materials
Review
Modeling Stone Columns
Jorge Castro
Department of Ground Engineering and Materials Science, University of Cantabria, Avda. de los Castros s/n,
39005 Santander, Spain; castrogj@unican.es; Tel.: +34-942-201813
Received: 4 May 2017; Accepted: 7 July 2017; Published: 11 July 2017
Abstract: This paper reviews the main modeling techniques for stone columns, both ordinary stone
columns and geosynthetic-encased stone columns. The paper tries to encompass the more recent
advances and recommendations in the topic. Regarding the geometrical model, the main options
are the “unit cell”, longitudinal gravel trenches in plane strain conditions, cylindrical rings of
gravel in axial symmetry conditions, equivalent homogeneous soil with improved properties and
three-dimensional models, either a full three-dimensional model or just a three-dimensional row or
slice of columns. Some guidelines for obtaining these simplified geometrical models are provided
and the particular case of groups of columns under footings is also analyzed. For the latter case,
there is a column critical length that is around twice the footing width for non-encased columns in a
homogeneous soft soil. In the literature, the column critical length is sometimes given as a function
of the column length, which leads to some disparities in its value. Here it is shown that the column
critical length mainly depends on the footing dimensions. Some other features related with column
modeling are also briefly presented, such as the influence of column installation. Finally, some
guidance and recommendations are provided on parameter selection for the study of stone columns.
Keywords: stone columns; encased stone columns; geosynthetic; numerical modeling; critical length;
parameter selection
1. Introduction
Ground improvement using stone columns, also known as granular piles or aggregate piers,
is one of the most popular techniques to improve soft soils for the foundation of embankments or
structures. These are vertical boreholes in the ground, filled upwards with gravel compacted by means
of a vibrator.
The idea of improving soft soils for foundation purposes using granular inclusions is relatively
old. It is documented [1] that in 1839 in Bayonne (France), the French colonel Burbach used for the
first time sand piles as deep foundations instead of the classical wood piles that rapidly degrade with
fluctuations of the ground water level. However, it was not until the 50 s of the last century when stone
columns started to be used. The ground improvement technique started as an extension of traditional
vibro-compaction (deep compaction) to non-granular soils, whose low permeability and cohesion do
not allow for a quick rearranging of soil particles in a denser configuration.
Stone columns act mainly as inclusions with a higher stiffness, shear strength and permeability
than the natural soil. Consequently, they improve the following aspects:
•
•
•
•
•
The bearing capacity
The stability of embankments and natural slopes
Final settlement
Degree of consolidation
Liquefaction potential
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Materials 2017, 10, 782
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The reduction of the liquefaction potential is beyond the scope of the present paper. Please refer
to [2,3] for further information on that topic. The present paper focuses on the other four improvements,
particularly, the settlement reduction. The modeling strategy for stone columns should be chosen
depending on which of the previous improvements is to be analyzed.
In extremely soft soils, stone columns are not suitable because their continuity, stability, geometric
shape, etc. cannot be guaranteed. The undrained shear strength (cu ) of natural soft soil is generally
used as the limiting parameter for stone column feasibility. A limiting value around 5–15 kPa [4] may
be adopted. To improve the lateral confinement of stone columns in those extremely soft soils, encasing
the columns with geotextiles or other geosynthetics, such as geogrids, has proven to be a successful
technique in recent times (e.g., [5]). Rigid or semi-rigid inclusions (e.g., adding lime or cement) is
another common alternative (e.g., [6,7]). Han [8] summarizes different ground column technologies.
Ground improvement using stone columns, either ordinary or encased stone columns, requires a
considerable amount of columns or, at least, a group of columns. This implies a complex modeling
process of the real geometry. This paper provides some guidelines for obtaining these simplified
geometrical models. Some other features related with column modeling are also briefly presented,
such as the critical length of the column and the influence of column installation. Additionally, some
guidance and recommendations are provided on parameter selection to study stone columns. Here,
the word “modeling” is understood in a broad sense, covering geometrical, mechanical, geotechnical
and installation features of stone columns.
The increase in computer power and the availability of finite element codes makes numerical
analyses very appealing in geotechnical design. They usually lead to more detailed studies but
require a clear conception of the modeling techniques. Ground improvement techniques, such as
stone columns, are more and more popular due to the increasing occupation of natural soft soils and
environmental concerns [9,10]. Within these current trends, the review of the modeling techniques
for stone columns seems interesting and useful. Besides, in some instances, there is some confusion
about geometrical models and their application. For example, the results for an isolated column
under concentrated load just on top of the column cannot be directly extrapolated for a large group of
columns under distributed uniform load.
2. Geometrical Models
To simplify the real geometry of the problem that usually involves a considerable amount of
columns (e.g., Figure 1a) and to be able to deal with the problem, the following simplified geometrical
models are usually adopted:
•
•
•
•
•
•
•
•
“Unit cell” in axial symmetry (Figure 1b). Only one column and its corresponding surrounding
soil are studied. It may be useful to study just a horizontal slice of the unit cell, rather than the
whole length.
Longitudinal gravel trenches (Figure 1c). The stone columns are transformed into longitudinal
gravel trenches to study the problem in plane strain conditions.
Cylindrical rings of gravel (Figure 1d). The columns are transformed into cylindrical rings of
gravel to study the problem in axial symmetry.
Homogenization or equivalent homogeneous soil (Figure 1e). The columns and the surrounding
soil are transformed into a homogeneous soil with equivalent improved properties.
Three-dimensional (3D) model of a row or slice of columns (Figure 1f).
Geometrical models for small groups of stone columns.
“Unit cell” with constant lateral pressure (triaxial conditions).
Isolated column.
Materials 2017, 10, 782
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Materials 2017, 10, 782
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(a)
(b)
(c)
(d)
(e)
(f)
Figure 1. Main geometrical models for stone column studies: (a) Full 3D model; (b) Unit cell;
Figure 1. Main geometrical models for stone column studies: (a) Full 3D model; (b) Unit cell; (c)
(c) Longitudinal gravel trenches; (d) Cylindrical gravel rings; (e) Equivalent homogenous soil; (f) 3D slice
Longitudinal
gravel trenches; (d) Cylindrical gravel rings; (e) Equivalent homogenous soil; (f) 3D slice
of columns.
of columns.
The two latter cases, namely “unit cell” under triaxial conditions and isolated column, do not
usually
real problems,
but they
have
been used
for conditions
laboratory tests
[11–13]).
The case
The twoappear
latterincases,
namely “unit
cell”
under
triaxial
and(e.g.,
isolated
column,
do not
of
an
isolated
column
is
widely
used
for
field
tests
(usually
plate
load
tests)
for
the
sake
of
simplicity
usually appear in real problems, but they have been used for laboratory tests (e.g., [11–13]). The case
(e.g., [14]).
of an isolated column is widely used for field tests (usually plate load tests) for the sake of simplicity
As a simple introductory comparison between the different geometrical models, Table 1
(e.g., [14]).
summarizes the suitability of some of these models to study the improvements achieved with a stone
As
a simple
introductory
comparison
between the different geometrical models, Table 1
column
treatment
for the foundation
of an embankment.
summarizes the suitability of some of these models to study the improvements achieved with a
Tabletreatment
1. Suitability
of the
simplified
geometrical
to study different features of a stone column
stone column
for
foundation
of anmodels
embankment.
treatment for the foundation of an embankment.
Table 1. Suitability of simplified geometrical models to study different features of a stone column
Geometrical Model
Final Settlement Consolidation Stability
treatment for the foundation of an embankment.
Unit cell
***
***
Gravel trenches
**
**
**
Geometrical Model
Final Settlement
Consolidation
Stability
Homogenization
**
*
*
Unit cell 3D slice
***
*** ***
***
*** Gravel
trenches
**
**
**
*** Completely suitable, ** Moderately suitable, * Slightly suitable, - Not suitable.
Homogenization
**
*
*
3D slice
***
***
***
All the geometrical models listed above are valid for non-encased stone columns. However, for
Completely
suitable,
Moderately
suitable, * way
Slightly
suitable, -the
Notcylindrical
suitable. encasement
encased stone***
columns,
there
is not**yet
any satisfactory
to convert
(geosynthetic) that surrounds the column for the cases of longitudinal gravel trenches and cylindrical
ringsthe
of gravel.
All
geometrical models listed above are valid for non-encased stone columns. However,
for encased stone columns, there is not yet any satisfactory way to convert the cylindrical encasement
3. Unit Cell
(geosynthetic)
that surrounds the column for the cases of longitudinal gravel trenches and cylindrical
The unit cell model is the most widely used for theoretical analyses and it is reviewed in detail,
rings of gravel.
for example, in [15]. The basis for the simplified model is the usage of a great number of columns,
3. Unit
Cell distributed in a wide area under a uniform load. This is the case, for example, in the central
uniformly
part of an embankment on soft ground improved with stone columns. In these situations, the
The unit cell model is the most widely used for theoretical analyses and it is reviewed in detail,
behavior of all the columns is the same and then, it is enough to study the behavior of just one column
for example,
in [15]. The basis
for thesoil
simplified
is the
usage of a
great number
of columns,
with the corresponding
surrounding
(tributarymodel
area). Due
to symmetry
conditions,
at the external
uniformly
a wide
under
a uniform
load. Thisand
is the
case,
for example,
in theare
central
lateraldistributed
boundary ofinthe
unit area
cell, only
vertical
displacements
only
vertical
water seepage
part of
an embankment on soft ground improved with stone columns. In these situations, the behavior
allowed.
Stone columns
are generally
uniformly
distributed
in the
triangular
or of
square
grids.
Thus, with
the the
of all the columns
is the same
and then,
it is enough
to study
behavior
just one
column
tributary
area
of
natural
soil
for
each
column
is
a
hexagon
or
a
square,
respectively.
To
allow
for
axial
corresponding surrounding soil (tributary area). Due to symmetry conditions, at the external lateral
symmetry
is transformed
a circle
(cylinder)
of the same
boundary
of theconditions,
unit cell, the
onlytributary
verticalarea
displacements
andinto
only
vertical
water seepage
are (crossallowed.
sectional)
area.
Therefore,
the
diameter
of
the
unit
cell
is
equal
to
=
1.05
−
1.13
for
triangular
Stone columns are generally uniformly distributed in triangular or square grids.
Thus, the
and square grids, respectively, where is the centre-to-centre distance between columns (Figure 2).
tributary area of natural soil for each column is a hexagon or a square, respectively. To allow for
Figure 3 shows the unit cell model with axial symmetry that can be studied in two dimensions.
axial symmetry conditions, the tributary area is transformed into a circle (cylinder) of the same
(cross-sectional) area. Therefore, the diameter of the unit cell is equal to de = 1.05 − 1.13 s for triangular
and square grids, respectively, where s is the centre-to-centre distance between columns (Figure 2).
Figure 3 shows the unit cell model with axial symmetry that can be studied in two dimensions.
Materials 2017, 10, 782
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Materials 2017, 10, 782
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Materials 2017, 10, 782
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s
s
rc
rc
re
re
rc
rc
re
re
s
s
Figure 2. Simplification of the unit cell to axial symmetry conditions.
Figure 2. Simplification of the unit cell to axial symmetry conditions.
Figure 2. Simplification of the unit cell to axial symmetry conditions.
