [kPa] 1
Strain [-] 1
Unit weight [N/m3] n
Force [N] 1/n2
Time (diffusion) [s] n2
3.2.2 Advantages and disadvantages of physical modelling under enhanced
gravity
Mayne et al. (2009) give an overview of the advantages and limitations of physical modelling
under enhanced gravity. The main advantages are listed here:
- the stress levels present in reality can be reproduced using models with smaller
dimensions,
- key mechanisms of the behaviour of soil may be revealed,
- the model build-up and the loading systems allow for knowledge of the
characteristics of the subsoil and for the possibility to validate predictions,
- the testing time is reduced significantly (see Table 3.1), which is of particular
interest when modelling diffusion processes in low permeability soils,
- the costs are relatively low, especially when compared to full-scale tests,
- the user is able to witness the deformation and failure mechanisms while these
are taking place.
3 Centrifuge modelling
95
Some issues however remain, which should not be forgotten when using centrifuge
modelling for the investigation of static boundary value problems (dynamic situations are not
considered here):
- the factor of increase of the Earth’s gravity n is not constant with depth, so that the
vertical stresses vary with the depth of the sample (Figure 3.4). The reference
radius, at which the nominal acceleration acts, is usually set at two thirds of the
depth of the sample. The vertical stresses in the centrifuge are underestimated
above the reference radius and overestimated underneath the reference radius
(see Figure 3.4),
- the use of reconstituted soils is more appropriate for this testing procedure. The
use of natural soils is possible, however, it does not make much sense as the
features are scaled up by a factor n and the stress history is inconsistent. This
limits the possibility to investigate the effect of fabric on the behaviour of natural
clays,
- the influence of the Coriolis effect on falling particles in the centrifuge must be
accounted for planning the test and analysing the data,
- the dimensions of shear surfaces may not be scaled correctly,
- boundary effects might appear if sufficient care has not been taken when
determining the boundary conditions,
- the size of any instrumentation (sensors or embedded) may be excessive and
cause the outcome of the test to be affected,
- the stress history of the soil may not be equal to that encountered in in-situ
situations,
- the construction methods usually differ from those imposed in the field.
Figure 3.4: Distribution of the vertical stress with depth in a prototype situation and in the
centrifuge (zs denotes the depth of the sample) (after Taylor, 1995).
3.3 Centrifuge modelling of ground improvement measures
96
3.3 Centrifuge modelling of ground improvement measures
Numerous studies dealing with the modelling of ground improvement measures under
enhanced gravity, using drains and granular inclusions, can be found in the literature.
Sharma & Bolton (2001) present the results of centrifuge tests modelling embankments
constructed on soft clay (Figure 3.5 a) and show the positive influence of wick drains
installed in the host soil on the dissipation of excess pore water pressures (Figure 3.5 b), as
the time needed for 90 % of the excess pore water pressures to dissipate is reduced by
about 50 %.
(a) (b)
Figure 3.5: (a) Cross-section of the centrifuge model of a clay sample reinforced by wick
drains and basal reinforcement loaded by an embankment and (b) influence of
the drains on the dissipation of excess pore water pressures during and after
embankment construction (Sharma & Bolton, 2001).
Other studies focus on the self-weight consolidation of clay layers improved by vertical
inclusions. Kitazume et al. (1993) investigate the behaviour of soft clay improved with so-
called fabri-packed sand drains (sand drains coated with a plastic material) for the needs of a
land reclamation project in Haneda, Japan. Studies considering the influence of uncoated
stone and sand columns in soft soils are presented now.
Almeida et al. (1985) investigate the behaviour of embankments constructed on normally
consolidated kaolin clay, with and without reinforcement by stone columns. The installation of
stone columns in this case is again performed outside the centrifuge under 1 g by pouring
sand into bored holes. A decrease of the settlements by a factor 2 was observed while the
excess pore water pressures triggered by the embankment loading were significantly lower
with stone columns, than without.
Al-Khafaji & Craig (2000) studied the behaviour of a tank foundation on soft clay reinforced
by sand columns installed under 1 g by pouring and vibrating sand in pre-bored holes with
area replacement ratios varying from 10% to 40%. The container used was a rigid strongbox
3 Centrifuge modelling
97
560 mm square and 460 mm deep while the clay specimen had a thickness of 200 mm. The
clay was loaded by a 325 mm diameter tank with a flexible base. Figure 3.6 shows a
comparison of the settlement improvement ratios obtained by using the analytical solution
presented in Priebe (1995) with measurements from centrifuge tests. A replacement ratio
smaller than 25% does not seem to have a major influence on the settlement performance of
the composite foundation. This comparison suggests that the angle of friction of the
compacted columns (denoted as Φc in Figure 3.6) is only 30°. However, no direct
measurement of the sand density or angle of friction has been performed during these tests,
which makes any definitive conclusion difficult. The ratios of the stiffness of the stone column
(denoted as Ec in Figure 3.6) to the stiffness of the untreated host soil (denoted as Es in
Figure 3.6) seem to be realistic.
Figure 3.6: Comparison between settlement improvement ratios obtained with the solution of
Priebe (1995) solution and from centrifuge tests (Al-Khafaji & Craig, 2000).
Zwanenburg et al. (2002) conducted centrifuge tests of loading of sand piles and sand walls
installed in clay under 1 g by pouring dry sand into pre-drilled holes. The length of the
inclusions is varied, so that both floating and end-bearing columns are investigated. The
results of these tests show a strong influence of the length of the inclusions, as end-bearing
columns are much more effective than floating columns in achieving significant settlement
reduction (41.5 % opposed to 6.1 %). The outcomes of the centrifuge investigations are used
to back-calibrate a numerical model.
Numerous other references can be found in the literature. The behaviour of soft soils
improved by stone columns installed by pouring and compacting sand into pre-bored
holes under 1 g was investigated by e.g. Terashi et al. (1991), Stewart & Fahey (1993),
and Jung et al. (1998). Another method was adopted by Huat & Craig (1994),
Priebe (1995), φc = 30°, Ec/Es = 7
Priebe (1995), φc = 35°, Ec/Es = 7
Priebe (1995), φc = 40°, Ec/Es = 7
Centrifuge results, Ec/Es = 7
3.3 Centrifuge modelling of ground improvement measures
98
Kitazume et al. (1998), Rahman et al. (2000) and Lee et al. (2006), who installed granular
columns by inserting a frozen sand cylinder into the soft soil bed under 1 g, and
subsequently loaded the composite foundation under enhanced gravity.
The major issue in the research on centrifuge modelling of ground improvement mentioned
above is, however, the fact that the construction of the inclusions is conducted under the
Earth’s natural gravity field, which means that the stress states are actually at a prototype
scale. The consequences of the displacement induced by the insertion of the mandrel can
not to be modelled by installing a stone column under 1 g by drilling a hole in the soft soil bed
and filling it with coarse grained material due to the fact that the actual stress state present
in-situ is not modelled correctly. The influence of the installation technique of the inclusion on
the loading behaviour can be highlighted by consideration of a similar boundary value
problem, e.g. by Dyson & Randolph (1998), who compared the behaviour under lateral
loading of piles installed by different techniques: pre-installed, jacked at 1 g, jacked at 160 g
and driven at 160 g (Figure 3.7). The influence of the installation technique can be seen as
the installation in-flight leads to significantly stiffer behaviour under loading.
Figure 3.7: Pile lateral pressure as a function of the lateral displacement y normalised by the
pile radius d (Dyson & Randolph, 1998).
A major step forward to solve the issue of constructing stone columns in flight was made by
Ng et al. (1998), who presented a setup developed for the beam centrifuge at the National
University of Singapore (Figure 3.8). The installation tool is pushed into the soft soil while
sand is fed through an Archimedes’ screw driven by a hydraulic motor. The SCP is
constructed during the withdrawal of the tool and its diameter can be varied by adjusting the
speed at which the installer is withdrawn from the model. The transport through the
Archimedes’ screw tends to crush the SCP material, which might be a concern, as
uncontrolled modification of the physical properties of the inclusion occurs. The rapid delivery
Normalised lateral displacement y / d
0 0.05 0.1 0.15 0.2
0
0.1
0.2
0.3
0.4
Pile
la
tera
l p
ressu
rep
[M
Pa
]
Driven
Jacked at 160 g
Jacked at 1 g
Pre-installed
asd
asd
asd
asd
3 Centrifuge modelling
99
of sand in-flight through the screw results in the development of significant heat, which leads
to the necessity to immerse the clay bed under a thin layer of water prior to the column
installation, in order to prevent the clay model from losing moisture and cracking.
Figure 3.8: Sand compaction pile installation tool used at the National University of
Singapore. All dimensions are in mm (Ng et al., 1998).
Lee et al. (2001) investigate the influence of constructing stone columns in flight by
conducting centrifuge tests modelling the behaviour of an embankment constructed on a soft
clay bed. Unimproved ground (denoted as U2 in Figure 3.9 b) is compared with a host soil
improved by inserting frozen SCPs under 1 g (denoted as R1_20 in Figure 3.9 b) and by
installing SCPs in-flight using the installation process described in Ng et al. (1998) (denoted
as D50_20 in Figure 3.9 b). Several observations could be made:
- the heave of the surface was significantly more important in the case of SCPs
built in-flight than when installing frozen sand samples under 1 g,
- the negative consequences of the thawing process of the frozen SCPs on the
clayey host soil could be avoided by the in-flight installation,
- the SCPs installed in-flight behave in a stiffer manner (Figure 3.9 c). This was to
be expected due to the compaction taking place during the insertion of the
installation tool.
(1)(1)
(1) Hydraulic cylinder
(2) Hydraulic motor
(3) Storage hopper
(4) XY table
(5) Soil sample
(6) Strongbox
(2)(2)
(2)(3)
(2)(4)
(5)
(2)(6)
640
560
430
3.3 Centrifuge modelling of ground improvement measures
100
(a) (b)
Figure 3.9: Embankment constructed on soft clay (U2) and when improved by SCPs installed
at 1g (R1_20) or at 50 g (D50_20) (a) deformation grid lines in clay improved
with SCPs built in-flight (b) maximum lateral displacement (in mm) of the grid
line L2 with g-level (Lee et al., 2001).
Again, using the in-flight installation setup presented in Ng et al. (1998), Lee et al. (2004)
conducted centrifuge tests to model the installation of SCPs in clay while recording pore
pressures and total stresses by means of pore pressure transducers and total stress
transducers, respectively (Figure 3.12). The goal was to investigate the impact of the
installation of SCPs on the host soil in terms of increase of pore water pressure and stresses.
Figure 3.10 and Figure 3.11 show the layout of the experimental set-ups used.
Figure 3.10: Layout of SCPs and transducers for the installation of SCPs in test T7, D 20 mm
(D: SCP diameter) (Lee et al., 2004).
G-level
Ma
xim
um
la
tera
l d
isp
lace
me
nt
U2
R1_20
D50_20
40.0
20.0
0.0
0 50 100 150
3 Centrifuge modelling
101
(a) (b) ©
(d) (e) (f)
Figure 3.11: Layout of SCPs and transducers for tests: (a) T1, D 18 mm; (b) T2, D 20 mm;
(c) T3, D 16 mm; (d) T4, D 17 mm; (e) T5, D 20 mm; (f) T6, D 20 mm (D SCP
diameter) (Lee et al., 2004).
3.3 Centrifuge modelling of ground improvement measures
102
The strongest reaction of the subsoil at a given depth occurs when the tip of the casing
reaches that depth and only very little excess pore pressure dissipation occurs during the
withdrawal of the mandrel (Figure 3.12). This is consistent with the low coefficient of
consolidation of 1 m2/year measured in the host soil used. It can also be seen that, other
than reported in Watts et al. (2000) from field tests (Figure 2.56), the total stress level does
not return to its original state immediately upon extraction of the poker. This might also be
due to the low coefficient of consolidation of the clay sample.
(a)
(b)
Figure 3.12: (a) Total horizontal stress at 60 mm depth and (b) pore pressures at 80 mm
depth during SCP installation in clay. Line 1: time at which the casing tip
reaches the depth of the transducers. Line 2: time at which the casing tip
reaches the full penetration and withdrawal starts. Line 3: time at which the
casing tip reaches the depth of the transducers during withdrawal. Line 4: end of
the SCP installation (Lee et al., 2004).
Figure 3.13 shows ratios of measured (Δσ, respectively Δu) to calculated ( , respectively
) total stresses and pore pressures plotted as function of the depth of the transducers dt
(Figure 3.13 a) and of the radial distance rt of the transducers from the axis of the SCP
(Figure 3.13 b), each normalised with the SCP diameter D. The calculated stresses and pore
3 Centrifuge modelling
103
pressures were computed here with a modified formulation of the cavity expansion theory
according to Vesic (1972).
Figure 3.13 (a) shows that the ratios ⁄ and ⁄ increase with increasing depth of the
transducers, although a significant scatter of the measurements is noted. This is not
surprising as plane-strain conditions in cavity expansion problems are approached when
depth increases. The measurements presented also support the suggestion formulated by
Randolph & Wroth (1979) that plane-strain conditions are reached for a ratio ⁄ equal to 5.
Figure 3.13 (b) shows that the ratios ⁄ and ⁄ increase with decreasing distance to
the column axis. Finally, the ratios ⁄ and ⁄ rise towards a steady value of 1 when
the ratio dt / rt increases (Figure 3.14).
(a) (b)
Figure 3.13: Ratios of measured to calculated horizontal stresses and pore pressures plotted
against (a) dt / D and (b) rt / D (Lee et al., 2004).
Figure 3.14: Ratios of measured to calculated horizontal stresses and pore pressures plotted
against the ratio of the depth of the transducers dt to the radial distance of the
transducers rt (Lee et al., 2004).
dt / D rt / D
,
,
,
dt / rt
3.3 Centrifuge modelling of ground improvement measures
104
Yi et al. (2013) present the results of investigations conducted using the Archimedes’ screw
(Ng et al., 1998) in order to assess the effect of the construction of sand compaction piles on
the undrained shear strength of soft soils. The T-Bar (Yi et al., 2010) was used to investigate
groups of 2 and 4 columns (Figure 3.15 a and b), built either in rapid succession (denoted as
I2 in Figure 3.16) or in a time interval allowing for the excess pore water pressures generated
by the installation to dissipate before the installation of the consecutive inclusion (denoted as
I 45 in Figure 3.16).
(a) (b)
Figure 3.15: Layout of sand compaction piles (P1 to P4) and location of the T-Bar test
(denoted as s) for pile group tests featuring either a) 2 piles or b) 4 piles (all
dimensions in mm) (Yi et al., 2013).
It was observed that the installation of a second pile in quick succession (denoted as 2P-I2 in
Figure 3.16) did not add any strength to the soil compared to the situation in which a single
pile was installed. An increase in the measured shear strength was observed after the
installation of a column group, which is thought to be due to the inevitable dissipation of the
excess pore water pressures taking place during the time necessitated for the repositioning
of the SCP installation tool.
However, the dissipation of the excess pore water pressure between the installation of the
SCPs has a strong influence on the measured undrained shear strength (Figure 3.16). A
substantial increase (about 50 %) compared to the single pile situation, and to the situation in
which the SCPs were installed in quick succession could be observed both in the case
featuring 2 piles (denoted as 2P-I45 in Figure 3.16) and in the case featuring 4 piles (denoted
as 4P-I45 in Figure 3.16).
This shows that the strength increase triggered by sand piles is not cumulative if the piles are
installed in a quick succession. If sufficient time is allowed for the dissipation between the
different installation phases, the increases can be added, by considering the increased
undrained shear strength after each pile has been constructed.
3 Centrifuge modelling
105
Figure 3.16: Undrained shear strengths measured in the centrifuge for different tests
(Yi et al., 2013).
Although the installation technique developed by Ng et al. (1998) represents an extremely
valuable improvement in modelling in the centrifuge, compared to the installation of granular
inclusions under 1 g, the two drawbacks brought up earlier through the Archimedes’ screw:
particle crushing and heat creation might influence the results, especially due to the need to
submerge the model below water.
These two issues were solved by Weber (2004) by developing a stone column installation
tool for the ETH Zürich geotechnical drum centrifuge (Springman et al., 2001) in order to
model the bottom feed construction technique. This tool consists of an open-ended filling
tube attached to the working arm of the centrifuge and which is driven into the clay model
(Figure 3.17). A drawing pin is pushed into the surface of the clay bed in order to seal the tip
of the tube in order to prevent the filling tube from clogging, and remains in the soil model
after the installation of the granular inclusion (denoted as Lost tip in Figure 3.17). In the case
this has not been effective, the rotations of the drum and of the tool platform can be
decoupled, allowing the tool platform to be stopped in order to unclog the filling tube without
disturbing the stress history of the soil sample in the drum.
In situ
1 pile, post-installation
long-term strength
1 pile, post-installation,
long-term strength
2 piles, 2 min interval,
long-term strength
2 piles, 45 min interval,
long-term strength
4 piles, 2 min interval,
long-term strength
4 piles, 45 min interval,
long-term strength
1P-LT
2P-I2
2P-I45
4P-I2
4P-I45
Undrained shear strength su [kPa]
0 5 10 15 20 25
Depth
[m]
0
1
2
3
4
5
3.3 Centrifuge modelling of ground improvement measures
106
The granular column material is fed from outside the centrifuge through a flexible sand feed
pipe once the desired installation depth is reached, and the filling tube can either be
withdrawn in one extraction, thus creating an uncompacted stone column, or a compaction
regime can be applied by withdrawing the tube by a certain distance before pushing it into
the model again. The compaction process causes the stone column diameter to increase
from 10 mm (outer diameter of the installation tool) to about 12 mm.
Figure 3.17: Experimental setup for the in-flight installation of stone columns (Weber et al.,
2005).
Figure 3.18 presents a detailed view of the installation tool. The flexible sand feed pipe (3)
has an inner diameter of 28 mm, the transition unit (5) an inner diameter of 7.5 mm and the
filling tube an inner diameter of 8.4 mm.
This method solves the issue of grain crushing that was encountered by Ng et al. (1998),
however there are some limitations in terms of the range of grain sizes that can be used
without clogging the filling tube. Moreover, no heat is created by pushing a tube into the host
soil, which means that the height of the groundwater table can be varied to suit the
requirements of the test.
There are two main differences between the bottom feed construction technique used in situ
and the experimental setup proposed by Weber (2004). Compressed air is used in the field
to prevent the tip of the installation mandrel from clogging, as opposed to a lost tip in the
centrifuge. Also, no vibratory movements can be achieved in the centrifuge, as opposed to
the field situation.
Chair for Geotechnics
Construction of sand compaction pile in soft soils
Dr. Jan Laue Construction Processes in Geotechnical Engineering
Axis
r
z
w
q
Pore pressuretransducers
Flexible sandfeed pipe
r-z-working
arm
Tool platform
Filling tube
Soil model
Lost tip
Laser distancegauge
Transition
unit
3 Centrifuge modelling
107
Figure 3.18: Detailed view of the stone column installation tool developed by Weber (2004).
Using his setup, Weber (2008) conducted centrifuge tests to model the behaviour of
embankments built on soft clay, with and without stone columns. Improving the ground
reduced the settlements observed on top of the embankment (Figure 3.19), as well as
accelerating dissipation of the excess pore water pressures generated by the installation of
the embankment, by a factor of four, for an area replacement ratio as of 10 % (Figure 3.20).
Figure 3.19: Settlements measured with and without stone columns at the toe of the
embankment (1), and on top of the embankment (2) (after Weber 2008).
1
2
3
45
6
7
1 Working arm
2 Tool clamp
3 Flexible sand feed pipe
4 Inlet unit
5 Transition unit
6 Filling tube
7 Laser distance gauge
(1)(2)
(1)
(2)
0 200 400 600 800 1000 1200
Time [min]
8
7
6
5
4
3
2
1
0
-1
Sett
lem
ents
[m
m]
Time [min]unimproved
improved
3.3 Centrifuge modelling of ground improvement measures
108
Figure 3.20: Evolution of the pore water pressure after embankment construction in the
improved ground within the sand pile grid (---) in comparison with unimproved
ground (––) at three depths in the model with a groundwater table located at the
surface of the model in the middle of the container: P1 = 120 mm, P2 = 70 mm,
P3 = 25 mm equivalent to prototype depths of 6 m, 3.5 m and 1.25 m
respectively (after Weber, 2008).
Moreover, due to the effective reproduction of the construction process, Weber (2008) could
also investigate the installation effects of stone columns in soft soils in more detail at
micromechanical scale. The development of different zones around the granular inclusions
could be observed (Figure 3.21):
- zone 1, where the coarse grains and clay are mixed,
- zone 2, where a clear reorganisation of the clay platelets parallel to the column
axis can be seen,
- zone 3, where a reduction of the void ratio and of the porosity could be measured
(Figure 2.69), and
- zone 4, where no clear installation effects can be noted.
Figure 3.21: Installation effects around a stone column at a model depth of 40 mm @ 50 g
(Weber, 2008).
0 200 400 600 800 1000 1200 1400
P1
P2
P3
Time [min]
0
20
40
60
80
100
120
140
Pore
wate
rpre
ssure
[kP
a]
P1
P2
P3
3 Centrifuge modelling
109
3.4 Techniques adopted
3.4.1 ETH Zürich geotechnical drum centrifuge and equipment
The ETH Zürich geotechnical drum centrifuge (Springman et al., 2001) has a diameter of 2.2
m and can achieve a maximal acceleration of 440 g, while the maximal weight of the model
in the drum is 2 tons. The axis of rotation of the drum is vertical, which means that the
centripetal acceleration acts in a horizontal plane and that the models have to be mounted on
a vertical plane. Figure 3.22 shows a cross-section of the machine.
The equipment implemented comprised:
- the stone column installation tool: the in-flight stone column installation is
performed using the experimental setup described in Weber (2004), Weber et al.
(2005) and Weber (2008), as well as in Section 3.3,
- aluminium footings: a 28-mm radius circular stiff aluminium footing was used in
order to load a single stone column and a 56 x 56 mm square stiff aluminium plate
was used to load a stone column group.
Figure 3.22: Cross-section of the ETH Zürich geotechnical drum centrifuge (Springman et al.,
2001).
3.4 Techniques adopted
110
3.4.2 Pore pressure transducers (PPTs)
The pore water pressures are recorded in the centrifuge using DRUCK PDCR 81 (König et
al., 1994) transducers. These sensors measure the pressure difference between the
atmospheric air pressure and the fluid pressure acting through the porous stone on the
silicon diaphragm (Figure 3.23). Transducers are produced for different operational pressure
ranging from 75 mbar up to 35 bar, and can all measure positive and negative pore water
pressure. The sensors used here have a maximal operational pressure of 7 bar and can
measure suction up to 1 bar. The saturation of the ceramic filter fitted to the PPTs is
extremely important in order to obtain reliable measurements. This saturation is obtained
using a pressure chamber with de-aired water under varying cycles of vacuum and pressure.
The pressure chamber is also used to conduct the calibration (R2 ranging from 0.99992 to
0.99999).
The form of the PPTs causes a sensitivity of the measurements to the orientation of the filter
(Figure 3.23). Thus good care during the installation of the PPTs in the soil model is
therefore necessary in order to obtain reliable results.
Figure 3.23: Cross-section of the transducer DRUCK PDCR 81 (König et al., 1994).
3.4.3 Load cells
Load cells manufactured by the company Hottinger Baldwin Messtechnik GmbH
(Figure 3.24) were used in this research. The maximum capacity of the load cells was 2 kN,
respectively 10 kN. The producer predicts a linear deviation of the results of 0.011 % for a
2 kN load cell and of 0.061 % for a 10 kN load cell. The maximum loads recorded during the
footing loadings conducted during the centrifuge tests were of about 350 N, which
corresponds to an error of 0.0385 N for a 2 kN load cell and of 0.2135 N for a 10 kN load cell.
This error was considered irrelevant in the present research, and was discarded for the
interpretation of the results.
The calibration was conducted in the same manner as for the T-Bar penetrometer
(Section 3.4.4) and load cells were mounted on the arm supporting the aluminium footing
3 Centrifuge modelling
111
and on the stone column installation tool. This measured the footing load as well as the
vertical load applied during the construction and compaction of the stone column.
Figure 3.24: Load cell produced by Hottinger Baldwin Messtechnik GmbH (Arnold, 2011).
3.4.4 T-Bar penetrometer
A T-Bar penetrometer is used to measure the undrained shear strength of the soil. The
penetrometer used in the centrifuge at ETH Zürich (Figure 3.26) is based on the tool
presented in Stewart & Randolph (1991) and in Stewart & Randolph (1994).
Stewart and Randolph (1994) formulate the undrained shear strength as:
3.4
with P force per unit length acting on the T-Bar
d diameter of the T-Bar
Nb T-Bar factor
The T-Bar factor Nb is dependent on the surface roughness of the material. Its analytical
value (Stewart & Randolph, 1991) varies from 9.0 (smooth surface) to 12.0 (fully rough
surface). Adhesion factors of 0 and 1 are extremely unlikely to be reached, therefore
intermediate values of Nb are used and must be combined with the effect of smooth ends of
the circular bar, which is not infinitely long. Randolph & Houlsby (1984) suggest that a value
of 10.5 is implemented for general use. This value was used for the assessment of the
undrained shear strength with the T-bar penetrometer, thus neglecting the effects of the
smooth ends of the tool (Figure 3.25) and the influence of the penetrometer shaft on the flow
mechanism around the T-Bar.
Figure 3.25 shows the dimensions of the penetrometer used in this test series (Figure 3.26
and Figure 3.27). The force P (Equation 3.4) is measured using four strain gauges installed
on a round cylinder at the end of the penetrometer shaft (Figure 3.25).
The calibration is conducted using the setup shown in Figure 3.28 (a). The penetrometer is
inverted and clamped vertically, and loaded in compression through a saddle placed on the
3.4 Techniques adopted
112
bar, which is connected by a rigid string to single weights placed on a horizontal plate
hanging beneath the device. The values measured by the strain gauges are recorded (Figure
3.28 b).
(a) (b)
Figure 3.25: T-Bar penetrometer (a) front view and (b) side view.
Figure 3.26: T-Bar penetrometer (after Weber, 2008).
Figure 3.27: T-Bar penetrometer mounted on the working arm of the tool platform in the
centrifuge (after Weber, 2008).
Strain gauges Shaft
Shaft
Strain gauges
3 Centrifuge modelling
113
Figure 3.28: T-Bar calibration setup (after Weber, 2008).
3.4.5 Electrical impedance needle
This tool was presented in Gautray et al. (2014). An assessment of the density changes
within a soil mass can be achieved by means of measuring resistance. Such methods can be
divided in measurements of the thermal resistance of the soil (Shublaq, 1992) and
measurements of the electrical resistance (Cho et al., 2004; Shin et al., 2009; Dijkstra et al.,
2012).
As shown e.g. by Weber (2008), the installation of stone columns in soft clay compacts the
host soil around the inclusion (Figure 3.21). In order to assess the spatial distribution of the
zone affected, an electrical impedance needle was developed at ETH Zürich,
which measures the impedance (electrical resistivity) of the soft soil in-flight
(Gautray et al., 2014), under the assumption that an increased density of the host soil
increases the electrical impedance recorded. Such measurements enable an insight to be
obtained into the variations of density around granular inclusions. This represents a step
forward compared to 1 g tests, presented e.g. in Shin et al. (2009), due to the more accurate
reproduction of the stress field, which is particularly important in terms of the resistance to
radial compaction (Section 3.2).
The design of the electrical impedance needle used in this research was inspired by Cho et
al. (2004), who inserted such a tool into the host soil, while measuring the electrical resistivity
at the tip. Cho et al. (2004) used 3 needles with diameters of 2.108 mm, 1.270 mm and 2.159
mm respectively. However, due to the scaling of the dimensions in the centrifuge, the
diameter of the impedance needle had to be reduced to 1 mm, in order to keep the difference
between diameter of the impedance needle and medium grain size at prototype scale within
acceptable limits.
Vesic (1977) highlighted the problems encountered during the insertion of an inclusion with a
flat tip into the soil, namely the formation of a dense plug, which then moves with the tip.
However, practical considerations prevailed for the choice of the tip shape, which was set to
be flat (Figure 3.29).
Halterung
Waagschale mit Gewichten
T-Bar Penetrometer
Messwerterfassung
Support
Value recording Weighting scale
T-Bar Penetrometer
T-Bar Penetrometer
Weight
3.4 Techniques adopted
114
(a) (b)
Figure 3.29: Electrical impedance needle (a) side view and (b) tilted view of the tip (outer
diameter 1 mm).
Although a wedged tip shape seems more adequate at first glance, its implementation in the
centrifuge may not be straightforward. The main challenge is the construction of such a
wedged form for the small diameter (1 mm). Geometrical imperfections at model scale will
also be strongly amplified at prototype scale due to the scaling factors. This might drastically
reduce the quality of the results, mainly due to any uncontrollable changes of direction of
penetration, which could be caused by an imperfect wedge.
Figure 3.30 shows a schematic view of the needle, which is made of a 1 mm diameter
stainless steel tube (inner diameter 0.8 mm) in which an electrical cable was incorporated
and glued. The electrical impedance (Z) is measured at the tip of the cable. A cover had to
be developed to make sure that the wind induced by the rotational movement of the
centrifuge would not trigger vibrations of the needle, which might provoke a distortion of the
actual location of the measurement performed, or even breakage of the needle. Due to the
centripetal acceleration, this cover is maintained over the needle (Figure 3.30 a) until the soil
surface is reached and then remains at the surface while the needle is pushed into the host
soil (Figure 3.30 b).
The development of the electrical impedance needle was an iterative process. The use of the
tool and the implementation process are presented in Section 4.7. This tool was first inserted
with a velocity of 3 mm /s into a soil specimen during the centrifuge test JG_v1, where
clogging of the tip prevented accurate measurement of the impedance. A water container
was built into the model used for a subsequent centrifuge test in order to clean up the tip in-
flight, but this procedure was not successful.
The tip of the needle was cleaned most efficiently using an ultrasonic bath (model Emmi 4,
produced by EMAG AG, Figure 3.31), which was mounted in the centrifuge drum and
subsequently filled with water. Such a bath generates ultrasound waves in order to clean
delicate items. The needle was immersed in the basin of the ultrasonic bath in flight, which
enabled the clay obstructing the tip of the tool to be removed, as the ultrasonic waves
destroyed the bonds that had formed between the clay particles and the electrical impedance
needle. The results of the measurements conducted with the electrical impedance needle
can be found in Section 4.7 and Appendices 8.6 and 8.7.
3 Centrifuge modelling
115
(a)
(b)
(c)
Figure 3.30: Schematic views of the electrical impedance needle (a) covered, (b) with the
cover retracted and (c) cross-section A-A (Gautray et al., 2014).
Figure 3.31: Ultrasonic bath Emmi 4, produced by EMAG AG (Gautray et al., 2014).
200 mm
90 mm
165 mm
50 mm
3.5 Model soils
116
3.5 Model soils
3.5.1 Birmensdorf clay
Three and a half tons of natural clay were extracted in 1999 during the construction of the
Birmensdorf traffic interchange near Zürich, Switzerland. This material has been stored since
then at ETH Zürich as a laboratory material and was used in several research projects
(Trausch-Giudici, 2003; Nater, 2005; Messerklinger, 2006; Weber, 2008). Many researchers
have investigated a range of properties, based on intact and remoulded specimens
(Fauchère, 2000; Fleischer, 2000; Panduri, 2000; Züst, 2000; Basler, 2002; Küng, 2003;
Messerklinger et al., 2003; Plötze et al., 2003; Trausch-Giudici, 2003; Nater, 2005;
Messerklinger, 2006). A summary of the mineralogical composition of Birmensdorf clay can
be found in Plötze et al. (2003).
This clay was also used as a soft soil bed for the centrifuge tests conducted in this research.
Applications of remoulded clays as soil models, including the effect of stress history, have
been discussed in Springman (2014), with specific reference to remoulded Birmensdorf clay.
A summary of these investigations is given in Table 3.2.
The soil models for the centrifuge tests are prepared using a clay suspension featuring a
water content higher than 100 %, which is then consolidated under a hydraulically loaded
piston in an oedometer (Section 3.6). This enables, in addition to the preparation of the
sample, the determination of different soil mechanical properties, some of which are
presented in Table 3.3.
Table 3.2: Classification and selected mechanical properties of Birmensdorf clay (after Weber, 2008).
USCS classification CH
Clay particle content from
sedimentation analysis < 2μm [%] 42
Liquid limit wl [%] 58
Plastic limit wp [%] 19
Plasticity index Ip [%] 39
Critical state effective angle of friction
φ’cv [°] 24.5
Effective cohesion c’ [kPa] 0
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117
Table 3.3: Selected properties of Birmensdorf clay, determined from oedometer tests.
Specific density of the saturated soil ρg – normally
consolidated under σ’v = 100 kPa [g/cm3] 1.85
Specific density of the saturated soil ρg – normally
consolidated under σ’v = 200 kPa [g/cm3] 2.01
Void ratio e under σ’v = 100 kPa [-] 1.06
Void ratio e under σ’v = 200 kPa [-] 0.73
Vertical permeability k under σ’v = 100 kPa [m/s] 2.1 10-9
Vertical permeability k under σ’v = 200 kPa [m/s] 6.25 10-10
Vertical coefficient of consolidation cv for a load
increment from 50 kPa to 100 kPa [m2/s] 2.5 10-7
Confined stiffness modulus ME for a load increment
from 50 kPa to 100 kPa [kPa] 1410
Vertical coefficient of consolidation cv for a load
increment from 100 kPa to 200 kPa [m2/s] 1.5 10-7
Confined stiffness modulus ME for a load increment
from 100 kPa to 200 kPa [kPa] 2230
3.5.2 Quartz sand
The quartz sand used to construct the columns in this research is exactly the same as that
used by Weber (2008). Properties of this soil were investigated by Weber (2008) and are
presented in Table 3.4.
Table 3.4: Parameters for the quartz sand used as stone column material (Weber, 2008).
USCS classification SP
Grain shape Semi-angular slightly rounded
Coefficient of uniformity [-] 1.4
Coefficient of gradation [-] 1.0
Specific density ρs [g/cm3] 2.65
Maximum bulk density ρd,max [g/cm3] 1.62
Minimum bulk density ρd,min [g/cm3] 1.50
Critical state effective angle of friction φ’cv [°] 37
Compression index Cc [-] 0.088
Swelling index Cs [-] 0.007
Coefficient of permeability k [m/s] 2 10-3
3.6 Soil model
118
3.5.3 Perth sand
The same Perth sand as used by Weber (2008) was used as a drainage bed and filling
material depending on the experimental setup (Section 3.6). Selected properties of this
material are listed in Table 3.5.
Table 3.5: Selected properties of Perth Sand (Nater, 2005).
USCS classification SP
Grain size d10 [mm] 0.14
Coefficient of uniformity [-] 2.2
Coefficient of gradation [-] 1.0
Specific density ρs [g/cm3] 2.65
Maximum bulk density ρd,max [g/cm3] 1.60
Minimum bulk density ρd,min [g/cm3] 1.50
Critical state effective angle of friction φ’cv [°] 30
Void ratio e [-] 0.5 – 0.7
Coefficient of permeability k under σ’v = 200 kPa [m/s] 2 – 4 10-5
The coefficient of permeability is significantly higher than that of Birmensdorf clay. Even
though the filter criteria according to Terzaghi (1925) are not fulfilled, there did not appear to
be any infiltration of the clay suspension into the sand. It could be observed that the drainage
function was maintained during the entire consolidation phase, and during the duration of the
centrifuge tests.
3.6 Soil model
Table 3.6 gives a summary of the containers used for the preparation of the soil models used
for centrifuge tests.
Table 3.6: Summary of the containers used for the preparation of soil models for centrifuge
tests.
Consolidation
stress Preparation container
Number of specimens
tested
100 Cylindrical strongbox (Ø 400 mm) 6
100 Oedometer container (Ø 250 mm) 1
100 Adapted oedometer container (Ø 250 mm) 2
200 Oedometer container (Ø 250 mm) 2
200 Adapted oedometer container (Ø 250 mm) 2
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119
3.6.1 Preparation of Birmensdorf clay
The first step of the preparation of the models used in this work is homogenisation of the
clayey material stored in transport containers. Clay with a water content ranging from 100 %
to 120 % is placed into a vacuum mixer (Figure 3.32). Once a homogeneous suspension is
obtained, the vacuum is activated for a period of 12 hours to remove any air bubbles.
Figure 3.32: Vacuum mixer.
3.6.2 Preparation of a soil model in a cylindrical strongbox
Centrifuge tests were conducted using a 400 mm diameter cylindrical strongbox. The
advantage of this method is that the model can be prepared outside the centrifuge while the
instrumentation for the specimen can be inserted relatively easily.
The base plates of all containers are equipped with channels (Figure 3.33 a), which are then
filled with Perth sand (Figure 3.33 b) to prevent the drainage channels from clogging during
the filling of the strongbox with the clay suspension (Figure 3.33 c). This permits the model to
drain in both vertical directions, reducing the consolidation time under the press and the
testing time in the centrifuge. The channels also allow water to be fed into the base of the
model during the test under enhanced gravity. Once the mixing process is over
(Section 3.6.1), the clay suspension obtained is poured into the container directly from the
vacuum mixer (Figure 3.33 c).
3.6 Soil model
120
(a) (b)
(c)
Figure 3.33: General view (a) cylindrical strongbox used for the consolidation of Birmensdorf
clay under the hydraulic press, (b) view of the channels filled with Perth sand
and (c) filling with clay suspension.
A hydraulic press (Figure 3.34) is used in order to consolidate the sample under oedometric
conditions. The consolidation takes place with progressive vertical loading steps of 6, 12, 25,
50, and 100 kPa. The preloading of the sample under the press then determines the OCR of
the model in-flight, which is presented in detail for each test in the subsequent sections.
Once the consolidation under a load of 100 kPa is achieved, the model height is manually
reduced to 160 mm using a cutting blade, the specimen is unloaded, the water valves are
closed, the PPTs are installed (Section 3.6.5.1), the model is mounted into the centrifuge and
the water supply is connected to the plate of the container.
3 Centrifuge modelling
121
Figure 3.34: Hydraulic press used for the consolidation of clay.
3.6.3 Preparation of a soil model in an oedometer container
Clay specimens for centrifuge tests were also prepared in smaller oedometer containers
(250 mm diameter, Figure 3.35 a). The motivation for the use of such containers was the
opportunity to install instrumentation very close to the inclusion.
The drainage occurs in the model container both at the surface and at the bottom of the soil
model. As the containers were not equipped with a riled plate similar to the cylindrical
centrifuge strongbox mentioned above, a filter plate, covered with filter paper to prevent
clogging, was installed at the bottom of the specimen in order to ensure there would be
drainage capacity at the bottom of the sample.
The container is filled with slurry up to a height of 360 mm and the consolidation is conducted
using a different oedometer press (Figure 3.35 a) with similar progressive loading steps of 6,
12, 25, 50, 100 kPa and 200 kPa, respectively. One of the two samples used for the test
JG_v5 was only consolidated up to 100 kPa.
As the 250 mm diameter oedometer container was originally not equipped with ports on the
side walls so that PPTs can be installed in the clay, the sample had to be removed from the
container before this was possible. Cylindrical rupture zones would be possible during the
extraction of the soil model from the oedometer container due to the adhesion of clay to the
container (Figure 3.36 a). Therefore a plastic sheet was installed around the circumference
of the oedometer container (Figure 3.35 a, Figure 3.36 b) in order to avoid adhesion between
container and sample. This allows for the consolidated sample to be removed, using the
crane at disposal in the laboratory next to the ETH Zürich drum centrifuge (Figure 3.37 a).
The container is lifted vertically at a constant rate (approximately 4 cm/min) and the clay
sample does not experience any lateral movement during this phase, thus reducing the
disturbance of the sample during the removal of the sample from the container to a minimum.