Figure 3. Unit cell model for analytical solutions.
Figure3.3.Unit
Unit cell
cell model
solutions.
Figure
modelfor
foranalytical
analytical
solutions.
The unit cell model is used for most of the existing analytical solutions (e.g., [16,17]). In the
analytical
solutions,
thereisisused
usually
further
simplifying
assumption
for the geometry,
which
to
The unit
cell model
for amost
of the
existing analytical
solutions
(e.g., [16,17]).
In isthe
The unit cell model is used for most of the existing analytical solutions (e.g., [16,17]). In the
assume that
the behavior
horizontal
slice
is independent,
i.e., the
stresses arewhich
neglected.
analytical
solutions,
there of
is each
usually
a further
simplifying
assumption
forshear
the geometry,
is to
analytical
solutions,
there
isofusually
a further
simplifying
assumption
for the
which
Numerical
[17,18]
show
that
shear
stresses
are usually
for geometry,
distributed
loads. is to
assume
that analyses
the behavior
each
horizontal
slice
is independent,
i.e.,negligible
the shear stresses
are neglected.
assume
that
the
behavior
of
each
horizontal
slice
is
independent,
i.e.,
the
shear
stresses
are
neglected.
Balaam andanalyses
Booker [17,18]
[19] is ashow
notable
to theare
simplifying
assumption
neglectingloads.
shear
Numerical
thatexception
shear stresses
usually negligible
for of
distributed
Numerical
analyses
[17,18]
show
that
shear
stresses
are
usually
negligible
for
distributed
loads.
Balaam
stressesand
because
they
study
total exception
length of the
unitsimplifying
cell as a whole.
Nevertheless,
the solution
Balaam
Booker
[19]
is a the
notable
to the
assumption
of neglecting
shear
and stresses
Booker
[19]
is athey
notable
to theof
simplifying
assumption
of neglecting
stresses
requiresbecause
numerical
integration,
makes
the
solution
complex
for practical
purposes.
studyexception
the which
total length
the
unit cell
as a whole.
Nevertheless,
theshear
solution
Most
analytical
solutions
focus
on
reduction
caused
bypractical
stone columns
[16,17,19]),
requires
numerical
integration,
which
makes
thecell
solution
for
purposes.
because
they
study
the
total length
ofthe
thesettlement
unit
as a complex
whole.
Nevertheless,
the(e.g.,
solution
requires
but stone
columns also
act as
vertical
drains
and, complex
therefore,
they
dissipate
excess
pore(e.g.,
pressures.
The
Most
analytical
solutions
focus
on
the
settlement
reductionfor
caused
by stone
columns
[16,17,19]),
numerical
integration,
which
makes
the
solution
practical
purposes.
consolidation
process
may
be
studied
independently
from
the
settlement
analysis
using
solutions
for
but
stone
columns
also
act
as
vertical
drains
and,
therefore,
they
dissipate
excess
pore
pressures.
The
Most analytical solutions focus on the settlement reduction caused by stone columns
vertical drainsprocess
[20,21].may
Hanbeand
Ye [22]
and Castro and
Sagaseta
[23] showed
that
the consolidation
consolidation
studied
independently
from
the settlement
analysis
using
solutions for
(e.g., [16,17,19]), but stone columns also act as vertical drains and, therefore, they dissipate excess pore
processdrains
around
stone Han
columns
is slightly
because
of the
vertical
stresses
vertical
[20,21].
and Ye
[22] and different
Castro and
Sagaseta
[23]distribution
showed thatofthe
consolidation
pressures. The consolidation process may be studied independently from the settlement analysis using
betweenaround
soft soil
and columns
stone columns
and different
they proposed
specific
solutions
to study
the
process
stone
is slightly
because
of the analytical
distribution
of vertical
stresses
solutions
forsoft
vertical
drains
[20,21].
and Yeproposed
[22]
andspecific
Castro
andLogar
Sagaseta
[23] to
showed
that the
consolidation
process
around
stoneHan
columns.
recently,
Pulkoanalytical
and
[24] have
developed
between
soil
and stone
columns
and theyVery
solutions
study thea
consolidation
process
around
stone
columns
is
slightly
different
because
of
the
distribution
of
vertical
fully coupled process
solutionaround
for thestone
consolidation
assuming
the and
soil Logar
as a poroelastic
and
consolidation
columns.process
Very recently,
Pulko
[24] have medium
developed
a
stresses
soft soil
and
stone
columns
and
they
proposed
specific
solutions
to study
the between
stone
column
as an
elastoplastic
material.
The
solution
is the
very
accurate
but requires
numerical
fully
coupled
solution
for
the
consolidation
process
assuming
soil
as a analytical
poroelastic
medium
and
inversion
of the
Laplace
transform.
the consolidation
process
stonematerial.
columns.
Very
recently,
Pulko
and Logar
[24] have
developed
the
stone column
as anaround
elastoplastic
The
solution
is very
accurate
but requires
numerical
Extending
analytical
solutions
for
ordinary
stone
columns
to
encased
stone
columns
is
quite and
inversion
of
the
Laplace
transform.
a fully coupled solution for the consolidation process assuming the soil as a poroelastic medium
straightforward
(e.g.,
[25]).
Equilibrium
and
compatibility
conditions
of
the
geosynthetic
encasement
are
Extending
analytical
solutions
for
ordinary
stone
columns
to
encased
stone
columns
is
quite
the stone column as an elastoplastic material. The solution is very accurate but requires numerical
those
of
a
thin-walled
tube,
or
better
said,
those
of
a
flexible
membrane
because
the
encasement
does
straightforward
(e.g.,
[25]).
Equilibrium
and
compatibility
conditions
of
the
geosynthetic
encasement
are
inversion of the Laplace transform.
not usually
support compressive
stresses.
The of
internal
andmembrane
external pressures
areencasement
here denoted
as
those
of a thin-walled
tube, or better
said, those
a flexible
because the
does
Extending analytical solutions for ordinary stone columns to encased stone columns is quite
not usually support compressive stresses. The internal and external pressures are here denoted as
straightforward (e.g., [25]). Equilibrium and compatibility conditions of the geosynthetic encasement
are those of a thin-walled tube, or better said, those of a flexible membrane because the encasement
does not usually support compressive stresses. The internal and external pressures are here denoted
Materials 2017, 10, 782
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10, 782
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as σrcMaterials
and σ2017,
(Figure 4). Thus, the equilibrium and compatibility conditions
of the
rs , respectively
geosynthetic encasement are the following, respectively:
and
, respectively (Figure 4). Thus, the equilibrium and compatibility conditions of the
geosynthetic encasement are the following, respectively:
Tg
σrc =
=
rc
+ σrs
+
Tg = Jg
=
(1)
(1)
sr
rc
(2)
(2)
where sr is the radial displacement of the column and the encasement, Jg is the hoop or circumferential
where
is the radial displacement
the force
column
andgeosynthetic.
the encasement,
is the
stiffness
of the geosynthetic
and Tg is the of
hoop
at the
The units
of Jhoop
Tg are
g andor
stiffness
of thethe
geosynthetic
is the hoop force
the geosynthetic.
The
units
of
force circumferential
per length (F/L)
because
thicknessand
of the geosynthetic
is at
usually
negligible.
The
equation
and the are
forcestress
per length
because
thickness
of the
negligible.
The
that relates
radial
of the(F/L)
column
(σrcthe
) with
that of
thegeosynthetic
soft soil (σrsis) usually
and allows
the confining
equation that relates the radial stress of the column ( ) with that of the soft soil ( ) and allows the
effect of the encasement to be included is obtained by combining Equations (1) and (2):
confining effect of the encasement to be included is obtained by combining Equations (1) and (2):
σrc ==
Jg sr
+
+ σrs
rc2
(3)
(3)
rs
sr
rc
Tg
rc
Tg
Figure 4. Equilibrium and compatibility conditions for the geosynthetic surrounding an encased
Figure 4. Equilibrium and compatibility conditions for the geosynthetic surrounding an encased
stone column.
stone column.
The unit cell model is also very useful for numerical analyses because numerical simulations
allowunit
more
complex
be studied
the analytical
solutions,
suchnumerical
as layered ground,
The
cell
modelfeatures
is alsotovery
usefulthan
for with
numerical
analyses
because
simulations
advance
constitutive
models
or
cyclic
loads
(e.g.,
[26–29]).
Nowadays,
3D
numerical
codes
are
fairly
allow more complex features to be studied than with the analytical solutions, such as layered
ground,
easy
to
access
and
to
use;
so,
there
are
some
authors
that
study
the
unit
cell
as
a
full
3D
problem,
i.e.,fairly
advance constitutive models or cyclic loads (e.g., [26–29]). Nowadays, 3D numerical codes are
considering the hexagonal or square prism. Nevertheless, the differences between the full 3D prism and
easy to
access and to use; so, there are some authors that study the unit cell as a full 3D problem, i.e.,
the two-dimensional (2D) model (cylinder) in axial symmetry are negligible [30,31]. The simplicity of
considering the hexagonal or square prism. Nevertheless, the differences between the full 3D prism
the unit cell model has recently led to highly advanced numerical models, such as that presented by
and the
two-dimensional (2D) model (cylinder) in axial symmetry are negligible [30,31]. The simplicity
Indraratna et al. [32], where the column is modelled using 2D discrete elements to represent the gravel
of theparticles.
unit cellThis
model
recentlyanalysis
led to highly
advanced
models,
as that
presented
type has
of numerical
can be regarded
onlynumerical
as explorative
and forsuch
research
purposes.
by Indraratna
etmodel
al. [32],
where
theofcolumn
2Ddepth
discrete
elements
represent the
The unit cell
or the
model
a slice of is
themodelled
unit cell atusing
a specific
are also
used fortosmall-scale
gravellaboratory
particles.studies
This type
numerical analysis can be regarded only as explorative and for research
(e.g., of
[33,34]).
The
unit
cell
also
allows
floating
columns
to be
that do
not reach
a rigid
purposes. The unit cell model or the
model
of a slice
ofstudied,
the uniti.e.,
cellcolumns
at a specific
depth
are also
used for
substratum.
In thisstudies
case, the(e.g.,
settlement
due to column punching into the underlying layer and the
small-scale
laboratory
[33,34]).
deformation
the underlying
layer should
be considered
[35]. The
cell model
only
valid
when
The
unit cellofalso
allows floating
columns
to be studied,
i.e.,unit
columns
thatisdo
not
reach
a rigid
the ratio between the width of the loaded area and the thickness of the soft soil layer ( / ) is high
substratum. In this case, the settlement due to column punching into the underlying layer and the
enough to ensure confined (oedometric) conditions. On the contrary, the applied load is distributed
deformation of the underlying layer should be considered [35]. The unit cell model is only valid when
with depth, in a roughly trapezoidal manner, its value decreasing with depth. Figure 5 presents some
the ratio
between
the of
width
of the loaded
area
andfor
theend-bearing
thickness and
of the
soft soil
layer[36–38].
(B/H)The
is high
specific
proposals
load distribution
width
depth
floating
columns
enough
to
ensure
confined
(oedometric)
conditions.