3.6 Soil model
122
(a) (b)
Figure 3.35: Preparation of the clay model (a) slurry inside the oedometer container (b) under
consolidation in the oedometer container.
(a) (b)
Figure 3.36: Schematic representation (a) of the possible cylindrical rupture zones when
extracting the clay sample from the container and (b) of the use of the plastic
sheet in order to prevent the adhesion between clay and oedometer container.
Once the sample has been extracted, the PPTs are then installed (Section 3.6.5.2), the clay
sample is put into the 400 mm diameter strongbox and the gap between sample
and strongbox is filled with Perth sand by means of dry pluviation with compaction
(Figure 3.37 b). A density index ID of approximately 70 % was reached.
Plastic sheet
3 Centrifuge modelling
123
(a) (b)
Figure 3.37: (a) Removal of the oedometer container from the sample (b) view of the model
with clay sample surrounded by Perth sand.
This preparation procedure causes additional disturbance to the specimen as a manual
manipulation of the sample is necessary to install it into the 400 mm diameter strongbox after
the installation of the PPTs, the effect of which is extremely difficult to quantify. Moreover, the
removal of the lateral support from the container affects the stress history. The coefficient of
earth pressure at rest K0 of Birmensdorf clay is:
3.5
with K0 coefficient of earth pressure at rest
φ’ effective angle of friction of Birmensdorf clay (Table 3.2)
Thus, for a principal vertical stress σ1’ of 200 kPa acting on the soil specimen in the
oedometer container, the horizontal principal stress σ3’ is equal to 116 kPa. The removal of
the soil model from the oedometer container causes a dissipation of the horizontal principal
stress, which is subsequently equal to 0 kPa. Thus the model can not only swell vertically,
but also horizontally. The filling of the gap between soil model and container wall with Perth
sand provided a lateral support to the clay sample, although it is not rigid. Thus the horizontal
principal stress σ3’ will be smaller than in an oedometer container. However, no stress
measurements were conducted within the soil, making a quantitative interpretation of the
changes of the horizontal principal stress impossible. A quantitative comparison of the lateral
stresses acting on the clay sample for specimens prepared in an oedometer container and
surrounded by sand and for specimens prepared in a rigid container (cylindrical strongbox or
adapted oedometer) is given in Section 3.7.4.
ClayPerth sand
Cables from PPTs
3.6 Soil model
124
These considerations led to the adaptation of the 250 mm oedometer containers so that they
could be mounted directly into the centrifuge, avoiding the need to remove the sample from
the stiff container (Figure 3.37 a) and the subsequent filling of the gap with compacted sand
(Figure 3.37 b). This avoided the soil model from being disturbed and the stress history from
being affected beyond the planned stress paths.
3.6.4 Preparation of a soil model in an adapted oedometer container
Adapted oedometer containers were used for centrifuge tests. Holes were drilled in the walls
of the containers presented in Section 3.6.3, so that PPTs could be installed in the
clay samples (Figure 3.38). Base plates were constructed similar to those used in the
strongbox (Section 3.6.2). The filling procedure is exactly the same as described in
Section 3.6.2: Perth sand is used as drainage bed at the bottom of the sample
(Figure 3.33 b) and the clay suspension is filled into the container directly from the vacuum
mixer (Figure 3.33 c) up to a height of 360 mm.
Figure 3.38: Ports for the installation of PPTs into the soil model prepared in 250 mm
diameter containers
After the consolidation phase, the height of the model is manually reduced to 160 mm using
a cutting blade, the PPTs are installed through the dedicated connections (Section 3.6.5.1),
the model is mounted into the centrifuge and the water supply is connected to the base plate
of the container.
3.6.5 Installation of the PPTs
The PPTs are installed into the consolidated clay models using a technique described in
König et al. (1994). Firstly holes (diameter 7 mm) are pre-drilled in the clay. These holes
need to be a few millimetres shorter than the intended penetration length of the PPTs in
order to ensure that the filters of the transducers (Figure 3.23) can be slightly pressed into
the host soil, guaranteeing a good connection in the planned location.
Base
plate
Ports for installation of the PPTs
3 Centrifuge modelling
125
The PPTs are then inserted into the pre-drilled hole using the installation tool shown in
Figure 3.39, which has a rill against which the PPT can be held while pushing it into the soil.
Once the desired penetration is reached, the tool is pulled back out of the model and the
transducer remains in the clay. A slight counter pressure applied on the cable of the PPT
is usually sufficient to separate the transducer from the installation tool. The pre-drilled
hole is sealed with a clay paste of the same origin as the host soil by using a syringe
(Figure 3.40 b).
Figure 3.39: PPT installation tool.
3.6.5.1 Installation of the PPTs into a cylindrical strongbox and an
adapted oedometer container
Insertion of the transducers into the models consolidated in the cylindrical strongbox and in
the adapted oedometer containers is conducted in the same manner. The installation tool
(Figure 3.39) is held horizontally and lateral movements are also prevented when passing it
through the port (Figure 3.40 a). The penetration length is marked on the installation tool,
taking the thickness of the container walls, as well as the length of the connections into
account.
(a) (b)
Figure 3.40: Installation of the PPTs in the cylindrical strongbox (a) introduction of the PPTs
through the dedicated ports into the pre-drilled hole (b) filling of the pre-drilled
hole with slurry (Weber, 2008).
Rill
3.6 Soil model
126
3.6.5.2 Installation of the PPTs into a model consolidated in an oedometer
container and installed in the 400 mm diameter strongbox
A special installation setup had to be developed (Figure 3.41) to install the pore pressure
transducers into a model that had been consolidated in an oedometer container. It consists
of a circular plate which is positioned at the upper surface of the cylindrical clay sample after
the oedometer ring had been removed and held in place by aluminium cylinders penetrating
slightly into the clay sample. Holes were drilled at the edge of the circular plate to achieve a
precise and stable positioning of the installation device, which is fixed to 2 vertical cylinders
and can be moved up and down depending on the depth at which the transducers have to be
installed.
Figure 3.41: PPT installation setup for a specimen consolidated in an oedometer container.
The installation tool (Figure 3.39) can then be positioned on the installation device, ensuring
that the tool is oriented horizontally, while preventing lateral movement (Figure 3.42).
The PPTs are then pushed into pre-drilled holes in a similar manner to that used for
installation in the cylindrical strongbox. The holes are sealed with a slurry of Birmensdorf clay
using a syringe.
Aluminium cylinders
holding the plate
in place
Installation
device
Holes used to
position the
installation
device
Consolidated
clay sample
3 Centrifuge modelling
127
Figure 3.42: Insertion of the PPT installation tool into the clay specimen using the installation
device.
3.6.6 Identification of the locations of the stone columns
The positions of the stone columns to be installed have to be marked before the start of the
centrifuge test. Drawing pins are used to seal the tip of the stone column installation tool and
are pushed into the planned locations on the surface of the clay model.
A difference was made between the models prepared in the cylindrical strongbox (Section
3.6.2), where four stone columns were installed and those consolidated using the oedometer
containers (Sections 3.6.3 and 3.6.4), where a single column was built. In the first case, the
model was installed in the centrifuge and the stone column installation tool, mounted on the
working arm of the tool platform, was used to mark the locations of the four inclusions to be
built. A pin was then positioned correspondingly at each location.
In the second case, it was of extreme importance to install the pin precisely in the middle of
the sample as PPTs are then installed very near to the column (Figure 3.53 and Figure 3.56)
and any error in the positioning of the pin can have important negative consequences (the
stone column installation tool could hit and destroy some of the transducers). One of the
aluminium cylinders holding the plate in place at the upper surface of the clay sample (Figure
3.41) was therefore positioned exactly in the middle of the plate, thus delivering a precise
location of where to install the pin.
The type of the drawing pins used in this research differ from the original industrial drawing
pins used by Weber (2008). There was only a very small margin of error in the positioning of
the stone column installation tool, otherwise the tip would clog. The pins used in this project
have a circular base and a conical cross-section (Figure 3.43 and Figure 3.44). This provided
a significantly higher degree of safety against clogging.
3.7 Centrifuge tests
128
(a) (b)
Figure 3.43: Pin used to mark the positions of the stone columns to be installed (a) plan view
and (b) side view.
Figure 3.44: Tilted view of the pin used to mark the positions of the stone columns to be
constructed with the stone column installation tool.
3.7 Centrifuge tests
3.7.1 Overview
Although different experimental setups were used, the main system parameters for the
centrifuge tests such as acceleration, height of the model and drainage conditions, were the
same for all the tests conducted, and are summarised in Table 3.7. The reference radius
(radius at which the nominal acceleration acts) is set at 2/3rds of the model height, as
suggested by Schofield (1980).
Vertical cross-sections of the different experimental setups used are shown in Figure 3.45
(detailed representations at scale are presented in Sections 3.7.2, 3.7.4 and 3.7.5). A
standpipe (9) was installed in order to control the position of the groundwater level. Table 3.8
gives an overview of the experimental setups (Sections 3.7.2, 3.7.4 and 3.7.5) used for the
test series.
7.2 mm
5 m
m
11
mm
1 m
m
2.5 mm
2 mm
Stone column installation tool
Pin
3 Centrifuge modelling
129
Table 3.7: Summary of the main system parameters.
Model height [mm] 160
Stone column length [mm] 120
Nominal acceleration [g] 50
Reference radius 2/3rds of the model height
Drainage conditions Double-sided
Insertion depth of the T-Bar [mm] 140
Insertion depth of the electrical
impedance needle [mm] 115
Full cylindrical
strongbox
Specimen prepared in an
oedometer container
Adapted oedometer
container
1 Water supply 6 Drainage bed
2 Drum 7 Standpipe
3 Strongbox with base plate 8 Water escape valve
4 Groundwater level 9 Adjustment base
5 Soil model 10 Perth Sand
Figure 3.45: Vertical cross-section of the experimental setup in the centrifuge for the
specimens prepared in a cylindrical strongbox (Section 3.6.2), in an oedometer
container and in an adapted oedometer container (Section 3.6.4) (after Weber,
2008).
3.7 Centrifuge tests
130
Table 3.8: Overview of the experimental setup used for the different tests.
Test Full cylindrical
strongbox
Specimen
prepared in an
oedometer
container
Adapted
oedometer
container
Pre-
consolidation
stress σc [kPa]
JG_v1 X (1 container) 200
JG_v2 X (1 container) 100
JG_v3 X (1 container) 100
JG_v5 X (2 containers) 100 / 200
JG_v6 X (1 container) 100
JG_v7 X (2 containers) 200
JG_v8 X (1 container) 100
JG_v9 X (2 containers) 100
JG_v10 X (2 containers) 100
3.7.2 Groundwater level
The position of the groundwater level is a key parameter for the tests conducted. The
saturation of the clay model has to be guaranteed during the whole duration of the centrifuge
tests, i.e. about 36 hours. Water is therefore supplied from the bottom of the model, as
shown in Figure 3.45, and the position of the groundwater level in the model is controlled by
a standpipe.
The surface of the clay model is vertical in the drum while the groundwater level is curved
due to the acceleration field in the centrifuge. The position of the water level in the soil and
in the standpipe is shown for the different setups used in Figure 3.46, Figure 3.47 and
Figure 3.48.
Figure 3.46: Position of the water level in the soil and in the standpipe for specimens
prepared in a cylindrical strongbox (tests JG_v2, JG_v3, JG_v6, JG_v8 and
JG_v10).
3 Centrifuge modelling
131
Figure 3.47: Position of the water level in the soil and in the standpipe for specimens
prepared in an oedometer container (tests JG_v1 and JG_v5).
Figure 3.48: Position of the water level in the soil and in the standpipe for specimens
prepared in an adapted oedometer container (tests JG_v7 and JG_v9).
3.7.3 Tests conducted with specimens prepared in a cylindrical strongbox
3.7.3.1 Loading of a single stone column (JG_v2, JG_v3 and JG_v6)
This setup was used for the tests JG_v2, JG_v3 and JG_v6. The boundary conditions can be
attributed to those of an oedometer, as the radial strains are restricted on the sides. Seven
PPTs, the locations of which can be seen in Figure 3.49 (a) and (b), were installed in the soil
model. A goal of these tests was to determine whether the main part of the installation effects
appeared during the insertion of the mandrel, during its removal or during the compaction of
the granular inclusion.
To this means, four columns (Figure 3.49 a) were installed in the following sequence:
- column A was built without compaction,
- columns B and C were installed with a compaction regime of 15/10,
- column D was left unfilled. The stone column installation tool was inserted, the
centrifuge was stopped and the installation tool was pulled out.
3.7 Centrifuge tests
132
Although the installation tool was pulled out of column D, the fact that this was done under 1
g and not under enhanced gravity limited the impact of the removal of the mandrel in terms of
radial relaxation of the host soil. The length of the stone columns installed in the centrifuge is
linked to the length of the sand feed pipe of the filling tube, which has to be stretched in order
to prevent the filling material from clogging in the feed pipe (Weber, 2008). As a
consequence, the sand feed pipe had to be changed after the installation of the columns A
and B.
A compaction regime of 15/10 means that once the tip of the stone column installation tool
has reached the desired depth of 120 mm, the tool is pulled out 15 mm before being inserted
by 10 mm again. This process is iterative until the tip of the tool reaches the surface of the
soil model. The stone column diameter increases from 10 mm (outer diameter of the
installation tool) to 12 mm during the compaction.
A T-Bar test, the location of which can be seen in Figure 3.49 (a) was conducted in order to
determine the undrained shear strength of the host soil. The compacted column B was
subjected to a displacement-controlled loading (v = 0.02 mm/s up to a depth of 17 mm) using
a stiff circular aluminium footing (diameter 56 mm). Table 3.9 contains a summary of the
testing procedure.
Table 3.9: Testing procedure for tests conducted using the full cylindrical strongbox (loading
of a single stone column, tests JG_v2, JG_v3 and JG_v6).
Step Nr. Activity Duration
1 Start of the centrifuge and consolidation of
the model under its self-weight under
increased gravity
10 h
2 Installation of columns A and B 2 h
3 Dissipation of the excess pore water
pressures triggered by the installation of the
stone columns
3.5 h
4 Displacement-controlled loading of column
B
40 min
5 Dissipation of the excess pore water
pressures triggered by the footing load
10 h
6 Installation of column C 1 h
7 T-Bar test 1 h
8 Installation of column D 1 h
9 Centrifuge stopped -
3 Centrifuge modelling
133
(a)
(b)
Figure 3.49: Specimens prepared in a cylindrical strongbox: (a) plan view and (b) cross-
section of the soil model with positions of the PPTs and of the stone columns.
3.7.3.2 Loading of a stone column group (JG_v8, JG_v10)
This setup was used for the tests JG_v8 and JG_v10. Seven PPTs, the locations of which
can be seen in Figure 3.51 and Figure 3.50, were installed in the soil model. Five compacted
stone columns (compaction regime 15/10) were built over a time frame of 1 hour and the
A
3.7 Centrifuge tests
134
stone column group was then subjected to a displacement-controlled loading (v = 0.02 mm/s
up to a depth of 17 mm) with a square footing (56 mm x 56 mm).
Figure 3.50: Specimens prepared in a cylindrical strongbox: cross-section of the soil model,
with positions of the PPTs and of the stone columns (a / dsc = 2 [-]).
Table 3.10: Overview of the experimental setups used for the different tests.
Test Stone column diameter
dsc [mm]
Distance a between the
axis of the columns
[mm]
Ratio a / dsc [-]
JG_v8 12 24 2
JG_v10 12 24 2
JG_v10 12 30 2.5
3 Centrifuge modelling
135
Figure 3.51: Specimens prepared in a cylindrical strongbox: plan view of the soil model with
positions of the PPTs and of the stone columns (a / dsc = 2 [-]).
The electrical impedance needle (Section 3.4.5) was used between the installation of the
columns and the footing load. Two measurements were first conducted at reference points 1
and 2 (RP1 and RP2 in Figure 3.52), located 100 mm away from the axis of the stone
column A, both laterally and vertically. The needle was then inserted at decreasing distances
towards the axis of the stone column A (points A2 to J2 in Figure 3.52).
3.7 Centrifuge tests
136
Figure 3.52: Specimens prepared in a cylindrical strongbox: insertion points of the electrical
impedance needle: positions of the reference points RP1 and RP2 and the
points A2 to J2 (a / dsc = 2 [-]).
Figure 3.52 illustrates a situation with a ratio a / dsc equal to 2, as used during the test JG_v8.
Although this ratio was modified during the test JG_v10, the distances of the needle points to
the axis of the column A were kept unchanged. The distance a between the axis of the
columns was varied from twice to 2.5 times the diameter of the column, as described in
Table 3.10. Table 3.11 shows a summary of the testing procedure.
3 Centrifuge modelling
137
Table 3.11: Testing procedure for tests conducted using the full cylindrical strongbox (loading
of a stone column group, tests JG_v8 and JG_v10).
Step Nr. Activity Duration
1 Start of the centrifuge and consolidation of
the model under its self-weight under
increased gravity
10 h
2 Installation of the stone columns in rapid
sequence
1 h
3 Dissipation of the excess pore water
pressures caused by the installation of the
stone columns, T-Bar test and
implementation of the electrical impedance
needle
4 h
4 Displacement-controlled loading of the
stone column group
40 min
5 Dissipation of the excess pore water
pressures triggered by the footing load
7 h
6 Centrifuge stopped -
3.7.4 Tests conducted with specimens prepared in an oedometer container
(JG_v1, JG_v4, JG_v5)
This setup was used for the tests JG_v1, JG_v4 and JG_v5. One single sample, pre-
consolidated up to 200 kPa, was examined in tests JG_v1 and JG_v4. Two specimens were
installed symmetrically in the centrifuge for JG_v5, one of which was pre-consolidated up to
100 kPa and the other up to 200 kPa. Although Perth sand used to fill the gap between the
clay specimen and the container wall was compacted (Section 3.6.3), the boundary
conditions were not equivalent to those of an oedometer, as the clay / sand interface is not
rigid.
Seven PPTs, the locations of which can be seen in Figure 3.53, were installed in the soil
model. A compacted stone column was installed in the middle of the model (Figure 3.53).
The compaction regime was again 15/10.
3.7 Centrifuge tests
138
(a)
(b)
Figure 3.53: Specimens prepared in an oedometer container and surrounded by Perth sand:
(a) plan view and (b) cross-section of the soil model with positions of the PPTs
and of the stone column.
The tool used for compacting Perth sand around the clay sample had a length of 40 mm and
a width of 10 mm. Assuming a depth of influence of the compaction of 20 mm, the profile of
the horizontal stresses acting on the clay sample at the interface with Perth sand can be
A
A
3 Centrifuge modelling
139
calculated (with the effective angle of friction of Perth sand equal to 30°) using the increased
coefficient of earth pressure at rest:
3.6
with increased coefficient of earth pressure at rest
K0 coefficient of earth pressure at rest
φ’ effective angle of friction
The profile of the lateral stresses acting on clay specimens prepared in a rigid container
(cylindrical strongbox or adapted oedometer) can be assessed using the coefficient of earth
pressure at rest of an over-consolidated soil:
( )
3.7
with K0OC coefficient of earth pressure at rest of an over-consolidated soil
K0NC coefficient of earth pressure at rest of a normally consolidated soil
OCR over-consolidation ratio
φ’ effective angle of friction
The profiles of the over-consolidation ratios used for the calculation of the horizontal stresses
are shown in Figure 4.3. Figure 3.54 shows a comparison of the lateral stresses acting on
the clay sample for specimens prepared in an oedometer container and surrounded by Perth
sand (denoted as σ’h Perth sand) and for specimens prepared in a rigid container with a pre-
consolidation of σ’v = 100 kPa (denoted as σ’h,clay, 100 kPa) or of σ’v = 200 kPa (denoted as
σ’h,clay, 200 kPa).
The lateral support by Perth sand is calculated based on the silo theory. Perth sand offers an
approximately similar support of the sample to the rigid container (for a pre-consolidation of
σ’v = 100 kPa) up to 20 mm (Figure 3.54), which corresponds to the assumed influence depth
of the compaction. However, below the depth of influence of the compaction, the horizontal
stress acting on the clay sample at the sand/clay boundary remains constant at 32 kPa, while
the horizontal stress acting on specimens prepared in rigid containers would rise up to 93
kPa (for a pre-consolidation of σ’v = 100 kPa) and to 137 kPa for a pre-consolidation of σ’v =
200 kPa) at a depth of 160 mm.
A T-Bar test, the location of which is given in Figure 3.53 (a) was conducted in order to
determine the undrained shear strength of the host soil. Seven PPTs (Figure 3.53 a and b)
were installed in the soil model. The column was then subjected to a displacement-controlled
loading (v = 0.02 mm/s up to a depth of 17 mm) using a stiff aluminium footing (diameter
56 mm).
3.7 Centrifuge tests
140
Figure 3.54: Comparison of the lateral stresses acting on the clay sample for specimens
prepared in an oedometer container and surrounded by Perth sand (σ’h Perth sand,
calculated based on the silo theory) and for specimens prepared in a rigid
container (cylindrical strongbox or adapted oedometer) with a pre-consolidation
of σ’v = 100 kPa (σ’h clay, 100 kPa) or of σ’v = 200 kPa (σ’h clay, 200 kPa).
The last step of the tests consisted of measurements using the electrical impedance needle
(Section 3.4.5). Two measurements were first conducted in the far field, namely at the
reference points 1 and 2 (RP1 and RP2 in Figure 3.55), located 62.5 mm away from the axis
of the stone columns both laterally and vertically. The needle was then inserted at
decreasing spacing towards the stone column axis (points A to F in Figure 3.55). A summary
of the testing procedure is shown in Table 3.12.
Table 3.12: Testing procedure for tests conducted using the specimens prepared in an
oedometer or in an adapted oedometer (tests JG_v1, JG_v4, JG_v5, JG_v7 and
JG_v9).
Step Nr. Activity Duration
1 Start of the centrifuge and consolidation of the
model under its self-weight under increased gravity
7 h
2 Installation of the stone column 1 h
3 Dissipation of the excess pore water pressures
caused by the installation of the stone column
3.5 h
4 Displacement-controlled loading of the column 40 min
5 Dissipation of the excess pore water pressures
caused by the footing load
10 h
6 T-Bar test 1 h
7 Insertion of the electrical impedance needle 5 h
8 Centrifuge stopped -
0
20
40
60
80
100
120
140
160
0 20 40 60 80 100 120 140
De
pth
[m
m]
σ'h [kPa]
σh' sand σh' 100 kPa σh' 200 kPaσ'h Perth sand [kPa] σ'h clay, 100 kPa [kPa] σ'h clay, 200 kPa [kPa]
3 Centrifuge modelling
141
Figure 3.55: Specimens prepared in an oedometer container and surrounded by Perth sand:
insertion points of the electrical impedance needle and positions of the reference
points RP1 and RP2 and the points A to F.
3.7.5 Tests conducted with specimens prepared in an adapted oedometer
container (JG_v7, JG_v9)
The experimental setup using specimens prepared in an adapted oedometer container is
similar to that using specimens prepared in an oedometer container. However, the boundary
conditions are, in this case, oedometric.
3.7 Centrifuge tests
142
Such a setup was used for the tests JG_v7 and JG_v9. A plan view and a cross-section of
the model can be seen in Figure 3.56. Seven PPTs, the locations of which can be seen in
Figure 3.56 were installed in the soil model.
The steps of the tests (column installation, footing load, T-Bar test and implementation of the
electrical impedance needle) are similar to those given in Table 3.12. The electrical
impedance needle was inserted at the same positions as those shown in Figure 3.55.
(a)
(b)
Figure 3.56: Specimens prepared in an adapted oedometer: (a) plan view and (b) cross-
section of the soil model with positions of the PPTs and of the stone column.
4 Results from the centrifuge tests
143
4 Results from the centrifuge tests
The following chapter presents the results obtained from 8 centrifuge tests, during which a
total of 12 specimens were investigated. All tests were conducted under 50 g. Although the
nominal acceleration actually acts at a depth of 2/3rds of the model height, it is assumed that
the whole specimen is subjected to 50 times the Earth’s gravity. This limits the error, as the
g-level will be under-estimated above and overestimated below 2/3rds of the height.
Due to mechanical issues with the centrifuge, the specimen used for test JG_v3 was left
under the consolidation press for about 5 months, which led to aging effects. The results
from this test could not be exploited in the end.
4.1 Undrained shear strength
The clay slurry was consolidated up to vertical effective stresses of either 100 kPa or 200
kPa depending on the experimental setup used. The pre-consolidation determines the OCR
and the theoretical profile of the undrained shear strength with depth. A theoretical prediction
of the undrained shear strength was compared with the results obtained experimentally in-
flight using the T-Bar penetrometer (Section 3.4.4).
4.1.1 Theoretical prediction
Ladd et al. (1977) propose the following expression for the calculation of the undrained shear
strength su:
4.1
(
)
4.2
4.3
with su undrained shear strength
σ’v vertical effective stress
OCR over-consolidation ratio
a, b undrained shear strength parameters
nc normally consolidated
σ’v,max maximum vertical effective stress
Ladd et al. (1977) suggest values of b ranging between 0.75 and 0.85. Trausch-Giudici
(2003) and Küng (2003) conducted triaxial tests in order to propose the values of a and b for
the Birmensdorf clay used in this research (Table 4.1).
The over-consolidation ratio OCR was determined based on the pre-consolidation stress
applied during the preparation of the specimens (σ’v,max = σ’v,Press) and the vertical stress
profile obtained under a nominal constant acceleration of 50 g (σ’v, = σ’v,Centrifuge). Figure 4.1
and Figure 4.2 show the profiles of the effective vertical stresses under the press (σ’v,Press)
4.1 Undrained shear strength
144
and in-flight under 50g (σ’v,Centrifuge) for pre-consolidation stresses of 100 kPa and 200 kPa,
respectively. Figure 4.3 represents the corresponding profiles of the over-consolidation ratio.
Table 4.1: Values of a and b obtained by Trausch-Giudici (2003) and Küng (2003).
Trausch-Giudici Küng
a [-] 0.26 0.34
b [-] 0.73 0.73
Ip [%] 26-56 26-56
Figure 4.1: Profile of the vertical effective stress in the centrifuge (σ’v,centrifuge) and under the
press (σ’v,press) for a pre-consolidation of 100 kPa.
Figure 4.2: Profile of the vertical effective stress in the centrifuge (σ’v,centrifuge) and under the
press (σ’v,press) for a pre-consolidation of 200 kPa.
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120
De
pth
[m
m]
σ'v [kPa]
σ' press σ' centrifugeσ'v,Press σ'v,Centrifuge
0
20
40
60
80
100
120
140
0 50 100 150 200 250
De
pth
[m
m]
σ'v [kPa]
σ' press σ' centrifuge
σ'v,Press
σ'v,Centrifuge
4 Results from the centrifuge tests
145
Figure 4.3: Profiles of the over-consolidation ratio for pre-consolidation stresses of 100 kPa
and 200 kPa.
4.1.2 Shear strength profile for pre-consolidation up to σ’v = 100 kPa
Figure 4.4 shows the shear strength profiles obtained for specimens pre-consolidated up to
100 kPa. The profiles calculated using the predictions according to Trausch-Giudici (2003)
and Küng (2003) are denoted as su,TG and su,K, respectively. The profile measured in a full
cylindrical strongbox is represented by su,JG_v2 while the profiles measured in specimens
prepared in adapted oedometers are denoted as su,JG_v8, su,JG_v10,A and su,JG_v10,B, respectively.
An error in the software controlling the movement of the tool platform prevented steps 6 to 8
from being conducted (installation of column C, T-bar test, installation of column D,
Table 3.9) for test JG_v6. The undrained shear strength profiles obtained during the tests
JG_v2, JG_v8 and JG_v10 are presented in Figure 4.4. The steep increase of the undrained
shear strength near the surface measured in container B during test JG_v9 (denoted as
su,JG_v10,B in Figure 4.4) is due to the presence of clay on the T-bar after testing in container A,
which increased the values measured at the beginning of the penetration. This effect was
noted for all second T-bar measurements (denoted as e.g. su,JG_v9,A in Figure 4.5).
The results obtained with the values of a and b proposed by Trausch-Giudici (2003) and
Küng (2003) (denoted as su,TG and su,K, respectively, in Figure 4.4) overestimate the
undrained shear strength in this case. This might be due to the fact that Trausch-Giudici
(2003) and Küng (2003) investigated specimens with plasticity indexes Ip ranging from 26 %
to 56 % and back-calculated the parameters a and b in order to obtain the best fit of the
results, which might not correspond to the Ip of 39 % of the soil used in this research. The
results from the T-Bar were well fitted with a = 0.22 [-] and b = 0.65 [-] (curve denoted as su,JG
in Figure 4.4). This value of b is significantly lower than the suggestion made by Ladd et al.
(1977), which is entirely feasible and would indicate a small over-consolidation of the model
at the surface.
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50
De
pth
[m
m]
OCR [-]
Series1 Series2σ'v= 100 kPa σ'v = 200 kPa
4.1 Undrained shear strength
146
Figure 4.4: Profiles of the undrained shear strength obtained with the T-Bar during tests
JG_v2 (su,JG_v2), JG_v8 (su,JG,v8) and JG_v10 (su,JGv10,A and su,JG,v10,B) compared
with theoretical predications based on Trausch-Giudici (2003, su,TG) and
Küng (2003, su,K) and with the back-calculated values of the parameters a and b
(su,JG).
Figure 4.5: Profiles of the undrained shear strength obtained with the T-Bar during
test JG_v9 (su,JG_v9,A and su,JG_v9_B) compared with theoretical predictions based
on Trausch-Giudici (2003, su,TG) and on Küng (2003, su,K), and with the back-
calculated values of the parameters a and b (su,JG).
The two specimens (A and B) used for test JG_v9 were consolidated up to 100 kPa and the
boundary conditions are clearly defined, as the lateral movements at the boundary were
restricted by the stiff container (adapted oedometer). Figure 4.5 shows a comparison
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30
De
pth
[m
m]
su [kPa]
su, K su, TG su,JG_v2 su,JG_v8
su,JG_v10,A su,JG_v10,B su,JG
su,K su,TG su,JG_v2 su,JG_v8
su,JG_v10,A su,JG_v10,B su,JG
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30
De
pth
[m
m]
su [kPa]
su2, K su1, TG su,A su,B su,JGsu,K su,TG su,JG_v9,A su,JG_v9,B su,JG
4 Results from the centrifuge tests
147
between the measured values of the undrained shear strength (denoted as su,JG_v9,A and
su,JG_v9,B) with the theoretical predictions.
The values of a and b proposed by Trausch-Giudici (2003) and Küng (2003) (denoted as
su,TG and su,K, respectively, in Figure 4.5) tend to overestimate the undrained shear strength.
The results obtained from the T-Bar were well fitted with a = 0.22 [-] and b = 0.79 [-] (curve
denoted as su,JG in Figure 4.5). The back-calculated value of b lies between 0.75 and 0.85,
as suggested by Ladd et al. (1977), and is also close to the values suggested by
Trausch-Giudici (2003) and Küng (2003) (b = 0.73 [-]).
4.1.3 Shear strength profile for pre-consolidation up to σ’v = 200 kPa
Figure 4.6 shows a comparison of the shear strength profiles obtained during the tests
conducted with specimens prepared in oedometer containers and subsequently transferred
into the cylindrical strongbox. Tests JG_v1 (denoted as su,JG_v1) and JG_v5 for the specimen
consolidated up to 200 kPa (denoted as su,JG_v5), are reported with the profile obtained using
the back-calculated values of a and b for the specimen prepared in an adapted oedometer
container and pre-consolidated up to 100 kPa (Section 4.1.2).
Figure 4.6: Profiles of the undrained shear strength obtained with the T-Bar during tests
JG_v1 (su,JG_v1) and JG_v5 in the specimen consolidated up to 200 kPa (su,JG_v5)
compared with the profile obtained with back-calculated values of the
parameters of a and b (su,JG).
Test JG_v1 was the first test conducted using the experimental setup with a specimen
consolidated in an oedometer and manually transferred into the centrifuge strongbox. It is
therefore assumed that the lower undrained shear strength measured during test JG_v1 is
due to a greater disturbance of the specimen during the transfer than during test JG_v5
It is interesting that the profile of the undrained shear strength recorded during test JG_v5 is
very close to the back-calculated profile of test JG_v9. This indicates that the strengthening
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30
De
pth
[m
m]
su [kPa]
su,JG_v1 su_JG_v5 su,JGsu,JG_v5su,JG,v1 su,JG,v5 su,JG
4.1 Undrained shear strength
148
effect of the higher over-consolidation of the specimen (200 kPa for test JG_v5 opposed to
100 kPa for test JG_v9) on the undrained shear strength is opposed by the loss of constraint
of the flexible boundary conditions (interface sand / clay opposed to clay / steel) and by the
disturbance caused by the manual transfer of the specimen from the oedometer container to
the strongbox.
Figure 4.7 shows a comparison of the shear strength profiles obtained during test JG_v7
(denoted as su,JG_v7_A and su,JG_v7_B), conducted using a specimen prepared in an adapted
oedometer container, with the theoretical predictions according to Trausch-Giudici (2003)
and Küng (2003) (denoted as su,TG and su,K, respectively). The experimental results (denoted
as su,JG in Figure 4.7) are fitted reasonably well with a = 0.17 [-] and b = 0.79 [-].
Figure 4.7: Profiles of the undrained shear strength obtained with the T-Bar during
test JG_v7 (su, JG_v7,A and su, JG_v7_B) compared with theoretical predictions based
on Trausch-Giudici (2003, su,TG) and on Küng (2003, su,K) and with the back-
calculated values of the parameters a and b (su,JG).
4.1.4 Summary of the back-calculated values of the shear strength parameters
a and b
Figure 4.8 shows the profiles of the back-calculated undrained shear strength for specimens
prepared in a full cylindrical strongbox (su,JG,1, σc = 100 kPa) and in adapted oedometers
(su,JG,2, σc = 100 kPa and su,JG,3, σc = 200 kPa). Table 4.2 shows a comparison of the back-
calculated values of the parameters a and b with those proposed by Trausch-Giudici (2003)
and Küng (2003).
The influence of the pre-consolidation on the undrained shear strength is clear as peak
values of about 27 kPa are reached for a pre-consolidation stress of 200 kPa, as opposed to
about 20 kPa for a pre-consolidation stress of 100 kPa.
0
20
40
60
80
100
120
140
0 10 20 30 40 50 60
De
pth
[m
m]
su [kPa]
su2, K su1, TG su,A su,B su,Gsu,K su,TG su,JG_v7,A su,JG_v7,B su,JG
4 Results from the centrifuge tests
149
The impact of the vicinity of the rigid boundary to the location of the penetration of the T-Bar
is not as marked as that of the pre-consolidation stress but it can be noted as well, with an
increase in su of about 4 kPa from a specimen prepared in a full cylindrical strongbox to a
specimen prepared in an adapted oedometer. This can be explained by the fact that the
adapted oedometer has a smaller diameter (250 mm) than the strongbox (400 mm), inducing
a slightly higher constraint to the specimen. Although the over-consolidation of the soil is
taken into account by Equation 4.1, the pre-consolidation stress does seem to have an
influence on the value of factor a (Table 4.2).
Figure 4.8: Profiles of the back-calculated undrained shear strength for a specimen prepared
in a full cylindrical strongbox (su,JG,1) and in adapted oedometers (su,JG,2 and
su,JG,3).
Table 4.2: Comparison of the back-calculated values of the parameters a and b with those
proposed by Trausch-Giudici (2003) and Küng (2003).
a [-] b [-]
This work (σc = 100 kPa,
full cylindrical strongbox) 0.22 0.65
This work (σc = 100 kPa,
adapted oedometer) 0.22 0.79
This work (σc = 200 kPa,
adapted oedometer) 0.17 0.79
Trausch-Giudici (2003) 0.26 0.73
Küng (2003) 0.34 0.73
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30
De
pth
[m
m]
su [kPa]
su,JG su,JG su,Gsu,JG,1 su,JG,2 su,JG,3
4.2 Pore pressure measurements conducted during the installation of stone columns
150
4.2 Pore pressure measurements conducted during the installation
of stone columns
4.2.1 Measurements conducted during the installation of a single stone
column
The PPTs from which the data discussed in the upcoming section were obtained were
located as follows:
- P1, P2 and P3 at a depth of 48 mm below the surface of the soft soil and at a
radial distance of 12 mm, 18 mm and 30 mm, from the axis of the column,
respectively,
- P4, P5 and P6 at a depth of 96 mm below the surface of the soft soil and at a
radial distance of 12 mm, 18 mm and 30 mm, from the axis of the column,
respectively, and
- P7 at a depth of 140 mm below the surface of the soft soil and aligned with the
axis of the stone column.
The pore water pressures were measured continuously during the installation of stone
columns into the soft soil model. Both compacted and non-compacted columns were
installed, while pore pressure measurements were recorded at different radial distances to
the axis of the columns, depending on whether the test was conducted using a full cylindrical
strongbox or a specimen prepared in an oedometer container. The PPTs installed in
specimens prepared in (adapted) oedometers are significantly closer to the column than
those installed in the full cylindrical strongbox. Their reaction to the installation process is
therefore greater. Figure 4.9 presents the results from the installation of a compacted column
during test JG_v7 (specimen pre-consolidated up to 200 kPa in an adapted oedometer).
The transducers exhibit the greatest reaction during the installation phase when the tip of the
installation tool reaches the depth where they are installed (Figure 4.9 a, arrows 1 till 3). The
greatest reaction during the compaction phase occurs when the tip of the installation tool is
approximately 30 mm above the depth of the transducers (Figure 4.9 a, arrows 4 and 5). The
pore water pressures decrease rapidly once the tool is extracted from the model (Figure 4.9
a, arrow 6), as the load applied by the installation tool disappears.
Transducer P6 reacted slowly, which is the reason why no peak can be noted in Figure 4.9
(a). The arrows (7) till (9) in Figure 4.9 (b) mark the most important phases of the installation
process:
- the start of tool insertion into the model (arrow 7),
- the start of the compaction process (arrow 8),
- the removal of the tool from the model (arrow 9).
4 Results from the centrifuge tests
151
1,2,3: tip of
the installation
tool at the
depth of the
PPTs
4,5: tip of the
intsallation
tool about 30
mm above the
PPTs
6: extraction of
the installation
tool
(a)
7: start of the
insertion of the
installation
tool
8: start of the
compaction
phase
9: extraction of
the installation
tool
(b)
Figure 4.9: Installation of a compacted column in a specimen pre-consolidated up to 200 kPa
(test JG_v7) (a) pore water pressures (b) depth of the tip of the installation tool
with time.
The influence of the radial distance to the axis of the columns, as well as of the depth, can be
noted when considering the insertion phase (t = 0 s to t = 80 s in Figure 4.9) only
(Figure 4.10):
- the time needed for the transducers to react to column construction is longer with
increasing distance to the axis of the column. Figure 4.10 a (arrow 1) shows the
consecutive peak values of the excess pore water pressures measured by the
transducers P1, P2 and P3, marked with vertical dashed lines. This tendency is
more prone with increasing depth (arrow 2 for P4 and P5),
- the excess pore pressures caused by the installation of the stone column increase
with depth (Figure 4.10 a).
0
20
40
60
80
100
120
140
160
0 100 200 300 400 500
Po
re w
ate
r p
res
su
res
[k
Pa
]
Time [s]
P1 P2 P3 P4 P5 P6 P7
-140
-120
-100
-80
-60
-40
-20
0
0 100 200 300 400 500
De
pth
[m
m]
Time [s]
Depth P4 -P6
Depth P1 -P3
(1)
(2)
(3)
(4) (5)
(6)
(8)
(7) (9)
4.2 Pore pressure measurements conducted during the installation of stone columns
152
1: tip of the
installation
tool reaches
the depth of
P1
2: tip of the
installation
tool reaches
the depth of
P4
(a)
(b)
Figure 4.10: Insertion of the stone column installation tool in a specimen pre-consolidated up
to 200 kPa (tests JG_v7) (a) excess pore water pressures (b) depth of the tip of
the installation tool with time.
Figure 4.11 shows the dissipation with time of the excess pore water pressures caused
by the installation of a stone column in a specimen pre-consolidated up to 200 kPa
(test JG_v7). The excess pore water pressures are dissipated after about 1500 s, which
shows the impact of the stone column on the consolidation time of the specimen. The time
needed for consolidation of a clay specimen, without a stone column, in-flight under 50 g can
be calculated as:
4.4
with t90 time needed to reach a consolidation of 90 %
Tv dimensionless time factor (equal to 0.848)
d length of the drainage path (80 mm at model scale, corresponding to 4 m at
prototype scale in this case)
cv coefficient of consolidation (equal to 2.5 . 10-7 m2/s, Table 3.3)
0
10
20
30
40
50
60
0 20 40 60 80
Ex
ce
ss
po
re w
ate
r p
res
su
res
[k
Pa
]
Time [s]
P1 P2 P3 P4 P5 P6 P7
-140
-120
-100
-80
-60
-40
-20
0
0 20 40 60 80
De
pth
[m
m]
Time [s]
Depth P4 -P6
Depth P1 -P3
(1)
(2)
4 Results from the centrifuge tests
153
The stone column causes a reduction of the consolidation time by a factor of about 15, which
is consistent with the reduction of the drainage path from 80 mm to 6 mm for the PPTs
closest to the column and to 24 mm for the PPTs furthest away from the column.
Figure 4.11: Insertion of the stone column installation tool in a specimen pre-consolidated up
to 200 kPa (test JG_v7): excess pore water pressures.
The pre-consolidation stress plays a role regarding the magnitude of excess pore pressure
caused during the installation of the stone columns (Figure 4.10 a and Figure 4.12 a).
Figure 4.12 (a) shows that the general trend remains the same as for specimens with higher
OCRs. The time needed for the transducers to react is longer with increasing distance to the
axis of the column (arrows 1 and 2) and the time difference between the reactions of the
different sensors is more marked with increasing depth (arrows 1 and 2). The recorded peak
values by P4, P5 and P6 vary quite significantly even though the sequence of the reactions is
logical. This variation might be due to a lower stiffness of the back-fill of the cavity made for
the installation of the PPT in comparison with the host soil. The intensity of the excess pore
water pressures generated decreases with decreasing consolidation stress σc.
0
20
40
60
80
100
120
140
160
0 1000 2000 3000 4000 5000
Po
re w
ate
r p
res
su
res
[k
Pa
]
Time [s]
P1 P2 P3 P4 P5 P6 P7
4.2 Pore pressure measurements conducted during the installation of stone columns
154
1: tip of the
installation
tool reaches
the depth of
P1
2: tip of the
installation
tool reaches
the depth of
P4
(a)
(b)
Figure 4.12: Insertion of the stone column installation tool in a specimen pre-consolidated up
to 100 kPa (test JG_v9) (a) excess pore water pressures (b) depth of the tip of
the installation tool with time.
4.2.2 Measurements conducted during the installation of a stone column
group
The PPTs from which the data discussed in the upcoming section were obtained were
located as follows:
- P1, P2 and P3 at a depth of 30 mm below the surface of the soft soil and at a
radial distance of 12 mm, 18 mm and 30 mm, from the axis of the centre column,
respectively,
- P4, P5 and P6 at a depth of 80 mm below the surface of the soft soil and at a
radial distance of 12 mm, 18 mm and 30 mm, from the axis of the centre column,
respectively, and
0
10
20
30
40
50
60
0 20 40 60 80
Ex
ce
ss
po
re w
ate
r p
res
su
res
[k
Pa
]
Time [s]
P1 P2 P3 P4 P5 P6 P7
-140
-120
-100
-80
-60
-40
-20
0
0 20 40 60 80
De
pth
[m
m]
Time [s]
Depth P4 -P6
Depth P1 -P3
(1)
(2)
4 Results from the centrifuge tests
155
- P7 at a depth of 65 mm below the surface of the soft soil and at a radial distance
of 18 mm from the axis of the centre column.
1: Installation
of column A
2: Installation
of column B
3: Installation
of column C
4: Installation
of column D
5: Installation
of column E
(a)
(b)
Figure 4.13: Insertion of the stone column installation tool in a specimen consolidated up to
100 kPa (test JG_v10) (a) excess pore water pressures (b) location of the stone
columns installed.
Figure 4.13 shows the evolution of the excess pore water pressures over time during the
installation of a stone column group (test JG_v10). The reason for the development of
negative excess pore pressures at the location of P5 during the installation of the column D
is unknown, but might be due to movement of the columns A and C caused by the
compaction process of column D.
0
20
40
60
80
100
0 500 1000 1500 2000 2500 3000 3500
Po
re w
ate
r p
res
su
res
[k
Pa
]
Time [s]
P1 P2 P3 P4 P5 P6
aaa
aaa
(1) (2) (3) (4) (5)
4.3 Measurements conducted during the footing loading of a single stone column installed in
a specimen prepared in a full cylindrical strongbox
156
The transducers show a greater reaction to the installation of column A, which is due to the
vicinity of this inclusion and to the orientation of the filter plate of the PPTs towards the centre
of the specimen. The reaction of the PPTs to the installation of the subsequent columns,
as well as the rate of dissipation of the excess pore water pressures after the installation
(Figure 4.13 a), is similar for columns B, C, D and E. The excess pore water pressures
caused by the installation of the stone column group are dissipated after about 1000 s
(Figure 4.14), which is faster than in the case of a single stone column and shows the
increased drainage performance of a stone column group, compared to a single stone
column.
Figure 4.14: Insertion of the stone column installation tool in a specimen consolidated up to
100 kPa (test JG_v10): excess pore water pressures.
4.3 Measurements conducted during the footing loading of a single
stone column installed in a specimen prepared in a full
cylindrical strongbox
The measurements presented here were performed in specimens prepared in the full
cylindrical strongbox (400 mm diameter). The PPTs from which the data discussed in the
upcoming section were obtained were located as follows:
- P1, P2 and P3 at radial distance of 262 mm from the loaded column and a depth
of 60 mm, 85 mm and 110 mm from the surface of the soft soil specimen,
respectively,
- P4 and P6 at a radial distance of 56 mm from the loaded column and a depth of
85 mm from the surface of the soft soil specimen,
- P5 and P7 at radial distance of 56 mm from the loaded column and a depth of
60 mm and 110 mm from the surface of the soft soil specimen, respectively.
0
20
40
60
80
100
0 1000 2000 3000 4000 5000 6000 7000
Po
re w
ate
r p
res
su
res
[k
Pa
]
Time [s]
P1 P2 P3 P4 P5 P6
4 Results from the centrifuge tests
157
The values recorded by transducers P1, P2 and P3 are not discussed here due to the
distance of the PPTs from the footing. Figure 4.15 (a), (b) and (c) show the recordings of the
PPTs P4 to P7, the load-settlement curve and the settlement-time curve recorded during test
JG_v2. The data obtained during test JG_v6 are presented graphically in Appendix 8.3. The
quantitative values are shown in Table 4.3 and Table 4.4.
The ratios of the peak value of the recorded excess pore pressure to the peak value of the
applied footing load (Δumax / Pmax) recorded by the transducers P6 and P7 remain remarkably
constant between test JG_v2 and test JG_v6 (12.52 % and 13.47 % for P6 and 6.17 % and
6.42 % for P7).
Table 4.3: Response of the PPT to the applied footing load on a single stone column during
test JG_v2.
PPT
Maximal excess
pore water pressure
Δumax [kPa]
Maximal load
applied on the
footing Pmax [kPa]
Δumax / Pmax [%]
P4 (z = 85 mm) 7.64 94.72 8.07
P5 (z = 60 mm) 6.02 94.72 6.36
P6 (z = 85 mm) 11.86 94.72 12.52
P7 (z = 110 mm) 5.84 94.72 6.17
Table 4.4: Response of the PPT to the applied footing load on a single stone column during
the test JG_v6.
PPT
Maximal excess
pore water pressure
Δumax [kPa]
Maximal load
applied on the
footing Pmax [kPa]
Δumax / Pmax [%]
P4 (z = 85 mm) 16.59 115.22 14.40
P5 (z = 60 mm) 3.50 115.22 3.04
P6 (z = 85 mm) 15.52 115.22 13.47
P7 (z = 110 mm) 7.40 115.22 6.42
4.3 Measurements conducted during the footing loading of a single stone column installed in
a specimen prepared in a full cylindrical strongbox
158
(a)
(b)
(c)
Figure 4.15: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v2) (a) excess pore water pressures (b) evolution of the footing load
(c) deformation controlled footing settlement.
0
2
4
6
8
10
12
14
0 200 400 600 800 1000
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P4 P5 P6 P7
0
10
20
30
40
50
60
70
80
90
100
0 200 400 600 800 1000
Fo
oti
ng
lo
ad
ing
[k
Pa
]
Time [s]
-20
-15
-10
-5
0
0 200 400 600 800 1000
Fo
oti
ng
se
ttle
me
nt
[mm
]
Time [s]
4 Results from the centrifuge tests
159
4.4 Measurements conducted during the footing loading of a single
stone column installed in a specimen prepared in an (adapted)
oedometer container
The PPTs from which the data discussed in the upcoming section were obtained were
located at the same positions as presented in Section 4.2.
4.4.1 Measurements conducted in a specimen consolidated up to
σ’v = 100 kPa
Figure 4.16 and Table 4.5 present the results of the measurements conducted during the
loading of a single column installed in a clay specimen that has been consolidated up to
100 kPa in an adapted oedometer (test JG_v9). The influence of the installation depth can be
noted as a decrease of the measured excess pore water pressures ranging from 40 % to
55 % can be observed between the PPTs installed at 48 mm depth (P1, P2 and P3) and
those installed at a depth of 96 mm (P4, P5 and P6).
More interesting is the influence of the radial distance to the axis of the column. P1 and P2
show a similar response to the applied load (0.35 Pmax and 0.37 Pmax, respectively) while P3
displays a significantly smaller reaction (0.27 Pmax). The same trend can qualitatively be
observed at a greater depth, although the quantitative reactions are less pronounced, as P4
and P5 record a peak excess pore pressure of about 0.24 Pmax and P6 only of 0.20 Pmax.
P7, installed directly below the column at a depth of 140 mm, records a peak excess pore
pressure of 0.11 Pmax. However, the influence of the lost tip of the stone column onto the
drainage is very difficult to quantify.
Table 4.5: Response of the PPT to the applied footing load on a single stone column during
the test JG_v9.
PPT
Maximal excess
pore water pressure
Δumax [kPa]
Maximal load
applied on the
footing Pmax [kPa]
Δumax / Pmax [%]
P1 (z = 48 mm) 42.23 119.67 35.28
P2 (z = 48 mm) 44.41 119.67 37.11
P3 (z = 48 mm) 33.33 119.67 27.85
P4 (z = 96 mm) 29.21 119.67 24.41
P5 (z = 96 mm) 28.67 119.67 23.96
P6 (z = 96 mm) 23.80 119.67 19.89
P7 (z = 140 mm) 13.46 119.67 11.25
4.4 Measurements conducted during the footing loading of a single stone column installed in
a specimen prepared in an (adapted) oedometer container
160
(a)
(b)
(c)
Figure 4.16: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v9) (a) excess pore water pressures (b) evolution of the footing load (c)
deformation controlled footing settlement.
0
10
20
30
40
50
0 500 1000 1500 2000 2500
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P1 P2 P3 P4 P5 P6 P7
0
20
40
60
80
100
120
140
0 500 1000 1500 2000 2500
Fo
oti
ng
lo
ad
ing
[k
Pa
]
Time [s]
-20
-15
-10
-5
0
0 500 1000 1500 2000 2500
Fo
oti
ng
se
ttle
me
nt
[mm
]
Time [s]
4 Results from the centrifuge tests
161
Figure 4.17 and Figure 4.18 show the dissipation with time of the excess pore water
pressures after reaching the peak footing load at depths of 48 mm and 96 mm, respectively.
In both cases, an accelerated dissipation can be noted 250 s after reaching the peak footing
load (which corresponds to t = 0 s). The influence of the stone column on the drainage length
is visible in the magnitude of the dissipation of the excess pore water pressure: a diminution
of approximately 17 kPa is measured, at a depth of 48 mm, 6 mm and 12 mm from the edge
of the stone column, while this reduction is only about 11 kPa, when measured 24 mm away
from the edge of the column.
(a)
(b)
Figure 4.17: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v9), dissipation with time of the excess pore water pressures at a depth
of 48 mm around the stone column (a) from 0 s to 2000 s and (b) from 3000 s to
9000 s after reaching the peak footing load (which corresponds to t = 0 s).
0
5
10
15
20
25
30
35
40
45
50
0 10 20 30 40Ex
ce
ss
po
re w
ate
r p
res
su
res
[k
Pa
]
Radial distance [mm]
t = 0 s t = 250 s t = 500 s t = 750 st = 1000 s t = 1500 s t = 2000 s
Edge of the stone column
0
1
2
3
4
5
6
7
8
9
10
0 10 20 30 40Ex
ce
ss
po
re w
ate
r p
res
su
res
[k
Pa
]
Radial distance [mm]
t = 3000 s t = 4000 s t = 5000 s t = 6000 st = 7000 s t = 8000 s t = 9000s
Edge of the stone column
4.4 Measurements conducted during the footing loading of a single stone column installed in
a specimen prepared in an (adapted) oedometer container
162
A similar observation can be made at a depth of 96 mm as a reduction of approximately
8.5 kPa is measured 6 mm and 12 mm from the edge of the column while this is only 7.5 kPa
at a location of 24 mm from the edge of the column. A pseudo-constant rate of dissipation is
reached 2000 s after the peak footing load has been applied.
(a)
(b)
Figure 4.18: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v9): dissipation of the excess pore water pressures at a depth of
96 mm with time around the stone column (a) from 0 s to 2000 s and (b) from
3000 s to 9000 s after reaching the peak footing load (which corresponds to
t = 0 s).
These observations are confirmed by consideration of the evolution of the rate of dissipation
of excess pore water pressures over time after the peak footing load has been applied
(Figure 4.19): the rate drops from approximately 0.07 kPa /s (transducers P1 and P2), 0.045
0
5
10
15
20
25
30
35
40
45
50
0 10 20 30 40Ex
ce
ss
po
re w
ate
r p
res
su
res
[k
Pa
]
Radial distance [mm]
t = 0 s t = 250 s t = 500 s t = 750 st = 1000 s t = 1500 s t = 2000 s
Edge of the stone column
0
1
2
3
4
5
6
7
8
9
10
0 10 20 30 40Ex
ce
ss
po
re w
ate
r p
res
su
res
[k
Pa
]
Radial distance [mm]
t = 3000 s t = 4000 s t = 5000 s t = 6000 st = 7000 s t = 8000 s t = 9000s
Edge of the stone column
4 Results from the centrifuge tests
163
kPa /s (transducer P3) and 0.03 kPa /s (transducers P4, P5 and P6) immediately after the
peak footing has been reached (t = 0 s) to values of approximately 0.01 kPa / s at t = 1000 s.
The rate of dissipation reaches a pseudo-constant state at about t = 2000 s after the peak
footing load was applied.
Figure 4.19: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v9): rate of dissipation of excess pore water pressures with time after
reaching the peak footing load (which corresponds to t = 0 s).
4.4.2 Measurements conducted in a specimen consolidated up to
σ’v = 200 kPa
Table 4.6 and Figure 4.20 present the results of the measurements conducted during the
loading of a single column installed in a clay specimen consolidated up to 200 kPa in an
adapted oedometer (250 mm in diameter, test JG_v7).
Similar loadings were conducted during tests JG_v1 and JG_v5, in which specimens were
pre-consolidated up to 200 kPa in an oedometer (250 mm in diameter) and subsequently
transferred into a cylindrical strongbox (400 mm in diameter). The gap was filled with
compacted sand. The results of the tests JG_v1 and JG_v5 are presented graphically in
Appendices 8.1 and 8.2. Quantitative results are summarised in Table 4.7 and Table 4.8.
The relative difficulty in assessing the influence of the manual steps of the preparation of the
specimen consolidated in an oedometer and installed in the cylindrical strongbox, as well as
of the boundary conditions on the loading behaviour, has to be kept in mind when
considering the results of tests JG_v1 and JG_v5.
Figure 4.20 (a) shows that the installation depth of the transducers and the radial distance to
the axis of the column have a great influence on the recorded excess pore water pressure
due to the footing load as the drop of the values of the excess pore water pressures caused
by the footing loading ranges from 20 % to 45 % between 48 mm and 96 mm. This drop is
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Δu
/ Δ
t [k
Pa
/s]
Time [s]
P1 P2 P3 P4 P5 P6
4.4 Measurements conducted during the footing loading of a single stone column installed in
a specimen prepared in an (adapted) oedometer container
164
less important than in the case of a specimen pre-consolidated up to 100 kPa, which
indicates a higher load transfer into depth with increasing pre-consolidation stress. This can
be explained by the higher stiffness of the host soil. However, as for the case of the loading
on a single stone column, installed in a specimen pre-consolidated up to 100 kPa, the radial
distance to the axis of the column plays an important role in the measurements, as the
excess pore water pressures are significantly higher near the edge of the stone column than
further off (Table 4.6).
The increase of pre-consolidation stress from 100 kPa (test JG_v9) to 200 kPa (test JG_v7)
causes a jump of the maximal applied load of 21.5 %. The observations made during the
loading of a stone column installed in a specimen consolidated up to 100 kPa are also valid
here. The ratios Δumax / Pmax are remarkably similar (Table 4.5 and Table 4.6), independently
of the over-consolidation of the specimen and of the load applied at the surface.
Table 4.6: Response of the PPT to the applied footing load on a single stone column during
test JG_v7 (adapted oedometer).
PPT
Maximal excess
pore water pressure
Δumax [kPa]
Maximal load
applied on the
footing Pmax [kPa]
Δumax / Pmax [%]
P1 (z = 48 mm) 51.31 145.44 35.28
P2 (z = 48 mm) 50.86 145.44 34.97
P3 (z = 48 mm) 35.62 145.44 24.49
P4 (z = 96 mm) 36.57 145.44 25.14
P5 (z = 96 mm) 35.24 145.44 24.23
P6 (z = 96 mm) 29.66 145.44 20.39
P7 (z = 140 mm) 24.00 145.44 16.50
Table 4.7: Response of the PPT to the applied footing load on a single stone column during
test JG_v1 (cylindrical strongbox with clay surrounded by sand).
PPT
Maximal excess
pore water pressure
Δumax [kPa]
Maximal load applied
on the footing Pmax
[kPa]
Δumax / Pmax [%]
P1 (z = 48 mm) 21.66 80.00 27.08
P2 (z = 48 mm) 16.72 80.00 20.90
P3 (z = 48 mm) 11.33 80.00 14.16
P4 (z = 96 mm) 18.41 80.00 23.01
P5 (z = 96 mm) 13.92 80.00 17.40
P6 (z = 96 mm) 17.67 80.00 22.09
P7 (z = 140 mm) 11.77 80.00 14.71
4 Results from the centrifuge tests
165
(a)
(b)
(c)
Figure 4.20: Loading of a single stone column in a specimen pre-consolidated up to 200 kPa
(JG_v7) (a) excess pore water pressures (b) evolution of the footing load (c)
deformation controlled footing settlement.
0
10
20
30
40
50
60
0 500 1000 1500 2000 2500
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P1 P2 P3 P4 P5 P6 P7
-20
0
20
40
60
80
100
120
140
160
0 500 1000 1500 2000 2500
Fo
oti
ng
lo
ad
ing
[k
Pa
]
Time [s]
-20
-15
-10
-5
0
0 500 1000 1500 2000 2500
Fo
oti
ng
se
ttle
me
nt
[m
m]
Time [s]
4.4 Measurements conducted during the footing loading of a single stone column installed in
a specimen prepared in an (adapted) oedometer container
166
Table 4.8: Response of the PPT to the applied footing load on a single stone column during
test JG_v5 (cylindrical strongbox with clay surrounded by sand).
PPT
Maximal excess
pore water pressure
Δumax [kPa]
Maximal load applied
on the footing Pmax
[kPa]
Δumax / Pmax [%]
P1 (z = 48 mm) 42.24 120.14 35.16
P2 (z = 48 mm) 35.05 120.14 29.17
P3 (z = 48 mm) 17.22 120.14 14.33
P4 (z = 96 mm) 23.46 120.14 19.53
P5 (z = 96 mm) 22.40 120.14 18.65
P6 (z = 96 mm) 19.67 120.14 16.37
P7 (z = 140 mm) 15.33 120.14 12.76
The difference between the applied footing loads is due to specimen disturbance during the
preparation. Test JG_v1 was the first to be conducted using specimens prepared in
oedometer containers and then transferred to the cylindrical centrifuge strongbox. Thus the
results need to be treated with caution. The specimen preparation procedure was not
completely controlled, which complicates any quantitative conclusion about the loading
behaviour during this test.
This disturbance was reduced to a minimum for test JG_v5. The ratios of the maximal
excess pore water pressures to the applied footing load for this test show a good agreement
with the results from test JG_v7. The peak load reaches values close to those obtained
during test JG_v9 (Pmax = 119.67 kPa), conducted with a specimen prepared in an adapted
oedometer and pre-consolidated up to 100 kPa.
The rigid, oedometric, lateral boundary conditions in test JG_v7 lead to an increase of the
maximal footing load applied by 21 %, compared to equivalent results from test JG_v5.
4.4.3 Comparison of the results
Similarities and differences appear when comparing the distribution of the excess pore water
pressure during footing loading for the tests JG_v1, JG_v5, JG_v7 and JG_v9. The influence
of the specimen disturbance on the results of test JG_v1 is significant when considering the
results obtained at a depth of 48 mm (Table 4.7), as the ratios of excess pore water pressure
to footing load are significantly lower than for the other tests at a radial distance of 12 mm
and 18 mm from the axis of the stone column. Although these ratios are close to those
measured during the other tests at higher depths, the quantitative influence of the specimen
disturbance is unclear and the results of test JG_v1 will not be discussed in greater detail.
The influence of the different drainage conditions is also noticeable. The ratio of excess pore
water pressure to footing load is similar for the tests JG_v5, JG_v7 and JG_v9 at a distance
of 12 mm from the axis of the stone column. This confirms the limited disturbance of the
specimen caused by the installation of the specimen from the oedometer container into the
4 Results from the centrifuge tests
167
strongbox for the preparation of test JG_v5. However, the ratios of excess pore water
pressure to applied load tend to remain constant between 12 mm and 18 mm from the axis
of the column for the cases of rigid and impermeable boundaries of the clay specimen
(tests JG_v7 and JG_v9), while this ratio drops significantly in case of non-rigid and
permeable boundaries (interface clay / sand for test JG_v5).
The impact of the over-consolidation on the ratio of excess pore water pressure to footing
load is, on the contrary very limited, as the difference between the results of tests JG_v7
(σc = 200 kPa) and JG_v9 (σc = 100 kPa) are very similar.
Figure 4.21: Distribution of the excess pore water pressure with increasing radial distance to
the axis of the stone column at a depth of 48 mm as a percentage of the applied
footing load P.
Figure 4.22 shows the distribution of the excess pore water pressure around the stone
column at a depth of 96 mm. The outcomes of tests JG_v7 (σc = 200 kPa) and JG_v9 (σc =
100 kPa) can be considered to be equal at this location as the difference in the ratios of
excess pore water pressure to applied footing load at a given radial distance from the axis of
the column does not exceed 5%, which confirms that the impact of the over-consolidation
ratio on the distribution of the excess pore water pressure is very limited. However, the
impact of the drainage conditions is clear as the normalised values recorded during
test JG_v5 lie about 5 % below those recorded during tests JG_7 and JG_v9.
0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30
No
rma
lis
ed
ex
ce
ss
po
re w
ate
r p
res
su
re [
% P
]
Radial distance [mm]
JG_v5 JG_v7 JG_v9
Edge of the stone column
4.4 Measurements conducted during the footing loading of a single stone column installed in
a specimen prepared in an (adapted) oedometer container
168
Figure 4.22: Distribution of the excess pore water pressure with increasing radial distance to
the axis of the stone column at a depth of 96 mm as a percentage of the applied
footing load P.
4.4.4 Load transfer around a single stone column
The load transfer around a single stone column can be investigated based on data collected
during the centrifuge tests. The first step is a comparison of the excess pore water pressures
recorded with the values obtained theoretically, for the case of a vertical footing load applied
on a homogeneous and isotropic host soil, in order to assess the influence of the granular
inclusion on the load spreading. A second step consists of a back-calculation of the vertical
total stress increments based on the pore water pressure measurements.
Grasshoff (1978) proposed a model to assess the distribution of the vertical stresses in soil
under a loaded circular plate. This model relies on the assumption that the soil behaviour
remains elastic throughout loading, that means the stresses are proportional to the strains,
and that the soil is homogeneous and isotropic. Grasshoff (1978) suggests that the vertical
stress increment can be calculated using a factor J4 (a table of the values of this factor is
given in Appendix 8.4), as follows:
4.5
with Δσz vertical stress increase
J4 depth factor according to Grasshoff (1978)
P footing load
The excess pore water pressures recorded during the centrifuge tests can be compared with
a back-calculation of the excess pore water pressures caused by a vertical load on a
homogeneous and isotropic host soil.
0
5
10
15
20
25
30
0 5 10 15 20 25 30No
rma
lis
ed
ex
ce
ss
po
re w
ate
r p
res
su
re [
% P
]
Radial distance [mm]
JG_v5 JG_v7 JG_v9
Edge of the stone column
4 Results from the centrifuge tests
169
Figure 4.23: Isobars of vertical stress increments under a vertically loaded quadratic plate
(Lang et al., 2007).
Skempton (1954) formulates the following expression for the assessment of the excess pore
water pressures in a known stress field:
( ) 4.6
with Δu excess pore water pressure
A, B pore pressure parameters according to Skempton (1954)
σr radial stress
σa axial stress
It is assumed that the radial stress σr is equal to the horizontal stress σh and that the axial
stress σa is equal to the vertical stress σv. Equation 4.6 can thus be formulated as:
( ) 4.7
with Δσh horizontal stress increment
Δσv vertical stress increment
The coefficient of earth pressure at rest K0 needs to be assessed in order to calculate the
horizontal stress increase Δσh. Mayne & Kulhawy (1982) suggest the following formula in
order to take the over-consolidation into account for the determination of K0:
4.8
with K0OC coefficient of earth pressure at rest of an over-consolidated soil
K0NC coefficient of earth pressure at rest of a normally consolidated soil
OCR over-consolidation ratio
φ’ effective angle of friction
4.4 Measurements conducted during the footing loading of a single stone column installed in
a specimen prepared in an (adapted) oedometer container
170
It is necessary to determine the over-consolidation ratio OCR to calculate the excess pore
water pressure at the installation depths of the PPTs. The values obtained are summarised
in Table 4.9, although they do not take the additional consolidation due to the installation of
the stone column into account. A linear distribution of stress over depth in the specimen is
assumed.
Table 4.9: Calculation of the over-consolidation ratio at the installation depths of the PPTs
depending on the pre-consolidation stress σc.
Depth
[mm]
Unit weight of saturated
host soil γg [kN/m3]
Vertical stress under
enhanced gravity
σ’z,centrifuge[kPa]
OCR [-]
σc = σ’v.max =
100 kPa
σc = σ’v.max =
200 kPa
σc = σ’v.max =
100 kPa
σc = σ’v.max =
200 kPa
σc = σ’v.max =
100 kPa
σc = σ’v.max =
200 kPa
48 18.5 20.1 20.4 24.24 4.90 8.25
96 18.5 20.1 40.8 48.48 2.45 4.13
140 18.5 20.1 59.5 70.70 1.68 2.83
The effective angle of friction of the Birmensdorf clay is =24.5° (Table 3.2) and K0NC = 1 -
sin . The values of K0OC are summarised in Table 4.10.
Table 4.10: Coefficients of earth pressure at rest of the over-consolidated Birmensdorf clay
with depth.
Depth
[mm]
OCR [-] K0
NC [-] K0
OC [-]
σc = 100 kPa σc = 200 kPa σc = 100 kPa σc = 200 kPa
48 4.90 8.25 0.585 1.13 1.40
96 2.45 4.13 0.585 0.848 1.05
140 1.68 2.83 0.585 0.725 0.901
With Δσh = K0OC Δσv’ + Δu, Equation 4.6 can be rewritten as:
[
( )
] 4.9
Equation 4.9 can be reformulated as:
(
( ))
4.10
Skempton (1954) suggests a value of B equal to 1 for fully saturated soils submitted to fully
undrained loading. The parameter B defines the portion of the total surcharge converted into
excess pore water pressure. The value of the parameter A should ideally be determined
experimentally. Good approximations are summarised in Table 4.11, depending on the type
of clay. The values of A used in Table 4.12, Table 4.13 and Table 4.14 were set in order to
take the influence of the variation of the over-consolidation ratio with depth into account.
4 Results from the centrifuge tests
171
Table 4.11: Values of the pore pressure parameter A depending on the type of clay
(Skempton, 1954).
Type of clay A
Clays of high sensitivity 0.75 – 1.5
Normally consolidated clays 0.5 – 1
Compacted sandy clays 0.25 – 0.75
Lightly over-consolidated clays 0 – 0.5
Compacted clay-gravels -0.5 – 0.25
Heavily over-consolidated clay -0.5 – 0
Equation 4.10 can thus be reformulated, using Equation 4.5 as:
( ) (
( ))
4.11
Reformulating Equation 4.11 gives an expression of the excess pore water pressure Δu:
(
( ))
( (
) ) 4.12
This formulation allows the excess pore water pressure caused by a vertical load applied at
the surface of a homogeneous isotropic host soil to be calculated. The results are presented
in Table 4.12, Table 4.13, Table 4.14 and Table 8.2. The parameter B was set equal to 0.8 in
order to take the partial dissipation of excess pore water pressures during the loading phase
into account. The parameter A was varied in order to take the variation of over-consolidation
with depth into account. The peak values of the excess pore water pressures estimated with
Equation 4.11 are denoted as ΔuSkempton, while the peak values measured during the
centrifuge tests are denoted as Δucentrifuge.
The manual manipulation necessary for the insertion of the PPTs and the installation of the
specimen into the strongbox during test JG_v1 was relatively cumbersome, which led to
significant disturbance of the specimen. The results are summarised in Appendix 8.5, but will
not be investigated in further detail in this section.
The influence of the stone column on the load distribution with depth can clearly be noted as
the ratio increases from values close to one at a depth of 48 mm, to values between 2 and 3
at a depth of 96 mm. A significantly higher part of the load is transmitted at depths of 96 mm
and 140 mm in the presence of the granular inclusion, compared to the case where a vertical
load is applied at the surface of a homogeneous isotropic fine-grained host soil.
4.4 Measurements conducted during the footing loading of a single stone column installed in
a specimen prepared in an (adapted) oedometer container
172
Table 4.12: Comparison of the peak analytical and measured excess pore water pressure at
end of the loading phase of a single stone column (test JG_v9, σc = 100 kPa,
P = 119.67 kPa).
PPT J4 [%]
Δσa = Δσz
= J4 P
[kPa]
A [-] ΔuSkempton [kPa] Δucentrifuge [kPa] Δucentrifuge /
ΔuSkempton [-]
P1 32.30 38.65 0.3 38.65 42.23 1.09
P2 29.13 34.86 0.3 34.86 44.41 1.27
P3 21.45 25.67 0.3 25.67 33.33 1.30
P4 11.45 13.70 0.4 13.70 29.21 2.13
P5 11.03 13.20 0.4 13.20 28.67 2.17
P6 9.73 11.64 0.4 11.64 23.80 2.04
P7 5.70 6.82 0.4 6.82 13.46 1.97
Table 4.13: Comparison of the peak analytical and measured excess pore water pressure at
end of the loading phase of a single stone column (test JG_v7, σc = 200 kPa,
P = 145.44 kPa).
PPT J4 [%]
Δσa = Δσz
= J4 P
[kPa]
A [-] ΔuSkempton [kPa] Δucentrifuge [kPa] Δucentrifuge /
ΔuSkempton [-]
P1 32.30 46.98 0.1 39.68 51.31 1.29
P2 29.13 42.37 0.1 35.79 50.86 1.42
P3 21.45 31.20 0.1 26.36 35.62 1.35
P4 11.45 16.65 0.2 13.42 36.57 2.73
P5 11.03 16.04 0.2 12.93 35.24 2.73
P6 9.73 14.15 0.2 11.41 29.66 2.60
P7 5.70 8.29 0.2 6.52 24.00 3.68
Table 4.14: Comparison of the peak analytical and measured excess pore water pressure at
end of the loading phase of a single stone column (test JG_v5, σc = 200 kPa,
P = 120.14 kPa).
PPT J4 [%]
Δσa = Δσz
= J4 P
[kPa]
A [-] ΔuSkempton [kPa] Δucentrifuge [kPa] Δucentrifuge /
ΔuSkempton [-]
P1 32.30 38.81 0.1 32.78 42.24 1.29
P2 29.13 35.00 0.1 29.57 35.05 1.19
P3 21.45 25.77 0.1 21.77 17.22 0.79
P4 11.45 13.76 0.2 11.09 23.46 2.12
P5 11.03 13.25 0.2 10.68 22.40 2.10
P6 9.73 11.69 0.2 9.42 19.67 2.09
P7 5.70 6.85 0.2 5.39 15.33 2.84
4 Results from the centrifuge tests
173
An influence of the over-consolidation ratio on the response of the composite foundation to
vertical loading can be detected. The ratios Δucentrifuge / ΔuSkempton range from 1.29 to 1.42 at a
depth of 48 mm for a specimen consolidated up to 200 kPa (test JG_v7, Table 4.13), while
they only reach values ranging from 1.09 and 1.30 at the same depth for a specimen
consolidated up to 100 kPa (test JG_v9, Table 4.12). This shows that the higher stiffness of
the host soil caused by the higher pre-consolidation of the specimen reduces the stress
concentration on top of the column and causes a higher load of the host soil. This effect
remains present with increasing depth as the ratios Δucentrifuge / ΔuSkempton range from 2.60 to
2.73 at a depth of 96 mm for a specimen consolidated up to 200 kPa (test JG_v7,
Table 4.13) while they only reach values comprised between 2.04 and 2.17 for a specimen
consolidated up to 100 kPa (test JG_v9, Table 4.12).
Although assumptions had to be made, these results provide a first insight into the load
distribution under a vertically loaded footing placed on stone column in comparison with a
uniform load applied on an isotropic soil. Back-calculation of the total vertical stress increase
based on the recorded excess pore water pressures was conducted to circumvent this issue.
Figure 4.24 shows isobars of the recorded peak excess pore water pressures during
application of the footing load onto the composite foundation, as percentage of the applied
footing load P. The isobars were evaluated as an average of the measurements recorded
during the centrifuge tests JG_v5, JG_v7 and JG_v9. Due to irregularities in the construction
of the columns, a direct evaluation of the percentage of load transmitted to the column and to
the host soil is not possible. A good example of the reasons for this can be seen in Figure
4.27:
- an excess of sand was poured in columns B, D and E, which could not be
compacted and remained on the clay surface as a heap of dry sand, which was
subsequently pushed into the host soil by the footing during loading,
- column C was not filled up to the surface, causing a closure of the unfilled cavity
during the loading phase.
As a consequence, no ideal case, in which the top of the column coincides with the surface
of the soft soil specimen, could be reached.
Although the stress distribution directly underneath the footing could not be accurately
measured, the impact of the stone column on the load distribution with depth can be
identified by comparing the isobars of the peak values of the excess pore water pressures
measured under a vertically loaded circular footing (Figure 4.24) with the distribution of the
vertical stress increments caused by loading a circular footing of same dimensions and
resting on a homogeneous soil (Figure 4.25). The stone column causes a transfer of the load
applied on the surface into greater depths compared to a homogeneous soil.
4.4 Measurements conducted during the footing loading of a single stone column installed in
a specimen prepared in an (adapted) oedometer container
174
Figure 4.24: Isobars of peak values of excess pore pressures measured in the centrifuge
under a vertically loaded circular footing resting on top of a stone column.
Figure 4.25: Isobars of vertical stress increments under a vertically loaded circular footing
(after Grasshoff, 1978).
The distribution of the total vertical stress increment can be reformulated, using Equation
4.12, as a function of the excess pore water pressure recorded with depth:
[ (
( ) )]
[ (
) ] 4.13
with Δσz vertical stress increase
A, B pore pressure parameters according to Skempton (1954)
Δu recorded excess pore water pressure during the centrifuge tests
K0OC coefficient of earth pressure at rest of an over-consolidated soil
4 Results from the centrifuge tests
175
K0NC coefficient of earth pressure at rest of a normally consolidated soil
The results obtained are presented in Table 4.15, both as absolute values and relative to the
applied footing load P. The values of the parameter A assumed are shown in Table 4.12 and
Table 4.13, and those of K0OC in Table 4.10. B was set to 0.8, as before.
Due to the influence of the lateral loading applied through the lateral deformation of the stone
column at the depth of the transducers P1 to P3, the back-calculation was limited to the
deeper PPTs (P4 to P6, installed at a depth of 96 mm below the surface). The radial
deformation of the granular inclusion due to the footing load may be assumed to be
negligible at that depth, so that the coefficient of earth pressure K0OC may be used, although
transducer P4 is within the compaction zone.
Table 4.15: Back-calculated values of the vertical stress increases at the locations of the
PPTs P4, P5, P6 and P7 for tests JG_v1, JG_v5, JG_v7 and JG_v9.
PPT
Δσa = Δσz
Test JG_v1
(P = 80 kPa)
Test JG_v5
(P = 120.14 kPa)
Test JG_v7
(P = 145.44 kPa)
Test JG_v9
(P =119.67 kPa)
[kPa] [% P] [kPa] [% P] [kPa] [% P] [kPa] [% P]
P4 22.83 28.54 29.10 24.22 45.36 31.19 37.38 31.24
P5 17.26 21.58 27.78 23.13 43.71 30.05 36.69 30.66
P6 21.92 27.40 24.40 20.31 36.79 25.30 30.46 25.45
P7 14.97 18.71 19.49 16.22 30.51 20.98 17.63 14.73
Figure 4.26: Distribution of the total vertical stress increase as a function of the radial
distance from the stone column at 96 mm depth as a percentage of the applied
footing load P, and in comparison with the depth factor J4 according to Grasshoff
(1978).
0
5
10
15
20
25
30
35
0 5 10 15 20 25 30
Δσ
z[%
P]
Radial distance [mm]
JG_v5 JG_v7 JG_v9 J4 J4J4
Edge of the stone column
4.5 Measurements conducted during the footing load of a stone column group
176
Figure 4.26 shows a graphical representation of the values obtained. The qualitative stress
distribution caused by the footing load (Figure 4.26) is similar in all cases and indicates the
transmission of load by the stone column into deeper areas of the host soil (Figure 4.26).
The load transfer behaviour becomes even more visible by comparing the coefficients J4 for
the locations of transducers P4, P5, P6 and P7 with the back-calculated values of the vertical
stresses (Figure 4.26). The load transmitted increases by a factor of almost 3 due
to the stone column when the boundary conditions of the specimen are rigid (tests JG_v7
and JG_v9). Comparing the results obtained from tests JG_v7 (σc = 200 kPa) and
JG_v9 (σc = 100 kPa) shows that the OCR does not affect the percentage of the load applied
on the surface which is transmitted into depth.