On
the
contrary,
the
applied
load
is
distributed
unit cell model for these cases usually overestimates the settlement because the reduction of the applied
with depth,
a roughly
trapezoidal
manner, its value decreasing with depth. Figure 5 presents some
verticalin
stress
with depth
is not reproduced.
specific proposals
load
width
depth for
end-bearing
floating
columnsand
[36–38].
Finally, theofunit
celldistribution
is an appropriate
simplified
geometrical
modeland
to study
the settlement
its evolution
with
time
at the
center
of an embankment,
obviously
it is because
not valid the
for studying
theof the
The unit
cell model
for
these
cases
usually
overestimatesbut
the
settlement
reduction
stability
of the
lateral
slopes
(Table
1). For
similar reasons, it does not allow the lateral spreading and
applied
vertical
stress
with
depth
is not
reproduced.
the
contribution
of
a
geosynthetic
reinforcement
that acts
as a “blanket”
or “bridging
layer”
over the and
Finally, the unit cell is an appropriate simplified
geometrical
model
to study the
settlement
columns and soft soil foundation (geosynthetic reinforced and column supported embankments,
its evolution with time at the center of an embankment, but obviously it is not valid for studying the
stability of the lateral slopes (Table 1). For similar reasons, it does not allow the lateral spreading
and the contribution of a geosynthetic reinforcement that acts as a “blanket” or “bridging layer” over
Materials 2017, 10, 782
6 of 23
Materials 2017, 10, 782
6 of 23
the columns
and soft soil foundation (geosynthetic reinforced and column supported embankments,
GRCSE) GRCSE)
to be studied
[31]. To
study
the lateral
spreading
of of
thetheembankment,
unitcell
cellmodel
model may
to be studied
[31].
To study
the lateral
spreading
embankment, the
the unit
Materials
2017,
10, 782 by substituting
6 of
23 [39].
be improved
by
substituting
the horizontally-fixed
external
lateral
elastic
springs
may be
improved
the horizontally-fixed
external
lateral boundary
boundary byby
elastic
springs
[39].
GRCSE) to be studied [31]. To study the lateral spreading of the embankment, the unit cell model
may be improved by substituting the horizontally-fixed external lateral boundary by elastic springs [39].
(a)
(b)
Figure 5. Stress distribution beneath footings: (a) End-bearing columns; (b) Floating columns.
Figure 5. Stress distribution beneath footings: (a) End-bearing columns; (b) Floating columns.
(a)
4. Longitudinal Gravel Trenches
(b)
Figure 5. Stress distribution beneath footings: (a) End-bearing columns; (b) Floating columns.
4. Longitudinal
Trenches
ManyGravel
geotechnical
problems fulfill plane strain conditions. Thus, when stone columns are used
those problems,
for example,
for the foundation of an embankment for a linear infrastructure, it is
4.inLongitudinal
Gravel
Trenches
Many
geotechnical
problems
fulfill
plane strain conditions. Thus, when stone columns are used
useful to study the problem under 2D plane strain conditions, transforming the columns into
in thoselongitudinal
problems,
for
example,
for
the
foundation
of
an embankment
for acolumns
linear infrastructure,
Many geotechnical
problems
fulfill
strainofconditions.
Thus,
stone
used
gravel trenches (Figure 6). plane
The width
the trenches
and when
the spacing
between are
them
are
in
those
problems,
for
example,
for
the
foundation
of
an
embankment
for
a
linear
infrastructure,
it is useful
to
study
the
problem
under
2D
plane
strain
conditions,
transforming
the
columns
part of the unknown parameters that should be properly estimated to be equivalent. Besides, itit isis into
useful
tonecessary
study
the
under
2D plane
conditions,
transforming
the columns
into
longitudinal
gravel
trenches
(Figure
6). The
widthstrain
of these
the
trenches
and
the trenches.
spacing
between
them are
usually
to problem
alter
the gravel
parameters
for
equivalent
gravel
Furthermore,
longitudinal
gravel
trenches
(Figure
6).
The
width
of
the
trenches
and
the
spacing
between
them
are
to appropriately
model the consolidation
process,
it is also estimated
necessary toto
modify
the permeability
of
part of the
unknown parameters
that should
be properly
be equivalent.
Besides,
it is
part
of the unknown
parameters that should be properly estimated to be equivalent. Besides, it is
the
natural
soft
soil.
usually usually
necessary
to alter
the the
gravel
parameters
theseequivalent
equivalent
gravel
trenches.
Furthermore,
necessary
to alter
gravel
parameters for
for these
gravel
trenches.
Furthermore,
to appropriately
modelmodel
the consolidation
process,
it itis isalso
to modify
modifythe
the
permeability
to appropriately
the consolidation
process,
alsonecessary
necessary to
permeability
of of the
natural the
softnatural
soil. soft soil.
Figure 6. Longitudinal gravel trenches to model a stone column treatment in plane strain.
4.1. Settlement
6. Longitudinal
gravel
trenchesto
tomodel
model aastone
treatment
in plane
strain. strain.
Figure Figure
6. Longitudinal
gravel
trenches
stonecolumn
column
treatment
in plane
The classical proposal for this simplified model is by Van Impe and De Beer [40], who developed
an
analytical
4.1. Settlementsolution to study the settlement for this case. Their proposal [40] is to keep the (drained)
4.1. Settlement
properties of the soil and column and transform each row of columns in a longitudinal gravel trench
The
classical
of the
same
area. proposal for this simplified model is by Van Impe and De Beer [40], who developed
Theanclassical
proposal
thisthe
simplified
model
by
Van Impe
and
De
[40],
who
developed
analytical
solution
tofor
study
settlement
forcolumn
thisis
case.
Their
proposal
[40]
is Beer
to keep
thefactor
(drained)
The most
important
parameter
in a stone
treatment
is the area
replacement
( ),
properties
of theto
soil
and
and
transform
each
row ofTheir
columns
in acolumns:
longitudinal
an analytical
study
the
settlement
fororthis
case.
[40] is togravel
keep trench
the (drained)
whichsolution
represents
the
areacolumn
of soft
soil replaced
displaced
by
theproposal
stone
of the
area.and column and transform each row of columns in a longitudinal gravel trench
properties
of same
the soil
The most important parameter in a stone column
=
=treatment is the area replacement factor ( (4)),
of the same area.
which represents the area of soft soil replaced or displaced by the stone columns:
The most important parameter in a stone column treatment is the area replacement factor (ar ),
The proposal by [40] seems reasonable because it allows the area replacement factor ( ) and the
which represents
the area
of softtosoil
replaced
or=displaced
by thedisadvantages:
stone columns:
(4)
= two major
soil and gravel
parameters
be kept.
However,
it has
The proposal by [40] seems reasonable because
Ac it allows
rc2 the area replacement factor ( ) and the
a
=
=
r
soil and gravel parameters to be kept. However,A
it has two
r2 major disadvantages:
e
(4)
e
The proposal by [40] seems reasonable because it allows the area replacement factor (ar ) and the
soil and gravel parameters to be kept. However, it has two major disadvantages:
Materials 2017, 10, 782
Materials 2017, 10, 782
•
•
7 of 23
7 of 23
The resulting thickness of the gravel trenches is usually small, and consequently, the gravel

The resulting thickness of the gravel trenches is usually small, and consequently, the gravel
trenches are very slender.
trenches are very slender.
The confining
conditions
of the
columns
thesame
sameasas
lateral
confinement
of the gravel
The confining
conditions
of the
columnsare
arenot
not the
thethe
lateral
confinement
of the gravel
trenches
[30]
(Figure
7).
trenches [30] (Figure 7).
Materials 2017, 10, 782


7 of 23
The resulting thickness of the gravel trenches is usually small, and consequently, the gravel
trenches are very slender.
The confining conditions of the columns are not the same as the lateral confinement of the gravel
trenches [30] (Figure 7).
Figure 7. Different confinement and seepage conditions for columns in axial symmetry and for
Figure 7.
Different
confinement
and seepage
conditions for columns in axial symmetry and for
longitudinal
gravel
trenches in plane
strain.
longitudinal gravel trenches in plane strain.
Regarding the first bullet point, there is no restriction on increasing the thickness of the gravel
trenches, while proportionally increasing the spacing between the longitudinal trenches to keep
Regarding the first bullet point, there is no restriction on increasing the thickness of the gravel
constant. A reasonable value for the thickness of the gravel trenches may be the diameter of the
trenches,
while proportionally increasing the spacing between the longitudinal trenches to keep
columns.
7. Differentvalue
confinement
and thickness
seepage conditions
forgravel
columns trenches
in axial symmetry
and the
for diameter of
ar constant.The
AFigure
reasonable
for the
of the
may
second
bullet point causes
that the matching
of the
settlement between
the be
real situation
longitudinal gravel trenches in plane strain.
and the plane strain model is not accurate enough, particularly when plastic strains appear in the
the columns.
and
then,
differences
thematching
column
confinement
are notable.
Plasticthe
in the
Thecolumns
second
bullet
point
causes
thatinthe
of theonsettlement
between
real
situation
and
Regarding
thethe
first
bullet point,
there
is no restriction
increasing
the
thickness
ofstrains
the
gravel
columns
arewhile
common
and, if theincreasing
column only
deforms
elastically,
the designtrenches
is overconservative.
trenches,
proportionally
the
spacing
between
the
longitudinal
to
keep
the plane strain model is not accurate enough, particularly when plastic strains appear in the columns
Tanconstant.
et al. [30]
analytical
proposals
equivalent
properties
of the
A presented
reasonable two
value
for the thickness
of to
theobtain
gravelthe
trenches
may be
the diameter
of gravel
the
and then,
the differences
in the column
confinement
are
notable.
Plastic An
strains
in the
columns are
trenches
and
the
surrounding
soil,
but
neither
of
them
is
totally
satisfactory.
analytical
equation
columns.
common
and,
if the
column
only
elastically,
the
design
is overconservative.
Tan
may
be The
derived
for bullet
the equivalent
elastic
of the
trenches
[41], but the
thatreal
is only
validet
if al. [30]
second
point deforms
causes
thatmodulus
the matching
ofgravel
the
settlement
between
situation
presented
analytical
proposals
to obtain
the
equivalent
of the
gravel
trenches
and
the
plane
strain
model
is not
accurateFor
enough,
particularly
when plastic
strains
ininthe
the two
column
and the
soil
deform
elastically.
general
cases,properties
comparing
the
unit
cellappear
model
axialand the
columns
andthe
then,
in the
column
are
notable.
Plasticstarting
strains
symmetry
unitthecell
of the is
gravel
trenches
isconfinement
very useful
(Figure
8).equation
Thus,
from
an
surrounding
soil,and
but
neither
ofdifferences
them
totally
satisfactory.
An analytical
mayin
bethe
derived
for
columnsproposal
are common
and,
if the column
only
deforms
elastically,
the design
is overconservative.
analytical
for
the
parameters
of
the
gravel
trenches,
they
may
be
further
adjusted
or
tuned
the equivalent elastic modulus of the gravel trenches [41], but that is only valid if the column and the
Tan et the
al. [30]
presented
twounit
analytical
to obtainTo
thefitequivalent
properties
the gravel
to match
settlement
of the
cell inproposals
axial symmetry.
the settlement
in theofplastic
range,
soil deformtrenches
elastically.
For
general cases,
comparing
the unit
cellsatisfactory.
model in An
axial
symmetry
and the unit
and
the
surrounding
soil,
but
neither
of them
totally
analytical
equation
adjusting the
friction
angle of the
gravel
trenches
seemsisthe
most
appropriate alternative
[41]. Once
cell of the
gravel
trenches
is
very
useful
(Figure
8).
Thus,
starting
from
an
analytical
proposal
may
be
derived
for
the
equivalent
elastic
modulus
of
the
gravel
trenches
[41],
but
that
is
only
valid
the parameters have been calibrated using the unit cell case as an auxiliary problem, they mayif be for the
the
column
and the
soil deform
elastically.
generaladjusted
cases, comparing
theto
unit
cell model
in axial
parameters
of
the
they
mayHowever,
beFor
further
oristuned
match
the settlement
used
for
thegravel
full
2Dtrenches,
plane
strain
model.
the calibration
tailor-made
or specific
for each of the
symmetry and the unit cell of the gravel trenches is very useful (Figure 8). Thus, starting from an
Thus,symmetry.
it varies, forTo
example,
the value
theplastic
appliedrange,
load [42].
unit cellcase.
in axial
fit the with
settlement
inofthe
adjusting the friction angle of the
analytical proposal for the parameters of the gravel trenches, they may be further adjusted or tuned
gravel trenches
seems
the mostofappropriate
[41].ToOnce
parameters
have been
to match
the settlement
the unit cell inalternative
axial symmetry.
fit thethe
settlement
in the plastic
range,calibrated
using the unit
cell the
case
as anangle
auxiliary
problem,
they
may
used
for the alternative
full 2D plane
strain model.
adjusting
friction
of the gravel
trenches
seems
thebe
most
appropriate
[41]. Once
the parameters
have
been calibratedorusing
the unit
case
as anThus,
auxiliary
problem,for
they
may be with the
However, the
calibration
is tailor-made
specific
forcell
each
case.
it varies,
example,
used for the full 2D plane strain model. However, the calibration is tailor-made or specific for each
value of the applied load [42].
case. Thus, it varies, for example, with the value of the applied load [42].
Figure 8. Calibration of the parameters of longitudinal gravel trenches using the unit cell model as an
auxiliary problem.
4.2. Consolidation
TheFigure
coefficient
of horizontal permeability
(or the coefficient of consolidation
) of the
8. Calibration of the parameters of longitudinal gravel trenches using the unit cell model as an
Figure
8. Calibration
of
parameters
of longitudinal
trenches using
the under
unit cell
model
as an
natural
soil should
bethe
adjusted
to properly
reproduce gravel
the consolidation
process
plane
strain
auxiliary
problem.
conditions
because, similarly to the column confinement, the seepage problem is different in axial
auxiliary
problem.
4.2. Consolidation
symmetry
and plane strain conditions (Figure 7). Hird et al. [43] and Indraratna and Redana [44]
The coefficient of horizontal permeability
4.2. Consolidation
(or the coefficient of consolidation
) of the
natural soil should be adjusted to properly reproduce the consolidation process under plane strain
The coefficient
of horizontal
permeability
consolidation
cvhin) axial
of the natural
conditions because,
similarly
to the column kconfinement,
the seepageofproblem
is different
h (or the coefficient
and plane
strain conditions
(Figure
Hird et al. [43]process
and Indraratna
Redana
[44]conditions
shouldsymmetry
be adjusted
to properly
reproduce
the 7).
consolidation
under and
plane
strain
soil
because, similarly to the column confinement, the seepage problem is different in axial symmetry and
plane strain conditions (Figure 7). Hird et al. [43] and Indraratna and Redana [44] proposed analytical
Materials 2017, 10, 782
8 of 23
expressions to modify k h , based on the comparison between analytical solutions for vertical drains for
both cases (axial symmetry and plane strain).
When using numerical analyses, the analytical values of k h [43,44] may be tuned using the unit
cell as an auxiliary problem (Figure 8) to have a better matching of the results. As it is difficult to
match the whole consolidation curve, it may be convenient to match the values of the degrees of
consolidation that are relevant for the study, usually around 80–95%.
4.3. Stability
The plane strain model is useful for studying the global stability, or, for example, the stability
of the lateral slopes of an embankment (Table 1). In this case, the plane strain model should match
the average shear strength along the slip line of the real situation. Here, it is important to distinguish
between limit equilibrium analyses (e.g., method of slices) or full stress-strain analyses (e.g., finite
differences or finite element method).
Stress-strain analyses are able to capture the stress concentration on the columns. Thus, if the
stress concentration on the columns is properly reproduced (depending on the rigidity of soil and
gravel), the results are satisfactory just keeping constant the area replacement ratio ar and the resistance
properties (shear strength) of soil and gravel (e.g., [45]).
For limit equilibrium analyses, the stress concentration on the longitudinal gravel trenches
should be artificially generated because the vertical stress is directly the weight above the studied
point. Priebe [35] proposed altering the real profile of the embankment, so the modeled height of the
embankment is the real one times the stress concentration either on the columns or on the soft soil.
Another alternative is to modify the friction angle of the longitudinal gravel trenches to match the
average shear strength along the slip line [15].
A very difficult task is to evaluate the stress concentration on the columns because it depends
on many factors, such as the drainage conditions (undrained, partial drainage or fully drained), the
depth and inclination of the slip line, the specific position of the columns beneath the slope and the
applied load. Recent studies [45] show that in situations close to failure, there is no stress concentration
on the columns, i.e., the vertical stress on soil and columns is very similar. Thus, assuming no stress
concentration may be generally advisable and in any case, it is an assumption on the safe side.
Finally, rigid inclusions, such as deep mixing columns, may fail not only by shearing as stone
columns (shear failure along the slip line), but they may also fail by tilting or bending [46]. For widely
spaced columns, soil may also extrude between columns [47]. When columns break by bending,
it is necessary to consider their moment of inertia [48] or the measured bending failure for encased
stone columns.
5. Homogenization
The homogenization method consists in replacing the stone columns and the soft soil by an
equivalent homogeneous soil with improved properties. This equivalent homogeneous soil occupies
the zone treated with stone columns. This model simplifies enormously the geometry of the problem.
For example, when the problem itself has a highly complex geometry and the zone treated with
columns is just a part of the problem, this method is highly advisable (e.g., [49]). For the design stage,
this method allows the area replacement ratio (ar ) to be varied without changing the geometry of the
model, just the material parameters.
5.1. Equivalent Parameters
The most straightforward proposal for obtaining the improved parameters of the equivalent
homogeneous soil is just to average the soil and column parameters weighted by their corresponding
areas though ar . Thus, for the elastic modulus, the weighted average is:
Em = Es (1 − ar ) + Ec ar
(5)
Materials 2017, 10, 782
9 of 23
Materials 2017, 10, 782
9 of 23
However, this can only be regarded as a first approximation. A more detailed analysis shows
that the equivalent
homogenous
soil[51–53])
shoulduse
be anisotropic
naturemedia
to account
for the
column
orientation
[50]. Some
authors (e.g.,
theories for by
periodic
to propose
analytical
Materials 2017, 10, 782
9 of 23
orientation [50]. from
Some
authors
(e.g.,
for periodic
media
to propose
analytical
transformations
the
columns
and[51–53])
the soft use
soil theories
(heterogeneous
periodic
composite
material
at the
transformations
from
the
columns
and
the
soft
soil
(heterogeneous
periodic
composite
material
microscale)
to
a
homogeneous
equivalent
soil
(homogenous
material
at
a
macroscale).
They
a
orientation [50]. Some authors (e.g., [51–53]) use theories for periodic media to propose analyticaluse at
the
microscale)
to
a
homogeneous
equivalent
soil
(homogenous
material
at
a
macroscale).
They
use
a
macroscopic
strengthfrom
criterion
of the and
anisotropic
homogeneous
equivalent
soil to evaluate
the
transformations
the columns
the soft soil
(heterogeneous
periodic composite
material
at bearing
the
microscale)
to a homogeneous
equivalent
(homogenous
material
atand
a macroscale).
Theythe
use
a for
macroscopic
criterion
of the
anisotropic
homogeneous
equivalent
soil
to evaluate
bearing
capacity.
Thestrength
practical
application
of thesesoiltechniques
is complex
they
are only
valid
macroscopic
strength
criterion
the
anisotropic
homogeneous
equivalent
soil
evaluate
bearing
capacity.
Thesmall
practical
application
these
techniques
is complex
and theythe
areto
only
valid
for
sufficiently
sufficiently
values
of the ofof
scale
factor,
i.e.,
the
ratio between
micro
andthe
macroscales,
capacity.
The
practical
application
of these
techniques
is characteristic
complex
and they
are
only
validproblem
for ratio
small
values of
the
scale
factor,
i.e.,
ratio
between
the micro
and macroscales,
forthe
example,
the
for
example,
the
ratio
between
the the
column
spacing
and
the
size of
whole
sufficiently small values of the scale factor, i.e., the ratio between the micro and macroscales,
between
the column spacing and the characteristic size of the whole problem to be studied.
to
be studied.
for example, the ratio between the column spacing and the characteristic size of the whole problem
as for
gravelgravel
trenches,
a simplea method
using
numerical
In aasimilar
similarmanner
manner
as the
for longitudinal
the longitudinal
trenches,
simple when
method
when
using
to be studied.
analyses is
tuning or
the improved
parameters
of the equivalent
soil usingsoil
theusing
unit cell
numerical
analyses
is adjusting
tuning or adjusting
the improved
parameters
of the equivalent
the
In a similar manner as for the longitudinal gravel trenches, a simple method when using
as ancell
auxiliary
problem
(Figure
9).
Ng
and
Tan
[54]
have
calibrated
the
improved
parameters
unit
as
an
auxiliary
problem
(Figure
9).
Ng
and
Tan
[54]
have
calibrated
the
improved
parameters
numerical analyses is tuning or adjusting the improved parameters of the equivalent soil using the for
several
cases
and
tabulated
their
results
asas
a reference.
The
accuracy
of
the
matching
may
for several
cases
tabulated
their
results
aand
reference.
The
accuracy
ofthe
the
matching
may be high
unit
cell
as
anand
auxiliary
problem
(Figure
9).
Ng
Tan [54]
have
calibrated
improved
parameters
and, yet
the specific
of the their
excess
poreaspressure
between
columns
may
not bemay
modeled
using
for several
cases values
and tabulated
results
a reference.
The accuracy
of the
matching
be high
and, yet the specific
values of the
the
pore
pressure
columns
be modeled
using
homogenization
technique,
the excess
average
degree
of between
consolidation
ormay
thenot
settlement
rate
may be
the homogenization
technique,
average
degree
of
consolidation
or
the
settlement
rate
may
the homogenization
technique,
the average
degree
of consolidation
or the settlement
rate maysoil
be [54].
proper
calibration
of the
the
parameters
of the
theequivalent
equivalent
homogeneous
soil
[54].
correctly
capture after a proper
calibration
of
parameters
of
homogeneous
correctly
capture
after
a
proper
calibration
of
the
parameters
of
the
equivalent
homogeneous
soil
[54].
cases, it may be enough
enough to match just the values of the degree of consolidation
consolidation that are
For most cases,
For most cases, it may be enough to match just the values of the degree of consolidation that are
relevant
for the study, usually around 80–95%. Figure 10 shows a simple example of application for
relevant for the study, usually around 80–95%. Figure 10 shows a simple example of application for
the extension of an
an airport
airport runway,
runway, whose embankment had to be founded
founded on ground
ground improved
improved with
the extension of an airport runway, whose embankment had to be founded on ground improved with
floating
stone
columns.
floating stone columns.