4.5 Measurements conducted during the footing load of a stone
column group
Three tests have been conducted with groups of stone columns: tests JG_v8 (1 container)
and JG_v10 (2 containers). However, some issues occurred:
- an uneven sand layer formed between the surface of the soft soil specimen and
the footing during construction of the stone columns in test JG_v8 (Figure 4.27,
Section 4.5.1). Thus, the quantitative results of this test should be considered with
care, as the exact stress distribution directly under the footing is not clear,
- the stone column installation tool clogged in one specimen during test JG_v10
(a / dsc = 2.5) during the installation of the first two columns of the group (columns
A and B, Figure 3.51). The installation of the rest of the inclusions was abandoned
and no loading phase was conducted. The specimen was solely used to conduct
site investigation using a T-Bar (curve denoted as su,JG_v10,B in Figure 4.4).
The PPTs from which the data discussed in the upcoming section were obtained were
located as follows:
- P1, P2 and P3 at a depth of 30 mm below the surface of the soft soil and at a
radial distance of 12 mm, 18 mm and 30 mm, from the axis of the centre column,
respectively,
- P4, P5 and P6 at a depth of 80 mm below the surface of the soft soil and at a
radial distance of 12 mm, 18 mm and 30 mm, from the axis of the centre column,
respectively, and
- P7 at a depth of 55 mm below the surface of the soft soil and at a radial distance
of 18 mm from the axis of the centre column.
4.5.1 Test JG_v8 (a / dsc = 2 [-])
The results of the measurements conducted, while the stone column group (a / dsc = 2 [-])
was loaded during test JG_v8, are presented in Figure 4.28 and in Table 4.16. Transducer
4 Results from the centrifuge tests
177
P4 was located 150 mm away from the axis of the centre column in test JG_v8, as opposed
to 12 mm from the axis of the centre column during test JG_v10.
The load-settlement behaviour (Figure 4.28 b) turns out to be pseudo-linear. This can be
explained by the fact that too much sand was poured into the installation tool for columns B,
D and E (Figure 4.27), which meant that the excess that could not be compacted into the
cylindrical hole in the host soil remained on the surface as a heap of dry sand on top of these
three columns. These heaps were pressed into the clay by the footing, thus forming a sand
layer between the footing and the clay. On the contrary, not enough sand was poured in for
the construction of column C, which caused an empty cavity between top of the column and
surface of the soft soil specimen.
Figure 4.27: Excess sand shown on top of columns B, D and E within the footprint of the
footing on the surface of the clay model after test JG_v8.
The installation depth of the transducers influences the response time of their reaction to
loading. P1, P2 and P3 exhibit the fastest reaction, while the time needed for P5 and P6 to
react is significantly longer. The response time of P7 is between that of P2 and that of P5,
which makes sense given the fact that P2 is installed 30 mm, P7 55 mm and P5 80 mm
under the surface of the clay specimen.
Column B Column C
Column DColumn E
Column A
4.5 Measurements conducted during the footing load of a stone column group
178
(a)
(b)
(c)
Figure 4.28: Loading of a stone column group in a specimen pre-consolidated up to 100 kPa
(JG_v8) (a) excess pore water pressures (b) evolution of the footing load
(c) deformation controlled footing settlement.
0
10
20
30
40
50
0 500 1000 1500Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P1 P2 P3 P4 P5 P6 P7
0
20
40
60
80
100
120
140
0 500 1000 1500
Fo
oti
ng
lo
ad
ing
[k
Pa
]
Time [s]
-20
-15
-10
-5
0
0 500 1000 1500
Fo
oti
ng
se
ttle
me
nt
[mm
]
Time [s]
4 Results from the centrifuge tests
179
Table 4.16: Response of the PPT to the applied footing load on a stone column group during
test JG_v8.
PPT
Maximal excess
pore water pressure
Δumax [kPa]
Maximal load
applied on the
footing Pmax [kPa]
Δumax / Pmax [-]
P1 (z = 30 mm) 43.52 124.01 35.09
P2 (z = 30 mm) 31.44 124.01 25.35
P3 (z = 30 mm) 34.14 124.01 27.53
P4 (z = 80 mm) 5.39 124.01 4.35
P5 (z = 80 mm) 27.98 124.01 22.56
P6 (z = 80 mm) 27.22 124.01 21.95
P7 (z = 55 mm) 33.05 124.01 26.65
Table 4.16 shows that the differences in magnitude of excess pore water pressure generated
by the footing load are less distinct with depth than when loading a single stone column
(Section 4.4). The ratio Δumax / Pmax drops in this case solely by 15 % to 25 % between
a depth of 30 mm and 80 mm, as opposed to a drop of about 40 to 55 % from a depth of
48 mm to 96 mm for a single stone column (Table 4.5). This would suggest a different load
transfer mechanism for a single stone column compared with a group of stone columns as a
larger part of the load applied to the surface is transferred to depth in this particular group of
stone columns. Due to the issues encountered during the installation of the stone columns
during this test, however, the interpretation will be conducted based on the results from
test JG_v10.
4.5.2 Test JG_v10 (a / dsc = 2 [-])
The results of the measurements conducted during the loading phase of the stone column
group (a / dsc = 2 [-]) for test JG_v10 are presented in Figure 4.30 and in Table 4.17. The
testing procedure (Table 3.11) was slightly amended, as the centrifuge was stopped after the
installation of the stone columns in order to remove the sand particles from the surface of the
clay so that the surface would be level and the stress distribution could be measured.
Unfortunately, the tool platform was positioned incorrectly with respect to the drum, which
caused the footing used for the loading phase not to be in the position planned (Figure 4.29).
The results shown in Table 4.17 indicate that the load transfer behaviour with depth in case
of a stone column group is similar to the case of a single stone column. The ratio Δumax / Pmax
drops by 80 to 95% for an increase of depth of 50 mm (from 30 mm to 80 mm below the
surface of the clay specimen)as opposed to test JG_v8. This would indicate that a lower part
of the load applied on the surface is transmitted into depth. However, the increased drainage
capacity of the stone column group compared to the case of a single stone column causes
an accelerated rate of dissipation of the excess pore water pressures (Figure 4.34). This
complicates significantly an accurate estimation of the stress levels based on the recorded
4.5 Measurements conducted during the footing load of a stone column group
180
excess pore water pressures as it is not possible to assess which part of the excess pore
water pressure was already dissipated when the peak footing load was reached.
Figure 4.29: Position of the footing used for the loading phase during test JG_v10 (a = 24
mm).
Table 4.17: Response of the PPT to the applied footing load on a stone column group during
test JG_v10.
PPT
Maximal excess
pore water pressure
Δumax [kPa]
Maximal load
applied on the
footing Pmax [kPa]
Δumax / Pmax [%]
P1 (z = 30 mm) 44.82 142.01 31.56
P2 (z = 30 mm) 35.65 142.01 25.10
P3 (z = 30 mm) 23.30 142.01 16.41
P4 (z = 80 mm) 22.36 142.01 15.75
P5 (z = 80 mm) 19.86 142.01 13.98
P6 (z = 80 mm) 25.64 142.01 18.06
P7 (z = 55 mm) 29.35 142.01 20.67
The drainage effect of the stone column group can be observed as the magnitude of the
ratios Δumax / Pmax is significantly lower in case of the stone column group (Figure 4.31) than
in case of a single stone column (Figure 4.21 and Figure 4.22). The effect of the incorrect
positioning of the platform can be noted when considering the drop of normalised excess
pore water pressure from P2 to P3 (Figure 4.31). The reason why the measurements made
by transducer P6 are higher than those made by transducer P3 is unknown.
4 Results from the centrifuge tests
181
(a)
(b)
(c)
Figure 4.30: Test JG_v10: (a) excess pore water pressures (b) evolution of the footing load
(c) footing settlement during the loading phase of a stone column group.
0
10
20
30
40
50
0 500 1000 1500Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P1 P2 P3 P4 P5 P6 P7
0
20
40
60
80
100
120
140
160
0 500 1000 1500
Fo
oti
ng
lo
ad
ing
[k
Pa
]
Time [s]
-20
-15
-10
-5
0
0 500 1000 1500
Fo
oti
ng
se
ttle
me
nt
[mm
]
Time [s]
4.5 Measurements conducted during the footing load of a stone column group
182
Figure 4.31: Distribution of the excess pore water pressure with increasing radial distance to
the axis of the centre stone column at depths of 30 mm and of 80 mm as a
percentage of the applied footing load P (test JG_v10).
Figure 4.32 and Figure 4.33 show the dissipation with time of the excess pore water
pressures after the peak footing load at depths of 30 mm and 80 mm, respectively. In both
cases, an accelerated dissipation can be noted 250 s after the peak footing load has been
reached (which corresponds to t = 0 s). The influence of the stone column group on the
drainage length is visible in the magnitude of the dissipation of the excess pore water
pressure at a depth of 30 mm as a diminution of approximately 23 kPa is measured 6 mm
away from the edge of the centre column, while this drop is only of 17 kPa and 12 kPa when
measured 12 mm and 24 mm away from the edge of the centre column, respectively.
A similar observation to that made at a depth of 96 mm around a single stone column can be
made at a depth of 80 mm at distances of 6 mm and 12 mm from the edge of the centre
column: the excess pore water pressure dissipation remains constant up to a distance of
12 mm from the edge of the column and reaches approximately 8 kPa. The uncertainty about
the measurements made by transducer P6 prevents a quantitative analysis of the results
from being carried out.
0
5
10
15
20
25
30
35
0 5 10 15 20 25 30
No
rma
lis
ed
ex
ce
ss
po
re w
ate
r p
res
su
re [
% P
]
Radial distance [mm]
z = 30 mm z = 80 mm
Edge of the stone column
P2
P1
P3
P4P5
P6
4 Results from the centrifuge tests
183
(a)
(b)
Figure 4.32: Loading of a stone column group in a specimen pre-consolidated up to 100 kPa
(test JG_v10), dissipation with time of the excess pore water pressures at a
depth of 30 mm around the stone column (a) from 0 s to 2000 s and (b) from
3000 s to 7000 s after reaching the peak footing load (which corresponds to
t = 0 s).
0
5
10
15
20
25
30
35
40
45
50
0 10 20 30 40Ex
ce
ss
po
re w
ate
r p
res
su
res
[k
Pa
]
Radial distance [mm]
t = 0 s t = 250 s t = 500 s t = 750 st = 1000 s t = 1500 s t = 2000 s
Edge of the stone column
0
1
2
3
4
5
6
7
8
9
10
0 10 20 30 40Ex
ce
ss
po
re w
ate
r p
res
su
res
[k
Pa
]
Radial distance [mm]
t = 3000 s t = 4000 s t = 5000 s t = 6000 s t = 7000 s
Edge of the stone column
4.5 Measurements conducted during the footing load of a stone column group
184
(a)
(b)
Figure 4.33: Loading of a stone column group in a specimen pre-consolidated up to 100 kPa
(test JG_v10), dissipation with time of the excess pore water pressures at a
depth of 80 mm around the stone column (a) from 0 s to 2000 s and (b) from
3000 s to 7000 s after reaching the peak footing load (which corresponds to
t = 0 s).
The increased drainage performance of a stone column group compared to a single stone
column can be noted by comparing the rates of dissipation of the excess pore water
pressures with time. The values reached in case of a stone column group are significantly
higher as they range from 0.045 kPa / s to 0.09 kPa / s at a depth of 30 mm and reach
approximately 0.035 kPa / s at a depth of 80 mm. Moreover, a pseudo-constant rate of
dissipation is reached at t = 1500 s after the peak footing load has been applied, opposed to
t = 2000 s in case of a single stone column.
0
5
10
15
20
25
30
35
40
45
50
0 10 20 30 40Ex
ce
ss
po
re w
ate
r p
res
su
res
[k
Pa
]
Radial distance [mm]
t = 0 s t = 250 s t = 500 s t = 750 st = 1000 s t = 1500 s t = 2000 s
Edge of the stone column
0
1
2
3
4
5
6
7
8
9
10
0 10 20 30 40Ex
ce
ss
po
re w
ate
r p
res
su
res
[k
Pa
]
Radial distance [mm]
t = 3000 s t = 4000 s t = 5000 s t = 6000 s t = 7000 s
Edge of the stone column
4 Results from the centrifuge tests
185
Figure 4.34: Loading of a stone column group in a specimen pre-consolidated up to 100 kPa
(test JG_v10): rate of dissipation of excess pore water pressures with time after
reaching the peak footing load (which corresponds to t = 0 s).
4.6 Comparison of the measurements around a single stone
column and inside a stone column group
Although the transducers have not been installed at similar locations for all of the
investigations, a comparison between the behaviour under loading of a single stone column
and of a group of stone columns reveals some interesting insights. Figure 4.35 and
Figure 4.36 show the evolution of the excess pore water pressures during the loading
of a single stone column (on the example of JG_v9) and of a stone column group
(test JG_v10), respectively.
For clarity, the installation depths of the different transducers are recalled:
- in case of a single stone column (test JG_v9):
o P1, P2 and P3 at a depth of 48 mm under the surface of the clay
specimen,
o P4, P5 and P6 at a depth of 96 mm,
o P7 at a depth of 140 mm directly under the column,
- in case of a stone column group (test JG_v10):
o P1, P2 and P3 at a depth of 30 mm under the surface of the clay
specimen,
o P4, P5 and P6 at a depth of 80 mm,
o P7 at a depth of 55 mm.
The magnitudes of the excess pore water pressures recorded by transducers P1, P2 and P3
are very similar in both cases, although the installation depths and the applied footing load
(119.67 kPa for test JG_v9 and 142.01 kPa for test JG_v10) are different. This shows the
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 1000 2000 3000 4000 5000 6000 7000
Δu
/ Δ
t [k
Pa
/ s
]
Time [s]
P1 P2 P3 P4 P5 P6
4.6 Comparison of the measurements around a single stone column and inside a stone
column group
186
efficiency of a stone column group in terms of dissipation of excess pore water pressures.
The shallower installation depth of the transducers, coupled with the higher footing load
acting on the stone column group should otherwise have led to the appearance of higher
excess pore water pressure than in the test featuring a single stone column.
Figure 4.35: Excess pore water pressures during the footing load test on a single stone
column (test JG_v9).
Figure 4.36: Excess pore water pressures during the footing load test on a stone column
group (test JG_v10). The maximum load was reached at 1000 s.
The drainage performance of a stone column group is highlighted by a comparison of the
rates of dissipation of the excess pore water pressures for a single inclusion (Figure 4.39)
and for a group of inclusions (Figure 4.38). These rates are significantly higher in the case of
a stone column group as they reach values of up to 0.09 kPa / s immediately after the peak
footing load has been applied, while the maximal rate of dissipation around a stone column is
0
10
20
30
40
50
0 1000 2000 3000 4000 5000 6000 7000 8000
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P1 P2 P3 P4 P5 P6 P7
0
10
20
30
40
50
0 1000 2000 3000 4000 5000 6000 7000 8000Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P1 P2 P3 P4 P5 P6 P7
4 Results from the centrifuge tests
187
only approximately 0.07 kPa / s in case of a single stone column, which corresponds to a
29 % higher performance of the stone column group. Moreover, a pseudo-constant value of
the rate of dissipation is reached after 1500 s in the case of a group, as opposed to 2000 s
for a single inclusion.
Figure 4.37: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v9): rate of dissipation of excess pore water pressures with time after
reaching the peak footing load (which corresponds to t = 0 s).
Figure 4.38: Loading of a stone column group in a specimen pre-consolidated up to 100 kPa
(test JG_v10): rate of dissipation of excess pore water pressures with time after
reaching the peak footing load (which corresponds to t = 0 s).
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 1000 2000 3000 4000 5000 6000 7000
Δu
/ Δ
t [k
Pa
/s]
Time [s]
P1 P2 P3 P4 P5 P6
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 1000 2000 3000 4000 5000 6000 7000
Δu
/ Δ
t [k
Pa
/ s
]
Time [s]
P1 P2 P3 P4 P5 P6
4.7 Electrical impedance measurements
188
4.7 Electrical impedance measurements
Electrical impedance measurements were conducted to determine the distribution of the
installation effects around stone columns with depth. The electrical impedance needle was
inserted at a speed of 3 mm /s. The measurements were conducted using a frequency of 200
kHz. Other frequencies were tried as well (50 kHz and 100 kHz), but the precision of the
measurements was best for a frequency of 200 kHz. The voltage was 1 V. The different
zones identified by Weber (2008) are already described in Section 3.3, but Figure 4.39 is
included here to recall the distribution of the 4 zones detected.
Figure 4.39: Installation effects around a stone column at a model depth of 40 mm @ 50 g
(Weber, 2008).
4.7.1 Measurements around a single stone column
The first successful implementation of the needle for electrical impedance measurements
occurred during test JG_v5. The results obtained are presented in the following section,
while those from test JG_v9 can be found in Appendix 8.6.
Figure 4.40: Positions of the needle insertion points around a single stone column, and
extent of the zones 2 and 3 according to Weber (2008).
Figure 4.40 shows the positions of the points where the electrical impedance needle was
inserted around a single stone column, together with the extent of zones 2 and 3, according
4 Results from the centrifuge tests
189
to Weber (2008). Figure 3.55 shows the positions of the Reference Points (denoted as RP1
and RP2 in Figure 4.41 and Figure 4.44). The PPTs were installed in the upper half of the
specimen (Figure 3.53) and have thus no influence on the measurements conducted with the
electrical impedance needle. The order in which the measurements were conducted was the
same for all tests: the needle was first inserted at reference points RP1 and RP2 and
subsequently at points A, B, C, D, E and F in this order. The tip of the needle was immersed
for 5 minutes in the ultrasonic bath (Section 3.4.5) after each insertion. Due to the scatter of
the results, a moving average (interval of 6 measurements with 4 measurements per
seconds) was calculated in order to improve the readability of the graphs.
4.7.1.1 Measurements conducted in specimens consolidated up to
σ’v = 100 kPa (test JG_v5)
The needle was inserted after the installation of the stone column and subsequent
consolidation of the specimen in this case. Figure 4.41 shows the measurements at the two
reference points RP1 and RP2, located 88 mm away from the axis of the stone column. No
significant differences could be found in the two profiles.
Figure 4.41: Impedance recorded at reference points RP1 and RP2 during test JG_v5.
Figure 4.42 shows the impedance values measured at points A, B and C. No significant
difference between the recorded values at these three points can be noted. Unlike the profile
obtained at the Reference Points, the slight decrease of the measured impedance between
10 mm and 40 mm depth (denoted as A in Figure 4.41) cannot be seen. It is rather surprising
that the measurements conducted at the three points A, B and C show similar values and
distribution, as point C is located only 15 mm away from the axis of the column. This location
would be expected to be on the borderline of the compaction zone (zone 3 according to
Weber (2008), Figure 4.39). Thus the impedance would be expected to rise in accordance
with a higher density in the host soil.
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
RP1 RP2
A
4.7 Electrical impedance measurements
190
Figure 4.42: Impedance recorded at the points A, B and C during test JG_v5.
Figure 4.43 shows the impedance values measured at point D, E and F. A significant
increase in the impedance can be noted in the upper third of the zones 2 and 3 around the
stone column. This indicates that compaction of the host soil was greater near to the surface
due to the installation of the stone column. The recorded values also diminish below about
40 mm at point F, which is in zone 2 closest to the stone column. This might indicate a
stronger vertical reorganisation of the clay particles.
Figure 4.43: Impedance recorded at the points D, E and F during the test JG_v5.
The analytical solutions proposed by the Strain Path Method (Baligh, 1985) and the Shallow
Strain Path Method (Sagaseta & Whittle, 2001) (Section 2.9.2) suggest that a reorganisation
of the clay platelets could occur up to a distance of 6 times the radius of the inclusion. The
impedance measured at point F (Figure 4.43) is significantly lower than at the other points
investigated. Located 7 mm away from the axis of the stone column, point F is within zone 2
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
Point A Point B Point C
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
Point D Point E Point F
4 Results from the centrifuge tests
191
(Figure 4.39), whereas Weber (2008) showed a clear reorganisation of the clay platelets
parallel to the axis of the inclusion. The absence of any obvious increase in the recorded
impedance at point C is also thought to be due to a vertical reorganisation of the clay
particles. The results show that the impedance measured with the electrical impedance
needle is obviously sensitive to the organisation of the particles.
4.7.1.2 Measurements conducted in a specimen consolidated up to
σ’v = 200 kPa (test JG_v5)
The results obtained from the insertion of the electrical impedance needle during test JG_v5,
in the specimen consolidated up to 200 kPa, are presented in this section. In this case, a
footing load of 120.14 kPa was applied after the installation of the stone column. The needle
was inserted after the footing loading phase and subsequent dissipation of the excess pore
water pressures.
Figure 4.44 shows the measurements taken at the two reference points RP1 and RP2,
located 88 mm away from the axis of the stone column. No significant variation of the
measured impedance, either over the depth or between the two points, can be noted.
The values are higher than in the specimen consolidated up to 100 kPa (Figure 4.41),
which is consistent with the assumption that the impedance increases in denser soils
(Cho et al., 2004).
Figure 4.44: Impedance recorded at the reference points RP1 and RP2 during test JG_v5.
Figure 4.45 and Figure 4.46 show the results of the impedance measurements conducted at
points A, B, C, D, E and F. These points, with the exception of point A, are located within the
imprint area of the footing (Figure 4.40). Therefore the measured profiles start at a depth of
17 mm, as the electrical impedance needle was inserted after the footing loading (Table
3.12).
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
RP1 RP2
4.7 Electrical impedance measurements
192
No significant difference can be noted between the measurements conducted at points A, B
and C (Figure 4.45). The reason for the drop in measured impedance at point C at a depth of
60 mm (Figure 4.45) could be a local reorganisation of the clay platelets. The explanation of
the lack of increase in the impedance at points B and C is the same as in Section 4.7.1.1,
which is a vertical reorganisation of the clay platelets as suggested by Baligh (1985) and
Sagaseta & Whittle (2001). However, no influence of the footing load can be seen either.
This tends to indicate that the measured impedance is more sensitive to the micro-
mechanical reorganisation of the clay platelets than to the macro-mechanical increase of
density. Points B and C are located within the area where a footing load was applied, which
caused compaction of the host soil. However, point B is located 18 mm, and point C 15 mm
away from the axis of the column. This implies that the effect of bulging of the stone column
on the density of the host soil is not as extreme as at point D, E and F (Figure 4.46) and the
vertical reorganisation of the clay platelets has a higher influence on the measured
impedance than the increase in density due to the vertical loading. This interpretation could
be confirmed by Environmental Scanning Electron Microscope (ESEM) observations
(Section 5.4).
Figure 4.45: Impedance recorded at the points A, B and C during test JG_v5.
A clear influence of the footing load can be noted at a distance of 12 mm from the axis of the
stone column (Figure 4.46, point D). The depth of the increase of the recorded values of the
impedance is significantly higher than in the case of an unloaded inclusion (Section 4.7.1.1).
Even though the footing loading has not been carried out up to failure, the load will have
caused lateral deformation of the column. This leads to a further compression of the host soil
and to an increase in the measured impedance.
The measured impedance at point F below a depth of 90 mm is remarkably constant, and
is significantly lower than the values measured at the reference points RP1 and RP2
(Figure 4.44). As point F is located within zone 2 according to Weber (2008), where a vertical
reorganisation of the clay platelets is observed, this tends to indicate that the bulging of the
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
Point A Point B Point C
4 Results from the centrifuge tests
193
stone column due to vertical loading caused a reorganisation of the clay platelets up to a
depth of 90 mm, and that the particles are vertical below that depth.
Figure 4.46: Impedance recorded at the points D, E and F during test JG_v5.
4.7.2 Measurements around a stone column group (test JG_v8)
In addition to the measurements around single stone columns, impedance measurements
were also conducted around a stone column group (test JG_v8) after installation of the stone
columns and dissipation of the excess pore water pressures and prior to the footing loading.
The needle was first inserted at reference points RP1 and RP2 and subsequently at points
A2, B2, C2, D2, E2, F2, G2, H2 and I2 (Figure 3.52), in that order. The tip of the needle was
immersed for 5 minutes in the ultrasonic bath (Section 3.4.5) after each insertion. All
investigations were conducted in specimens that had been pre-consolidated up to σ’v = 100
kPa.
Figure 4.48 shows the impedance recorded at the reference points RP1 and RP2, located
100 mm away from the axis of the column (Figure 3.52), during tests JG_v8. No clear pattern
can be identified in the results. This is thought to be due to the fact that the needle was not
inserted into the soft soil vertically, but at an angle of 7°. As a consequence, the sensitivity of
the measurements to the micro-mechanical organisation of the clay platelets had a high
impact on the measured results.
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
Point D Point E Point F
4.7 Electrical impedance measurements
194
Figure 4.47: Positions of the needle insertion points around a stone column group
(test JG_v8, a / dsc = 2 [-].
Figure 4.48: Impedance recorded at the reference points RP1 and RP2 during test JG_v8.
The impedance recorded at points A2, B2 and C2, located outside the imprint area of the
footing are presented in Figure 4.49. The impedance measured at point A2 is higher than at
the other two points below 40 mm. The recordings at point C2 are closer to those at point B2,
and the values of the impedance at point B2 are comparable with those obtained at points A,
B and C for the investigations around a single stone column (Figure 4.45). The tip of the
needle clogged partially, although the tool was routinely cleaned in the ultrasonic bath, for
the test at point B2, which may explain the smaller impedance values. The light decrease of
the impedance recorded between points A2 and C2 below 20 mm is thought to be due to a
0
20
40
60
80
100
120
0 0.005 0.01 0.015 0.02
De
pth
[m
m]
Impedance [Ohm]
RP1 RP2
4 Results from the centrifuge tests
195
vertical reorganisation of the clay platelets caused by the installation of the stone column
group.
Figure 4.49: Impedance recorded at the points A2, B2 and C2 during test JG_v8.
The results of the impedance measurements conducted at points D2, E2 and F2 are
presented in Figure 4.50. The measurements at point D2 (25 mm away from the axis of the
column A) are similar to the values obtained at point E for the investigations around a single
stone column (9 mm away from the axis of the column, Section 4.7), with impedance values
of about 0.006 Ohm. This indicates a significantly wider extent of the reorganisation of the
clay platelets in the case of a stone column group.
Figure 4.50: Impedance recorded at the points D2, E2 and F2 during test JG_v8.
The wide spatial extent of particle reorganisation is confirmed by the impedance values
recorded at points E2, F2, G2, H2, I2 and J2 (Figure 4.50 and Figure 4.51). These are similar
0
20
40
60
80
100
120
0 0.005 0.01 0.015 0.02
De
pth
[m
m]
Impedance [Ohm]
Point A2 Point B2 Point C2
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
Point D2 Point E2 Point F2
4.8 Summary of the conducted modelling under enhanced gravity
196
in magnitude and variation to the recordings conducted at point F around a single stone
column (Figure 4.46), located in zone 2 (Figure 4.39), where Weber (2008) found the clay
platelets to be organised parallel to the stone column axis.
Figure 4.51: Impedance recorded at the points G2, H2, I2 and J2 during test JG_v8.
4.8 Summary of the conducted modelling under enhanced gravity
Centrifuge tests were carried out in order to model boundary value problems with stone
columns and represent a central part of this work. This section presents a short summary of
the insights gained about the influence of the OCR, on the load transfer behaviour of stone
columns, and on the microscopic phenomena caused by the installation of granular
inclusions.
It has been shown that the over-consolidation ratio of the soft soil plays a role in the
magnitude of the excess pore water pressures generated during the installation phase of the
inclusions, as a greater OCR leads to higher excess pore water pressures. The depth under
the surface of the soft soil also plays a role, as the deeper the elements the higher the
excess pore water pressures.
The interpretation of the results obtained from the footing loads showed that there was load
transfer to depth in the presence of a stone column, as the back-calculated vertical load
increased at a depth of 96 mm under the surface of the clay to a factor of about three times
greater with a granular inclusion, than without (Figure 4.26).
A consistent trend between a wide range of measurements with the electrical impedance
needle indicates that the clay particles underwent microscopic reorganisation up to a
distance of 30 mm from the axis of a stone column (corresponding to 5 times the radius of
the inclusion). The extent of the macroscopic mechanical installation effects will be
investigated in more detail in the following chapter.
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
Point G2 Point H2 Point I2 Point J2
5 Complementary investigations
197
5 Complementary investigations
5.1 Oedometer tests conducted on samples extracted from the soil
model used for the centrifuge test JG_v9
Four samples were extracted from the soft soil bed, two each in a horizontal and a vertical
direction, after test JG_v9 (σc = 100 kPa), in order to conduct oedometer tests and
to investigate the anisotropy of the host soil. The oedometer cells have a diameter of
56.419 mm and a height of 20 mm.
The extraction positions are shown in a plan view in Figure 5.1. The specimens
JG_v9 – Oedo 1 and JG_v9 – Oedo 3 were extracted vertically from a maximum depth of
120 mm (denoted as [1] in Figure 5.2) while the samples JG_v9 – Oedo 2 and JG_v9 – Oedo
4 were extracted horizontally from a maximum depth of 90 mm (denoted as [2] in Figure 5.2).
The specimens were extracted from such locations so that they were not influenced by the
installation or by the loading of the stone column (Figure 5.2). Metallic sampling cylinders
were therefore pushed into the clay sample approximately 2 hours after the end of the
centrifuge test, and the soil around the pots was cut using a thin metallic wire so that the
specimens were almost undisturbed. The sampling cylinders were subsequently directly
mounted into an oedometer cell.
Figure 5.1: Plan view of the extraction positions of the specimens for oedometer tests (test
JG_v9).
5.1 Oedometer tests conducted on samples extracted from the soil model used for the
centrifuge test JG_v9
198
Figure 5.2: Cross-section of the extraction positions of the specimens for oedometer tests
(test JG_v9).
Figure 5.3: Distribution of the over-consolidation ratio of the specimens used
for the oedometer tests during the centrifuge test.
Figure 5.3 shows the profiles of the over-consolidation ratios in the vertical and horizontal
directions (denoted as OCR (vertical) and OCR (horizontal), respectively) during the
centrifuge test. A constant g level throughout the specimen was assumed for the calculation
of the OCR. A maximal vertical effective stress σ’v,max of 100 kPa was used in the vertical
case and K0 conditions were assumed. With a K0 value of 0.585 (Table 4.10), a maximal
horizontal vertical stress σ’h,max of 58.5 kPa is obtained. The over-consolidation ratio of the
specimens extracted vertically ranged from 2.4 to 2 (denoted as V in Figure 5.3) of that of the
specimens extracted horizontally from 4.0 to 1.5 (denoted as H in Figure 5.3). The extraction
procedure caused relatively small disturbance of the specimens, and thus it was hoped that
the stress history would be conserved in the best possible manner. The specimens were
0
20
40
60
80
100
120
140
160
0 5 10 15 20
De
pth
[m
m]
OCR [-]
OCR (vertical) OCR (horizontal)
HH
V
5 Complementary investigations
199
loaded up subsequently to σ’v = 200 kPa in the oedometer, before undergoing two unloading-
reloading cycles from σ’v = 200 kPa to σ’v = 50 kPa and back to σ’v = 200 kPa.
The evolution of the void ratio under one dimensional vertical loading is shown in Figure 5.4.
The quantitative difference in void ratio between the two samples extracted in the vertical
direction (denoted as JG_v9 – Oedo 1 and JG_v9 – Oedo 3 in Figure 5.4) remains more or
less constant throughout the loading steps, and is only about 7 %. This may be due to the
higher g level acting during the centrifuge test at the extraction position of the specimen
JG_v9 – Oedo 1, due to the greater radial distance from the centre of the centrifuge. The
results from the two samples extracted in the horizontal direction almost superpose in both
figures.
Figure 5.4: Evolution of the void ratio with one dimensional loading in an oedometer.
As long as the specimens were pre-consolidated vertically up to 100 kPa (which corresponds
to a horizontal load of 58 kPa assuming K0 conditions), the compression index CC is to be
calculated for the load step ranging from 100 kPa to 200 kPa. The values obtained are
presented in Table 5.1.
Table 5.1: Compression indexes CC obtained from the oedometer tests.
Load
interval
[kPa]
CC, vertical [-] CC, horizontal [-]
OCR [-] JG_v9 –
Oedo 1
JG_v9 –
Oedo 3
JG_v9 –
Oedo 2
JG_v9 -
Oedo 4
100 - 200
(loading) 0.045 0.044 0.043 0.045 1.0
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1 10 100 1000
e [
-]
Vertical effective stress [kPa]
JG_v9 - Oedo 1 (Vertical) JG_v9 - Oedo 2 (Horizontal)
JG_v9 - Oedo 3 (Vertical) JG_v9 - Oedo 4 (Horizontal)
σ'v,max
σ'h,max
5.1 Oedometer tests conducted on samples extracted from the soil model used for the
centrifuge test JG_v9
200
The swelling index for the unloading-reloading path CS can be calculated for the two
unloading-reloading cycles for 200 kPa to 50 kPa and back to 200 kPa. The results obtained
are presented in Table 5.2. The values of the compression indexes CC and CS are
remarkably similar for all tests. The impact of the over-consolidation ratio on the loading
behaviour can be noted as the compression indexes CS of the samples extracted horizontally
are lower than those of the samples extracted vertically, which is due to the higher over-
consolidation of the specimens (Figure 5.3).
Table 5.2: Compression indexes CS obtained from the oedometer tests.
Load
interval
[kPa]
CS, vertical [-] CS, horizontal [-]
OCR [-] JG_v9 –
Oedo 1
JG_v9 –
Oedo 3
JG_v9 –
Oedo 2
JG_v9 -
Oedo 4
200 - 50
(unloading) 0.012 0.012 0.011 0.011 4.0
50 - 200
(reloading) 0.015 0.014 0.013 0.013 1.0
200 - 50
(unloading) 0.012 0.012 0.011 0.011 4.0
50 - 200
(reloading) 0.016 0.015 0.014 0.014 1.0
Figure 5.5: Distribution of the confined stiffness moduli as a function of the vertical effective
stress.
The values obtained of the confined stiffness moduli ME, for the different load intervals, are
presented in Table 5.3. The vertical and horizontal confined stiffness moduli are denoted as
ME, v and ME,h, respectively.
0
500
1000
1500
2000
2500
3000
0 50 100 150 200
ME
[kP
a]
Vertical effective stress σ'v [kPa]
JG_v9 - Oedo 1 (Vertical) JG_v9 - Oedo 2 (Horizontal)
JG_v9 - Oedo 3 (Vertical) JG_v9 - Oedo 4 (Horizontal)
σ'v,maxσ'h,max
5 Complementary investigations
201
The calculated stiffnesses in the horizontal direction tend to be higher than the values
calculated in the vertical direction. As stated earlier, this is due to the greater over-
consolidation of the samples extracted horizontally (Figure 5.3) compared to those extracted
vertically. However, this difference is only about 10 % and may be considered irrelevant for
engineering purposes (Figure 5.5 and Figure 5.6).
Table 5.3: Vertical (ME,v) and horizontal (ME,h) confined stiffness moduli obtained from the
oedometer tests and values of the over-consolidation ratios for samples
extracted vertically (OCRv) and horizontally (OCRh).
Load
interval
[kPa]
ME, v [kPa] OCRv
[-]
ME, h [kPa] OCRh
[-] JG_v9 –
Oedo 1
JG_v9 –
Oedo 3 Average
JG_v9 –
Oedo 2
JG_v9 -
Oedo 4 Average
0 - 12.5
(loading) 982.7 755.6 869.1 8.0 1082.8 1164.4 1123.6 4.64
12.5 - 25
(loading) 774 871.1 822.5 4.0 841.8 892.9 867.3 2.32
25 - 50
(loading) 1057.1 1073.0 1065.0 2.0 1037.3 1077.6 1057.5 1.16
50 - 100
(loading) 1490.3 1652.9 1571.6 1.0 1432.7 1567.4 1500.0 1.0
100 - 200
(loading) 2430.1 2164.5 2297.3 1.0 2433.1 2472.2 2452.6 1.0
200 - 50
(unloading) 12448.1 12345.7 12396.9 4.0 14018.7 13157.9 13643.6 4.0
50 - 200
(reloading) 10416.7 10380.6 10398.6 1.0 11673.2 11278.2 11475.7 1.0
200 - 50
(unloading) 12244.9 12244.9 12244.9 4.0 13953.5 13333.3 13643.4 4.0
50 - 200
(reloading) 9460.7 9434.0 9448.8 1.0 10526.3 10238.9 10382.6 1.0
The tangent stiffness for primary oedometer loading for a reference effective stress of
100 kPa is of particular interest for the numerical modelling using the Hardening Soil Model
(Section 6.3.2.6). Figure 5.7 shows the distribution of the mean vertical strain with increasing
vertical loading. The definition of the tangent stiffness for primary oedometer loading Eoedref is
displayed in Figure 5.8. The slopes of the tangents to the load-strain curve were determined
to be 1900 kPa in both vertical and horizontal cases, for a vertical effective stress of 100 kPa
(Figure 5.7). Another important parameter for the numerical modelling is the unloading-
reloading stiffness Eurref, which is set to an average value of 13000 kPa (Table 5.3).
5.1 Oedometer tests conducted on samples extracted from the soil model used for the
centrifuge test JG_v9
202
Figure 5.6: Distribution of the mean vertical (ME, v, average) and horizontal (ME, h, average) confined
stiffness moduli as a function of the vertical effective stress.
Figure 5.7: Distribution of the mean settlements for the samples extracted in the vertical and
horizontal directions with one-dimensional loading in an oedometer.
The calculated values of the coefficient of permeability k, computed based on the time-
settlement curve of each load step (as suggested by Lang et al., 2007), are displayed in
Figure 5.9. The distribution of the permeability (Figure 5.9) has a similar form in the
horizontal and vertical directions. No significant difference could be measured in terms of
coefficient of permeability between the horizontal and the vertical directions. The first point
obtained for specimen JG_v9 – Oedo 4 may be regarded as an outlier, inherent to the
uncertainty of laboratory investigations.
These investigations lead to the conclusion that although some anisotropy could be detected
in terms of stiffness, no significant influence of the fabric on the void ratio distribution, or on
the values of the coefficient of permeability, could be ascertained.
0
500
1000
1500
2000
2500
3000
0 50 100 150 200
ME
[kP
a]
Vertical effective stress σ'v [kPa]
ME vertical ME horizontalME, v, averageME, h, average
σ'v,maxσ'h,max
0
2
4
6
8
10
12
14
0 50 100 150 200
Ve
rtic
al s
tra
in Δ
h /
h0
[%]
Vertical effective stress σ'v [kPa]
Settlement (vertical) Settlement (horizontal)
5 Complementary investigations
203
Figure 5.8: Definition of Eoedref from oedometer test results (after Brinkgreve & Broere, 2008).
Figure 5.9: Evolution of the permeability with one dimensional loading in an oedometer.
5.2 Oedometer tests conducted on samples extracted from soil
models after consolidation
Due to problems encountered during the installation of the PPTs, the first attempt to conduct
the centrifuge test JG_v10 failed, which gave the opportunity to extract samples from the soil
models prepared in cylindrical strongboxes (σc = 100 kPa) without being exposed to
enhanced gravity. Four samples were withdrawn, 15 days after the removal of the
consolidation load, in the horizontal direction at two different depths and were submitted to a
vertical load up to 200 kPa, and to unloading – reloading cycles.
The specimens JG_v10_Oedo 1 and JG_v10_Oedo 3 were extracted from a depth of up to
75 mm, while the samples JG_v10_Oedo 2 and JG_v10_Oedo 4 were extracted from up to
σ’v
pref
Δh/h0
4E-10
6E-10
8E-10
1E-09
1.2E-09
1.4E-09
1.6E-09
1.8E-09
2E-09
0 50 100 150 200
k [
m/s
]
Vertical effective stress σ'v [kPa]
JG_v9 - Oedo 1 (Vertical) JG_v9 - Oedo 2 (Horizontal)
JG_v9 - Oedo 3 (Vertical) JG_v9 - Oedo 4 (Horizontal)
5.2 Oedometer tests conducted on samples extracted from soil models after consolidation
204
120 mm below the surface (Figure 5.10). The oedometer cells and the extraction procedure
used are the same as those described in the previous section.