Figure 9. Calibration of the parameters of the equivalent homogeneous soil using the unit cell model
Figure 9.
9. Calibration of
of the
the parameters
parameters of
of the
the equivalent
equivalent homogeneous
homogeneous soil
soil using
using the
the unit
unit cell
cell model
model
Figure
as anCalibration
auxiliary problem.
as
an
auxiliary
problem.
as an auxiliary problem.
Figure 10. Example of calibration of the parameters of the equivalent homogeneous soil using the unit
cell model as an auxiliary problem.
Figure 10. Example of calibration of the parameters of the equivalent homogeneous soil using the unit
method
reproduce
physical
phenomena
at the local
level, the
such
as
FigureThe
10. homogenization
Example of calibration
of cannot
the parameters
of the
equivalent
homogeneous
soil using
unit
cell model as an auxiliary problem.
stress
concentration
on
the
column
or
radial
drainage
towards
the
column,
but
the
overall
response
cell model as an auxiliary problem.
The homogenization method cannot reproduce physical phenomena at the local level, such as
stress concentration on the column or radial drainage towards the column, but the overall response
Materials 2017, 10, 782
10 of 23
The homogenization method cannot reproduce physical phenomena at the local level, such as
stress concentration on the column or radial drainage towards the column, but the overall response
of the system on a larger scale may be correctly reproduced after calibrating the parameters of the
equivalent homogeneous soil.
Materials 2017, 10, 782
10 of 23
5.2. Specific Constitutive Model
of the system on a larger scale may be correctly reproduced after calibrating the parameters of the
The idea of developing an advanced, specific constitutive model to represent the behavior of
equivalent homogeneous soil.
the periodic material formed by the columns and the corresponding surrounding soil was initially
formulated
by Schweiger
and
Pande [55]. The constitutive model assumed that both soil and column
5.2. Specific
Constitutive
Model
undergo theThe
same
strains,
there
slippage
between
soil and
column
and the
internally
uses
idea of developing is
anno
advanced,
specific
constitutive
model
to represent
themodel
behavior
of the
independent
and
existing
constitutive
models
for
the
soil
and
the
column
and
ensures
that
there
is
periodic material formed by the columns and the corresponding surrounding soil was initially
formulated
by Schweiger
and Pande
[55].
The constitutive
model assumed
that
both soil
and
column
equilibrium
of horizontal
stresses
at the
soil-column
interface
[55]. The
model
was
later
improved,
undergo
same strains,
there is no slippage
between
soil andthe
column
thebeen
modelapplied
internallytouses
particularly
its the
numerical
implementation
[56].
Recently,
ideaand
has
deep soil
and existing
constitutive
modelsmodels
for the soil
and
the column
is
mixing independent
columns using
advanced
constitutive
and
with
the aimand
of ensures
using itthat
forthere
complex
3D
equilibrium of horizontal stresses at the soil-column interface [55]. The model was later improved,
geometries [57]. However, the numerical stability of its formulation and its range of application are
particularly its numerical implementation [56]. Recently, the idea has been applied to deep soil
still limited.
mixing columns using advanced constitutive models and with the aim of using it for complex 3D
geometries [57]. However, the numerical stability of its formulation and its range of application are
6. Gravel
stillRings
limited.
The idea of transforming the stone columns into gravel rings, tubes or cylindrical trenches
6. Gravel Rings
follows the same concept as for the longitudinal gravel trenches (Section 4) but for problems with
The idea of transforming the stone columns into gravel rings, tubes or cylindrical trenches
axial symmetry rather than plane strain conditions. Thus, the model is used for problems with
follows the same concept as for the longitudinal gravel trenches (Section 4) but for problems with
axial symmetry,
suchrather
as circular
embankments
or Thus,
circular
storage
tanks.
The model
the
axial symmetry
than plane
strain conditions.
the model
is used
for problems
with implies
axial
transformation
of
each
group
of
columns
into
an
equivalent
gravel
ring
with
the
same
area
to
keep
symmetry, such as circular embankments or circular storage tanks. The model implies the
ar constant
(Figure 11).
In the
caseofofcolumns
Figureinto
11, an
theequivalent
8 neighboring
columns
of same
the central
column are
transformation
of each
group
gravel ring
with the
area to keep
constant
(Figure
11). In the
case
of Figure
11, the 8(tneighboring
columns
of thearea.
central
column arethat tr is
transformed
into an
equivalent
ring
with
a thickness
the same
Assuming
r ) that gives
transformed
into an
small enough,
its value
is:equivalent ring with a thickness ( ) that gives the same area. Assuming that
is small enough, its value is:
tr = d2c /rr
(6)
=
/
(6)
The distance of the gravel ring to the central point (rr ) is the weighted average of the distance of
The distance of the gravel ring to the central point ( ) is the weighted average of the distance of
the 8 columns
to the center.
√ the 8 columns to the center.
rr = 1 + 2 s/2
(7)
= 1 + √2 s/2
(7)
For a group
of onlyof4only
columns,
the weighted
average
proposalofof other
For a group
4 columns,
the weighted
averageisisdirectly
directly rr =
= ss [58].
[58]. The
The proposal
authors other
[59] is
slightly
different
and
implies
that
the
gravel
ring
surrounds
the
same
area
as that
authors [59] is slightly different and implies that the gravel ring surrounds the same area
as of the
that of the
by the 8 columns:
square formed
bysquare
the 8formed
columns:
√
rr ==2s/
(8)
(8)
2s/√ π
For practical
purposes,
differencesare
arenegligible.
negligible.
For practical
purposes,
the the
differences
Figure 11. Transformation of stone columns in equivalent gravel rings.
Figure 11. Transformation of stone columns in equivalent gravel rings.
Materials 2017, 10, 782
11 of 23
Mitchel and Huber [58] seem to be the first authors to use this model. They used it in combination
with theMaterials
finite2017,
element
10, 782 method to study a field case (the foundation of a water treatment
11 of 23 plant).
10, 782
of 23 gravel
ContraryMaterials
to the2017,
model
of the longitudinal gravel trenches, here the confining conditions of11the
and Huber
[58] seem
to be the
authorscolumns
to use thisand
model.
usedto
it in
combination
rings seem toMitchel
be somehow
similar
to those
of first
the stone
it isThey
enough
maintain
the values
Mitchel
Hubermethod
[58] seem
be theafirst
to use
this model.
usedtreatment
it in combination
with the
finiteand
element
totostudy
fieldauthors
case (the
foundation
of They
a water
plant).
of ar and of the drained properties of the soil and the columns to obtain satisfactory results [60,61].
with the to
finite
element
method
to studygravel
a field
case (the
foundation
of a conditions
water treatment
plant).
Contrary
the model
of the
longitudinal
trenches,
here
the confining
of the gravel
As for the
longitudinal
gravel
isofnot
yet
any
satisfactory
to convert
thegravel
cylindrical
Contrary
of trenches,
thesimilar
longitudinal
gravel
trenches,
here
theand
confining
conditions
of the
rings
seemtotothe
bemodel
somehow
tothere
those
the
stone
columns
it way
is enough
to maintain
the
encasement
(geosynthetic)
that
surrounds
the
columns
for
this
case.
rings seem
to
be of
somehow
similar
to those
stone
and to
it obtain
is enough
to maintain
the
values
of
and
the drained
properties
ofof
thethe
soil
and columns
the columns
satisfactory
results
values As
of for and
of the drained
properties
of the
soilisand
to obtainway
satisfactory
results
[60,61].
the longitudinal
gravel
trenches,
there
notthe
yetcolumns
any satisfactory
to convert
the
7. Three-Dimensional
of Columns
[60,61]. Asencasement
for theSlice
longitudinal
gravelthat
trenches,
therethe
is columns
not yet any
cylindrical
(geosynthetic)
surrounds
for satisfactory
this case. way to convert the
cylindrical encasement (geosynthetic) that surrounds the columns for this case.
There
are problems that
fulfill
plane strain conditions but for the stone column treatment, such
7. Three-Dimensional
Slice
of Columns
as an embankment
for
a
linear
infrastructure.
In these cases, analyzing a slice or a row of columns
7. Three-Dimensional Slice of Columns
There are problems that fulfill plane strain conditions but for the stone column treatment, such
is very useful
when
3D that
numerical
analyses
(Figure
12).
Nowadays,
the highofcomputer
power
areusing
problems
fulfill
plane
strain
conditions
for the astone
such
as an There
embankment
for a linear
infrastructure.
In these
cases,but
analyzing
slicecolumn
or a rowtreatment,
columns
is
and the very
availability
of
3D
numerical
codes
makes
this
simplified
geometrical
model
very
appealing
as anuseful
embankment
for 3D
a linear
infrastructure.
In these12).
cases,
analyzingthe
a slice
a row ofpower
columns
when using
numerical
analyses (Figure
Nowadays,
highor
computer
andis
useful when
3D numerical
analyses
(Figure
12).
Nowadays,
theorhigh
computer
power
and
because the
itvery
isavailability
relatively
simple
and
it does
not
require
any
transformation
determination
of equivalent
of using
3D numerical
codes
makes
this
simplified
geometrical
model
very appealing
the
availability
of
3D
numerical
codes
makes
this
simplified
geometrical
model
very
appealing
because
it
is
relatively
simple
and
it
does
not
require
any
transformation
or
determination
of a stone
parameters [48,62]. Furthermore, it allows any of the features or improvements achieved with
because
it
is
relatively
simple
and
it
does
not
require
any
transformation
or
determination
equivalent
parameters
[48,62].
Furthermore,
it
allows
any
of
the
features
or
improvements
achieved
column treatment to be studied, such as consolidation, deformations and stability (Table 1), of
and it is
equivalent
[48,62].toFurthermore,
it allows
any of the deformations
features or improvements
achieved
with
a stone parameters
column treatment
be studied, such
as consolidation,
and stability (Table
1),
valid for both encased and non-encased columns.
withitaisstone
treatment
benon-encased
studied, suchcolumns.
as consolidation, deformations and stability (Table 1),
and
validcolumn
for both
encased to
and
For and
theFor
ofcase
aforsquare
gridgrid
or
mesh
enough
to model
justofhalf
of of
a row of
itcase
is the
valid
both
encased
andor
non-encased
columns.
of
a square
meshofofcolumns,
columns,
ititisisenough
to model
just half
a row
columnscolumns
dueFor
todue
symmetry
conditions
(Figure
13).
On
the
contrary,
for
a
triangular
grid
of
columns,
the to
case
of a square
grid or (Figure
mesh of13).
columns,
is enough
model justgrid
halfofofcolumns,
a row of
symmetry
conditions
On the itcontrary,
for to
a triangular
to study
symmetry
conditions
(Figure
13).
the contrary,
for
a triangular
grid
of columns,
itcolumns
is necessary
at least
two
different
rows
ofOn
columns
(a(ahalf
of of
each
is enough)
(Figure
13). 13).
it is necessary
to due
study
at least
two
different
rows
of
columns
half
each
is enough)
(Figure
it is necessary to study at least two different rows of columns (a half of each is enough) (Figure 13).