Figure 5.10: Extraction positions of the samples for oedometer tests (test JG_v10).
Figure 5.11 shows the distribution of the over-consolidation ratio in the horizontal direction
(σ’h,max = 58.5 kPa) with depth for the sample from which the specimen used for the
oedometer tests were extracted. The OCR ranges from 50 to 28 for specimens extracted
from a depth up to 120 mm (denoted as 120 mm in Figure 5.11) and from 170 to 40 for
specimens extracted from a depth up to 75 mm (denoted as 75 mm in Figure 5.11).
Figure 5.11: Distribution of the over-consolidation ratio in the horizontal direction of the
specimens used for the oedometer tests.
The evolution of the void ratio with one-dimensional vertical loading is shown in Figure 5.12.
The results of the samples extracted from 75 mm (denoted as JG_v10 - Oedo 1 - z = 75 mm
and JG_v10 - Oedo 3 - z = 75 mm) and 120 mm (denoted as JG_v10 - Oedo 2 - z = 120 mm
and JG_v10 - Oedo 4 - z = 120 mm) are remarkably close (Figure 5.12).
0
20
40
60
80
100
120
140
160
0 50 100 150 200 250 300
Dep
th [
mm
]
OCR [-]
75 mm
120 mm
5 Complementary investigations
205
Figure 5.12: Evolution of the void ratio with one dimensional loading in an oedometer.
As long as the specimens were pre-consolidated vertically up to 100 kPa (which corresponds
to a horizontal load of 58 kPa assuming K0 conditions), the compression index CC is to be
calculated for the load step ranging from 100 kPa to 200 kPa. The obtained values are
presented in Table 5.4. No significant variation of the results can be noted between the two
extraction depths.
Table 5.4: Compression indexes CC obtained from the oedometer tests.
Load
interval
[kPa]
CC, z = 75 mm [-] CC, z = 120 mm [-]
OCR [-] JG_v10 -
Oedo 1 - z
= 75 mm
JG_v10 -
Oedo 3 - z
= 75 mm
JG_v10 -
Oedo 2 - z
= 120 mm
JG_v10 -
Oedo 4 - z
= 120 mm
100 - 200
(loading) 0.048 0.043 0.046 0.047 1.0
The calculated values of the compression index CS for the unloading-reloading cycles are
presented in Table 5.5. No significant impact of the over-consolidation can be noted as the
values of CS for samples extracted up to a depth of 75 mm and those for samples extracted
up to a depth of 120 mm are very similar (Table 5.5). The higher values of the compression
index CS for the bigger load steps (marked with an asterisk * in Table 5.5) can be explained
by the accumulation of plastic deformation during cycling loading, or ratcheting, as shown
e.g. in Alonso-Marroquin et al. (2008).
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1 10 100 1000
e [
-]
Vertical effective stress σ'v [kPa]
JG_v10 - Oedo 1 - z = 75 mm JG_v10 - Oedo 2 - z = 120 mm
JG_v10 - Oedo 3 - z = 75 mm JG_v10 - Oedo 4 - z = 120 mm
σ'v,maxσ'h,max
5.2 Oedometer tests conducted on samples extracted from soil models after consolidation
206
Table 5.5: Compression indexes CS obtained from the oedometer tests.
Load interval [kPa]
CS, z = 75 mm [-] CS, z = 120 mm [-]
OCR
[-]
JG_v10 -
Oedo 1 - z
= 75 mm
JG_v10 -
Oedo 3 - z
= 75 mm
JG_v10 -
Oedo 2 - z
= 120 mm
JG_v10 -
Oedo 4 - z
= 120 mm
200 – 100 (unloading) 0.005 0.003 0.004 0.004 2.0
100 – 200 (reloading) 0.007 0.005 0.005 0.006 1.0
200 - 50 (unloading) * 0.013 0.009 0.010 0.011 4.0
50 – 100 (reloading) 0.006 0.004 0.004 0.005 2.0
100 – 50 (unloading) 0.005 0.003 0.003 0.004 4.0
50 - 200 (reloading) * 0.015 0.011 0.012 0.013 1.0
200 – 100 (unloading) 0.005 0.003 0.003 0.003 2.0
100 – 200 (reloading) 0.007 0.005 0.005 0.005 1.0
Figure 5.13 shows the distribution of the horizontal confined stiffness moduli obtained from
the oedometer tests as a function of the applied vertical stress. No significant variation of the
stiffness with depth can be noted. Figure 5.14 shows that no significant variation of the
permeability of the sample with depth could be measured.
Figure 5.13: Distribution of the horizontal confined stiffness moduli as a function of the
vertical stress.
0
500
1000
1500
2000
2500
3000
0 50 100 150 200
ME
, h
[kP
a]
Vertical effective stress σ'v [kPa]
JG_v10 - Oedo 1 - z = 75 mm JG_v10 - Oedo 2 - z = 120 mm
JG_v10 - Oedo 3 - z = 75 mm JG_v10 - Oedo 4 - z = 120 mm
σ'h,max σ'v,max
5 Complementary investigations
207
Figure 5.14: Evolution of the coefficient of permeability with one dimensional loading in an
oedometer.
5.3 Electrical impedance measurement under 1 g
In order to assess the accuracy of the electrical impedance to density changes, it was
decided to test the tool developed for the centrifuge (Figure 3.30) under 1 g conditions in the
laboratory. A soil model was prepared in the same way as for a centrifuge test using an
oedometer container (Sections 3.6.3 and 3.6.4) and the electrical impedance needle was
inserted up to a depth of 115 mm using the setup shown in Figure 5.15 at three different
points (Figure 5.16) after the completion of each consolidation step (Table 5.6). The model
was therefore unloaded, the top plate was removed and the needle was inserted manually
into the specimen. This operation took approximately 5 minutes, which allows fully undrained
behaviour to be assumed for the soil specimen. The needle was guided vertically in order to
prevent any corruption of the results through an unanticipated change of direction. The
position of the setup was changed for two subsequent consolidation stages: the insertion
points at the end of the consolidation stages 1, 3 and 5, respectively 2 and 4, are located at
the same positions.
0
5E-10
1E-09
1.5E-09
2E-09
2.5E-09
3E-09
0 50 100 150 200
k [
m/s
]
Vertical effective stress σ'v [kPa]
JG_v10 - Oedo 1 - z = 75 mm JG_v10 - Oedo 2 - z = 120 mm
JG_v10 - Oedo 3 - z = 75 mm JG_v10 - Oedo 4 - z = 120 mm
5.3 Electrical impedance measurement under 1 g
208
(a) (b)
Figure 5.15: Setup for the insertion of the electrical impedance needle under 1 g in the
laboratory (a) schematic view (b) picture.
Table 5.6 shows an overview of the load steps and of the density and void ratio of the clay at
the end of the corresponding consolidation stages. Figure 5.17 and Figure 5.18 show the
results of the investigation conducted after the first and after the fifth consolidation stage. The
other outcomes are shown in Appendix 8.7.
Table 5.6: Overview of the consolidation stages for the implementation of the electrical
impedance needle under 1 g.
Consolidation stage Consolidation effective
stress [kPa] Density [kN/m3] Void ratio [-]
1 12.5 16.8 1.53
2 25 17.4 1.30
3 50 18.1 1.20
4 100 18.7 1.06
5 200 20.4 0.73
20
0 m
m
115 mm Clay
Container
Needle
5 Complementary investigations
209
Figure 5.16: Positions of the insertion points of the electrical impedance needle under 1 g. All
dimensions in mm.
Figure 5.17: Impedance recorded under 1 g after completion of the first consolidation stage.
The density changes do not significantly affect the measured impedance values under 1 g.
However, the values obtained are one order of magnitude higher at 1 g than under 50g,
which is unexpected as the unit weight of the soil (kN/m3) should be n times higher under
enhanced gravity than under Earth’s gravity (n being the factor of increase of the Earth’s
gravity in the centrifuge), triggering higher impedance values. These observations confirm
that the electrical impedance needle should be used in-flight as a tool for complementary
qualitative investigations rather than to obtain quantitative results.
Clay
Insertion
points
1 2
3
0
20
40
60
80
100
120
0.043 0.044 0.045 0.046 0.047
De
pth
[m
m]
Impedance [Ohm]
Point 1 (Stage 1) Point 2 (Stage 1) Point 3 (Stage 1)
5.4 Microscopic investigations
210
Figure 5.18: Impedance recorded under 1 g after completion of the fifth consolidation stage.
5.4 Microscopic investigations
In order to investigate the spatial distribution of the compaction zone, samples were taken at
a radial distance of 11 mm from the axis of the compacted column C that was installed during
test JG_v2 at different depths: 20 mm, 60 mm and 100 mm at model scale. Even though an
optical microscope is a useful tool in many research areas, the maximum resolution is
500 nm (5.10-7 m), which means that the clay particles (grain size < 2.10-6 m) can hardly be
detected. This is the reason why it was decided to conduct only investigations using an
Environmental Scanning Electron Microscope.
5.4.1 Description of the Scanning Electron Microscope
A Scanning Electron Microscope (SEM) circumvents the issue of the resolution, as it can
reach resolutions of 3 nm (3.10-9 m). The major difference between optical microscopes and
SEMs is that the sample is illuminated by electrons rather than a light beam (i.e. photons).
The maximal resolution of SEMs depends on diverse factors, the most important being the
spot diameter of the electron beam on the surface of the sample: an SEM cannot detect
elements smaller than the “spot diameter” (Figure 5.19). An SEM, in contrast to an optical
microscope, does not deliver a real picture of the sample but generates a virtual illustration.
The electron beam illuminates a point of the sample, creating a pixel of the virtual illustration,
before being displaced. The combination of the pixels obtained forms the illustration of the
sample.
An SEM illustration is generated by the signals produced by the interaction between the
incident electron beam and the sample, which is recorded by detectors mounted in the
sample chamber. The interaction volume between sample and electron beam (Figure 5.20)
depends on the velocity of the primary electrons within the beam – the higher the speed the
0
20
40
60
80
100
120
0.043 0.044 0.045 0.046 0.047
De
pth
[m
m]
Impedance [Ohm]
Point 1 (Stage 5) Point 2 (Stage 5) Point 3 (Stage 5)
5 Complementary investigations
211
bigger the interaction volume – and on the spot diameter (Figure 5.19). The different
products generated within the electron interaction volume are illustrated in Figure 5.21.
Figure 5.19: Illustration of the contact between the electron beam and the surface of the
sample (Peschke, 2013).
Figure 5.20: Electron interaction volume within a sample
(after Science Education Resource Center, 2013).
Although SEMs are a major step forward in terms of resolution, compared to optical
microscopes, they still suffer from two major limitations:
- the electrical conductibility of the sample.
If electrical non-conductive samples are investigated, the interaction between the
electron beam and the surface of the sample triggers a significant electrical
charge within the sample, which disturbs the signal coming from the electron-
sample interaction and produces bright surfaces in the illustration.
sample surface
Auger electrons
secondary electrons
characteristics X-rays
back-scattered electrons
Bremsstrahlung X-rays
secondary fluorescence
Spot diameter
5.4 Microscopic investigations
212
- the need for vacuum.
A high quality electron beam can only be achieved under a very high vacuum of
10-6 to 10-7 torr (Donald, 2003). As a consequence, vacuum-incompatible
specimens, such as humid or outgassing samples, cannot be investigated using
an SEM.
These two limitations reduce the field of application of SEMs to electrical conductive, vacuum
compatible samples.
Figure 5.21: Types of interaction between electrons and a sample
(Science Education Resource Center, 2013).
5.4.2 Description of the Environmental Scanning Electron Microscope
Environmental Scanning Electron Microscopes (ESEM) sidestep these two problems by
enabling the presence of gas (mostly water vapour) in the sample chamber, while
maintaining the high resolution of the SEMs. If electrical non-conductive samples are
investigated, the gas interacts with the sample surface in order to dissipate the negative
charge.
The limitation concerning the high vacuum is circumvented by using differential pumps along
the column by means of a series of different pressure zones with increasing pressure
towards the sample chamber. The high vacuum needed for the generation of a high quality
electron beam can be conserved in this way, while pressures up to 10 torr can be reached in
proximity to the sample (Figure 5.22). This way, hydrated samples can be investigated in the
original state, if water vapour is used as a gas in the specimen chamber. The specific ESEM
model used for the investigations conducted here was a Quanta 600 produced by FEI.
characteristic X-rays
Bremsstrahlung X-rays
visible light (cathodoluminescence)
heat
diffracted electrons
transmitted electrons
sample surface
Auger
electrons
secondary
electrons
back-scattered
electrons
incident electron beam
5 Complementary investigations
213
Figure 5.22: Schematic of an ESEM illustrating the different pressures zones (Donald, 2003).
5.4.3 Results obtained
Figure 5.23 shows the results of the ESEM investigation conducted by Weber (2008) in
zone 2 (magnification 800 times), in which the clay particles have been reorganised
vertically, parallel to the axis of the stone column.
The investigations using the electrical impedance needle (Section 4.7) showed that the
distribution of zone 2 is homogeneous over the whole depth of the column. The spatial
distribution of zone 2 was therefore not investigated further at the microscopic scale.
Figure 5.24 shows the result of the ESEM investigation of zone 3 (specimen located at a
radial distance of 5 mm from the edge of the column) at a depth of 20 mm (magnification
1500 times). A start of the vertical reorganisation of the clay particles can be observed as the
organisation of the longer particles has a vertical trend, although most of the clay platelets
are still organised randomly.
Figure 5.25 shows the results of the ESEM investigation into zone 3 (specimen located at a
radial distance of 5 mm from the edge of the column, as illustrated in Figure 5.24) at depths
of 60 mm and 100 mm (magnification 1500 times). The same mechanisms, as at a depth of
20 mm, can be observed here as the longer particles are organised with a vertical trend.
5.4 Microscopic investigations
214
These observations confirm the interpretation of the results of the investigations conducted
with the electrical impedance needle according to which the reduction of the measured
impedance is due to a vertical reorganisation of the clay particles. This reorganisation is a
progressive mechanism starting in zone 3, in which the longer clay particles start to exhibit a
vertical trend and ending in zone 2, in which both the longer particles and the clay platelets
are reorganised vertically.
Figure 5.23: ESEM picture of zone 2, located a radial distance of 1 mm from the edge of the
column and at a depth of 40 mm below the surface, with the radial axis
horizontal (Weber, 2008).
Figure 5.24: ESEM picture of the zone 3 located at a radial distance of 5 mm from the edge
of the column and at a depth of 20 mm below the surface, with the radial axis
horizontal.
50 μm
30 μm
Longer
particles
5 mm
Sto
ne
co
lum
n
5 Complementary investigations
215
(a) (b)
Figure 5.25: ESEM pictures of the zone 3 at a radial distance of 5 mm from the edge of the
column and at (a) 60 mm depth and (b) 100 mm depth, with the radial axis
horizontal.
5.5 Mercury Intrusion Porosimetry (MIP)
MIP investigations were conducted to investigate the distribution of the density changes
caused by the installation of the stone columns with depth.
5.5.1 General principle
As mentioned in Section 2.10.2.2, the Mercury Intrusion Porosimetry was developed to
assess the pore diameter in a non-destructive manner. The pore diameter is inversely related
to the mercury insertion pressure by the equation proposed by Washburn (1921):
2.80
with σ surface tension
θ wetting angle for mercury, assumed to be equal to 130° for clay minerals at
room temperature (Diamond, 1970)
p mercury pressure
The porosity is obtained by dividing the total pore volume calculated, based on the pore
diameter, by the total volume of the sample investigated.
20 μm 20 μm
Longer particles
5.5 Mercury Intrusion Porosimetry (MIP)
216
5.5.2 Sample preparation
42 undisturbed samples were extracted from the soft clay bed at depths of 20 mm, 60 mm
and 100 mm (model scale) at distances of 7 mm, 13 mm, 18 mm, 23 mm, 31 mm and 36 mm
from the axis of the stone column after the centrifuge tests had been carried out. The
samples were then freeze-dried before applying vacuum by means of a vacuum pump
(Figure 5.26).
Figure 5.26: Vacuum pump.
5.5.3 Apparatus used
The apparatuses used were the models Pascal 140 and Pascal 440, produced by
CE Instruments Ltd (2014), to measure pore diameters ranging from 1.8 nm to 58 μm. The
specimens were first placed in a volume calibrated glass vessel (dilatometer, Figure 5.27 a)
and evacuated to approximately 0.03 kPa using the Pascal 140. The dilatometer was then
filled with mercury up to a given level (Figure 5.27 b) and the pressure was raised
continuously from vacuum to 375 kPa in the macro pore unit. Once this pressure was
reached, the dilatometer containing the specimen was transferred to the Pascal 440 and the
pressure was raised stepwise up to 400 MPa. The measurement of the electrical capacity
along the capillary tube of the dilatometer allows the penetration of the mercury into the
pores of the specimen to be determined.
5 Complementary investigations
217
(a) (b)
Figure 5.27: Dilatometer (a) containing the soil specimen before and (b) containing mercury
after the investigation using the macro pore unit Pascal 140.
5.5.4 Results obtained
The first observation is that the edge of the zone in which the host soil becomes denser
through the installation of the stone column remains constant with depth (Figure 5.28). This
compaction zone reaches 13 mm from the axis of the inclusion, corresponding to about twice
the radius of the column, which confirms the observations made by Weber (2008), who
identified that zone 3 extended to 15 mm from the axis of the stone column (Figure 3.21,
Figure 5.28).
The impact of the over-consolidation on the porosity can be noted as the porosity of the host
soil outside the compaction zones rises from approximately 33 % at a depth of 20 mm
(OCR = 11.8 [-]) to 34 % at depths of 60 mm and 100 mm (OCR = 3.9 [-] and OCR = 2.4 [-],
respectively).
The influence on the porosity of the greater number of extraction-reinsertion cycles that the
host soil undergoes near the surface is highlighted by comparing the results of the
measurements at the edge between the smear and compaction zones. A value of 29 % was
obtained at a depth of 20 mm (Figure 5.28 a), while values of 30 % and of 31 % are reached
at depths of 60 mm, respectively 100 mm (Figure 5.28 a and b).
The MIP investigations indicate that the extent of installation effects of stone columns
probably remains constant with depth while the compaction of the inclusion causes a more
significant reduction of porosity at the zone 2-3 boundary at shallower depths than at the tip
of the column.
Soil
specimen
Mercury
5.5 Mercury Intrusion Porosimetry (MIP)
218
(a)
(b)
(c)
Legend:
Figure 5.28: Porosity as a function of the radial distance from the axis of the stone column at
a depth of (a) 20 mm (b) 60 mm (c) 100 mm.
26
28
30
32
34
36
38
40
0 5 10 15 20 25 30 35 40
Po
ros
ity [
%]
Distance from stone column axis [mm]
Stone
column
Edge of densification OCR = 11.8 [-]
26
28
30
32
34
36
38
40
0 5 10 15 20 25 30 35 40
Po
ros
ity [
%]
Distance from stone column axis [mm]
Edge of densificationStone
column
OCR = 3.9 [-]
26
28
30
32
34
36
38
40
0 5 10 15 20 25 30 35 40
Po
ros
ity [
%]
Distance from stone column axis [mm]
Measured data
Hyperbolic trend function
Edge of densificationStone
column
Smear zone (zone 2 in Weber, 2008)
Zone 3 (Weber, 2008)
OCR = 2.4 [-]
Compaction zone
Measured data
Hyperbolic trend function
26
28
30
32
34
36
38
40
0 5 10 15 20 25 30 35 40
Po
ros
ity [
%]
Distance from stone column axis [mm]
Measured data
Average
Edge of densificationStone
column
Smear zone (zone 2 in Weber, 2008)
Zone 3 (Weber, 2008)
OCR = 2.4 [-]
Compaction zone
6 Numerical modelling
219
6 Numerical modelling
The rapid development of computer science and increase of computing power over the past
decades have led to an extended use of numerical modelling, in order to solve engineering
problems in general, and geotechnical issues in particular. These numerical analyses are
conducted using the Finite Element Method (FEM), the principles of which are presented e.g.
in Zienkiewicz (1977) and Bathe (1996). The calculations conducted herein are performed
with the commercial code Plaxis (2D Version 2012.2 and 3D Version 2013.1).
6.1 Principles of numerical modelling of ground improvement
Schweiger & Schuller (2005) present an overview of the principles of numerical modelling for
ground improvement. These methods can be divided into improvement through compaction
(e.g. dynamic compaction or embankment preloading with vertical drains), through material
addition without displacement (e.g. mixed-in-place and injections), through material addition
with displacement (e.g. stone columns) and through structural elements (e.g. cemented
columns). Only cases of improvement through compaction, and through material addition
with displacement, are considered here due to the focus of this research. These techniques
can be implemented in the modelling of improved soft soil loaded by an embankment or a
foundation. Different approaches are available to convert an axisymmetric unit cell into a
plane-strain boundary value problem, which are presented in this section.
6.1.1 Improvement through compaction (embankment loading with installation
of vertical drains)
A combination of embankment loading and vertical drains is often used in order to preload
and consolidate soft soils. These boundary value problems can be modelled with software
that couples mechanical and hydraulic behaviour. This work focuses on stone columns, thus
the modelling of drains only will be treated in this review. The permeability properties of the
host soil have to be modified correspondingly, in order to take the installation effects of the
drains into effect.
Indraratna & Redana (1997) propose a conversion from an axisymmetric unit cell to a plane-
strain unit (Figure 6.1). The width bw of a drain, as well as the width bs of the smear zone
under plane-strain conditions, can be assessed for a square pattern as:
6.1
6.2
and for a triangular pattern as:
6.3
6.1 Principles of numerical modelling of ground improvement
220
6.4
with bw width of the drain in plane-strain conditions
bs width of the smear zone in plane-strain conditions
rw radius of the drain
rs radius of the smear zone
S spacing (axis to axis) between two adjacent drains
(a) (b)
Figure 6.1: Conversion of axisymmetric unit cell into plane-strain for drains (a) axisymmetric
radial flow (b) plane-strain (Indraratna & Redana, 1997).
Unlike Onoue et al. (1991) and Weber (2008), Indraratna & Redana (1997) assume that the
smear zone is homogeneous across the shear and radial compaction zones. The decrease
of the horizontal permeability of the undisturbed host soil in plane-strain conditions kh,p,
neglecting the smear effect, can be formulated as:
( ) 6.5
6.6
with kh,p coefficient of horizontal permeability of the undisturbed host soil in plane strain
conditions
kh horizontal permeability of the undisturbed host soil
R radius of the axisymmetric unit cell
rw radius of the drain
n radius ratio of the unit cell to the drain well
6 Numerical modelling
221
Under the assumption that the radius of the axisymmetric unit cell R and the width of the
corresponding zone in plane-strain B (Figure 6.1) are equal, the decreased horizontal
permeability within the smear zone in plane-strain conditions k’h,p can be expressed as:
( )
( )
6.7
6.8
6.9
6.10
with k’h coefficient of horizontal permeability in the smear zone
kh horizontal permeability of the undisturbed host soil
k’h,p coefficient of horizontal permeability in the smear zone in plane-strain
conditions
rw radius of the drain
rs radius of the smear zone
s radius ratio of the smear zone to the drain well
n radius ratio of the unit cell to the drain well
B width of the zone of influence in plane-strain conditions
Indraratna et al. (2005) refine the formulation proposed by Indraratna & Redana (1997) in
order to circumvent the assumption that the radius of the unit cell R is equal to the width of
the corresponding zone in plane-strain B. While the formulation of the horizontal permeability
of the host soil in plane-strain kh,p is still expressed as in Equation 6.5, the equivalent
horizontal permeability within the smear zone k’h,p is defined as (with n and s according to
Equation 6.6 and Equation 6.8, respectively):
[ (
)
]
6.11
( )
( ) 6.12
( )
( ) [ ( )
( )] 6.13
6.1.2 Discrete modelling of improvement through material addition with
displacement
The soil-structure interaction can be modelled efficiently by using the unit cell approach while
the load transfer underneath composite foundations can also be replicated accurately. A
6.1 Principles of numerical modelling of ground improvement
222
practical advantage of this approach is the limited time needed for a calculation, which
simplifies investigations with parametric studies.
The actual geometry of the foundation can be modelled more accurately using a three-
dimensional (3D) approach, while constitutive models for stone columns and host soil can be
differentiated, as required. Stress paths, (differential) settlements and the different
interactions that develop underneath a loaded composite foundation, as shown e.g. in Figure
2.4, can be reproduced more accurately.
Two-dimensional (2D) modelling of stone columns remains, however, very common and can
be conducted either by converting the mechanical and permeability properties of the soil
(Section 6.1.1), or by adapting the geometry of the boundary value problem. Tan et al. (2008)
present two methods to convert the axisymmetric model to an equivalent plane-strain model.
The effect of smear is neglected in this basic approach.
(a) (b) (c)
Figure 6.2: Cross-sections of the stone column (a) unit-cell; and plane-strain conversions
according to (b) method 1 and (c) method 2 (Tan et al., 2008).
Method 1 adapts the stiffness and permeability parameters of the unit-cell (Figure 6.2 a) to fit
the plane-strain conditions (Figure 6.2 b). The validity of the approach according to Method 1
was confirmed by comparison with results obtained from field testing (Ng & Tan, 2012). The
width of the plane-strain column (denoted as bc in Figure 6.2) is equal to the radius of the
unit-cell stone column (denoted as rc in Figure 6.2) and the width B of the zone considered in
plane-strain is also equal to the radius R of the unit-cell.
The plane-strain permeability is adjusted according to the following set of equations:
( )
( )
[
( )
⁄
( )
⁄
⁄
]
[ ( )
⁄
⁄
( )
⁄
]
6.14
( ) (
)
6.15
for axisymmetric conditions 6.16
R
Legend:
Flow path
Rrc
2B 2B
BB
bc bc
R
Legend:
Flow path
Rrc
2B 2B
BB
bc bc
R
Legend:
Flow path
Rrc
2B 2B
BB
bc bc
6 Numerical modelling
223
in plane-strain conditions 6.17
with Esc Young’s modulus of the stone column material
as replacement ratio
Es Young’s modulus of the host soil
kh, p coefficient of horizontal permeability of the undisturbed host soil in plane-strain
conditions
kh coefficient of horizontal permeability of the undisturbed host soil
B width of the zone of influence in plane-strain conditions
p plane-strain conditions
R radius of the unit cell
n radius ratio of the unit cell to the drain well
Method 2 converts the unit-cell geometry (Figure 6.2 a) to plane-strain conditions (Figure 6.2
c). This approach assumes that the drainage capacity of the column and the replacement
ratio remain constant in plane-strain and axisymmetric conditions. The width of the stone
column in plane-strain conditions is given by:
6.18
Under the assumption that the total area for a square pattern of column is equivalent (Barron,
1948), the relationship between R (Figure 6.2 a) and B (Figure 6.2 c) can be expressed as:
6.19
Chan & Poon (2012) suggest that the mechanical properties of the composite foundation
should be adapted. They consider strips each featuring a width equal to an equivalent square
for the cross-sectional area (Figure 6.3 a) and a depth equal to that of the inclusions. Under
the assumption that the spacing between the strips is equal to the spacing b between the
discrete columns for a square pattern and to √ b/2 for a triangular pattern (Figure 6.3 b) they
formulate an equivalent Young’s modulus Eeq and cohesion c’eq as well as an equivalent
angle of friction ’eq for the strips, while the calculation of the equivalent angle of friction
requires an assumption of the stress concentration on top of the inclusions, which is
summarised by the parameter m (Equation 6.22):
6.20
6.21
6.22
with Eeq equivalent Young’s modulus
c’eq equivalent cohesion
’eq equivalent angle of friction
6.2 Literature review of numerical modelling of ground improvement through stone columns
and prefabricated vertical drains
224
c’s effective cohesion of the host soil
c’sc effective cohesion of the stone column material
’s effective angle of friction of the host soil
’sc effective angle of friction of the stone column material
m stress concentration ratio
(a) (b)
Figure 6.3: Plan view of 2D stone columns strips (a) width of an equivalent strip (b) strip
spacing (Chan & Poon, 2012).
Two-dimensional modelling in plane-strain enables a good approximation to be made of the
serviceability limit state provided that the stiffness chosen for the host soil and stone column
material are appropriate for the loading cases considered. However, parameter conversion
from 3D to 2D geometry leads to an alteration of the stress path within the inclusions. This
can cause challenges for the estimation of the ultimate limit state (Schweiger & Schuller,
2005).
6.2 Literature review of numerical modelling of ground
improvement through stone columns and prefabricated vertical
drains
6.2.1 Numerical modelling of ground improvement with stone columns and
prefabricated vertical drains
Rujikiatkamjorn et al. (2007) present the results of the finite element analysis of a case study
at Tianjin Port in Bejing, China, of an embankment stabilised by vertical drains combined with
vacuum loading to accelerate dissipation of excess pore pressure. Figure 6.4 shows the soil
profile and properties at the location of the case study. A vacuum pressure of 80 kPa was
applied and a 3 m high embankment was built in order to reach a loading of 120 kPa (Figure
6.5 a). The settlements were monitored with settlement gauges installed below the top of the
embankment and were measured at depths of 3.5 m, 7.0 m, 10.5 m and 14.5 m below the
surface of the host soil.
d = diameter of
stone column
a = width of equivalent
strip in 2D FEA
b cos30°
2D strip
Asoil
Acolumn b
6 Numerical modelling
225
Figure 6.4: Soil profile and properties at Tianjin Port in Beijing, China
(Rujikiatkamjorn et al., 2007).
The conversion presented in Indraratna & Redana (1997) was implemented to conduct a
plane-strain calculation and the modified Cam-Clay theory (Roscoe & Burland, 1968). Figure
6.5 shows that the results of the numerical analysis are in good agreement with the
observations in the field. However, various assumptions had to be made to calibrate the
numerical model, such as the ratio of the horizontal permeability of the undisturbed host soil
kh to the permeability within the smear zone k’h. This ratio was assumed to be equal to 3. The
settlements were slightly over-estimated by the numerical model up to a depth of 7.0 m
below the ground surface, and slightly underestimated at a depth 10.5 m. However, the
overall match was very good.
Atterberg limits [%]Vane shear strength
[kPa] Void ratio [-] Description
of soil
Silty clay(dredged from sea bed)
Muddy clay
Soft silty clay
Stiff silty clay
Depth
[m]
Plastic limit
Water content
Liquid limit
6.2 Literature review of numerical modelling of ground improvement through stone columns
and prefabricated vertical drains
226
Figure 6.5: Case study at Tianjin Port in Beijing, China: embankment and vacuum loading on
soft soil stabilised by drains (a) loading history and (b) comparison of the
predicted (FEM) and measured (Field) consolidation settlements
(Rujikiatkamjorn et al., 2007).
Indraratna et al. (2009) conducted both two- and three-dimensional analyses of the Tianjin
Port in Beijing, China, using the modified Cam-Clay theory (Roscoe & Burland, 1968). The
soil properties determined at the test site are presented in Figure 6.4. A vacuum pressure of
80 kPa was applied before a 3 m high embankment was built in order to reach a load of
120 kPa. The settlements were monitored with settlement gauges installed below the top of
(a)
(b)
Vacuum pressure under membrane
Pre
load
pre
ssure
[kP
a]
Surface (Field)
3.5 m (Field)
7.0 m (Field)
10.5 m (Field)
14.5 m (Field)
Surface (FEM)
3.5 m (FEM)
7.0 m (FEM)
10.5 m (FEM)
14.5 m (FEM)
0 40 80Time [days]
120Time [days]
160 200
1.6
0
40
80
120
160
Settle
ment [m
m]
1.2
0.8
0.4
Settle
ment [m
m]
Pre
load
pre
ssure
[kP
a]
a aVacuum plus preloading
6 Numerical modelling
227
the embankment and were measured at depths of 1.0 m and 5.0 m below the surface of the
host soil. The plane-strain study was conducted by implementing the permeability conversion
presented in Indraratna (2005), which leads to good agreement between the two-dimensional
modelling with field data, as well as with the three-dimensional analysis (Figure 6.6).
Figure 6.6: Embankment pre-loading at Tianjin Port in Beijing, China (a) loading history (b)
comparison of the results obtained via 2D and 3D modelling with field
observations (Indraratna et al., 2009).
Weber et al. (2009) present a numerical back-calculation of the centrifuge tests conducted by
Weber (2008) on 1.85 m high embankments placed in-flight (using balls of lead shot) on
over-consolidated remoulded Birmensdorf clay, reinforced with stone columns that had been
constructed in-flight. Birmensdorf clay was modelled using the Hardening Soil Model (Schanz
et al., 1999), stone column material with the Mohr-Coulomb model and the installation
process is modelled by the application of outwards and downwards prescribed
displacements on the edge of an initial cavity. The settlements computed with this procedure
showed a good agreement with the observations made during centrifuge tests (Figure 6.7).
Vacuum pressure under membrane
Vacuum plus preloading
5 m
1 m
120400
Time [days]80
Time [days]
(a)
(b)
0.8
0
Tim
e [
da
ys]
Tim
e [
da
ys]
Pre
loa
dp
ressu
re[k
Pa
]
0.6
0.4
0.2
Se
ttle
me
nt
[mm
]
40
80
120
160
Field
3D
2D
6.2 Literature review of numerical modelling of ground improvement through stone columns
and prefabricated vertical drains
228
Figure 6.7: Development of settlement at the crest of an embankment constructed in-flight on
remoulded Birmensdorf clay reinforced with stone columns – comparison
between numerical model and centrifuge results (Weber et al., 2009).
Basu et al. (2010) make a distinction between a smear and a transition zone, in a similar way
to Onoue et al. (1991) (Figure 2.60) and investigate the effect of soil disturbance around
Prefabricated Vertical Drains (PVD). In this case, the drains have a rectangular cross-section
(a x d), thus the extent of the disturbed zone cannot be expressed in terms of equivalent
mandrel diameter. The decrease of horizontal permeability within the transition zone (i.e. for
⁄⁄ , Figure 6.8 a). is expressed as a function of the permeability of the
undisturbed soil and of the smear zone:
( )
(
) lx/2 2x tx/2 6.23
with kht(x) horizontal permeability within the transition zone for a horizontal flow in the x
direction.
A similar expression is obtained for the variation of kht in the y direction by substituting x with
y in Equation 6.23. The results obtained from an experimental study with small-scale model
tests conducted by Indraratna & Redana (1998) (Figure 2.61 and Figure 2.62) and the
numerical analysis correlate well with results obtained by Basu et al. (2010) using the
modified Cam-Clay theory (Figure 6.8 b). It is surprising, however, that the settlement
calculated without transition zone become significantly higher than those obtained with a
smear zone when the load is increased from 100 kPa to 200 kPa. This could be due to an
over-estimation of the excess pore pressure dissipation caused by the absence of smear
zone with reduced permeability around the drain, which would cause an over-estimation of
the settlements at the beginning of the loading phase. However, the experiment was stopped
10 days after the increase of the load, thus preventing the analysis of the long-term
behaviour of the improved ground under loading. It would also be interesting to confront the
0 50 100 150 200 250 300 350 400 450 500
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Zeit [d]
Se
tzu
ng
en
[m
]
Numerische Berechnung
ZentrifugenversuchFEMFEM
Centrifuge
Settle
ment
[m
m]
Time [d]
(2)
(2)
6 Numerical modelling
229
model with field measurements or data from centrifuge modelling in order to overcome the
limitations of small-scale tests concerning the reproduction of the stress levels at prototype
scale (Section 3.2).
(a) (b)
Figure 6.8: Numerical and experimental study of PVDs installed in clay (a) plan view with
dimensions of the smear and transition zones in terms of mandrel size (b)
comparison of settlement obtained with results by Indraratna & Redana (1998)
(Basu et al., 2010).
Castro & Karstunen (2010) investigate the installation effects around stone columns installed
in over-consolidated Bothkennar Clay. The boundary conditions are such that the excess
pore water pressures can only dissipate towards the column and the surface. The column
material is not modelled and the cavity is maintained as a hole with infinite permeability
throughout the consolidation process. This does not allow for the interaction between column
material and surrounding soil to be taken into account. The authors assume that the
permeability of the column material is high enough compared to that of the surrounding soil
to be considered infinite and that the displacement of the soil-column interface after
installation only has minor consequences on the soil properties.
The constitutive models S-CLAY1 (Wheeler et al., 2003) and S-CLAY1S (Karstunen et al.,
2005) were used and the results were compared with those obtained using the modified
Cam-Clay model (denoted as MCC in Figure 6.10). Good agreement of the excess pore
water pressures generated by the column installation normalised by the undrained shear
strength (denoted as cu in Figure 6.10) can be observed between the models at all depths.
The normalisation enables a direct comparison to be made between data obtained from
different depths, soil models and field measurements.
mandrel
transition zone
boundary
Smear Zone: 2 p 3
Transition Zone: 6 p 12
transition
transitsmear zone
boundary
Time [days]
Settle
ments
[m
m]
Str
ess [kP
a]
0 10 20 30 40 50
200
100
0
40
80
120
160
Measured (Indraratna & Redana 1998)
Present Analysis
(Smear Zone = 2d, Transition zone
= 5d, khs/kh0 = 0.2)
Analysis Without Transition Zone
(Smear Zone = 2d, khs/kh0 = 0.2)
6.2 Literature review of numerical modelling of ground improvement through stone columns
and prefabricated vertical drains
230
Figure 6.9: Model geometry and axisymmetric finite element mesh with applied radial
deformation of the stone column wall (Castro & Karstunen, 2010).
Figure 6.10: Normalised excess pore pressures generated by the stone column installation
(Castro & Karstunen, 2010).
Impermeable boundary
6 Numerical modelling
231
Figure 6.11: Decrease of the undrained shear strength after column or pile installation
(Castro & Karstunen, 2010).
The variation of the undrained shear strength (denoted as cu in Figure 6.11) obtained
numerically was compared with the data obtained by Roy et al. (1981) for the installation of
piles in field tests. Good agreement between the field observations and the outcomes of the
calculations using the constitutive model S-CLAY1S can be observed for the estimation of
the reduction of the undrained shear strength (normalised with the initial undisturbed shear
strength, denoted as cu0 in Figure 6.11). The results are consistent with the observations
made by Roy et al. (1981), who determined that the radial extent of destructuration of the
host soil was smaller than the radius of influence of the excess pore water pressures. Roy et
al. (1981) however conducted field tests in sensitive clay with a water content noticeably
higher than the liquid limit. Thus, the results may not be generalised.
Indraratna et al. (2013) propose a numerical model using the finite-difference method and
based on a unit cell approach for the boundary value problem of soft soil reinforced by stone
columns, and loaded uniformly by an embankment (Figure 6.12 a and b). The cross-section
of the unit cell is subdivided into 4 different zones: the unclogged column zone (denoted as
rc’), the clogged column zone (denoted as rc), the smear zone (denoted as rs) and the
undisturbed zone (denoted as re).
A soft clay layer with a thickness H overlays an impervious rigid boundary and is improved by
a group of stone columns resting on the rigid boundary. The constitutive model used has
been presented by Indraratna et al. (2013). A range of assumptions were made in this model:
- Darcy’s law is valid and no vertical water flow occurs within the soil mass. No flow
occurs through the cylindrical boundary and the base of the unit cell,
- the flow remains at steady state,
- all compressive strains are vertical,
6.2 Literature review of numerical modelling of ground improvement through stone columns
and prefabricated vertical drains
232
- the elastic part of the settlement may be neglected compared to the plastic
consolidation settlement,
- the degree of saturation of the soil mass is 1 and the water is incompressible,
- the coefficients of permeability and compressibility of the soil mass remain
constant throughout the consolidation process.
Limited information is available in the literature for the determination of the value of the
clogging parameters α and of αk. Mays (2010) suggests values ranging for 0 (representing
total clogging of the column) to 1 (representing a totally unclogged column). The effective
radius r’sc of the column with clogging is expressed as:
6.24
with r’sc effective radius of the stone column (denoted as r’c in Figure 6.12)
rsc radius of the stone column (denoted as rc in Figure 6.12)
α non-dimensional factor ranging for 0 to 1 ( representing fresh and totally
unclogged columns)
The coefficient of horizontal permeability in the clogged column zone may be written as:
6.25
with kh,cl coefficient of horizontal permeability in the clogged zone
k’h coefficient of horizontal permeability in the smear zone
αk ratio of horizontal permeability of the clogged column zone to that of the
smear zone
This approach allows Indraratna et al. (2013) to investigate the influence of clogging on the
dissipation velocity of the generated excess pore water pressures (Figure 6.13). The
clogging reduces the settlements for Tr < 1 and delays the dissipation of excess pore water
pressure. However, as the stress distribution remains the same, the same total settlements
occur.
Although this model opens up some interesting perspectives, it also features some
limitations. Vertical flow is neglected although it can influence the performance of the
improved foundation system. This can be exacerbated when there is a sand bed underneath
the clay layer or for short stone columns. The loading is also assumed to be steady and
uniform, which means that the model is not able to cope with cyclic or time-dependent
loadings.
6 Numerical modelling
233
Figure 6.12: Unit cell (a) typical stone column–reinforced soft clay deposit supporting an
embankment; (b) unit cell idealisation; (c) cross-section (Indraratna et al., 2013).
6.2 Literature review of numerical modelling of ground improvement through stone columns
and prefabricated vertical drains
234
Figure 6.13: Influence of clogging on the normalised average excess pore water pressure
and on the normalised average ground settlement (Indraratna et al., 2013).
6.2.2 Analogy to installation of rigid inclusions
The similarity between the installation processes for piles and stone columns has already
been mentioned (Section 2.8). Dijkstra et al. (2011) propose to simulate the installation
phase in axisymmetric conditions using a fixed pile and a moving pile approach (Figure 6.14)
with a hypoplastic constitutive model. The soil moves around the inclusions in the first case
(Figure 6.14 a) while the pile is inserted into the ground in the second (Figure 6.14 b). Due to
the immobility of the pile in the first approach (Figure 6.14 a), the initial stiffness response
during the insertion depends on the location in the mesh where the displacements are
plotted. This issue is solved by the second approach (Figure 6.14 b).
a
a
Time factor Tr
10.80.60.40.20
0
0.03
0.06
0.09
0.12
0.15
0.18
No
rma
lize
da
ve
rag
eg
rou
nd
se
ttle
me
nt
No
rma
lize
da
ve
rag
ee
xce
ss
po
rep
ressu
re
0.0
0.2
0.4
0.6
0.8
No clogging
α = 0.75; αk = 1.0
α = 0.75; αk = 0.5
α = 0.5; αk = 1.0
α = 0.5; αk = 0.5
No clogging
α = 0.75; αk = 1.0
α = 0.75; αk = 0.5
α = 0.5; αk = 1.0
α = 0.5; αk = 0.5
6 Numerical modelling
235
(a) (b)
Figure 6.14: Boundary conditions for (a) the fixed pile approach (b) the moving pile approach
(Dijkstra et al., 2011).
The fixed pile and moving pile approaches are used to simulate centrifuge tests during which
piles were installed in sand. The measured effective stress in the centrifuge (denoted as
meas. in Figure 6.15 and Figure 6.16) is compared with the stress at the pile base calculated
with the fixed pile approach (denoted as FP in Figure 6.15) and with the moving pile
approach (denoted as MP in Figure 6.16) for different values of the initial porosity n0 of the
host soil. Loose conditions are modelled with n0 = 0.439 [-], medium dense conditions with
n0 = 0.415 [-] and dense conditions with n0 = 0.389 [-]. The fixed pile approach tends to
overestimate the stress response while the moving pile approach tends to underestimate the
response, in comparison with the measured tip resistance.
q
pile
L
R
soil
H
inflow of material with prescribed velocity
W
geometry lineinfluence radius of
geometry line == pile radius
node in pile, v = vpile
node in soil
v not prescribed
6.2 Literature review of numerical modelling of ground improvement through stone columns
and prefabricated vertical drains
236
Figure 6.15: Comparison of calculated and measured stress response at the tip of the pile
during installation for the fixed pile approach (after Dijkstra et al., 2011).
Figure 6.16: Comparison of calculated and measured stress response at the tip of the pile
during installation for the moving pile approach (after Dijkstra et al., 2011).
-14000 -12000 -10000 -8000 -6000 -4000 -2000 00
-16000
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
σyy;base [kPa]
Pile
dis
pla
ce
me
nt[m
]
0
FP; n0=0.439
FP; n0=0.415
FP; n0=0.389
meas.; n0=0.389
meas.; n0=0.439
meas.; n0=0.415
FP; n0=0.439
FP; n0=0.415
FP; n0=0.389
meas.; n0=0.389
meas.; n0=0.415
meas.; n0=0.415
-14000 -12000 -10000 -8000 -6000 -4000 -2000 00
-16000
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
σyy;base [kPa]
Pile
dis
pla
cem
ent[m
] 0
MP; n0=0.439
MP; n0=0.415
MP; n0=0.389
meas.; n0=0.389
meas.; n0=0.439
meas.; n0=0.415
meas.; n0=0.389
MP; n0=0.439
MP; n0=0.415
MP; n0=0.389
meas.; n0=0.439
meas.; n0=0.415
6 Numerical modelling
237
Grabe & Pucker (2012) modelled the construction of displacement piles using finite
elements. They built construction aids (denoted as Pipe in Figure 6.17) into the mesh and
inserted the piles into the soil continuum along the aids. Although this way of modelling the
insertion of an inclusion into the soil is interesting, the results presented by Grabe & Pucker
(2012) are specifically related to the most relevant mechanisms for piles. This study focuses
on the influence of the rotating speed of the mandrel, as well as its form.
Figure 6.17: Modelling technique for the simulation of the pile insertion
(after Grabe & Pucker, 2012).
6.3 Constitutive models
The granular inclusions were modelled using the Mohr-Coulomb (MC) model, while the soft
soil was idealised with the Hardening Soil Model (HSM) in Plaxis
(Brinkgreve & Broere, 2008).
6.3.1 Mohr-Coulomb model
6.3.1.1 Description
The MC model is a linear elastic perfectly plastic model in which the strains are decomposed
into an elastic and a plastic part (εe and εp respectively, Figure 6.18). Plasticity is associated
with the development of irreversible strains. A yield function f is used to assess whether
plasticity occurs in a calculation: plastic yielding is associated with the condition f = 0.
Pile
Pipe,
R=1 mm
Soil continuumSoil continuum
6.3 Constitutive models
238
Figure 6.18: Elastic perfectly plastic model.
The two parameters controlling the yield surface are the effective cohesion c’ and the
effective angle of friction . The effective cohesion can be used to model approximately the
effect of suction or cementation on the intersection of the failure line with the vertical axis in a
– σ’ diagram (Figure 6.19).
Figure 6.19: Impact of the effective cohesion on the failure line in a – σ’ diagram.
Special attention should be given to the definition of the Young’s modulus E, as this
parameter controls the development of strains. The initial stiffness modulus Ei (Figure 6.21)
can be used for soils with a large elastic compression domain. It is more common to use the
secant stiffness in a consolidated drained triaxial test E50 (Figure 6.21) to represent the
loading of soils. In case of unloading and reloading, e.g. for applications in tunnelling, the
stiffness for unloading – reloading (Figure 6.21) should be used. Unlike the Hardening Soil
Model (HSM, Section 6.3.2), the stiffness moduli used in the Mohr-Coulomb model are not
stress-dependent.
f = 0
’
0
c’
τ [kPa]
Failure line
σ’ [kPa]
’
6 Numerical modelling
239
The tangent stiffness for primary oedometer loading Eoed can be calculated as:
( )
( ) ( ) 6.26
with Eoed tangent stiffness for primary oedometer loading
E Young’s modulus
Poisson’s ratio
6.3.1.2 Limitations of the Mohr-Coulomb model
Although Mohr-Coulomb is a well-proven failure criterion, it exhibits some limitations, two of
which are exposed here. First, a linear elastic stiffness behaviour is assumed up to the failure
surface, which limits the ability of the model to predict deformations before yielding. Second,
the use of effective strength parameters in undrained analysis leads to an overestimation of
the undrained shear strength by Δsu due to the different stress paths followed in reality and
by the FEM model (Figure 6.20).
Figure 6.20: Effective stress paths followed real soil and FEM prediction using the
Mohr-Coulomb model.
6.3.1.3 Input parameters of the Mohr-Coulomb model
The Mohr-Coulomb model requires five parameters, as summarised in Table 6.1.
Table 6.1: Input parameters for the Mohr-Coulomb model.
E Young’s modulus [kPa]
’ Poisson’s ratio [-]
c’ Cohesion [kPa]
Effective angle of friction [°]
Angle of dilatancy [°]
6.3.2 Hardening Soil Model
The Hardening Soil Model (HSM) is an elasto-plastic model, formulated in 3D, which differs
from the Mohr-Coulomb model as the soil behaviour before failure can be better modelled
q
p’
FEM prediction with
Mohr-Coulomb
Real soil behaviour
Δsu
φ’
6.3 Constitutive models
240
using three input stiffness moduli (Figure 6.21). The HSM is based on the Mohr-Coulomb
failure criterion, but the shear strain hardening (Figure 6.22) caused by deviatoric loading,
and the volumetric hardening (Figure 6.23) induced by volumetric strains, are both accounted
for. Schanz et al. (1999) present a detailed description of the HSM.
6.3.2.1 Stiffness moduli
The HSM uses stiffness moduli determined in two-dimensional, axisymmetric, triaxial
compression tests. Three stiffness moduli are used in order to describe the loading
behaviour of the soil. Ei is the initial tangent stiffness modulus and E50 is the secant stiffness,
both for primary axial loading and Eur is the stress-dependent stiffness for unloading-
reloading paths. All stiffnesses are those obtained from a consolidated drained triaxial test in
axial compression (CIDC or CADC). While the differentiation between isotropic and
anisotropic consolidation is not often made in practice, both kinds of triaxial tests will be
denoted from here on as CDC. The different stiffness moduli (Figure 6.21) used in the HSM
are listed in Table 6.2.
Table 6.2: Stiffness moduli used in the HSM.
Ei Initial tangent stiffness modulus
E50 Secant stiffness for primary loading in a CDC triaxial test
Eur Stress-dependent stiffness for unloading – reloading in a CDC triaxial test
Figure 6.21: Hyperbolic stress-strain relation in primary loading and unloading-reloading in a
CDC triaxial test (Schanz et al., 1999).
E50 is obtained from a CDC triaxial test for a mobilisation of 50 % of the failure shear strength
qf (Figure 6.21). It is preferred as stiffness for initial loading to the initial stiffness modulus Ei,
Asymptote
E50
1Ei
1
Eur
1
qf
qa
q
ε1
6 Numerical modelling
241
which is more difficult to determine experimentally, and gives a good approximation of the
initial behaviour of the soil. E50 is defined as:
(
)
6.27
with E50 secant stiffness for primary loading in a CDC triaxial test
E50ref secant stiffness for primary loading in a CDC triaxial test corresponding to the
reference stress pref
pref reference stress, usually set equal to 100 kPa
σ’3 principal effective stress
c’ effective cohesion
φ’ effective angle of friction
m power for stress-level dependency of stiffness. This factor should be equal to
1.0 in the case of soft clays [-] (Schanz et al., 1999).
The unloading – reloading path (for OCR > 1) is purely non-linear elastic and Eur is
formulated as:
(
)
6.28
with Eur unloading / reloading stiffness
Eurref unloading / reloading stiffness corresponding to the reference stress pref
6.3.2.2 Yield surfaces
The two yield surfaces considered by the HSM are shown in Figure 6.22 (shear strain
hardening) and Figure 6.23 (volumetric hardening). The shear strain hardening corresponds
to the rotation of the yield loci with increasing plastic shear strain γp, while the volumetric
hardening denotes the expansion of the cap yield surface with growing volumetric strains.
Figure 6.22: Successive yield loci for shear strain hardening for various values of the plastic
shear strain γp and failure surface for m = 0.5 [-] (Brinkgreve & Broere, 2008).
Deviatoric stress q [kPa]
Mean effective stress p’ [kPa]
Mohr-Coulomb failureMohr-Coulomb failure line
6.3 Constitutive models
242
Figure 6.23: Cap yield surface of the HSM for volumetric hardening in the - plane
(after Brinkgreve & Broere, 2008).
6.3.2.3 Shear strain hardening
Shear strain hardening (Figure 6.22) is controlled by two yield functions, f12 and f13
(Equations 6.29 and 6.30), which are defined as:
(
)
⁄ (
)
(
)
6.29
(
)
⁄ (
)
(
)
6.30
(
) 6.31
6.32
⁄
6.33
with f12, f13 yield surfaces in σ1, σ2 and σ1, σ3 planes, respectively [-]
E50 secant stiffness for primary loading in a CDC triaxial test
σ’1, σ’2, σ’3 principal effective stresses
Eur unloading / reloading stiffness in a CDC triaxial test
γp plastic shear strain [-]
Rf failure ratio, usually set equal to 0.9 [-]
c’ effective cohesion
φ’ effective angle of friction
ε1p, ε2
p principal plastic strains [-]
εvp plastic volumetric strain [-]
Ei initial stiffness modulus in a CDC triaxial test
q deviatoric stress
qf deviatoric stress at failure
‘
Cap yield surface
p’
pp’c’ cotφ’
α pp’
6 Numerical modelling
243
Schanz et al. (1999) assume that the approximation made in Equation 6.32 is valid under the
assumption that the plastic volumetric strains are equal to zero. Although this is never
precisely the case, Schanz et al. (1999) declare that the plastic volumes changes in “hard”
soils tend to be small compared to the axial strain, which they contend makes this
approximation acceptable. This is discussed in Section 6.3.2.5.
The yield loci can be visualised in a p’-q plane (Figure 6.22) for a given constant value of the
plastic shear strain γp. Plotting such loci, which satisfy the condition f12 = f13 = 0, implies using
Equations 6.27, 6.28, 6.29 and 6.30. The shape of the curves depends on the value selected
for m. Straight lines are obtained for m = 1.0 (as for example for the Mohr-Coulomb failure
line, Figure 6.22), and curves for lower values of this parameter (Figure 6.22). The failure
criterion is given by the MC-failure condition.
6.3.2.4 Volumetric hardening
The volumetric hardening corresponds to the expansion of the cap yield surface with
increasing volumetric strains (Figure 6.23). The formulation of a cap yield surface is
necessary for a model with independent input of E50ref
, mainly controlling the shear yield
surface and the plastic strains related to it, and of Eoedref, controlling the cap yield surface and
the plastic strains originating from it. The cap yield surface can be expressed as:
6.34
6.35
( )
6.36
6.37
with fc cap yield surface [-]
special stress measure for deviatoric stresses
p’ mean effective stress
σ’1, σ’2, σ’3 principal effective stresses
φ’ effective angle of friction
α cap parameter relating to K0NC [-]
pp pre-consolidation stress
δ parameter depending on the effective angle of friction [-]
is a 3D generalisation of the formulation of the deviatoric stress q for a triaxial compression
stress state (q = σ1’ – σ3’), which allows for the principal effective stress σ’2 to be considered
by using a factor δ. is equal to q for a triaxial compression (σ’2 = σ’3). A 3D formulation of
the deviatoric stress is necessary in order to obtain yield contours in effective principal stress
space (Figure 6.24).
6.3 Constitutive models
244
Figure 6.24: Representation of yield contours of the HSM in effective principal stress space
(Schanz et al., 1999).
The magnitude of the yield cap is mainly controlled by the pre-consolidation stress pp
(Equation 6.34). The hardening function relating pp to the volumetric cap strain εvpc is:
(
)
6.38
with εvpc volumetric cap strain [-]
cap parameter relating to Eoedref [-]
m power for stress-level dependency of stiffness [-]
pp pre-consolidation stress
pref reference stress
The cap parameters α and β are not direct input parameters, but their magnitude can be
defined by the values of K0NC and Eoed
ref (Brinkgreve & Broere, 2008). Figure 6.23 shows the
yield surfaces of the HSM model in - space, taking the effect of any effective cohesion c’
into account, and displaying the location of the cap yield surface.
6.3.2.5 Limitations of the Hardening Soil Model
The assumption made by Schanz et al. (1999) is only valid for “hard” soils, i.e. rocks, and
could be assumed to be representative in the highly over-consolidated zones of the clay
samples used in this research (OCR > 20, Figure 4.3), that is up to a depth of 15 mm (σc =
100 kPa), respectively 20 mm (σc = 200 kPa). This assumption is however violated in both
fine grained and coarse grained soils and in the lower part of the soil samples used in this
research, where OCR < 20.
In p’-q space, neglecting the volumetric part of the plastic strain εvp causes a rotation of the
incremental plastic strain d εp, which becomes vertical (Figure 6.25) instead of being normal
to the critical state line (associated flow rule). This rotation results in a state that violates the
σ’2
σ’3
σ’1
6 Numerical modelling
245
consistency condition during plastic loading, according to which the stress states before and
after plastic loading have to be on the failure surface, which can be expressed as:
6.39
with F failure surface
p’ mean effective stress
q deviatoric stress
This challenges the theoretical evolution of the distribution of the yield loci during shear strain
hardening (Figure 6.22).
Figure 6.25: Representation of the associated flow rule in triaxial space.
6.3.2.6 Input parameters of the Hardening Soil Model
Some of the parameters of the HSM (the failure parameters , c’ and ) coincide with those
of the Mohr-Coulomb model (Table 6.1). The required input parameters for the soil stiffness
are summarised in Table 6.3.
Table 6.3: Input parameters for the soil stiffness in the HSM.
E50ref
Secant stiffness for primary loading in a CDC triaxial test
corresponding to the reference stress pref [kPa]
Eoedref
Tangent stiffness for primary oedometer loading corresponding to
the reference stress pref [kPa]
Eurref
Unloading / reloading stiffness in a CDC triaxial test corresponding
to the reference stress pref [kPa]
m Power for stress-level dependency of stiffness [-]
’ur Poisson’s ratio for unloading - reloading [-]
pref Reference stress [kPa]
K0NC Coefficient of earth pressure at rest of a normally consolidated soil [-]
Rf Failure ratio, usually set equal to 0.9 [-]
0
q, dεsp
p’, dεvp
1
M
dεvp
dεsp
dεp
Critical state line
dεp (dεvp=0)
6.4 Axisymmetric numerical modelling
246
6.4 Axisymmetric numerical modelling
The numerical modelling was first conducted using an axisymmetric 2D model and
subsequently in 3D. This section presents the ideas and options discarded (Section 6.4.1)
that led to the model used (Section 6.4.2), as well as the results obtained (Section 6.4.3) with
the axisymmetric numerical modelling.
2-dimensional axisymmetric numerical modelling of the centrifuge tests has been conducted
using specimens prepared in adapted oedometer containers (Section 3.7.5), namely tests
JG_v7 (pre-consolidation up to 200 kPa) and JG_v9 (pre-consolidation up to 100 kPa). The
modelling was carried out using the code Plaxis 2D Version 2012.2, with an axisymmetric
model and 15-noded elements.
6.4.1 Options discarded
Modelling the installation phase of inclusions into soil is cumbersome. A relatively elegant
solution using the commercially available code Abaqus is the method proposed by Grabe &
Pucker (2012), which was presented in the previous section. However, the use of Plaxis 2D
restricts the possibilities of modelling the installation phase to the application of radially
outwards and vertically downwards prescribed displacements applied on the vertical limits of
an initial cavity (Weber, 2008; Figure 6.26).
Figure 6.26: Modelling of the insertion of the stone column installation tool by means of
application of prescribed displacements on the wall of an initial cavity (Weber,
2008).
The modelling of the installation phase in a 2D axisymmetric model conducted by Weber
(2008) delivered some results in good agreement with the results obtained from centrifuge
tests, in respect of the stress paths and excess pore pressures. However, the stiffness of the
6 Numerical modelling
247
soft clay bed had to be adapted in the numerical analysis to achieve a good fit to the load-
settlement behaviour during embankment loading.
Moreover, the modelling of the installation phase is very cumbersome in practical cases,
which has stimulated the research towards the development of a simple and realistic
numerical modelling approach by taking the installation effects into account. The goal is to
develop an approach that a practical engineer might be willing and able to use. This has led
to the decision to adopt a “wished-in-place” approach, in order to model both the inclusions
and any installation effects.
The influence of the over-consolidation of the soft soil can be taken into account by defining
a Pre-Overburben-Pressure (POP) in the numerical parameters describing the soil
properties, when calculating the initial stress field. The Plaxis code (Brinkgreve & Broere,
2008) recalculates the coefficient of earth pressure at rest K0, taking the pre-consolidation
into account by using the following formula:
( )
| |
6.40
with K0OC coefficient of earth pressure at rest of an over-consolidated soil
K0NC coefficient of earth pressure at rest of a normally consolidated soil
OCR over-consolidation ratio
ur Poisson’s ratio for the unloading-reloading
POP Pre-Overburden-Pressure
| | absolute value of the vertical total stress
Although this approach looks tempting, it leads to an overestimation of the horizontal stress
near the surface due to the very high values reached in predicting the OCR, and thus to an
unrealistic failure state because of the generation of the stresses during the initial phase.
The initial stress state was calculated by modelling the consolidation under 1 g. This was
done by implementing a soil featuring the properties of the slurry used in the preparation of
the soil models (Section 3.6) and loading it by 100 kPa or 200 kPa, depending on the
centrifuge test modelled. The calculation phases are listed in Table 6.4.
Table 6.4: Description of the calculation phases used in the axisymmetric model.
Phase Action
Initial
phase Generation of an initial stress state with the slurry (clay with reduced unit weight)
1 Loading of the slurry
2 Unloading of the slurry
3 Replacement of the slurry by clay with properties determined in the laboratory
4 Activation of the clusters corresponding to the stone columns and to the
installation effects
5 Loading of the stone columns using prescribed displacements
6 Dissipation of the excess pore water pressures generated
6.4 Axisymmetric numerical modelling
248
6.4.2 Model
The numerical modelling was conducted at prototype scale. The mesh for the axisymmetric
model (Figure 6.27) is 6.25 m wide and 8 m high, corresponding to the prototype dimensions
of the adapted oedometer containers (Sections 3.6.3 and 3.7.5). The stone column is
modelled by a cluster featuring a width of 0.3 m and a height of 6 m. The water table is
located 0.5 m below the surface of the model.
The Hardening Soil parameters used to model the clay are listed in Table 6.5. The soil
properties implemented in the numerical model were determined in laboratory experiments
(Sections 3.5.1 and 5.1). The unloading / reloading stiffness had to be amended
in order to improve the match between recorded and modelled load-settlement curves
(Figure 6.28), but the difference between the value used in the numerical model and the
value determined in the laboratory is acceptable (15000 kPa instead of 13000 kPa).
A value of 50 kPa was assigned to represent cohesion in the 0.2 m thick layer at the top of
the model, in order to prevent the appearance of very large unrealistic soil movements at the
surface during loading. The displacement-controlled footing loading was modelled by
applying prescribed displacements at the surface of the model. A displacement of 850 mm
(17 mm @ 50 g) was applied over a time of 2.125.106 s (850 s @ 50 g).
Table 6.5: Summary of the Hardening Soil parameters for the clay.
Parameter σc [kPa] Value
Unit weight γsat [kN/m3] 100 18.50
200 20.10
Coefficient of horizontal permeability kx [m/s] 100 1.10-9
200 5.10-10
Coefficient of vertical permeability ky [m/s] 100 5.10-10
200 2.5.10-10
Secant stiffness for primary loading in CDC triaxial test E50ref
[kPa] 100 / 200 2362
Tangent stiffness for primary oedometer loading Eoedref [kPa] 100 / 200 1900
Unloading / reloading stiffness Eurref [kPa] 100 / 200 15000
Reference stress for stiffness pref [kPa] 100 / 200 100
Poisson´s ratio for unloading / reloading ur [-] 100 / 200 0.2
Power for stress-level dependency of stiffness m [-] 100 / 200 1
Effective cohesion c’ [kPa] 100 / 200 1
Effective angle of internal friction φ’cv [°] 100 / 200 24.5
Angle of dilatancy y [°] 100 / 200 0
6 Numerical modelling
249
The Mohr-Coulomb parameters used for the stone column material are summarised in
Table 6.6.
Table 6.6: Mohr-Coulomb parameters for the stone column material.
Parameter Value
Unsaturated unit weight γunsat [kN/m3] 15.00
Saturated unit weight γsat [kN/m3] 20.00
Isotropic coefficient of permeability [m/s] 1.10-5
Young´s modulus E [kPa] 20000
Poisson’s ratio [-] 0.3
Effective cohesion c’ [kPa] 1
Effective angle of internal friction φ’ [°] 37.00
Angle of dilatancy y [°] 10.00
The influence of the installation effects is taken into account by means of a wished-in-place
procedure. The dimensions of the corresponding zones must be defined. Weber (2008)
showed the appearance of a smear zone, in which the clay platelets display a vertical
reorganisation. This zone is 2 mm wide at model scale under 50 g, i.e. 0.1 m at prototype
scale (Figure 3.21). This was modelled numerically by a cluster extending radially outwards
and vertically downwards 0.1 m from the edge of the stone column (Figure 6.27). The
extensive destructuring undergone by the clay within the smear zone is taken into account by
a reduction of the stiffness by a factor of 0.8 (Table 6.7).
Table 6.7: Summary of the Hardening Soil parameters for the smear zone.
Parameter σc [kPa] Value
Unit weight γsat [kN/m3] 100 18.50
200 20.10
Coefficient of horizontal permeability kx [m/s] 100 / 200 1.10-10
Coefficient of vertical permeability ky [m/s] 100 / 200 1.10-10
Secant stiffness for primary loading in CDC triaxial test E50ref
[kPa] 100 / 200 1889
Tangent stiffness for primary oedometer loading Eoedref [kPa] 100 / 200 1520
Unloading / reloading stiffness Eurref [kPa] 100 / 200 12000
Effective angle of internal friction φ’cv [°] 100 / 200 24.5
6.4 Axisymmetric numerical modelling
250
Onoue et al. (1991) identified a reduction of the horizontal permeability by a factor of about
0.8 in the so-called disturbed zone (Figure 2.60). Such an approach was implemented here,
while the extent of this zone is different to that suggested by Onoue et al. (1991).The radial
extent of this zone is 2.5 times the radius of the stone column, which takes Weber’s (2008)
findings, as well as the results of the investigations conducted with the electrical impedance
needle (Section 4.7), into account. Its vertical extent from the toe of the stone column is 1.5
times the diameter of the inclusion (Figure 6.27), which is in good agreement with the
observations made by Linder (1977) for inclusions installed in dense sand (Figure 2.50). This
annular zone will be described from now on as the compaction zone. The permeability of the
compaction zone was reduced by a factor of 0.8, compared to that of the surrounding soft
clay, while the stiffness was increased by a factor of 1.2 in order to take the strength increase
caused by the stone column installation into account (Table 6.8).
Table 6.8: Summary of the Hardening Soil parameters for the compaction zone.
Parameter σc [kPa] Value
Unit weight γsat [kN/m3] 100 22.20
200 24.10
Coefficient of horizontal permeability kx [m/s] 100 8.10-10
200 4.10-10
Coefficient of vertical permeability ky [m/s] 100 4.10-10
200 2.10-10
Secant stiffness for primary loading in CDC triaxial test E50ref
[kPa] 100 / 200 2834
Tangent stiffness for primary oedometer loading Eoedref [kPa] 100 / 200 2280
Unloading / reloading stiffness Eurref [kPa] 100 / 200 18000
Effective angle of internal friction φ’cv [°] 100 / 200 24.5
6 Numerical modelling
251
Figure 6.27: 2D axisymmetric numerical model for a unit cell including a single stone column.
6.4.3 Results
Figure 6.28 shows a comparison of the load-settlement curves for tests JG_v7
(σc = 200 kPa) and JG_v9 (σc = 100 kPa) obtained experimentally in the centrifuge (denoted
as JG_v7 and JG_v9, respectively) and numerically (denoted as
JG_v7 – Plaxis and JG_v9 – Plaxis, respectively). The settlement of the footing in the
centrifuge was scaled up to prototype scale. The maximum values of the footing load can be
modelled with satisfying precision (Table 6.9).
Table 6.9: Comparison of the experimental and numerical values of the maximum footing
loads.
Test Pmax, Centrifuge
[kPa]
Pmax, Plaxis
[kPa]
Difference
[%]
JG_v7 (σc = 200 kPa) 145.44 141.90 2.49
JG_v9 (σc = 100 kPa) 119.67 115.11 3.96
Stone column
Smear zone
Compaction zone
8 m
6m
6.1
m
7m
Clay
0.3 m
0.4 m
0.75 m
0.2 m
6.25 m
1.4 m
Prescribed displacements
x
y
6.4 Axisymmetric numerical modelling
252
Figure 6.28: Comparison of the experimental and numerical load-settlement curves for tests
JG_v7 and JG_v9.
Figure 6.29 and Appendix 8.11 show the deformed mesh obtained after modelling the footing
loading during tests JG_v7 and JG_v9, respectively. The differences are minimal and tend to
indicate that the depth of the bulging deformation of the columns is independent of the pre-
consolidation stress. The deformation of the inclusion is concentrated in both cases near the
surface, while the displacements of the toe are negligible. This is confirmed by the
distribution of the vertical strain increments computed numerically for test JG_v7, for a
footing settlement of 850 mm (Figure 6.30). The vertical and shear strain increment
computed for tests JG_v7 and JG_v9 are presented in Appendices 8.8 and 8.9, and
Appendices 8.13 and 8.14, respectively. Both vertical and shear strain increments are
concentrated near the surface up to a depth of 2.0 m. The deformations below that depth are
negligible and the total vertical stress increase at the toe of the inclusion is minimal
(Figure 6.31). The sudden vertical stress increase within the stone column at a depth of
approximately 3 m cannot be explained either by any physical mechanisms or by the
distribution of the vertical strain increments (Figure 6.30) and is thus assumed to be due to a
local numerical instability.
0
20
40
60
80
100
120
140
160
0 100 200 300 400 500 600 700 800 900
Fo
oti
ng
lo
ad
ing
[k
Pa
]
Footing settlement [mm]
JG_v7 JG_v7 - Plaxis JG_v9 JG_v9 - Plaxis
6 Numerical modelling
253
Figure 6.29: Deformed mesh obtained for test JG_v7 for a settlement of 850 mm and a
footing load of 145.44 kPa.
Figure 6.30: Vertical strain increment computed numerically for test JG_v7 for a settlement of
850 mm and a footing load of 145.44 kPa.
Stone column
Smear zone
Compaction zone
8 m
6m
6.1
m
7m
Clay
0.3 m
0.4 m
0.75 m
0.2 m
6.25 m
1.4 m
0.8
5 m
x
y
6.4 Axisymmetric numerical modelling
254
Figure 6.31: Total vertical stress distribution computed numerically for test JG_v7 for a
settlement of 850 mm and a footing load of 145.44 kPa.
The development of plastic points during the loading phase is presented in Figure 6.32.
Failure points denote elements that are located on the Mohr-Coulomb failure line
(Figure 6.19, respectively Figure 6.22), while Cap points and Hardening points refer to
elements undergoing plastic shear strain hardening (Figure 6.22, Section 6.3.2.3) and plastic
volumetric hardening (Figure 6.23, Section 6.3.2.4), respectively. Tension cut-off points
denote elements which would be in tension if the value of cohesion was positive. The depth,
up to that elements undergo plastic shear hardening, ranges from 3.2 m for a settlement of
100 mm (P = 85 kPa) to 4.0 m for a settlement of 850 mm (P = 145.4 kPa), as illustrated in
Figure 8.19 to Figure 8.21. Failure points reach a depth of 2.0 m within the stone column for
a settlement of 100 mm (Figure 8.19) and of approximately 2.5 m for settlements of 400 mm
(Figure 8.20) and 850 mm (Figure 8.21). These points are present up to a depth of 1.5 m
within the host soil for a settlement of 400 mm (Figure 8.20) and up to a depth of 2.0 m for a
settlement of 850 mm (Figure 8.21), which corresponds to the zone where the excess pore
water pressures are the highest (Figure 6.46).
6 Numerical modelling
255
(a) (b) (c)
Figure 6.32: Development of plastic points during the loading phase for test JG_v7 (a) for a
settlement of 100 mm (P = 85 kPa), (b) for a settlement of 400 mm
(P = 115.2 kPa) and (c) for a settlement of 850 mm (P = 145.44 kPa).
Figure 6.28 shows that the initial stiffness in the load-settlement curve obtained in the
numerical model is noticeably higher than that measured in the physical model. Three
explanations can be proposed:
- the clay sample used for the centrifuge test was thicker than 160 mm after
consolidation under the press. The height of the sample was reduced manually,
which is thought to have caused a disturbed layer with reduced stiffness near the
surface of the sample,
- the assumption made for the numerical modelling is that of a full contact between
footing and soil specimen. However, the surface onto which the load was applied
during the centrifuge tests was not perfectly smooth,
- in combination with the development of the stress distribution under a rigid footing
(Nater, 2005), the actual stress distribution differs from the theoretical stress
distribution expected under a rigid footing (Figure 6.33), which causes a less stiff
response than was computed numerically. The latter can also be seen when
comparing field measurements with results from numerical modelling
(Arnold, 2011).
The first explanation is not easy to prove numerically. The attempts to model the disturbance
of the soil near the surface were not conclusive. However, the implementation of pressure
pads between the footing and the surface of the subsoil has given an insight into
6.4 Axisymmetric numerical modelling
256
the pressure distribution under the footing during loading for tests JG_v7 and JG_v9
(Figure 6.34, one pixel corresponds to a surface of approximately 1 mm2). The blue surfaces
in Figure 6.34 represent, for a footing settlement of 100 mm at prototype scale, the zones in
which a load is actually applied onto the soil, while the white areas correspond to the
unloaded zones. The location of the stone column is, in both cases, marked by a yellow
circle. However, the quantity of sand that was poured into the feed pipe during the installation
of the stone column was not sufficient for the top of the column to reach the surface of the
soft soil. This created a cavity, which can be seen in Figure 6.34 (a), as no load was applied
at the location of the stone column. This cavity was closed during loading as clay collapsed
into it (Figure 6.34 b). Thus the loading was actually applied on a clay surface, which covered
the stone column, and no significant pressure difference could be measured (Figure 6.34 b).
(a) (b) (c) (d) (e)
Figure 6.33: Vertical stress distribution from a line load below a rigid strip footing (a) for the
self-weight of the footing and for a global safety factor equal to (b) 3.0, (c) 2.0,
(d) 1.5 and (e) 1.0 (Jessberger, 1995).
The representation of the pressure distribution shows that the surface onto which a load is
actually applied is significantly smaller than in an ideal case. As the load is brought onto the
footing in a displacement-controlled manner, it builds up noticeably lower than as calculated
numerically.
Figure 6.35 shows the distribution of the loaded zones under the footing for a settlement of
400 mm at prototype scale for the tests JG_v7 and JG_v9. The location of the stone column
is, in both cases, that indicated in Figure 6.34. No significant pressure difference could be
measured as the load was actually applied on a clay surface, which covered the stone
column. The white vertical stripes on the right hand side are due to local failure of the row of
sensors in the pressure pads. Figure 6.35 (a) shows, without doubt, that full contact has
been reached between footing and clay surface during test JG_v7, for a settlement of 400
mm at prototype scale. Such an assessment is not as straightforward in the case of
test JG_v9. In addition to the white stripes on the right hand side, the white zones in the
bottom right corner indicate an unloaded zone.
6 Numerical modelling
257
(a) (b)
0 kPa 100 – 140 kPa 140 – 180 kPa
Figure 6.34: Pressure distribution under the footing for a settlement of 100 mm at prototype
scale (2 mm under 50 g) for (a) test JG_v7 (P = 55.6 kPa) and (b) test JG_v9 (P
= 46.1 kPa).
(a) (b)
0 kPa 100 – 140 kPa 140 – 180 kPa 180 – 220 kPa
Figure 6.35: Pressure distribution under the footing for a settlement of 400 mm at prototype
scale (8 mm under 50 g) for (a) test JG_v7 (P = 110.1 kPa) and (b) test JG_v9
(P = 90.1 kPa).
However, the fact that most of these apparently unloaded zones are still present for a
settlement of 800 mm at prototype scale (Figure 6.36), indicates that there might be a local
measurement issue of the pressure pads. As a consequence, full contact between footing
and clay surface may be assumed for a settlement of 400 mm at prototype scale during
test JG_v9.
Stone
column
Stone
column
56 m
m
56 m
m
6.4 Axisymmetric numerical modelling
258
0 kPa 100 – 140 kPa 140 – 180 kPa 180 – 220 kPa
Figure 6.36: Pressure distribution under the footing for a settlement of 800 mm at prototype
scale (16 mm under 50 g) for test JG_v9 (P = 118.1 kPa).
These observations confirm the second explanation for the initial difference between
centrifuge and numerical modelling of the load-settlement behaviour. They also show that the
results obtained with the proposed numerical model are in good agreement with the
measurements conducted in the centrifuge when full contact between the footing and the
clay surface is attained in the centrifuge.
Figure 6.37 and Figure 6.38 show the stress distribution under the footing, as computed
numerically for footing settlements of 100 mm, 400 mm and 850 mm for tests JG_v7 and
JG_v9, respectively. The differences between the measurements obtained with the pressure
pads (Figure 6.34, Figure 6.35 and Figure 6.36) and the stress distributions computed
numerically are significant as a strong stress concentration on top of the stone column is
obtained numerically, which was not the case during centrifuge tests due to the issues
encountered with the filling of the inclusion. However, the stress peaks that were measured
by the pressure pads around the edge of the footing, once full contact between footing and
clay was reached, were also observed in the numerical modelling.
The stress concentration factor m at the top of the stone column computed numerically
reaches a value of approximately 3 in both tests JG_v7 and JG_v9 for a footing settlement of
100 mm and decreases to approximately 2.6, for a footing settlement of 400 mm
and 2 (test JG_v7), respectively 2.3 (test JG_v9) for a footing settlement of 850 mm
(Figure 6.41 and Figure 6.42). This confirms the dependency of the stress concentration
factor on the load applied on the composite foundation and is consistent with the
measurements obtained by Greenwood (1991) at St. Helens (Figure 2.7).
The influence of the varying stiffnesses of the smear and compaction zones and of the host
soil can be noted when comparing the stress distribution under the footing. The compaction
zone, which has the highest stiffness (Table 6.8), attracts more load than the surrounding
host soil, the stiffness of which is lower (Table 6.5). The surrounding host soil is also carrying
56
mm
6 Numerical modelling
259
more load than the smear zone, which has the lowest stiffness (Table 6.7). The differences
between the three zones are less with decreasing pre-consolidation of the sample, which can
be noted when comparing the distribution computed for tests JG_v7
(σc = 200 kPa, Figure 6.37) and JG_v9 (σc = 100 kPa, Figure 6.38).
Figure 6.37: Total vertical stress distribution under the footing, as a function of the radial
distance, for settlements of 100 mm, 400 mm and 850 mm for test JG_v7
(σc = 200 kPa).
Figure 6.38: Total vertical stress distribution under the footing, as a function of the radial
distance, for settlements of 100 mm, 400 mm and 850 mm for test JG_v9
(σc = 100 kPa).