Figure
12. Finite
elementmodel
model of
of of
columns.
Figure
12. Finite
element
ofaa3D
3Dslice
slice
columns.
Figure 12. Finite element model of a 3D slice of columns.
Figure 13. Numerical model of a slice of columns for triangular and square grids.
Figure 13. Numerical model of a slice of columns for triangular and square grids.
Figure
13. Numerical model of a slice of columns for triangular and square grids.
8. Isolated
Column
8. Isolated Column
For the sake of simplicity, some experimental tests, either in the field or in the laboratory, are
8. Isolated
Column
For the
of simplicity,
some
experimental
tests,
eitherThe
in the
field
or in the
laboratory,
are
performed
onsake
an isolated
or single
stone
column (e.g.,
[63,64]).
load
is usually
applied
only on
performed
on an isolated
single
column
(e.g.,
[63,64]).
load is usually
only
on
the
column surface
(Figure or
14a)
or onstone
an area
slightly
higher
than The
the column.
In theseapplied
cases, the
area
For the sake of simplicity, some experimental tests, either in the field or in the laboratory, are
the column surface
or on an
higher the
thanarea
the of
column.
In these
cases,
area
replacement
ratio ( (Figure
) may 14a)
be defined
asarea
the slightly
ratio between
the column
and
the the
loaded
performed
on an isolated
or single
stone column
(e.g.,
[63,64]).
The load
is usually
applied
only on
replacement
ratio footprint
( ) may area
be defined
as the
between
the areaisofusually
the column
and
the which
loaded
area
or the footing
[61]. Thus,
forratio
isolated
columns,
around
100%,
the column
surface
(Figure
14a)
or
on
an
area
slightly
higher
than
the
column.
In
these
cases,
the
area
the footing
footprint
area [61].
isolated
columns,the loaded
is usually
isarea
notor
a realistic
situation
because
it is Thus,
more for
efficient
to increase
areaaround
( << 100%,
100%)which
[61].
replacement
ratio
(a
)
may
be
defined
as
the
ratio
between
the
area
of
the
column
and
the
loaded
is not a realistic
<< 100%) [61]. area
r situation because it is more efficient to increase the loaded area (
or the footing footprint area [61]. Thus, for isolated columns, ar is usually around 100%, which is not a
realistic situation because it is more efficient to increase the loaded area (ar << 100%) [61]. Loading
Materials 2017, 10, 782
12 of 23
the surroundings of the columns is beneficial because it increases the horizontal stresses at the lateral
Materials 2017, 10, 782
12 of 23
boundaries of the columns and increases their confinement.
TheLoading
resultsthe
ofsurroundings
research performed
on isolated
columns
sometimes
directly stresses
extrapolated
to
of the columns
is beneficial
becauseare
it increases
the horizontal
at
the lateral
boundaries
ofconfusion
the columnsand
and non-accurate
increases their confinement.
other cases,
leading
to some
predictions. Some authors (e.g., [15] (p. 28))
The
results bulges
of research
on isolated
columns
sometimes
directly
extrapolated
show that the
column
(or performed
notably expands
radially)
atare
their
upper part,
specifically
at to
an upper
other cases, leading to some confusion and non-accurate predictions. Some authors (e.g., [15] (p. 28))
zone with a length of two or three times the column diameter (2–3 dc ). Hughes and Withers [63]
show that the column bulges (or notably expands radially) at their upper part, specifically at an upper
measured
a length
of 4 column
diameters
forcolumn
the upper
bulging
for an
column
(ar = 100%)
zone
with a length
of two or three
times the
diameter
(2–3 zone
). Hughes
andisolated
Withers [63]
measured
throughalaboratory
tests. That
is valid
forupper
casesbulging
with load
the single
column,
i.e.,through
dc = B. In the
length of 4 column
diameters
for the
zone only
for anon
isolated
column
( = 100%)
laboratory
tests.
is valid
forfor
cases
withcases,
load only
the single
column, i.e.,
= .the
In the
next zone
next section,
it will
beThat
shown
that
other
it isonmore
meaningful
to define
bulging
section,
it will
be shown
that for other
cases,
it is more
meaningful
to define the
bulgingcases,
zone using
using the
footing
width
(B) instead
of the
column
diameter
[61] because
in those
the column
footing width ( ) instead of the column diameter [61] because in those cases, the column bulging
bulgingthe
may
be deeper [3] (p. 114) [15] (p. 28) (Figure 14). Besides, the failure mechanism of the
may be deeper [3] (p. 114) [15] (p. 28) (Figure 14). Besides, the failure mechanism of the columns may
columns may not be bulging, for example, it may be shearing [65].
not be bulging, for example, it may be shearing [65].
(a)
(b)
(c)
(d)
Figure 14. Column deformation for different configurations: (a) Isolated column with
= 100%;
Figure 14.
Column
deformation
different
configurations:
(a) Isolated
ar = 100%;
< for
100%;
(c) Groups
of columns beneath
footing; (d)column
Column with
treatment
(b) Isolated
column
with
(b) Isolated
column
with
a
<
100%;
(c)
Groups
of
columns
beneath
footing;
(d)
Column
treatment
r
beneath an embankment.
beneath an embankment.
Isolated columns are generally used to study their bearing capacity. The maximum vertical stress
that a stone column may bear is usually given as:
Isolated columns are generally used to study their bearing capacity. The maximum vertical stress
(9)
20
that a stone column may bear is usually given as:
where
is the undrained shear strength of the surrounding soft soil. Equation (9) is derived assuming
max
that the column is in an active state with an active
earth
pressure
coefficient of around
= 1/3 and that
σvc
≈ 20c
(9)
u
the radial cavity expansion factor is around 6–8 (e.g., [63,66]). Equation (9) is strictly valid for isolated
with
= 100%,
but
when used
for larger
loaded areas,
is conservative
it neglects
where cucolumns
is the undrained
shear
strength
of the
surrounding
softitsoil.
Equation because
(9) is derived
assuming
the
increase
in
the
radial
stress
due
to
vertical
loading
of
the
soil
surrounding
the
column.
Besides,
that the column is in an active state with an active earth pressure coefficient of around k ac = 1/3 and
Equation (9) assumes perfect undrained conditions for the soil surrounding the column and some
that the radial cavity expansion factor is around 6–8 (e.g., [63,66]). Equation (9) is strictly valid for
drainage could be expected near the granular column. For common cases, Equation (9) gives a vertical
isolatedload
columns
withforareach
= 100%,
when
usedtons
for [15]
larger
loaded
areas,field
it istests
conservative
because it
supported
stone but
column
of 20–50
(p. 6).
Sometimes,
are performed
neglectstothe
increase
invertical
the radial
duebytoone
vertical
of the soil
surrounding
theofcolumn.
check
the exact
load stress
supported
isolatedloading
column with
= 100%.
The approach
the assumes
bearing capacity
a footing asconditions
the sum of for
the the
contribution
of each column
may
Besides,considering
Equation (9)
perfectofundrained
soil surrounding
the column
and
usually
be
overconservative.
For
footings,
an
improvement
for
field
tests
(plate
load
tests)
on
an
some drainage could be expected near the granular column. For common cases, Equation (9) gives
isolated column may be achieved by using a larger loaded area, so
same as in the
a vertical
load supported for each stone column of 20–50 tons [15] (p. is6).theSometimes,
field tests
footing [3] (p. 165).
are performed to check the exact vertical load supported by one isolated column with ar = 100%.
The approach
of of
considering
9. Groups
Columns the bearing capacity of a footing as the sum of the contribution of each
column mayStone
usually
betreatments
overconservative.
Forused
footings,
improvement
for as
field
tests (plate load
column
are traditionally
beneathan
large
loaded areas, such
embankments,
tests) onbut
anthey
isolated
column
may
be
achieved
by
using
a
larger
loaded
area,
so
a
is
the
same
are also used beneath footings, when the applied loads are not high [67]. Forr these cases,
theas in the
footing homogenization
[3] (p. 165).
technique is valid [68]. Besides, the plane strain model using gravel trenches
9. Groups of Columns
Stone column treatments are traditionally used beneath large loaded areas, such as embankments,
but they are also used beneath footings, when the applied loads are not high [67]. For these cases,
Materials 2017, 10, 782
13 of 23
the homogenization technique is valid [68]. Besides, the plane strain model using gravel trenches
(Figure 1c)Materials
is also2017,
valid
for strip footings. On the other hand, when the footing could be13converted
into
10, 782
of 23
a circular footing (e.g., a square footing), the simplified geometrical model in axial symmetry that uses
(Figure 1c) is also valid for strip footings. On the other hand, when the footing could be converted
gravel rings
is also applicable [60,61] (Figure 15). For the latter case (i.e., axial symmetry conditions),
into a circular footing (e.g., a square footing), the simplified geometrical model in axial symmetry
the authorthat
hasuses
recently
proposed
an additional
model
thatcase
assumes
that
all the columns
gravel rings
is also applicable
[60,61] simplified
(Figure 15). For
the latter
(i.e., axial
symmetry
may be converted
into
central
the an
same
area (Figure
15d),
keeping
ar constant
conditions),
theaauthor
hascolumn
recently with
proposed
additional
simplified
model
that assumes
that all [61]. The
the columns
maymodel
be converted
intoitamay
central
the same area
(Figure solutions
15d), keeping
main advantages
of this
are that
becolumn
used towith
developed
analytical
[38] and that it
constant [61]. The main advantages of this model are that it may be used to developed analytical
is also applicable to encased columns. For encased columns, it is also necessary to transform the elastic
solutions [38] and that it is also applicable to encased columns. For encased columns, it is also
propertiesnecessary
of the encasement,
the
factor Jgof/rthe
kept constant
[69].
this central
c isencasement,
⁄ is of
to transform so
the that
elastic
properties
so that
the The
factorvalidity
kept
column model
for
small
groups
of
columns
is
based
on
the
small
influence
on
the
footing
constant [69]. The validity of this central column model for small groups of columns is based on thesettlement
small
influence
footing
settlement
(usually less
than
10%)position
of the number
of columns
and their [61]. This
(usually less
than
10%)onofthethe
number
of columns
and
their
beneath
the footing
position beneath the footing [61]. This simplified central column is only valid to study the footing
simplified central column is only valid to study the footing settlement and should not be used for
settlement and should not be used for other features, such as the consolidation process or the bending
other features,
such
consolidation process or the bending moments in the footing.
moments
in as
the the
footing.
(a)
(b)
(c)
(d)
Figure 15. Simplified geometrical model for groups of stone columns beneath a square footing:
Figure 15.(a)Simplified
for groups
of stone
columns beneath a square footing:
3D model; (b)geometrical
Gravel rings; (c)model
Homogenization;
(d) One
central column.
(a) 3D model; (b) Gravel rings; (c) Homogenization; (d) One central column.
10. Critical Column Length
The concept
of critical length is used for stone columns as in piles (e.g., [65]). Thus, for columns
10. Critical Column
Length
longer than the critical length, the settlement reduction or the bearing capacity of the footing does
The concept
of change
criticalorlength
used
for stone
columns
incritical
pileslength
(e.g., of
[65]).
Thus, for
for columns
not notably
improve.isFor
non-encased
stone
columns,asthe
the columns
settlement
reduction
is around
1.5 − 2 [61,70].
The critical
length
for encased
columns
usually
longer than
the critical
length,
the settlement
reduction
or the
bearing
capacity
ofisthe
footing does
higher,
around
2 − 3.5 for
situations
[69]. The
main controlling
parameter
of the
critical
not notably
change
or improve.
Forcommon
non-encased
stone
columns,
the critical
length
of the
columns for
column length is the extension of the load, i.e., , but the specific value of the critical column length
settlementalso
reduction
is around 1.5 − 2 B [61,70].
The critical length for encased columns is usually
depends on other parameters, such as
and the soil and column properties [61,65,69,70].
higher, aroundIt 2is−
3.5
B
for
common
situations
[69].
Thegive
main
controlling
worth noting that many authors (e.g., [71–74])
critical
lengths as parameter
a function ofof
thethe critical
column
diameter,
but
the
column
length
to
diameter
ratio
has
a
minor
influence
by
itself
[61,69].
The
column length is the extension of the load, i.e., B, but the specific value of the critical column length
origin
the misinterpretation
factthe
thatsoil
the first
were for[61,65,69,70].
the case of
also depends
onofother
parameters, could
such lie
asinarthe
and
andproposals
column[63,64]
properties
an isolated column with load only on the column surface, i.e.,
= .
It is worthFor
noting
that many authors (e.g., [71–74]) give critical lengths as a function of the column
settlement reduction, the critical column length is related to the pressure bulb that the
diameter, but
thegenerates
column(Figure
length16a),
to diameter
ratio bearing
has a minor
influence
itself
[61,69].
footing
while for footing
capacity,
the criticalby
length
depends
on The
the origin of
failure mechanism
(Figure
thethat
critical
settlement[63,64]
reduction
is longer,
thatcase
is usually
the misinterpretation
could
lie in16b).
theAs
fact
thelength
firstfor
proposals
were
for the
of an isolated
the considered
Forcolumn
large loaded
areas (high
of ), the critical column length is higher
column with
load onlyvalue.
on the
surface,
i.e., dvalues
c = B.
than the soft layer thickness, and consequently, the concept of critical or optimal length does not apply.
For settlement
reduction, the critical column length is related to the pressure bulb that the footing
The mechanisms in Figure 16 also explain the influence of additional columns outside the
generates footing.
(FigureColumns
16a), while
forthefooting
capacity,
theline
critical
length
depends
outside
footing bearing
are crossed
by the slip
or failure
mechanism;
so, on
theythe failure
contribute
to
the
bearing
capacity.
On
the
contrary,
they
hardly
influence
the
settlement
reduction
if
mechanism (Figure 16b). As the critical length for settlement reduction is longer, that is usually
the
considered value. For large loaded areas (high values of B), the critical column length is higher than
the soft layer thickness, and consequently, the concept of critical or optimal length does not apply.
The mechanisms in Figure 16 also explain the influence of additional columns outside the footing.
Columns outside the footing are crossed by the slip line or failure mechanism; so, they contribute to
the bearing capacity. On the contrary, they hardly influence the settlement reduction if they are outside
the pressure bulb. The only minor beneficial effect for the settlement reduction is that they distribute
Materials 2017,
10, 782
Materials 2017, 10, 782
14 of 23
14 of 23
they are outside the pressure bulb. The only minor beneficial effect for the settlement reduction is
that
theyover
distribute
the applied
load
over Therefore,
a slightly wider
area.the
Therefore,
when
the bearing
capacityin terms of
the applied
load
a slightly
wider
area.
when
bearing
capacity
is defined
is defined in the
terms
of a critical
settlement,
theoutside
effect of the
extrafooting
columnsisoutside
the footing[65].
is only
a critical settlement,
effect
of extra
columns
only marginal
marginal [65].
The presented
analysis on critical column length is based on a homogeneous soft soil layer;
The presented analysis on critical column length is based on a homogeneous soft soil layer; for
for layered
soils,
this
analysis
directly
applicable.
layered soils, this
analysisisisnot
not directly
applicable.
16. Conceptual justification of critical column length in a homogenous soil layer: (a) for
Figure 16.Figure
Conceptual
justification of critical column length in a homogenous soil layer: (a) for
settlement reduction; (b) for bearing capacity.
settlement reduction; (b) for bearing capacity.
11. Column Installation Effects
11. Column Installation
Effects
Column installation
alters the surrounding soil, especially when the columns are installed by
vibrodisplacement. However, it is commonly accepted that in a stone column treatment, installation
Column
installation alters the surrounding soil, especially when the columns are installed by
effects are less significance and the main improvement is caused by the inclusion of gravel. The search
vibrodisplacement.
However,
commonly
accepted
that
in installation
a stone column
treatment,
for more accurate
designs it
hasisled
to more detailed
analyses
of the
effects [75–78].
Notableinstallation
pore pressuresand
are consistently
in the field
due to column
[79].
effects areexcess
less significance
the main measured
improvement
is caused
by theinstallation
inclusion
ofHowever,
gravel. The search
most cases,
these pore
quicklydetailed
dissipate analyses
[80]. Column
especially
by
for more inaccurate
designs
has pressures
led to more
of installation,
the installation
effects
[75–78].
vibrodisplacement, also leads to an increase in the horizontal stresses [75]. This is usually quantified
Notable excess
pore
pressures
are
consistently
measured
in
the
field
due
to
column
installation
[79].
through the earth pressure coefficient,
[3,75,76]. The increase in effective mean stress leads, in turn,
However,toinanmost
cases,
these
pore
pressures
dissipate [80]. Column installation, especially by
increase
in the
soft soil
stiffness
[3] (p. quickly
138).
The excess
pressures
and the in
remolding
caused by
column [75].
installation
to an quantified
vibrodisplacement,
alsopore
leads
to an increase
the horizontal
stresses
This islead
usually
instantaneous
reduction
of the undrained
shear strength
just afterin
installation
the pore
through the
earth pressure
coefficient,
K [3,75,76].
The increase
effective[77,81].
meanAs
stress
leads, in turn,
pressures dissipates and the effective mean stress increases, the value of the undrained shear strength
to an increase
in the soft soil stiffness [3] (p. 138).
recovers and usually increases above its initial value [77,81]. Special care must be taken with soil
The remolding
excess pore
pressures
and Poker
the vibration
remolding
caused
column
in sensitive
soils [67,77].
is necessary
forby
proper
column installation
compaction, butlead to an
excessive
use of the of
poker
many repenetrations
(“overworking”)
may
result in high[77,81].
excess pore
instantaneous
reduction
thewith
undrained
shear strength
just after
installation
As the pore
pressures
and
disturbance
of
sensitive
or
overconsolidated
soil
layers,
such
as
a
dry,
stiff
upper
crust
pressures dissipates and the effective mean stress increases, the value of the undrained shear strength
that is common is soft soils and is beneficial for load distribution [79].
recovers and The
usually
increases above its initial value [77,81]. Special care must be taken with soil
remolding caused by column installation also alters the permeability of the soft soil in the
remoldingvicinity
in sensitive
soils [67,77].
isisnecessary
propersoft
column
compaction, but
of the column.
This zone,Poker
whosevibration
permeability
lower than for
the natural
soil, is usually
“smear”
[82,83].
Therepenetrations
properties of the smear
zone, specially its
sizeresult
and permeability,
excessive called
use ofthethe
pokerzone
with
many
(“overworking”)
may
in high excess pore
largely studied
for verticalor
drains
[84–86]. Another related
problemsuch
is column
[83].
pressures have
andbeen
disturbance
of sensitive
overconsolidated
soil layers,
as aclogging
dry, stiff
upper crust
Gravel compaction during column installation and the high hydraulic gradients at the soil-column
that is common
is soft soils and is beneficial for load distribution [79].
interface inevitably produce a migration of clay particles into the pores of the granular column,
The remolding
caused
by column
installation
alsoofalters
the permeability
soft soil in the
notably reducing
the permeability
of an
external annulus
the column.
This phenomenonofis the
normally
important
in geosynthetic
because the
acts as asoft
filter.soil, is usually
vicinity ofless
the
column.
This zone,encased
whosecolumns
permeability
is geosynthetic
lower thanusually
the natural
The introduction
of installation
effects in the
modeling
stone columns
is a complex
called the “smear”
zone [82,83].
The properties
ofnumerical
the smear
zone, of
specially
its size
and permeability,
process. Some attempts have been made using field measurements and back-fitting the results [75]
have been largely studied for vertical drains [84–86]. Another related problem is column clogging [83].
Gravel compaction during column installation and the high hydraulic gradients at the soil-column
interface inevitably produce a migration of clay particles into the pores of the granular column, notably
reducing the permeability of an external annulus of the column. This phenomenon is normally less
important in geosynthetic encased columns because the geosynthetic usually acts as a filter.
The introduction of installation effects in the numerical modeling of stone columns is a complex
process. Some attempts have been made using field measurements and back-fitting the results [75]
or using previous numerical analyses that simulate column installation as a cylindrical cavity
expansion [78]. From a practical point of view, column installation may be considered altering
the at-rest earth pressure coefficient of the natural soft soil. Priebe’s theoretical solution [16] considers,
Materials 2017, 10, 782
15 of 23
for example, a value of 1. Published values [59] vary between 0.4 and 2.5, with average values slightly
above 1. Obviously, the earth pressure coefficient after column installation should depend on the
initial value, the construction method and the type of soil. Finally, if the numerical model does
not consider a stress-dependent constitutive model for the soft soil, its stiffness should be increased
correspondingly [76].
12. Properties of the Columns
The modeling process of the soft soil is beyond the scope of this review and it is broadly
analyzed elsewhere (e.g., [87,88]). Some authors [79,89] warn about the importance of secondary
compression in some soils and the limited capacity of stone columns to reduce it. Gravel properties
are also well-analyzed (e.g., [90–92]), but the properties of the gravel for stone columns are not so
readily available nor are they generally measured for each project. For stone columns, the gravel
should be clean, preferably crushed stone, hard, unweathered, free from organics or other deleterious
materials and its degradation using the Los Angeles testing machine should be less than a 45%
of loss [15] (p. 117). Its grain size should be between 12 and 75 mm. For bottom feed columns,
the maximum size is usually limited to 50 mm to avoid obstructions in the feeding tube.
The relative density of the gravel in the stone columns is not usually measured and it may vary
along the length of the column, in a similar manner as the column diameter. A proper stone column
construction should achieve relative densities of the gravel above 75% [15]. Herle et al. [93] measured
values close to 100%.
The friction angle of the columns (φc ) has a notable influence on the results of a stone column
treatment. Its value decreases with the confining pressure (σc ). Using samples in a dense state,
Herle et al. [93] measure values up to 60◦ for low confining pressures (σc = 50 kPa) and values around
50◦ for medium confining pressures (σc = 200 kPa) (Table 2). However, when choosing a value of the
friction angle to model stone columns, it is necessary to consider the following points:
•
•
•
•
Columns are usually under triaxial conditions and for granular soils, the friction angle measured
by direct shear tests is slightly higher than that measured using triaxial tests [94].
Values in Table 2 are peak values, but plastic strains in the columns may be important; so, it is
recommended to reduce peak friction values by 5–7% to obtain approximate residual values [3].