Figure 6.39 and Figure 6.40 present the distribution of the total vertical stress computed
numerically for tests JG_v7 and JG_v9, respectively, as a function of the radial distance from
0
50
100
150
200
250
300
350
400
450
0 0.5 1 1.5
To
tal ve
rtic
al s
tre
ss
σv
[kP
a]
Radial distance [m]
100 mm (P = 85 kPa) 400 mm (P = 115.2 kPa)850 mm (P = 145.44 kPa)
Edge of stone column
Sm
ear
zon
e
Compactionzone
Edge of footing
0
50
100
150
200
250
300
350
400
450
0 0.5 1 1.5
To
tal ve
rtic
al s
tre
ss
σv
[kP
a]
Radial distance [m]
100 mm (P = 70.8 kPa) 400 mm (P = 94.3 kPa)850 mm (P = 119.67 kPa)
Edge of stone column
Sm
ear
zon
e
Compactionzone
Edge of footing
6.4 Axisymmetric numerical modelling
260
the axis of the stone column at depths of 0 m, 2 m, 4 m and 6 m and for a footing settlement
of 100 mm. A dissipation of the stress peak measured at the edge of the footing with depth is
observed. The load transferred to the stone column does not vary significantly between 0 m
and 2 m and between 4 m and 6 m depth. The decrease of load transferred to the column
between depths of 2 m and 4 m can be explained by the increase of the load transferred to
the surrounding host soil. The differences between loads transferred to smear and
compaction zones and host soil become less marked with increasing depth.
Figure 6.39: Total vertical stress distribution, as a function of the radial distance, under the
footing (z = 100 mm) and at depths of 2 m, 4 m and 6 m for a footing settlement
of 100 mm during the footing loading for test JG_v7 (P = 85 kPa).
Figure 6.40: Total vertical stress distribution, as a function of the radial distance, under the
footing (z = 100 mm) and at depths of 2 m, 4 m and 6 m for a footing settlement
of 100 mm during the footing loading for test JG_v9 (P = 70.8 kPa).
0
50
100
150
200
250
300
0 0.5 1 1.5
To
tal ve
rtic
al s
tre
ss
σv
[kP
a]
Radial distance [m]
z = 100 mm z = 2 m z = 4 m z = 6 m
Edge of stone column
Sm
ear
zon
e
Compactionzone
Edge of footing
0
50
100
150
200
250
300
0 0.5 1 1.5
To
tal ve
rtic
al s
tre
ss
σv
[kP
a]
Radial distance [m]
z = 100 mm z = 2 m z = 4 m z = 6 m
Edge of stone column
Sm
ear
zon
e
Compactionzone Edge of
footing
6 Numerical modelling
261
Figure 6.41 and Figure 6.42 show the distribution with depth of the stress concentration
factor m for footing settlements of 100 mm, 400 mm and 850 mm for tests JG_v7 and JG_v9,
respectively. A decrease of the stress concentration at the top of the stone column (z = 0 m)
with increasing load can be noted. The stress concentration decreases with depth to reach a
value of approximately 1 at the toe of the stone column. However, a greater decrease of the
stress concentration factor with depth is observed for lower loads than for higher loads, as
shown in Figure 6.43 and Figure 6.44, in which the values of the factor m are normed with
the initial value.
Figure 6.41: Distribution of the stress concentration factor m over depth for footing
settlements of 100 mm, 400 mm and 850 mm for test JG_v7 (σc = 200 kPa).
Figure 6.42: Distribution of the stress concentration factor m over depth for footing
settlements of 100 mm, 400 mm and 850 mm for test JG_v9 (σc = 100 kPa).
The low values of the stress concentration in the lower third of the stone column (values
ranging from 1 to 1.5 for depths between 4 m and 6 m) could open up the way to a more
sustainable use of the stone column material as the diameter of the inclusions could be
reduced in the deeper zones, where its impact is less significant, and augmented near the
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5 3
Dep
th [
m]
m [-]
s = 100 mm (P = 85 kPa) s = 400 mm (P = 115.2 kPa)s = 850 mm (P = 141.9 kPa)
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5 3
De
pth
[m
]
m [-]
s = 100 mm (P = 85 kPa) s = 400 mm (P = 115.2 kPa)s = 850 mm (P = 141.9 kPa)
6.4 Axisymmetric numerical modelling
262
surface, where its influence is greater. This hypothesis should, however, be confirmed by
further numerical and physical modelling, and the influence of a variation of the stone column
diameter on the load transfer and on the drainage performance would need to be
investigated in detail.
Figure 6.43 and Figure 6.44 present the distribution of the normalised stress concentration
factor with depth. The decrease of the values is greater for lower loads. This can be
explained by the transmission of the normal loading in the stone column over depth by the
activation of skin friction around the stone column with increasing loading. This causes a
rotation of the direction of the principal stresses, as σ1 tilts from an angle of 90° to an angle of
approximately 45° (Figure 6.45), which causes an additional load to act on the surrounding
host soil.
Figure 6.43: Distribution of the normalised stress concentration factor m over depth for
footing settlements of 100 mm, 400 mm and 850 mm for test JG_v7
(σc = 200 kPa).
0
1
2
3
4
5
6
0 0.5 1 1.5
De
pth
[m
]
m / m0 [-]
s = 100 mm (P = 85 kPa) s = 400 mm (P = 115.2 kPa)s = 850 mm (P = 141.9 kPa)
6 Numerical modelling
263
Figure 6.44: Distribution of the normalised stress concentration factor m over depth for
footing settlements of 100 mm, 400 mm and 850 mm for test JG_v9 (σc = 100
kPa).
Figure 6.45: Direction of the total principal stress at the end of the loading phase for
test JG_v7 (P = 141.90 kPa).
0
1
2
3
4
5
6
0 0.5 1 1.5
De
pth
[m
]
m / m0 [-]
s = 100 mm (P = 85 kPa) s = 400 mm (P = 115.2 kPa)s = 850 mm (P = 141.9 kPa)
Stone column
Smear zone
Compaction zone
Clay
6 m
1.4 m
6.4 Axisymmetric numerical modelling
264
Figure 6.46 and Appendix 8.15 show the distribution of the excess pore water pressures
computed numerically for tests JG_v7 and JG_v9, respectively. Significant differences
between the measurements from centrifuge tests (Figure 4.24) and the results of the
numerical modelling can be noted, as the excess pore water pressures computed
numerically are concentrated in a zone extending from 0.8 m to 2.0 m depth, while the
outcomes from physical modelling under enhanced gravity indicate a distribution of the
excess pore water pressures along the stone column down to its toe.
A comparison of the excess pore water pressures measured in the centrifuge during
tests JG_v7 and JG_v9 from the PPTs inserted into the soil model, with the values obtained
numerically, is shown in Figure 6.48 and Appendices 8.19 and 8.20, respectively. The
difference between the values of the excess pore water pressures recorded during the tests,
and those modelled numerically, is significant. However, the pore water pressures are often
difficult to reproduce due to the insufficiencies of the constitutive models
(Brinkgreve & Broere, 2008). Therefore load-settlement curves were given higher priority and
the efforts to try and match the values of the excess pore water pressures were not pursued.
Figure 6.46: Distribution of the excess pore water pressures computed numerically for test
JG_v7 for a footing settlement of 850 mm and a footing load of 141.90 kPa.
6 Numerical modelling
265
Figure 6.47: Comparison of the values of the excess pore water pressures measured during
test JG_v7 with the values obtained numerically with Plaxis 2D (P1 till P3).
An investigation of the influence of the installation effects on the load-settlement behaviour is
made possible by assigning the material Clay to the clusters Compaction zone and Smear
zone (Figure 6.27) in order to neglect the impact of the installation phase of the stone column
onto the host soil. Figure 6.48 and Figure 6.49 show comparisons of the load-settlement
curves obtained from the physical modelling and from the numerical modelling with, and
without, installation effects for tests JG_v7 and JG_v9, respectively. Table 6.10 summarises
the maximum values of the footing load. The percentage differences of the maximum footing
loads remain in both cases under 10 %. It is interesting that this difference rises with
increasing pre-consolidation stress (200 kPa for test JG_v7, 100 kPa for test JG_v9). This is
consistent with the greater differences between the smear and compaction zones and clay
observed in stress distribution under the footing with increasing pre-consolidation stress
(Figure 6.37 and Figure 6.38).
Table 6.10: Comparison of the values of the maximum footing loads obtained numerically
with (Pmax, Plaxis) and without (Pmax, Plaxis, no smear) installation effects for a settlement
of 850 mm.
Test Pmax, Plaxis
[kPa]
Pmax, Plaxis, no smear
[kPa]
Difference
[%]
JG_v7 (σc = 200 kPa) 141.90 130.72 8.55
JG_v9 (σc = 100 kPa) 115.11 110.90 3.80
0
10
20
30
40
50
60
0 500000 1000000 1500000 2000000 2500000
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P1 P2 P3 P1 - Plaxis P2 - Plaxis P3 - Plaxis
6.5 3D numerical modelling
266
Figure 6.48: Comparison of the experimental load-settlement curves for test JG_v7 (σc = 200
kPa) with the numerical simulations, with and without installation effects.
Figure 6.49: Comparison of the experimental load-settlement curves for test JG_v9 (σc = 100
kPa) with the numerical simulations, with and without installation effects.
6.5 3D numerical modelling
6.5.1 Model
The modelling was conducted using the Plaxis 3D code Version 2013.1 with 15-noded
elements. Figure 6.50 shows a general view of the mesh used to model test JG_v10. The
model has a rectangular shape, the width of which was set equal to the diameter of the
strongbox at prototype scale. The groundwater table was located 0.5 m below the surface of
the model.
0
20
40
60
80
100
120
140
160
0 100 200 300 400 500 600 700 800 900
Fo
oti
ng
lo
ad
ing
[k
Pa
]
Footing settlement [mm]
JG_v7 JG_v7 - Plaxis JG_v7 - Plaxis (no smear)
0
20
40
60
80
100
120
140
0 100 200 300 400 500 600 700 800 900
Fo
oti
ng
lo
ad
ing
[k
Pa
]
Footing settlement [mm]
JG_v9 JG_v9 - Plaxis JG_v9 - Plaxis (no smear)
6 Numerical modelling
267
Figure 6.50: General view of the 3D mesh (groundwater table 0.5 m below the surface).
The experience of the 2D axisymmetric modelling was transferred into the three-dimensional
model and the dimensions of the stone columns, smear zones and compaction zones were
implemented accordingly (Figure 6.51 and Figure 6.52). The red points in Figure 6.51 are of
graphical nature and do not have any physical meaning. Figure 6.52 shows a side view of
the stone column group (Figure 6.51), in which clay and compaction zone (left hand side),
respectively clay, compaction zone and smear zone (right hand side), were hidden in order to
expose the smear zone, respectively the stone column. A value of 50 kPa was assigned to
model cohesion in the 0.2 m thick layer at the top of the model, in order to prevent the
appearance of very large unrealistic soil movements at the surface during loading. The
calculation phases (Table 6.4) and the materials properties used (Table 6.5, Table 6.6, Table
6.7, Table 6.8) are the same as for the axisymmetric modelling.
Figure 6.51: Plan on the stone column group.
8 m
20 m
Clay
0.2 m
y
x
z
Clay
Stone
column
Smear
zone
Compaction
zone
0.6 m
0.8 m1.5 m
0.85 mA
B C
DF
x
y
1
1
2 2
6.5 3D numerical modelling
268
Figure 6.52: Side view of the stone columns and zones created to represent the installation
effects (the base of the box is located 1 m below the toe of the compaction
zone), in which clay and compaction zone (left hand side), respectively clay,
compaction zone and smear zone (right hand side), were hidden in order to
expose the smear zone, respectively the stone column.
Figure 6.53: Plan on the stone column group with position of the square footing, as applied in
the centrifuge test JG_v10.
The actual position of the footing during the loading phase in the centrifuge (Figure 4.29) was
modelled numerically at a similar position by shifting the centre of the footing laterally 0.4 m
(corresponding to 8 mm under 50 g, Figure 4.29) and vertically 0.05 m (corresponding to
6 m6.1 m
7 m
0.2 m
Stone column
Smear zone Compaction zone
0.6 m0.8 m1.5 m
Clay
Stone
column
Smear
zone
Compaction
zone
2.8 m
2.8 m
Footing
0.4 m
0.05 m
x
y
6 Numerical modelling
269
1 mm under 50 g, Figure 4.29). Figure 6.53 shows a plan view of the stone column group
with the position of the square footing.
6.5.2 Results
Figure 6.54 shows a comparison between the load-settlement curves obtained
experimentally in the centrifuge (denoted as JG_v10) and numerically (denoted as
JG_v10 – Plaxis) for test JG_v10. As for the 2D numerical modelling, the final part of the
load-settlement curves measured during the centrifuge test and obtained through numerical
modelling matched quite well (Table 6.11). The reason for the initial difference between the
measured and computed load-settlement curves is the same as in the axisymmetric case
(Figure 6.55), although the centrifuge has been stopped after the stone column installation in
order to remove the sand particles from the clay surface, and to obtain as smooth a sample
surface as possible.
Table 6.11: Comparison of the experimental and numerical values of the maximum footing
loads for a footing settlement of 850 mm.
Test Pmax, Centrifuge
[kPa]
Pmax, Plaxis
[kPa]
Difference
[%]
JG_v10 (σc = 100 kPa) 142.01 138.66 2.42
Figure 6.54: Comparison of the experimental and numerical load-settlement curves for the
test JG_v10.
Figure 6.56 and Figure 6.57 show the distribution of the total vertical stresses with depth. A
similar mechanism to that observed in the 2D case (Figure 6.31) can be observed as the
load transfer to the toe of the inclusions is minimal. The distributions of the vertical stresses
for footing settlement of 100 mm, and 400 mm, are shown in Appendix 8.21.
0
20
40
60
80
100
120
140
160
0 100 200 300 400 500 600 700 800 900
Fo
oti
ng
lo
ad
ing
[k
Pa
]
Footing settlement [mm]
JG_v10 JG_v10 - Plaxis
6.5 3D numerical modelling
270
0 kPa 100 – 140 kPa 140 – 180 kPa
Figure 6.55: Pressure distribution under the square footing for a settlement of 100 mm at
prototype scale (2 mm under 50g) for test JG_v10.
Figure 6.56: Distribution of the total vertical stresses for test JG_v10 (σc = 100 kPa) for a
settlement of 850 mm and a footing load of 138.66 kPa (section 1-1, Figure
6.51). The dimensions are given in Figure 6.51 and Figure 6.52.
56
mm
56
mm
AB D
6 Numerical modelling
271
Figure 6.57: Distribution of the total vertical stresses for test JG_v10 (σc = 100 kPa) for a
settlement of 850 mm and a footing load of 138.66 kPa (section 2-2, Figure
6.51). The dimensions are given in Figure 6.51 and Figure 6.52.
The development of plastic points during footing loading for test JG_v10 is shown in
Appendix 8.26. A similar extent of the plastic (hardening) points is noted for section 1-1
(Figure 6.51), while the effect of the stone column group can be noted in section 2-2
(Figure 6.51), as the plastic points reach higher depths (approximately 4 m) than in the
axisymmetric case, for which plastic points were computed up to a depth of approximately
2 m under the surface of the host soil. The impact of this distribution can be noted by the
distribution of the excess pore water pressures (Figure 6.61), which are significantly less
concentrated outside the group than in the axisymmetric case (Figure 6.46). A similar
distribution of the excess pore water pressures is observed within the stone column group
(Figure 6.62) and around a single stone column (Figure 6.46).
Figure 6.58 and Figure 6.59 show the stress distribution under the footing, and at a depth of
6 m, respectively, for a settlement of 850 mm. The stress distributions below the footing at
depths of 2 m and 4 m under the surface of the model are shown in Appendix 8.24. A similar
overall development of the values of the stress concentration factor can be observed, as in
the axisymmetric modelling. For a footing settlement of 850 mm, the factor m for the whole
group has a value of approximately 2.5 directly under the footing Figure 6.58) and decreases
to approximately 1 at a depth of 6 m (Figure 6.59). Similar values are obtained for footing
settlements of 100 mm (Appendix 8.22), and of 400 mm (Appendix 8.23). The stone column
group can thus be considered as an equivalent pier at its toe.
The average stress acting on top of column A is of approximately 300 kPa to 325 kPa, while
that acting on top of columns C and D is of 225 to 250 kPa and that on columns B and E
of 125 kPa to 175 kPa. The variation between the total vertical stress acting on top of
6.5 3D numerical modelling
272
columns C and D, and columns B and E, respectively, is due to the non-centred position of
the footing during the centrifuge test (Figure 6.53).
The lower installation depths of the PPTs P1, P2 and P3 (30 mm at model scale, i.e. 1.5 m at
prototype scale) shows that an acceptable match of the excess pore water pressures caused
by the footing load can be achieved near the surface of the sample during the loading phase
with the numerical model presented (Figure 6.60). This appears to be random, as the PPTs
are located in the zone where the excess pore water pressures computed numerically, and
shown in Figure 6.61 and Figure 6.62, have been concentrated. The asymmetric distribution
of the excess pore water pressures is due to the position of the footing (Figure 6.53). A
comparison of the excess pore water pressures measured in the centrifuge with those
computed numerically is presented in Appendix 8.25.
Figure 6.58: Total vertical stress distribution under the footing for a settlement of 850 mm for
test JG_v10 (σc = 100 kPa, P = 138.66 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
A
B C
DE
6 Numerical modelling
273
Figure 6.59: Total vertical stress distribution at a depth of 6 m for a settlement of 850 mm for
test JG_v10 (σc = 100 kPa, P = 138.66 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
Figure 6.60: Comparison of the values of the excess pore water pressures measured during
test JG_v10, with the values obtained numerically with Plaxis 3D (P1 till P3).
A
B C
DE
0
5
10
15
20
25
30
35
40
45
50
0 500000 1000000 1500000 2000000 2500000
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P1 P2 P3 P1 - Plaxis P2 - Plaxis P3 - Plaxis
6.5 3D numerical modelling
274
Figure 6.61: Distribution of the excess pore water pressures computed numerically for
test JG_v10 (σc = 100 kPa) for a settlement of 850 mm and a footing load of
138.66 kPa (section 1-1, Figure 6.51).
Figure 6.62: Distribution of the excess pore water pressures computed numerically for
test JG_v10 (σc = 100 kPa) for a settlement of 850 mm and a footing load of
138.66 kPa (section 2-2, Figure 6.51).
Figure 6.63 shows the deformation of the columns A, C and D (Figure 6.51) caused by the
footing load of 138.66 kPa. Figure 6.63 is a side view of the model (similar to Figure 6.52), in
which clay, compaction zone and smear zone were hidden in order to expose the
deformation of the stone columns. The outer columns (C and D) deform laterally, while the
B A D
6 Numerical modelling
275
radial deformation of the centre column is marginal. As in the axisymmetric case, the
deformations do not extend to the bottom of the inclusions. The fact that the bulging
deformation of the columns C and D is is directed towards the outside of the group is due to
the compaction of the host soil caused by the installation of column A.
Figure 6.63: Deformed columns A, C and D for test JG_v10, for a settlement of 850 mm and
a footing load of 138.66 kPa, in which clay, compaction zone and smear zone
were hidden in order to expose the deformation of the stone columns.
The influence of the installation effects on the load-settlement behaviour was investigated by
assigning the material Clay to the clusters Compaction zone and Smear zone (Figure 6.51
and Figure 6.52). Figure 6.64 shows a comparison of the load-settlement curves obtained
from the physical modelling and from the numerical modelling with, and without, installation
effects for the tests. The installation effects do not influence the load-settlement curve at the
beginning of the loading, but cause a slight offset of the curve after the initial loading phase.
Table 6.12 summarises the maximum values of the footing load. The difference in the
maximum loads, with and without a smear and a compaction zone, is in the same order of
magnitude as in the axisymmetric case for a pre-consolidation stress σc = 100 kPa.
Table 6.12: Comparison of the values of the maximum footing loads obtained numerically
with (Pmax, Plaxis) and without (Pmax, Plaxis, no smear) installation effects.
Test Pmax, Plaxis
[kPa]
Pmax, Plaxis, no smear
[kPa]
Difference
[%]
JG_v10 (σc = 100 kPa) 138.66 135.10 2.64
0.85 m
6 m Stone column A
Stone column C
Stone column D
6.6 Summary of numerical modelling
276
Figure 6.64: Comparison of the experimental load-settlement curves for test JG_v10 with the
numerical simulations, with and without, installation effects.
6.6 Summary of numerical modelling
The numerical modelling performed aimed at simulating the tests conducted under enhanced
gravity. A wished-in-place approach was adopted in order to avoid modelling the entire
installation phase of the stone columns, and thus to obtain a numerical model that a practical
engineer might be willing to use. Weber’s (2008) findings about the extent of smear and
compaction zones were successfully implemented, which confirms that an approach
considering a homogeneous smear zone (e.g. Indraratna & Redana, 1997) is rather
simplistic, although it may deliver good results in some cases.
The model adopted for the numerical approach takes the installation effects into account, as
well as proposes simple conversion factors for the soil properties in the smear and
compaction zones, based on the properties of the host soil. The results obtained provide a
good match to the load-settlement behaviour in the centrifuge model and allow some insights
into the load-transfer behaviour of single stone columns and of stone column groups to be
gained. A significant decrease of the stress concentration factor with depth is observed in
both cases, as this factor reaches values ranging from 1.0 to 1.5 in the lower third of the
stone column. This could open up the way to a more economical and sustainable design of
granular inclusions as their diameter could be reduced in zones where the stress
concentration is low.
The outcomes of the numerical modelling show a very small load transfer to the toe of the
stone columns and thus a very little mobilisation of the tip resistance. This might also be a
possibility to achieve a more economical design as the length of the inclusions could be
reduced without losing lead bearing performance, if the drainage function is of secondary
importance.
0
20
40
60
80
100
120
140
160
0 100 200 300 400 500 600 700 800 900
Fo
oti
ng
lo
ad
ing
[k
Pa
]
Footing settlement [mm]
JG_v10 JG_v10 - Plaxis JG_v10 - Plaxis (no smear)
6 Numerical modelling
277
However, the values of the excess pore water pressures computed numerically do not
correspond well, and underestimate those measured during the centrifuge tests. Although
this is an issue for scientific work, it is of less relevance compared to the correct modelling of
the load-settlement behaviour in practical cases.
As a conclusion, the model proposed allows insights into the stress paths, although the
constitutive model used for the modelling of clay (HSM) assumes a fully elastic behaviour
within the yield surface, which is not fully realistic. The model also reaches its goal of
providing a practically applicable way of taking the installation effects of stone columns into
account and delivers a good prediction of the load-settlement behaviour of the composite
foundation. From a scientific point of view, improvements can still be reached in the
computation of excess pore water pressures.
6.6 Summary of numerical modelling
278
7 Summary
279
7 Summary
7.1 General considerations
Ground improvement aims to enhance the engineering properties of a soil in terms of bearing
capacity and / or stiffness in order to make it suitable for construction. The continuous
technological innovation of the past decades has enabled the development of machinery to
build stone columns efficiently. These granular inclusions both stiffen the soil and allow for
higher loads to be carried while reducing post-construction settlements. The consolidation
time is also shorter because the length of the drainage paths is decreased as well.
The bearing behaviour of stone columns differs from that of rigid inclusions, such as piles, as
the host soil plays a decisive role in providing lateral support to the stone column material.
The interactions between host soil, inclusions and supported structure are governed by the
difference between the characteristics of the host soil and influenced by the installation
effects, especially in clay. Although the development of installation effects is acknowledged,
different approaches exist to take them into account.
The installation effects are the cause of a decrease in the drainage performance of the stone
columns, as they cause radial compaction of the host soil. A reorientation of the clay platelets
reduces the permeability as well. Some researchers (e.g. Indraratna & Redana, 1998)
assume that the soil properties are constant over the whole extent of the installation effects;
the so-called smear zone. Others, such as Onoue et al. (1991) and Weber et al. (2010),
identify variation in the values of the permeability and porosity in the host soil around stone
columns and therefore subdivide the installation effects into smear and compaction zones.
Although the latter approach delivers more accurate results than the former one, the vertical
distribution of the two zones remains unknown.
The bearing behaviour of stone columns is further influenced by geometrical aspects such as
the spacing between the inclusions and their diameter. A re-distribution of the stresses under
a foundation is noted during footing loading, due to the significantly higher stiffness of the
stone columns compared with that of the host soil. The stress concentration ratio m
quantifies this effect and usually ranges from 2 to 6. The values are influenced by the
intensity and type of loading (flexible / stiff), and by the stress history and characteristics of
the host soil. A better knowledge of the installation effects might help understanding some of
the differences in stress concentration, which is the main focus of this study.
7.2 Findings from centrifuge modelling and complementary
investigations
The installation of stone columns into a clay specimen in-flight, in a geotechnical centrifuge,
enabled insights to be gained into the factors influencing the installation effects and into the
bearing behaviour of composite foundations with measurements of pore pressures, load-
settlement behaviour and electrical impedance measurements. It also offered the chance to
7.2 Findings from centrifuge modelling and complementary investigations
280
study the micro-mechanical impact of the installation on the host soil, with complementary
investigations conducted after the centrifuge tests.
The magnitude of the excess pore water pressures caused by the insertion of the installation
mandrel into the host soil is mostly governed by the over-consolidation ratio OCR, as it
becomes greater with an increasing OCR. A transfer of the load applied on the surface into
the host soil could be measured as the load-increase at a depth of 96 mm under the surface
(at 50 g) was shown to be 3 times higher with a stone column than without (Figure 7.1). This
could not be reproduced in the numerical modelling of the centrifuge test, but important
insights were gained about the stress distribution with depth, which are summarised in
Sections 7.3 and 7.4.
Figure 7.1: Distribution of the vertical stress increase as a function of the radial distance from
the stone column at 96 mm depth as a percentage of the applied footing load P
and comparison with the depth factor J4 according to Grasshoff (1978).
An electrical impedance needle (Figure 7.2) was developed in order to measure the electrical
impedance in the host soil surrounding the stone columns, to determine the extent of the
installation effects. A consistent trend observed during the investigations conducted with the
electrical impedance needle indicated a microscopic reorganisation of the clay particles up to
a distance of 5 times the radius of the stone column from the axis of the inclusion. The
outcomes also indicated a constant extent of the smear zone with depth, as long as no
significant bulging deformation occurs.
(a) (b)
Figure 7.2: Electrical impedance needle (a) side view and (b) tilted view of the tip (outer
diameter 1 mm).
0
5
10
15
20
25
30
35
0 5 10 15 20 25 30
Δσ
z[%
P]
Radial distance [mm]
JG_v5 JG_v7 JG_v9 J4 J4J4
Edge of the stone column
7 Summary
281
The interpretation of the impedance measurements was confirmed by observations using
Environmental Scanning Electron Microscopy (ESEM), which identified a progressive vertical
reorganisation of longer particles in the compaction zone with decreasing distance from the
edge of the stone column (Figure 7.3). Weber (2008) observed a reorganisation of all clay
platelets within the smear zone (ranging from the edge of the stone column to a radial
distance of 2 mm from the interface stone column / clay).
Figure 7.3: ESEM picture of zone 3, located at a radial distance of 5 mm from the edge of the
column and at a depth of 20 mm below the surface, with the radial axis
horizontal.
Complementary investigations to study the extent of the compaction zone around a stone
column were conducted using Mercury Intrusion Porosimetry (MIP) on samples extracted
from clay specimens used for the modelling under enhanced gravity at depths of 20 mm, 60
mm and 100 mm below the surface of the model. These investigations showed that the
macroscopic effects on the porosity of the host soil could only be detected up to a distance of
about twice the radius of the inclusions from its axis, which confirmed Weber’s (2008)
findings. The extent of the compaction zone was shown to remain constant over the depth of
the stone column (Figure 7.4). An effect of the compaction cycles on the porosity of the
surrounding host soil near the surface was identified: the porosity at the interface between
the smear zone and the compaction zone drops from 31 % at a depth of 100 mm below the
surface to 29 % at a depth of 20 mm. These findings were transferred into the numerical
model used in this research in order to achieve a usable model for design.
30 μm
Longer
particles
5 mm
Sto
ne
co
lum
n
7.3 Numerical modelling
282
(a)
(b)
(c)
Legend:
Figure 7.4: Porosity as a function of the radial distance from the axis of the stone column at a
depth of (a) 20 mm (b) 60 mm (c) 100 mm.
7.3 Numerical modelling
The aim of the numerical modelling conducted was to achieve a numerical model, which a
practical engineer may be willing to use, and which takes the installation effects of stone
columns into account. Such a model, using a “wished-in-place” approach, was developed
26
28
30
32
34
36
38
40
0 5 10 15 20 25 30 35 40
Po
ros
ity [
%]
Distance from stone column axis [mm]
Stone
column
Edge of densification OCR = 11.8 [-]
26
28
30
32
34
36
38
40
0 5 10 15 20 25 30 35 40
Po
ros
ity [
%]
Distance from stone column axis [mm]
Edge of densificationStone
column
OCR = 3.9 [-]
26
28
30
32
34
36
38
40
0 5 10 15 20 25 30 35 40
Po
ros
ity [
%]
Distance from stone column axis [mm]
Measured data
Hyperbolic trend function
Edge of densificationStone
column
Smear zone (zone 2 in Weber, 2008)
Zone 3 (Weber, 2008)
OCR = 2.4 [-]
Compaction zone
Measured data
Hyperbolic trend function
26
28
30
32
34
36
38
40
0 5 10 15 20 25 30 35 40
Po
ros
ity [
%]
Distance from stone column axis [mm]
Measured data
Average
Edge of densificationStone
column
Smear zone (zone 2 in Weber, 2008)
Zone 3 (Weber, 2008)
OCR = 2.4 [-]
Compaction zone
7 Summary
283
based on the findings from the centrifuge tests and from the complementary investigations
conducted.
A good prediction of the load-settlement behaviour was achieved, although the initial
response of the composite foundation computed numerically was significantly stiffer than that
measured during centrifuge tests. This was found to be due to a difference between the
stress distribution under the footing computed numerically and that measured in the
centrifuge. Such differences in the initial phase of loading could also be observed by Arnold
(2011), when comparing field measurements and results from centrifuge tests.
A significant decrease of the load transfer onto the stone columns could be measured below
4 m depth, as the stress concentration factor reached values close to unity. This could open
up new perspectives for a more sustainable construction of stone columns as the radius of
the inclusions could be increased near the surface, where the stress concentration factor
ranges between 2.5 and 3 (Figure 7.5), and reduced in zones where the concentration factor
is of 1.0 to 1.5 (Figure 7.6). This could allow for a reduction of the material needed and thus
for a lower impact on the environment and economy. Although the distribution of the
installation effects was successfully modelled, the influence of the diameter variation of the
stone column on the load transfer, and on the drainage performance, would have to be
investigated in detail before it can be implemented in practice.
The investigation of the total vertical stress distribution in 2D and 3D (Figure 7.7 and
Figure 7.8) shows clearly that there is very little load transfer to the toe of the stone column.
Thus, if tip resistance is needed, shorter columns would be more sensible than longer
inclusions. This should however be done taking the drainage performance needed into
account, which is of course smaller for shorter columns. A new design philosophy could be
achieved, implementing stone columns depending on the local needs: shorter inclusions with
constant diameter could be built in cases where tip resistance needs to be mobilised and
where the drainage performance is of less importance, while longer stone columns could be
reserved to the cases where a high drainage efficiency needs to be reached in order to
reduce consolidation time, while the diameter of the inclusions could be reduced in the lower
third, where the stress concentration is low.
The distribution of the computed values of the excess pore water pressures caused by the
footing loading (Figure 7.9) is significantly different from those measured in-flight
(Figure 7.10). However, the modelling of pore water pressures is a delicate subject, as the
constitutive models available still have some insufficiencies in that domain.
7.3 Numerical modelling
284
Figure 7.5: Total vertical stress distribution under the footing for a settlement of 850 mm for
test JG_v10 (σc = 100 kPa, P = 138.66 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
Figure 7.6: Total vertical stress distribution at a depth of 6 m for a settlement of 850 mm for
test JG_v10 (σc = 100 kPa, P = 138.66 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
A
B C
DE
A
B C
DE
7 Summary
285
Figure 7.7: Total vertical stress distribution computed numerically for test JG_v7 for a
settlement of 850 mm and a footing load of 145.44 kPa.
Figure 7.8: Distribution of the total vertical stresses for test JG_v10 (σc = 100 kPa) for a
settlement of 850 mm and a footing load of 138.66 kPa (section 1-1,
Figure 6.51). The dimensions are given in Figure 6.51 and Figure 6.52.
AB D
7.3 Numerical modelling
286
Figure 7.9: Distribution of the excess pore water pressures computed numerically for
test JG_v7 for a footing settlement of 850 mm and a footing load of 141.90 kPa.
Figure 7.10: Isobars of peak values of excess pore pressures measured in the centrifuge
under a vertically loaded circular footing resting on top of a stone column.
7 Summary
287
7.4 Outlook
This research has reached its aim to identify the spatial distribution of installation effects
around stone columns. The findings of the centrifuge modelling and subsequent
complementary investigations were successfully implemented in a numerical model. The
model proposed opens new perspectives in the consideration of installation effects around
stone columns in practical cases, as it shows that a practical, simplified modelling of granular
inclusions with installation effects is possible.
The variation of the values measured with the electrical impedance needle under Earth’s
gravity and under 50 g might be due to a scaling effect of the electrical waves under
enhanced gravity. An investigation of this scaling effect could open up new perspectives for
the implementation of the electrical impedance needle in a wider range of boundary value
problems, including in-situ tests.
This numerical model was developed based on tests and investigations conducted with one
type of host soil and a specific stone column material. A validation or optimisation ought to be
conducted considering other types of clays (and varying pre-consolidation stress) and a
variation of the granular material used for the stone column construction.
The literature review has shown that the stress concentration on top of stone columns
depends on the interaction between type of loading (flexible / stiff) and load intensity, on the
stress history, and on the characteristics of the host soil. A sensitivity analysis could
therefore be conducted in order to quantify the impact of these different factors (type of
loading, stress history and host soil) on the dissipation of the stress concentration with depth
observed in the numerical modelling.
Another interesting point for further research would be the conduction of full-scale
experiments in order to investigate the effect of the differences between the bottom field
installation technique used in practice and that used in the centrifuge. This could enable a
further optimisation of the numerical model to be achieved, which would allow for a wider use
of the numerical model in practice.
A topic of interest from a scientific point of view will be the development of numerical
constitutive models allowing for an accurate reproduction of the excess pore water pressures
in the host soil. The implementation of a bubble model (Stallebrass & Taylor, 1997) could
solve the issue of the unrealistic fully elastic behaviour of clay within the yield surface
observed with the HSM.
The implementation of interface elements between the zones representing the installation
effects in the numerical modelling could be investigated when using another numerical code
than Plaxis. The use of such elements with the numerical code Plaxis is not sensible in this
specific case as it would suppose that the zones are very clearly separated and thus not take
the progressive evolution of the soil characteristics with radial distance to the axis of the
inclusion into account.
7.4 Outlook
288
Finally, the influence of a diameter variation should be investigated in order to validate the
outcomes of the numerical model in terms of reduction of the stress concentration factor in
the lower third of the inclusions.
8 Appendices
289
8 Appendices
Appendices 8.1, 8.2 and 8.3 present the measurements (pore water pressure, footing load
and footing settlement) recorded during the loading phase of tests JG_v1, JG_v5 and JG_v6
respectively. Appendix 8.4 presents the values of the J4 factor according to Grasshoff (1978)
and Appendix 8.5 shows a comparison between the measured values of the excess pore
water pressure during the loading phase of test JG_v1 with back-calculated values (Section
4.4.4).
Appendices 8.6 and 8.7 show the electrical impedance measurements conducted in-flight
during test JG_v9 and under 1 g on soil prepared in the laboratory at successive phases to
achieve the same stress history, hence with varying soil densities (Section 5.3), respectively.
Appendices 8.8, 8.9 and 8.10 show the distribution of vertical strain increments, the
distribution of shear strain increments and the development of plastic points computed
numerically for test JG_v7.
Appendices 8.11 and 8.12 present the deformed mesh and the distribution of total vertical
stresses obtained at the end of the loading phase for the numerical modelling of test JG_v9,
respectively.
Appendices 8.13, 8.14 and 8.16 the distribution of vertical strain increments, the distribution
of shear strain increments and the development of plastic points computed numerically for
test JG_v9.
Appendix 8.15 presents the distribution of the excess pore water pressures computed
numerically for test JG_v9.
Appendices 8.17 and 8.18 show the distribution with depth of the total vertical stress
computed numerically with depth for footing settlements of 400 mm and 850 mm for
tests JG_v7 and JG_v9, respectively.
Appendices 8.19 and 8.20 present comparisons of the values of the excess pore water
pressure measured in the centrifuge with the values obtained from numerical modelling for
tests JG_v7 and JG_v9, respectively.
Appendix 8.21 presents the total vertical stress distribution for test JG_v10 for footing
settlements of100 mm and 400 mm.
Appendices 8.22, 8.23 and 8.24 show the distribution of the total vertical stress below the
footing with depth, for test JG_v10, for footing settlements of 100 mm, 400 mm and 850 mm.
Appendix 8.25 presents a comparison of the values of the excess pore water pressure
measured in the centrifuge with the values obtained from numerical modelling for
test JG_v10. Appendix 8.26 shows the development of plastic points during footing loading
for test JG_v10.
8.1 Pore pressure and load measurements conducted during loading with a footing on a
single stone column installed in a specimen prepared in an oedometer container (test JG_v1)
290
8.1 Pore pressure and load measurements conducted during
loading with a footing on a single stone column installed in a
specimen prepared in an oedometer container (test JG_v1)
A connection problem between the computers mounted on the tool platform and the
computer in the control room led to a false evaluation of the position of the footing relative to
the surface of the model during test JG_v1. As a consequence, the stone column underwent
preloading and the pore pressure measurements of the footing load could not be exploited
fully.
Figure 8.1 shows the measurements conducted during test JG_v1. The increase in pore
water pressures at approximately 500 s (Figure 8.1 a) is due to a manipulation of the
actuator arm, which caused an unintended load to be applied to the granular inclusion.
The applied load was not recorded and could not be visually controlled by the operative
personal due to a loss of the connection between the computers installed on the tool platform
of the centrifuge and those in the control room. Nevertheless, the data are given here for
reference and potential further analysis.
8 Appendices
291
(a)
(b)
(c)
Figure 8.1: Loading of a single stone column in a specimen pre-consolidated up to 200 kPa
(test JG_v1): (a) excess pore water pressures (b) evolution of the footing load
(c) footing settlement.
0
5
10
15
20
25
30
35
0 500 1000 1500 2000 2500 3000Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P1 P2 P3 P4 P5 P6 P7
-40
-20
0
20
40
60
80
100
0 500 1000 1500 2000 2500 3000
Fo
oti
ng
lo
ad
ing
[k
Pa
]
Time [s]
-18
-13
-8
-30 500 1000 1500 2000 2500 3000
Fo
oti
ng
se
ttle
me
nt
[mm
]
Time [s]
8.2 Pore pressure and load measurements conducted during loading a single stone column
installed in a specimen prepared in an oedometer container (test JG_v5) with a circular
footing
292
8.2 Pore pressure and load measurements conducted during
loading a single stone column installed in a specimen prepared
in an oedometer container (test JG_v5) with a circular footing
(a)
(b)
(c)
Figure 8.2: Loading of a single stone column in a specimen pre-consolidated up to 200 kPa
(test JG_v5): (a) excess pore water pressures (b) evolution of the footing load
(c) footing settlement.
0
5
10
15
20
25
30
35
40
45
50
0 500 1000 1500 2000 2500
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P1 P2 P3 P4 P5 P6 P7
-20
0
20
40
60
80
100
120
0 500 1000 1500 2000 2500Fo
oti
ng
lo
ad
ing
[k
Pa
]
Time [s]
-20
-15
-10
-5
0
0 500 1000 1500 2000 2500
Fo
oti
ng
se
ttle
me
nt
[mm
]
Time [s]
8 Appendices
293
8.3 Pore pressure and load measurements conducted during
loading a single stone column installed in a specimen prepared
in a full cylindrical stongbox (test JG_v6) with a circular footing
(a)
(b)
(c)
Figure 8.3: Loading of a single stone column in a specimen pre-consolidated up to 100 kPa
(test JG_v6): (a) Excess pore water pressures (b) evolution of the footing load
(c) footing settlement.
0
2
4
6
8
10
12
14
16
18
0 500 1000 1500 2000
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P4 P5 P6 P7
0
10
20
30
40
50
60
70
80
90
100
110
120
0 500 1000 1500 2000
Fo
oti
ng
lo
ad
ing
[k
Pa
]
Time [s]
-20
-15
-10
-5
0
0 500 1000 1500 2000
Fo
oti
ng
se
ttle
me
nt
[mm
]
Time [s]
8.4 Values of the J4 factor according to Grasshoff (1978)
294
8.4 Values of the J4 factor according to Grasshoff (1978)
Table 8.1: Values of the factor J4 in ‰ (Lang et al., 2007) (z: depth of the point considered;
R: radius of the circular footing; a: radial distance of the point considered from
the centre of the footing).
8.5 Comparison of the analytical and measured excess pore water
pressure around a single stone column when the maximum
load is applied (P = 80 kPa, test JG_v1)
Table 8.2 shows a comparison of the analytical and measured excess pore water pressure
during the footing loading phase for test JG_v1, as a complement to the data presented in
Section 4.4.4 about load transfer around a single stone column.
Table 8.2: Comparison of the analytical and measured excess pore water pressure during
the loading phase of a single stone column (test JG_v1, σc = 200 kPa, P = 80
kPa).
PPT J4 [%]
Δσa = Δσz
= J4 P
[kPa]
A [-] ΔuSkempton [kPa] Δucentrifuge [kPa] Δucentrifuge /
ΔuSkempton [-]
P1 32.30 25.84 0.1 21.83 21.66 0.99
P2 29.13 23.30 0.1 19.68 16.72 0.85
P3 21.45 17.16 0.1 14.50 11.33 0.78
P4 11.45 9.16 0.2 7.38 18.41 2.49
P5 11.03 8.82 0.2 7.11 13.92 1.96
P6 9.73 7.78 0.2 6.27 17.67 2.82
P7 5.70 4.56 0.2 3.59 11.77 3.28
8 Appendices
295
8.6 Electrical impedance measurements conducted during the test
JG_v9
The results of the investigations conducted with the electrical impedance needle during test
JG_v9 are presented here. Both container A and container B were pre-consolidated up to
100 kPa. The column installed in container A was loaded using a 56 mm diameter stiff
circular footing.
Figure 8.4: Impedance recorded at the reference points RP1 and RP2 during test JG_v9
(Container A).
Figure 8.5: Impedance recorded at the reference points RP1 and RP2 during test JG_v9
(Container B).
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
RP 1 RP 2
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
RP 1 RP 2
8.6 Electrical impedance measurements conducted during the test JG_v9
296
Figure 8.6: Impedance recorded at the points A, B and C during test JG_v9 (Container A).
Figure 8.7: Impedance recorded at the points D, E and F during test JG_v9 (Container A).
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
Point A Point B Point C
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
Point D Point E Point F
8 Appendices
297
Figure 8.8: Impedance recorded at the points A, B and C during test JG_v9 (Container B).
Figure 8.9: Impedance recorded at the points D, E and F during test JG_v9 (Container B).
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
Point A Point B Point C
0
20
40
60
80
100
120
0 0.002 0.004 0.006 0.008 0.01
De
pth
[m
m]
Impedance [Ohm]
Point D Point E Point F
8.7 Electrical impedance measurements conducted under 1 g
298
8.7 Electrical impedance measurements conducted under 1 g
The results of the insertion of the electrical impedance needle under 1 g (Section 5.3) are
presented here. For clarity, the different consolidation stages conducted, as well as the
location of the insertion points, are given in Table 8.3 and Figure 8.10, respectively.
Table 8.3: Overview of the consolidation stages conducted for the insertion of the electrical
impedance needle under 1 g.
Consolidation stage Consolidation effective
stress [kPa] Density [kN/m3] Void ratio [-]
1 12.5 16.8 1.53
2 25 17.4 1.30
3 50 18.1 1.20
4 100 18.7 1.06
5 200 20.4 0.73
Figure 8.10: Positions of the insertion points of the electrical impedance needle under 1 g. All
dimensions in mm.
Clay
Insertion
points
1 2
3
8 Appendices
299
Figure 8.11: Impedance recorded under 1 g after completion of the second consolidation
stage.
Figure 8.12: Impedance recorded under 1 g after completion of the third consolidation stage.
0
20
40
60
80
100
120
0.043 0.044 0.045 0.046 0.047
De
pth
[m
m]
Impedance [Ohm]
Point 1 (Stage 2) Point 2 (Stage 2) Point 3 (Stage 2)
0
20
40
60
80
100
120
0.043 0.044 0.045 0.046 0.047
De
pth
[m
m]
Impedance [Ohm]
Point 1 (Stage 3) Point 2 (Stage 3) Point 3 (Stage 3)
8.7 Electrical impedance measurements conducted under 1 g
300
Figure 8.13: Impedance recorded under 1 g after completion of the fourth consolidation
stage.
0
20
40
60
80
100
120
0.043 0.044 0.045 0.046 0.047
De
pth
[m
m]
Impedance [Ohm]
Point 1 (Stage 4) Point 2 (Stage 4) Point 3 (Stage 4)
8 Appendices
301
8.8 Vertical strain increments computed numerically for test
JG_v7
The distribution of the vertical strain increments computed numerically for test JG_v7 using
an axisymmetric model is presented in this section.
Figure 8.14: Vertical strain increments computed numerically for test JG_v7 for a settlement
of 100 mm and a footing load of 85 kPa.
8.8 Vertical strain increments computed numerically for test JG_v7
302
Figure 8.15: Vertical strain increments computed numerically for test JG_v7 for a settlement
of 400 mm and a footing load of 115.2 kPa
8 Appendices
303
8.9 Shear strain increments computed numerically for test JG_v7
The distribution of the shear strain increments computed numerically for test JG_v7 using an
axisymmetric model is presented in this section.
Figure 8.16: Shear strain increments computed numerically for test JG_v7 for a settlement of
100 mm and a footing load of 85 kPa.
8.9 Shear strain increments computed numerically for test JG_v7
304
Figure 8.17: Shear strain increments computed numerically for test JG_v7 for a settlement of
400 mm and a footing load of 115.2 kPa.
8 Appendices
305
Figure 8.18: Shear strain increments computed numerically for test JG_v7 for a settlement of
850 mm and a footing load of 145 kPa.
8.10 Development of plastic points (test JG_v7)
306
8.10 Development of plastic points (test JG_v7)
The development of plastic points during footing loading of a single stone column for test
JG_v7 obtained numerically using an axisymmetric model is presented in this section. The
development of plastic points during the loading phase is presented in Figure 6.32. Failure
points denote elements that are located on the Mohr-Coulomb failure line (Figure 6.19,
respectively Figure 6.22), while Cap points and Hardening points refer to element undergoing
plastic shear strain hardening (Figure 6.22, Section 6.3.2.3) and plastic volumetric hardening
(Figure 6.23, Section 6.3.2.4), respectively. Tension cut-off points denote elements which
would undergo tension if the value of cohesion was positive.
Figure 8.19: Development of plastic points during the loading phase for test JG_v7 for a
settlement of 100 mm (P = 85 kPa).
8 Appendices
307
Figure 8.20: Development of plastic points during the loading phase for test JG_v7 for a
settlement of 400 mm (P = 115.2 kPa).
8.10 Development of plastic points (test JG_v7)
308
Figure 8.21: Development of plastic points during the loading phase for test JG_v7 for a
settlement of 850 mm (P = 145.44 kPa).
8 Appendices
309
8.11 Deformed mesh (test JG_v9)
Figure 8.22 shows the deformed mesh at the end of the modelling of the footing load during
test JG_v9. The applied load was 115.11 kPa and the settlement reached 850 mm.
Figure 8.22: Deformed mesh obtained for test JG_v9 for a settlement of 850 mm and a
footing load of 115.11 kPa.
Stone column
Smear zone
Compaction zone8
m
6m
6.1
m
7m
Clay
0.3 m
0.4 m
0.75 m
0.2 m
6.25 m
1.4 m
0.8
5 m
x
y
8.12 Total stress distribution computed numerically for test JG_v9
310
8.12 Total stress distribution computed numerically for test JG_v9
Figure 8.23 presents the distribution of total vertical stresses computed numerically for test
JG_v9 for a settlement of 850 mm and a footing load of 119.67 kPa.
Figure 8.23: Total vertical stress distribution computed numerically for test JG_v9 for a
settlement of 850 mm and a footing load of 119.67 kPa.
8 Appendices
311
8.13 Vertical strain increments computed numerically for test JG_v9
The distribution of the vertical strain increments computed numerically for test JG_v9 using
an axisymmetric model is presented in this section.
Figure 8.24: Vertical strain increment computed numerically for test JG_v9 for a settlement of
100 mm and a footing load of 70.8 kPa.
8.13 Vertical strain increments computed numerically for test JG_v9
312
Figure 8.25: Vertical strain increment computed numerically for test JG_v9 for a settlement of
400 mm and a footing load of 94.3 kPa.
8 Appendices
313
Figure 8.26: Vertical strain increment computed numerically for test JG_v9 for a settlement of
850 mm and a footing load of 119.67 kPa.
8.14 Shear strain increments computed numerically for test JG_v9
314
8.14 Shear strain increments computed numerically for test JG_v9
The distribution of the shear strain increments computed numerically for test JG_v7 using an
axisymmetric model is presented in this section.
Figure 8.27: Shear strain increment computed numerically for test JG_v9 for a settlement of
100 mm and a footing load of 70.8 kPa.
8 Appendices
315
Figure 8.28: Shear strain increment computed numerically for test JG_v9 for a settlement of
400 mm and a footing load of 94.3 kPa.
8.14 Shear strain increments computed numerically for test JG_v9
316
Figure 8.29: Shear strain increment computed numerically for test JG_v9 for a settlement of
850 mm and a footing load of 119.67 kPa.
8 Appendices
317
8.15 Excess pore water pressures computed numerically for test
JG_v9
The distribution of the excess pore water pressures computed numerically for test JG_v9
using an axisymmetric model is presented in Figure 8.30.
Figure 8.30: Distribution of the excess pore water pressures computed numerically for
test JG_v9 for a footing settlement of 850 mm and a footing load of 115.11 kPa
8.16 Development of plastic points (test JG_v9)
318
8.16 Development of plastic points (test JG_v9)
The development of plastic points during footing loading of a single stone column for test
JG_v9 obtained numerically using an axisymmetric model is presented in this section. Failure
points denote elements that are located on the Mohr-Coulomb failure line (Figure 6.19,
respectively Figure 6.22), while Cap points and Hardening points refer to element undergoing
plastic shear strain hardening (Figure 6.22, Section 6.3.2.3) and plastic volumetric hardening
(Figure 6.23, Section 6.3.2.4), respectively. Tension cut-off points denote elements which
would undergo tension if the value of cohesion was positive.
Figure 8.31: Development of plastic points during the loading phase for test JG_v9 for a
settlement of 100 mm (P = 70.8 kPa).
8 Appendices
319
Figure 8.32: Development of plastic points during the loading phase for test JG_v9 for a
settlement of 400 mm (P = 94.3 kPa).
8.16 Development of plastic points (test JG_v9)
320
Figure 8.33: Development of plastic points during the loading phase for test JG_v9 for a
settlement of 850 mm (P = 119.67 kPa).
8 Appendices
321
8.17 Total vertical stress distribution as a function of the radial
distance at depths of 0 m, 2 m, 4 m and 6 m (test JG_v7)
Figure 8.34 and Figure 8.35 present the distribution of the total vertical stress computed
numerically for test JG_v7 as a function of the radial distance from the axis of the stone
column under the footing and at depths of 2 m, 4 m and 6 m and for footing settlements of
400 mm and 850 mm, respectively.
Figure 8.34: Total vertical stress distribution as a function of the radial distance under the
footing (z = 400 mm) and at depths of 2 m, 4 m and 6 m for a footing settlement
of 400 mm during the footing loading for test JG_v7 (P = 115.2 kPa).
Figure 8.35: Total vertical stress distribution as a function of the radial distance under the
footing (z = 850 m) and at depths of 2 m, 4 m and 6 m for a footing settlement of
850 mm during the footing loading for test JG_v7 (P = 145.44 kPa).
0
50
100
150
200
250
300
350
400
0 0.5 1 1.5
To
tal ve
rtic
al s
tre
ss
σv
[kP
a]
Radial distance [m]
z = 400 mm z = 2 m z = 4 m z = 6 m
Edge of stone column
Sm
ear
zon
e
Compactionzone
Edge of footing
0
50
100
150
200
250
300
350
400
0 0.5 1 1.5
To
tal ve
rtic
al s
tre
ss
σv
[kP
a]
Radial distance [m]
z = 850 mm z = 2 m z = 4 m z = 6 m
Edge of stone column
Sm
ear
zon
e
Compactionzone
Edge of footing
8.18 Total vertical stress distribution as a function of the radial distance at depths of 0 m, 2
m, 4 m and 6 m (test JG_v9)
322
8.18 Total vertical stress distribution as a function of the radial
distance at depths of 0 m, 2 m, 4 m and 6 m (test JG_v9)
Figure 8.36 and Figure 8.37 present the distribution of the total vertical stress computed
numerically for test JG_v9 as a function of the radial distance from the axis of the stone
column at depths of 0 m, 2 m, 4 m and 6 m and for footing settlements of 400 mm and 850
mm, respectively.
Figure 8.36: Total vertical stress distribution as a function of the radial distance under the
footing (z = 400 m) and at depths of 2 m, 4 m and 6 m for a footing settlement of
400 mm during the footing loading for test JG_v9 (P = 94.3 kPa).
Figure 8.37: Total vertical stress distribution as a function of the radial distance under the
footing (z = 850 m) and at depths of 2 m, 4 m and 6 m for a footing settlement of
850 mm during the footing loading for test JG_v9 (P = 119.67 kPa).
0
50
100
150
200
250
300
0 0.5 1 1.5
To
tal ve
rtic
al s
tre
ss
σv
[kP
a]
Radial distance [m]
z = 400 mm z = 2 m z = 4 m z = 6 m
Edge of stone column
Sm
ear
zon
e
Compactionzone Edge of
footing
0
50
100
150
200
250
300
350
0 0.5 1 1.5
To
tal ve
rtic
al s
tre
ss
σv
[kP
a]
Radial distance [m]
z = 850 mm z = 2 m z = 4 m z = 6 m
Edge of stone column
Sm
ear
zon
e
Compactionzone
Edge of footing
8 Appendices
323
8.19 Comparison of the measured and modelled excess pore water
pressures for the test JG_v7
Figure 8.38 and Figure 8.39 present a comparison of the excess pore water pressures
measured in the centrifuge with those computed numerically for test JG_v7 using an
axisymmetric model.
Figure 8.38: Comparison of the values of the excess pore water pressures measured during
test JG_v7 with the values obtained numerically with Plaxis 2D (P4 till P6).
Figure 8.39: Comparison of the values of the excess pore water pressures measured during
test JG_v7 with the values obtained numerically with Plaxis 2D (P7).
0
10
20
30
40
0 500000 1000000 1500000 2000000 2500000
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P4 P5 P6 P4 - Plaxis P5 - Plaxis P6 - Plaxis
0
5
10
15
20
25
30
0 500000 1000000 1500000 2000000 2500000
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P7 P7 - Plaxis
8.20 Comparison of the measured and modelled excess pore water pressures for the test
JG_v9
324
8.20 Comparison of the measured and modelled excess pore water
pressures for the test JG_v9
Figure 8.40, Figure 8.41 and Figure 8.42 present a comparison of the excess pore water
pressures measured in the centrifuge with those computed numerically for test JG_v9 using
an axisymmetric model.
Figure 8.40: Comparison of the values of the excess pore water pressures measured during
test JG_v9 with the values obtained numerically with Plaxis 2D (P1 till P3).
Figure 8.41: Comparison of the values of the excess pore water pressures measured during
test JG_v9 with the values obtained numerically with Plaxis 2D (P4 till P6).
0
5
10
15
20
25
30
35
40
45
50
0 500000 1000000 1500000 2000000 2500000
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P1 P2 P3 P1 - Plaxis P2 - Plaxis P3 - Plaxis
0
5
10
15
20
25
30
35
0 500000 1000000 1500000 2000000 2500000
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P4 P5 P6 P4 - Plaxis P5 - Plaxis P6 - Plaxis
8 Appendices
325
Figure 8.42: Comparison of the values of the excess pore water pressures measured during
test JG_v9 with the values obtained numerically with Plaxis 2D (P7).
0
2
4
6
8
10
12
14
16
0 500000 1000000 1500000 2000000 2500000
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P7 P7 - Plaxis
8.21 Distribution of the total vertical stresses for footing settlements of 100 mm and of 400
mm (test JG_v10)
326
8.21 Distribution of the total vertical stresses for footing settlements
of 100 mm and of 400 mm (test JG_v10)
Figure 8.43, Figure 8.44, Figure 8.45 and Figure 8.46 present the distribution of the total
vertical stresses for test JG_v10 for settlements of 100 mm and 400 mm.
Figure 8.43: Distribution of the total vertical stresses for test JG_v10 for a settlement of
100 mm and a footing load of 76 kPa (section 1-1, Figure 6.51). The dimensions
are given in Figure 6.51 and Figure 6.52.
8 Appendices
327
Figure 8.44: Distribution of the total vertical stresses for test JG_v10 for a settlement of
100 mm and a footing load of 76 kPa (section 2-2, Figure 6.51). The dimensions
are given in Figure 6.51 and Figure 6.52.
Figure 8.45: Distribution of the total vertical stresses for test JG_v10 for a settlement of
400 mm and a footing load of 108.60 kPa (section 1-1, Figure 6.51). The
dimensions are given in Figure 6.51 and Figure 6.52.
8.21 Distribution of the total vertical stresses for footing settlements of 100 mm and of 400
mm (test JG_v10)
328
Figure 8.46: Distribution of the total vertical stresses for test JG_v10 for a settlement of
400 mm and a footing load of 108.60 kPa (section 2-2, Figure 6.51). The
dimensions are given in Figure 6.51 and Figure 6.52.
8 Appendices
329
8.22 Total vertical stress distribution below the footing for a
settlement of 100 mm (test JG_v10)
Figure 8.47, Figure 8.48, Figure 8.49 and Figure 8.50 present the total vertical stress
distribution below the footing for a settlement of 100 mm under the footing and at depths of 2
m, 4 m and 6 m below the surface, respectively.
Figure 8.47: Total vertical stress distribution under the footing for a settlement of 100 mm for
test JG_v10 (σc = 100 kPa, P = 76.00 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
A
BC
DE
8.22 Total vertical stress distribution below the footing for a settlement of 100 mm (test
JG_v10)
330
Figure 8.48: Total vertical stress distribution at a depth of 2 m for a settlement of 100 mm for
test JG_v10 (σc = 100 kPa, P = 76.00 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
Figure 8.49: Total vertical stress distribution at a depth of 4 m for a settlement of 100 mm for
test JG_v10 (σc = 100 kPa, P = 76.00 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
A
B C
DE
A
B C
DE
8 Appendices
331
Figure 8.50: Total vertical stress distribution at a depth of 6 m for a settlement of 100 mm for
test JG_v10 (σc = 100 kPa, P = 76.00 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
A
B C
DE
8.23 Total vertical stress distribution below the footing for a settlement of 400 mm (test
JG_v10)
332
8.23 Total vertical stress distribution below the footing for a
settlement of 400 mm (test JG_v10)
Figure 8.51, Figure 8.52, Figure 8.53 and Figure 8.54 present the total vertical stress
distribution below the footing for a settlement of 100 mm under the footing and at depths of 2
m, 4 m and 6 m below the surface, respectively.
Figure 8.51: Total vertical stress distribution under the footing for a settlement of 400 mm for
test JG_v10 (σc = 100 kPa, P = 108.60 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
A
B C
DE
8 Appendices
333
Figure 8.52: Total vertical stress distribution at a depth of 2 m for a settlement of 400 mm for
test JG_v10 (σc = 100 kPa, P = 108.60 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
Figure 8.53: Total vertical stress distribution at a depth of 4 m for a settlement of 400 mm for
test JG_v10 (σc = 100 kPa, P = 108.60 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
A
B C
DE
A
B C
DE
8.23 Total vertical stress distribution below the footing for a settlement of 400 mm (test
JG_v10)
334
Figure 8.54: Total vertical stress distribution at a depth of 6 m for a settlement of 400 mm for
test JG_v10 (σc = 100 kPa, P = 108.60 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
A
B C
DE
8 Appendices
335
8.24 Total vertical stress distribution below the footing for a
settlement of 850 mm (test JG_v10)
Figure 8.55 and Figure 8.56 present the total vertical stress distribution below the footing for
a settlement of 850 mm at depths of 2 m and 4 m below the surface, respectively.
Figure 8.55: Total vertical stress distribution at a depth of 2 m for a settlement of 850 mm for
test JG_v7 (σc = 100 kPa, P = 138.66 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
A
B C
DE
8.24 Total vertical stress distribution below the footing for a settlement of 850 mm (test
JG_v10)
336
Figure 8.56: Total vertical stress distribution at a depth of 4 m for a settlement of 850 mm for
test JG_v7 (σc = 100 kPa, P = 138.66 kPa). Negative stresses denote
compression. The dimensions are shown in Figure 6.51 and Figure 6.53.
A
B C
DE
8 Appendices
337
8.25 Comparison of the measured and modelled excess pore water
pressures for the test JG_v10
Figure 8.57 and Figure 8.58 present a comparison of the excess pore water pressures
measured in the centrifuge with those computed numerically for test JG_v10 using a 3D-
model.
Figure 8.57: Comparison of the values of the excess pore water pressures measured during
test JG_v10 with the values obtained numerically with Plaxis 3D (P4 till P6).
Figure 8.58: Comparison of the values of the excess pore water pressures measured during
test JG_v10 with the values obtained numerically with Plaxis 3D (P7).
0
5
10
15
20
25
30
0 500000 1000000 1500000 2000000 2500000
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P4 P5 P6 P4 - Plaxis P5 - Plaxis P6 - Plaxis
0
5
10
15
20
25
30
35
0 500000 1000000 1500000 2000000 2500000
Ex
ce
ss
po
re w
ate
r p
res
su
re
[kP
a]
Time [s]
P7 P7 - Plaxis
8.26 Development of plastic points (test JG_v10)
338
8.26 Development of plastic points (test JG_v10)
The development of plastic points during footing loading of a single stone column for test
JG_10 obtained numerically using an axisymmetric model is presented in this section.
Failure points denote elements that are located on the Mohr-Coulomb failure line (Figure
6.19, respectively Figure 6.22), while Cap points and Hardening points refer to elements
undergoing plastic shear strain hardening (Figure 6.22, Section 6.3.2.3) and plastic
volumetric hardening (Figure 6.23, Section 6.3.2.4), respectively. Tension cut-off points
denote elements that would undergo tension, if the value of cohesion was positive.
Figure 8.59: Development of plastic points during the loading phase for test JG_v10 (section
1-1, Figure 6.51) for a settlement of 100 mm (P = 76 kPa).
8 Appendices
339
Figure 8.60: Development of plastic points during the loading phase for test JG_v10 (section
1-1, Figure 6.51) for a settlement of 400 mm (P = 108.60 kPa).
Figure 8.61: Development of plastic points during the loading phase for test JG_v10 (section
1-1, Figure 6.51) for a settlement of 850 mm (P = 138.66 kPa).
8.26 Development of plastic points (test JG_v10)
340
Figure 8.62: Development of plastic points during the loading phase for test JG_v10 (section
2-2, Figure 6.51) for a settlement of 100 mm (P = 76 kPa).
Figure 8.63: Development of plastic points during the loading phase for test JG_v10 (section
2-2, Figure 6.51) for a settlement of 400 mm (P = 108.60 kPa).
8 Appendices
341
Figure 8.64: Development of plastic points during the loading phase for test JG_v10 (section
2-2, Figure 6.51) for a settlement of 850 mm (P = 138.66 kPa).
8.26 Development of plastic points (test JG_v10)
342
9 List of subscripts and symbols
343
9 List of subscripts and symbols
Subscripts Description Unit
c Clay [-]
s Host soil [-]
sc Stone column [-]
p Plane-strain conditions [-]
Symbol Description Unit
2D Two-dimensional [-]
3D Three-dimensional [-]
a Centripetal acceleration [m/s2]
a Distance between the axis of the stone columns [m]
as Replacement ratio [-]
a √ ⁄ [-]
a Undrained shear strength parameter [-]
as, p Replacement ratio in plane-strain conditions [-]
A Footing area [m2]
A Pore pressure parameter according to Skempton (1954) [-]
Asc Stone column cross-section [m2]
As Soft soil cross-section in the unit cell (As = a2 – Asc) [m2]
b Undrained shear strength parameter [-]
bc Width of the equivalent plane-strain column [m]
bs Width of the smear zone in plane-strain conditions [m]
bsc Width of the stone-column in plane-strain conditions [m]
bw Width of a drain in plane-strain conditions [m]
B Width of loaded area [m]
B Foundation width [m]
B Width of the zone of influence in plane-strain conditions [m]
B Pore pressure parameter according to Skempton (1954) [-]
c’ Effective cohesion [kPa]
c’eq Equivalent effective cohesion [kPa]
ch Horizontal coefficient of consolidation [m2/year]
chm Modified horizontal coefficient of consolidation [m2/year]
c’s Effective cohesion of the host soil [kPa]
c’sc Effective cohesion of the stone column material [kPa]
cv Coefficient of consolidation [m2/s]
Cc Compression index [-]
Cf Conversion factor between coefficients of permeability
obtained in the laboratory and in the field - according to Chai &
Miura (1999)
[-]
9 List of subscripts and symbols
344
Cs Swelling index [-]
CPT Cone Penetration Test [-]
d Length of the drainage path [m]
d Diameter of the T-Bar [m]
d Pore diameter [m]
de Diameter of the unit cell considered [m]
dsc Stone column diameter [m]
dεed Elastic deviatoric volumetric strain increment [-]
dεev Elastic volumetric strain increment [-]
dεpd Plastic deviatoric strain increment [-]
dεpv Plastic volumetric strain increment [-]
D Diameter of the zone of influence of a stone column [m]
e Void ratio [-]
e Base of natural logarithms [-]
e0 Initial void ratio [-]
es Void ratio of the host soil [-]
esc Void ratio of the stone column material [-]
E Young’s modulus [kPa]
Eeq Equivalent Young’s modulus [kPa]
Ei Initial stiffness modulus [kPa]
Eoed Tangent stiffness for primary oedometer loading [kPa]
Eoedref Tangent stiffness for primary oedometer loading corresponding
to the reference stress pref (Hardening Soil Model)
[kPa]
Es Young’s modulus of the host soil [kPa]
Es, p Young’s modulus of the host soil in plane-strain conditions [kPa]
Esc Young’s modulus of the stone column material [kPa]
Esc, p Young’s modulus of the stone column material in plane strain
conditions
[kPa]
Eu Elasticity modulus of the undrained soil [kPa]
Eur Unloading / reloading stiffness in a CDC triaxial test [kPa]
Eurref Unloading / reloading stiffness in a CDC triaxial test
corresponding to the reference stress pref (Hardening Soil
Model)
[kPa]
E50 Secant stiffness for primary loading in a CDC triaxial test [kPa]
E50ref Secant stiffness for primary loading in a CDC triaxial test
corresponding to the reference stress pref (Hardening Soil
Model)
[kPa]
ESEM Environmental Scanning Electron Microscope [-]
f Yield function [-]
F Failure surface [-]
fsc Volume content of the stone column [m3]
f12, f13 Yield surfaces yield surfaces in σ1, σ2 and σ1, σ3 planes, [-]
9 List of subscripts and symbols
345
respectively (Hardening Soil Model)
F’c Cavity expansion parameter [-]
F’q Cavity expansion parameter [-]
FEM Finite Element Method [-]
g Earth’s gravity ( ) [m/s2]
G Shear modulus of the soil [kPa]
H Height of the unit cell considered [m]
H Thickness of the layer [m]
He Embankment height [m]
HSM Hardening Soil Model [-]
Ip Plasticity index [%]
Ir Stiffness index [-]
I’rr Reduced stiffness index for cylindrical cavity [-]
J4 Depth factor according to Grasshoff (1978) [-]
k Coefficient of permeability [m/s]
kh Coefficient of horizontal permeability of the undisturbed host
soil
[m/s]
kh, cl Coefficient of horizontal permeability in the clogged zone [m/s]
kh,p Coefficient of horizontal permeability of the undisturbed host
soil in plane-strain conditions
[m/s]
k’h Coefficient of horizontal permeability in the smear zone [m/s]
k’h,p Coefficient of horizontal permeability in the smear zone in
plane-strain conditions
[m/s]
kht Coefficient of horizontal permeability in the transition zone [m/s]
ks Coefficient of permeability of the undisturbed host soil [m/s]
ks’ Coefficient of permeability of the soil in the smear zone [m/s]
kv Coefficient of vertical permeability [m/s]
K Coefficient of earth pressure [-]
K0 Coefficient of earth pressure at rest [-]
Increased coefficient of earth pressure at rest [-]
K0,sc Coefficient of earth pressure at rest of the stone column
material
[-]
K0NC Coefficient of earth pressure at rest of a normally consolidated
soil
[-]
K0OC Coefficient of earth pressure at rest of an over-consolidated
soil
[-]
Ka Coefficient of active earth pressure [-]
Ka, sc Coefficient of active earth pressure of the stone column
material
[-]
Ks Coefficient of earth pressure of the host soil [-]
Ksc Coefficient of earth pressure of the stone column material [-]
Kp Coefficient of passive earth pressure [-]
9 List of subscripts and symbols
346
Kp, e Coefficient of passive earth pressure of the embankment [-]
l Half-length of drain [m]
ID Density index [%]
L Pile length [m]
L Length of stone column [m]
m Stress concentration ratio [-]
m Power for stress-level dependency of stiffness. This factor
should be equal to 1.0 in the case of soft clays (Hardening Soil
Model)
[-]
M Slope of critical state line (Cam Clay) [-]
ME Confined stiffness modulus [kPa]
ME, h Horizontal confined stiffness modulus [kPa]
ME, s Confined stiffness modulus of the undisturbed host soil [kPa]
ME, sc Confined stiffness modulus of the stone column material [kPa]
ME, v Vertical confined stiffness modulus [kPa]
MC Mohr-Coulomb [-]
MCC Modified Cam-Clay [-]
MIP Mercury Intrusion Porosimetry [-]
n Factor of increase of the Earth’s gravity in the centrifuge [-]
n Radius ratio of the unit cell to the drain well [-]
n0 Initial porosity of the host soil [-]
n0 Ground improvement factor [-]
n1 Ground improvement factor for compressibility [-]
n2 Depth-dependent ground improvement factor [-]
Nb T-Bar factor [-]
Nc, Nγ, Nq Dimensionless bearing capacity parameters [-]
OCR Over-consolidation ratio [-]
p’ Mean effective stress [kPa]
p Mercury pressure [kPa]
psc,u Stress acting within the column at the depth of the tip of the
column
[kPa]
pp Pre-consolidation stress [kPa]
pu Ultimate cavity pressure [kPa]
pref Reference stress (Hardening Soil Model) [kPa]
P Footing load [kPa]
P Force per unit length acting on the T-Bar [kN/m]
POP Pre-Overburden-Pressure [kPa]
PPT Pore pressure transducer [-]
PVD Prefabricated vertical drain [-]
(P0)V,s Initial effective vertical stress in the host soil [kPa]
(P0)V,SC Initial effective vertical stress in the stone column [kPa]
9 List of subscripts and symbols
347
(ΔP)*V Effective vertical stress increase averaged over horizontal
projected area of the unit cell
[kPa]
(ΔP)*V,s Effective vertical stress increase in the clay averaged over the
horizontal projected area of host soil
[kPa]
q Surcharge at the surface [kPa]
q Deviatoric stress [kPa]
Special stress measure for deviatoric stresses [kPa]
q(t) Average applied loading at time t [kPa]
q0 Overburden pressure [kPa]
q0 Ultimate loading [kPa]
qf Frictional resistance [kPa]
qf Deviatoric stress at failure [kPa]
qs Bearing capacity of the host soil [kPa]
qsc Bearing capacity of a stone column [kPa]
qsc, bulging Bulging failure load of a stone column [kPa]
qsc, shear Shear failure load of a single stone column [kPa]
qsc, shear, PW,i Shear strength of the i-th equivalent plane wall [kPa]
qt Tip resistance [kPa]
qw Discharge capacity of the drain [m3/year]
r Radius [m]
r Radius of stone column or pile [m]
r Distance from centreline [m]
r0 Pile radius [m]
rsc Radius of the stone column [m]
r’sc Effective radius of the stone column [m]
rs Radius of the smear zone [m]
rs Radius of the remoulded zone [m]
rw Radius of the drain [m]
R Radius of the unit cell considered [m]
R Electrical resistance [Ω]
R Global radial coordinate [m]
Req Equivalent radius of square pile [m]
Rf Failure ratio, usually set equal to 0.9 (Hardening Soil Model) [-]
Ri Initial radius of the cavity [m]
Rp Radius of the plastic zone [m]
Rs Radius of the smear zone [m]
Ru Final cavity radius [m]
s Total settlement [m]
s Radius ratio of the smear zone to the drain well [-]
s Distance between the axis
S Spacing (axis to axis) between two adjacent drains [m]
9 List of subscripts and symbols
348
ss Settlement of the host soil [m]
ssc Settlement of the stone column [m]
sc Settlement of the layer [m]
se Settlement due to the stress concentration at the tip of the
stone column
[m]
se’ Reduced settlement due to the stress concentration at the tip
of the stone column
[m]
si Settlement of the improved layer [m]
s(t) Average settlement at time t [m]
sr Radial deformation [m]
su Undrained shear strength of the host soil [kPa]
sv Vertical settlement [m]
sv Settlement of the composite foundation [m]
su, avg Composite undrained shear strength [kPa]
s0 Settlement of the treated layer without ground improvement [m]
Average final settlement [m]
S Spacing (axis to axis) between two adjacent drains [m]
SCP Sand Compaction Pile [-]
SEM Scanning Electron Microscope [-]
SLS Serviceability Limit State [-]
SPM Strain Path Method [-]
SPT Standard Penetration Test [-]
SSPM Shallow Strain Path Method [-]
t Consolidation time [s]
t90 Time needed to reach a consolidation of 90 % [s]
Th Dimensionless time factor for radial flow [-]
Thm Modified dimensionless time factor for radial flow [-]
Tv Dimensionless time factor [-]
TST Total stress transducer [-]
u Pore water pressure [kPa]
ui Pore water pressure at the slip surface for the i-th slice [kPa]
Average pore pressure throughout the soil-stone column
cylinder
[kPa]
U Overall average degree of consolidation [-]
Average degree of consolidation for a radial flow [-]
ULS Ultimate Limit State [-]
wl Liquid limit [%]
wp Plastic limit [%]
W Width of equivalent pile strip [m]
z Depth [m]
zc Depth to which the column has been compacted [m]
zf Depth of foundation [m]
9 List of subscripts and symbols
349
zi Depth of the i-th slice below the surface [m]
Z Electrical impedance [Ω]
α Electrode shape factor [1/cm]
α Henkel’s (1959) pore pressure parameter for the particular
stress level
[-]
α Non-dimensional factor for the consideration of the clogging of
a stone column
[-]
α Cap parameter relating to K0NC (Hardening Soil Model) [-]
αf Pressure parameter according to Skempton (1954) [-]
αi Inclination angle of the lower side of the i-th slice [°]
αk Ratio of horizontal permeability of the clogged column zone to
that of the smear zone
[-]
αvs Coefficient of compressibility of the host soil [m2/kN]
αvsc Coefficient of compressibility of the stone column material [m2/kN]
β Inclination of the failure surface [°]
β Settlement reduction factor [-]
β Cap parameter relating to (Hardening Soil Model) [-]
δ Inclination of the failure mechanism [°]
δri / r Radial strain [-]
δσr Radial total stress change [kPa]
δσθ Circumferential total stress change [kPa]
Δ Volumetric strain in the plastic zone [-]
ΔP Footing load increment [kPa]
Δt Layer thickness [m]
Δu0 Initial uniform excess pore water pressure [kPa]
Δumax Maximal excess pore water pressure [kPa]
Δur Excess pore water pressure due to radial flow [kPa]
Average excess pore water pressure due to radial flow [kPa]
Δσ Pressure difference on a footing [kPa]
Δσa Axial stress increase
Δσz Vertical stress increase [kPa]
εr Radial strain [-]
εv Vertical strain [-]
εrp Plastic volumetric strain [-]
εe Elastic strain [-]
εp Plastic strain [-]
εvpc Volumetric cap strain (Hardening Soil Model) [-]
ε1p, ε2
p Principal plastic strains [-]
φ’ Effective angle of friction [°]
Average mobilised effective angle of friction of the improved
ground [°]
9 List of subscripts and symbols
350
φ’cv Critical state effective angle of friction [°]
φ’eq Equivalent effective angle of friction [°]
φ’s Effective angle of friction of the host soil [°]
φ’sc Effective angle of friction of the stone column material [°]
γc Unit weight of clay [kN/m3]
γe Unit weight of fill [kN/m3]
γeq Equivalent unit weight of the plane wall [kN/m3]
γq Unit weight of saturated soil [kN/m3]
γs Unit weight of the host soil [kN/m3]
γsc Unit weight of the stone column material [kN/m3]
γw Unit weight of water [kN/m3]
γp Plastic shear strain [-]
Effective Poisson’s ratio [-]
u Undrained Poisson’s ratio (
u = 0.5 [-]) [-]
ur Poisson’s ratio for unloading - reloading [-]
Poisson’s ratio [-]
Slope of swelling line (Cam Clay) [-]
Slope of normal consolidation line (Cam Clay) [-]
μsc Stress concentration factor for the stone column [-]
Wetting angle for mercury [°]
ρ Electrical resistivity [Ω.m]
ρ Soil arching ratio [-]
ρd,max Maximum bulk density [g/cm3]
ρd,min Minimum bulk density [g/cm3]
ρg Specific density of the saturated soil [g/cm3]
ρs Specific density [g/cm3]
σ Surface tension (MIP) [N/m]
σ Total stress acting on the unit cell [kPa]
σa Axial stress [kPa]
σc Pre-consolidation stress [kPa]
σh Horizontal stress [kPa]
σ’n,PW,i Normal stress applied on the i-th equivalent plane wall [kPa]
σr Radial stress [kPa]
σr Normal stress in soil element along R-direction [kPa]
σr0 Ultimate total in-situ lateral stress [kPa]
σs Total stress acting on the soft soil surface [kPa]
σsc Total stress acting on the stone column [kPa]
σsc,i Normal stress acting on the i-th stone column [kPa]
σ*v Tip resistance [kPa]
σv Vertical total stress [kPa]
σ’v Vertical effective stress [kPa]
9 List of subscripts and symbols
351
σ’v,max Maximum vertical effective stress [kPa]
σ0 Average load intensity on a footing [kPa]
σ0 Load acting on the top of the embankment [kPa]
σ’1, σ’2, σ’3, Principal effective stresses [kPa]
σ1, σ2, σ3, Principal total stresses [kPa]
σ3 Average lateral confining pressure [kPa]
σθ Tangential total stress [kPa]
Angular velocity [rad/s]
Angle of dilatancy [°]
9 List of subscripts and symbols
352
10 References
353
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