The validity of some theoretical methods and their success in predicting field measurements may
be linked to the use of conservative values of the friction angle.
Published numerical analyses consider values of 40–45◦ (Table 3).
Table 2. Stress-dependent peak friction angles of dense gravel [93].
Type of Gravel
φc,max
(◦ )
σc,max
(kPa)
φc,min
(◦ )
σc,min
(kPa)
Remarks
Crushed lime stone
River gravel
River gravel, sub-round
Rivel gravel, sub-round
Rivel gravel, crushed
Dolomite
Dolomite
Sandstone
Basalt
Basalt
Basalt
63.1
58.8
57.1
59.2
60.4
64.0
54.0
60.1
64.2
71.8
70.0
50
50
50
50
50
15
15
27
27
8
8
53.8
51.9
50.9
53.2
55.2
43
40
45.6
45.6
45.6
51.1
200
200
200
200
200
500
500
695
695
240
120
DS, Vibro SC
DS, Vibro SC
DS, Vibro SC, d60 /d10 = 2.6
DS, Vibro SC, d60 /d10 = 2.1
DS, Vibro SC
TX, γ = 17 kN/m3 , [90]
TX, γ = 15 kN/m3 , [90]
TX, [92]
TX, [92]
TX, d50 = 30 mm, [91]
TX, d50 = 30 mm, [91]
DS: Direct shear test; TX: Triaxial test; Vibro SC: dense gravel for vibrated stone columns [93]; d60 /d10 :
uniformity coefficient.
Materials 2017, 10, 782
16 of 23
Table 3. Parameters used to model stone columns in numerical analyses.
Reference
φc (◦ )
ψc (◦ )
Ec (MPa)
ν (-)
m (-)
γd /γsat (kN/m3 )
[59]
[68]
[70]
[74]
[73]
[30]
[54]
[49]
[78]
[95]
[96]
41
35
40–35
45
48
40
40
35
42
46
45
3
0
26
0
5
12
10
15
29.2
67.5
50
100
2.5
30
30
25
35
22
70
0.2
0.3
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.15
0.2
0.59
0.3
0.25
0.3
18.6/21.6
19/19
16/15/15
-/15
20/23.5
16/19
19/19
The reference pressure for the stiffness is 100 kPa; m: Exponent of the power law used to reproduce the stress
dependent stiffness.
The Young’s modulus of the columns is usually between 25 and 100 MPa (Table 3) and it also
varies with the confining pressure. A hyperbolic power law is sometimes used to reproduce the stress
dependent stiffness of the gravel of the columns (e.g., [58,95,96]). The common value of the exponent of
the power law is around 0.3 (Table 3). The unit weight of the gravel does not require further comment,
except for the cases where the columns do not reach a rigid substratum and they are installed by
vibrodisplacement (e.g., [49]). In those cases, the addition of gravel and its corresponding weight
may cause additional deformation in the underlying layer that is not improved with stone columns.
For example, [49] used non-realistic high values, namely γsat = 23.5 kN/m3 to account for this effect.
Regarding encased stone columns, the material used to construct them is usually sand instead of
gravel. The available information is more limited, but it seems obvious to use lower values of both the
friction and dilatancy angles (Table 4).
Table 4. Parameters used to model encased stone columns in numerical analyses.
Reference
φc (◦ )
ψc (◦ )
Ec (MPa)
ν (-)
m (-)
γd /γsat (kN/m3 )
[73]
[97]
[98]
[28]
[5]
[31]
[48]
[69]
48
40
38
40
32.5
35
38
40
4
10
5
0
0
10
5
9
40
15.5
80
30
40
30
0.3
0.3
0.3
0.2
0.3
0.33
-
16/-/23
-/18
-/20
19/20
-/22
-/20
m: Exponent of the power law used to reproduce the stress dependent stiffness.
13. Modeling the Geosynthetic Encasement
As shown in Figure 4, the geosynthetic encasement may be modeled as a flexible membrane
that does not support compressive stresses, has a negligible thickness and behaves as a linear elastic
material with a modulus of Jg = 1000–5000 kN/m. Regarding its tensile strength, for modeling
purposes, it may usually be enough to verify that it is far from being reached. The tensile strength
is usually reached for circumferential strains of around 5–10%, which implies strength values of
100–300 kN/m [5]. It is common that geosynthetic may be anisotropic and then different properties
should be input for each direction.
Some other geosynthetic features, such as creep and damage during installation, are usually
considered through reduction factors. Recent numerical analyses [99] model numerically the creep
behavior of geosynthetics.
Materials 2017, 10, 782
17 of 23
In those numerical analyses where the geosynthetic is modeled as a continuum element of small
thickness, it is necessary to ensure that it does not support compressive stresses, and hence, bending
moments. Its Young’s modulus should be Jg divided by its thickness.
Little attention is usually paid to the Poisson’s ratio of the geosynthetic (νg ), but it may have
important consequences on the results [100]. When the encasement is modeled as a membrane, it is
common to study the two directions of the geosynthetic independently, without interaction between
them (Equation (3)). That means νg = 0.
On the other hand, when the geosynthetic is modeled as a continuum element, it is necessary
to specify a value for νg . In some cases (e.g., [31]), a common value of 0.3 is input. That implies
that, although the geosynthetic that surrounds the columns does not support compressive stresses
in the vertical direction, it compresses vertically (vertical strains) and then, that vertical compression
leads to a radial expansion of the geosynthetic (horizontal or circumferential strains) due to Poisson’s
effect. Thus, the geosynthetic encasement decreases its lateral confinement to the column due to this
radial expansion. Common geosynthetics for column encasement are woven geotextiles. For woven
geotextiles, the two directions work nearly independently; so, it seems logical to use values of the
Poisson’s ratio close to 0. Soderman and Giroud [101] propose values of νg = 0.1 for woven geotextiles
and νg = 0.35 for non-woven geotextiles.
14. Conclusions
This paper reviews the main modeling techniques for stone columns, both ordinary or
non-encased stone columns, and geosynthetic-encased stone columns. The paper tries to encompass
the more recent advances and recommendations in the topic.
There exist several simplified geometrical models. The suitability of each of them depends on
the type of process to be studied, e.g., bearing capacity or settlement, and the type of analyses, e.g.,
analytical or numerical in 2D or 3D. For numerical analyses of a problem that fulfils plane strain
conditions but for the stone column treatment, such as an embankment for a linear infrastructure,
the simplified geometrical model of a three-dimensional slice of columns is recommended because
it does not require any transformation of the problem parameters. For more simplified models for
the stone column treatment, such as gravel trenches or homogenization, calibrating or tuning the
parameters using the unit cell as an auxiliary problem is recommended. The model of an isolated
column with load just on top of it may be useful for field tests, but it is not a common situation
in real cases because loading the soil that surrounds the column is beneficial. The behavior of an
isolated column is different from that of a column within a large group under distributed load. Those
differences should be considered when extrapolating results.
For groups of columns, the critical column length depends mainly on the loading area.
For non-encased columns, its value is around twice the footing width and for encased columns
is slightly higher.
Some guidance is provided to consider installation effects and to model the column and the
geosynthetic encasement. Column installation usually increases the horizontal stresses and that is
often accounted for using a high value of the earth pressure coefficient, typically around 1. On the other
hand, the relevance of the Poisson’s ratio of the geosynthetic is taken into account and an appropriate
value for a woven geotextile could be close to 0, i.e., the two directions work nearly independently.
Finally, laboratory tests are not commonly performed to characterize stone column properties
and some column parameters published in the literature are tabulated as a reference.
Acknowledgments: The author thanks all the institutions that have allowed him to work on stone columns
and all the people that have collaborated in the research on stone columns. The author thanks the Spanish
Society for Soil Mechanics and Geotechnical Engineering (SEMSIG) for the invitation to give a lecture on
stone columns in the 10th National Symposium on Geotechnical Engineering, held in A Coruña in October
2016. The present paper is based on that lecture and the corresponding conference paper [102] (In Spanish).
The author acknowledges the funding received through the following projects: “Improvement of soft soils
with stone columns for foundation of embankments. Analysis of the process and design criteria” for
Materials 2017, 10, 782
18 of 23
the Spanish Ministry of Public Works (03-A634), “An integrated calculation procedure for stone columns,
considering the influence of the method of installation” for the Spanish Ministry of Science and Innovation
(BIA2009-13602) and “The critical distance in rock fracture” for the Spanish Ministry of Economy, Industry and
Competitiveness (BIA2015-67479-R). The author is very thankful to the Group of Geotechnical Engineering
of the University of Cantabria, particularly to C. Sagaseta, J. Cañizal, A. Da Costa, A. Cimentada and
M. Miranda. The research on stone columns has been contributed by the international cooperation of H.
Schweiger, B. McCabe, M. Karstunen and N. Sivasithamparam. The author acknowledges the participation
in the following projects supported by the European Commission: the Marie Cure Research Training Network
entitled “Advanced Modelling of Ground Improvement on Soft Soils (AMGISS)” (MRTN-CT-2004-512120) and
the Industry-Academia Partnerships and Pathways project on “Modelling Installation Effects in Geotechnical
Engineering (GEO-INSTALL)” (PIA-GA-2009-230638). The following companies have also contributed to the
research: Keller Group, mainly through E. Carvajal, G. Vukotić and J. Wehr; Huesker Inc., mainly through I. Diego;
and Geocisa, particularly for the field measurements carried out in Sueca, near Valencia (Spain).
Conflicts of Interest: The author declares no conflict of interest. The founding sponsors had no role in the design
of the review; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the
decision to publish the results.
List of Symbols
ar
cu
dc
d10 , d60
k
m
pa
pa (0)
pa (z)
r
s
sr
sz
sz0
t
u
A
B
E
H
Jg
K
K0
L
Tg
b
g
ν
σ
σc
ϕ
ψ
Area replacement ratio: Ac /Ae
Undrained shear strength
Column diameter
Diameters corresponding to 10% and 60% finer in the particle-size distribution, respectively
Hydraulic conductivity/permeability
Exponent of the power law used to reproduce the stress dependent stiffness
Uniform applied vertical pressure
Uniform applied vertical pressure at the surface
Reduced value using a trapezoidal distribution of the uniform applied vertical pressure at a depth z
Radius
Centre-to-centre column spacing
Radial displacement of the column
Settlement
Settlement without columns
Thickness
Pore water pressure
Cross-sectional area
Footing width
Young’s modulus
Soft soil layer thickness
Encasement stiffness
Coefficient of lateral earth pressure
Coefficient of lateral earth pressure at rest
(Column) length
Tensile hoop force at the encasement
Settlement reduction factor: β = sz /sz0
Unit weight
Poisson’s ratio
Stress
Confining pressure
Friction angle
Dilatancy angle
Materials 2017, 10, 782
19 of 23
Subscripts
c, s, g, e
d
m
max
min
r
r, θ, z
sat
x, y, z
¯ (upper bar)
column, soil, encasement or geosynthetic, loaded area or unit cell
dry
equivalent homogeneous soil
maximum value
minimum value
equivalent gravel ring
Cylindrical coordinates
Saturated
Cartesian coordinates
average value along the radius
Abbreviations
DS
G.W.T.
SC
TX
Direct shear test
Ground water table
Stone column
Triaxial test
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