Prediction of Stone Column Ultimate Bearing Capacity Using Expansion Cavity Model

Proceedings of the Institution of Civil Engineers

http://dx.doi.org/10.1680/grim.13.00045

Paper 1300045

Received 21/07/2013 Accepted 19/03/2014

Keywords: columns/geotechnical engineering/design methods & aids

ICE Publishing: All rights reserved

Ground Improvement

Prediction of stone column ultimate bearingcapacity using expansion cavity modelFrikha and Bouassida

Prediction of stone columnultimate bearing capacityusing expansion cavity modelj1 Wissem Frikha

Universite de Tunis El Manar/Ecole Nationale d’Ingenieurs de Tunis,Tunis, Tunisia

j2 Mounir BouassidaUniversite de Tunis El Manar/Ecole Nationale d’Ingenieurs de Tunis,Tunis, Tunisia

j1 j2

The ultimate bearing capacity of an isolated column was investigated by combining a state of stress with a failure

mechanism of lateral expansion in a cylindrical cavity which reproduces the stone column installation in a purely

cohesive soft soil. Some analytical and empirical methods were used to determine the limit pressure of a cylindrical

cavity with the assumption of varied behaviour laws of the medium around the cavity. These models permitted the

calculation of the limit pressure of an expanded cylindrical cavity from which the ultimate bearing capacity was

derived. Using recorded in situ data from load tests performed on 13 isolated column configurations, the model

parameters are identified. A comparison between experimental data and predictions generated from several models

is presented.

Notationci radius of plastic zone i

cU undrained shear strength

E Young’s modulus

Gl linear shear modulus

Gs secant shear modulus

KP coefficient of passive stress state

ki coefficient of compressibility of plastic zone i

L multiplier

p0 initial horizontal stress at rest

pl limit lateral pressure

p�l limit net lateral pressure

qu ultimate bearing capacity

R2 coefficient of determination

Æ ¼ c1=c2 ratio of the plastic radius variation

� ¼ p�l =cU coefficient

� non-elastic exponent

� Poisson’s ratio

� friction angle

ł dilation angle

1. IntroductionReinforcing soft clays by stone columns is becoming more

successful thanks to the use of advanced procedures of column

installation as well as higher interest by researchers on this topic.

The efficiency of this improvement technique consists in settle-

ment reduction and increase of bearing capacity of soft soils.

Furthermore, the rapid process of installation and inexpensive

cost make this technique quite competitive compared to other

types of foundations (Frikha et al., 2013).

The vibro-installation of stone columns was designed so that the

improvement of soft soil takes place by lateral expansion of the

added stone material. This expansion is induced by horizontal

vibrations powered by an eccentric motor within a vibro-probe.

Figure 1 illustrates that the diameter of installed stone columns

depends on the consistency of the soft soil layer. The lateral

expansion represents a loading subjected to the surrounding soft

soil from which results its primary consolidation.

Several investigations have been conducted to determine the

bearing capacity of an isolated column by considering several

methods, which are classified by Bouassida and Hadhri (1995)

into three categories. In the first approach, the state of stress was

considered (Aboshi et al., 1979). Second, a failure mechanism

was combined with a state of stress (Balaam and Booker, 1985;

Datye, 1982; Greenwood, 1970; Hughes et al., 1975; Van Impe

and De Beer, 1983). In the third type of approach a failure

mechanism only was used (Bouassida and Jellali, 2002). In the

present study, the second approach was considered to compute

the bearing capacity of an isolated column analytically. The

column installation was modelled in a similar manner to lateral

1

expansion of a cylindrical cavity by assuming an elastic–plastic

stone column behaviour. The stress distribution within the stone

column and surrounding soil was calculated using equilibrium

equations.

In the present study the bearing capacity of an isolated column

was determined from empirical predictions and analytical models

based on theoretical approaches of lateral expanded cylindrical

cavity. These models were calibrated from records of the ultimate

bearing capacity measured during in situ load testing of a column.

2. The problem of an expanded cylindricalcavity

The problem of cylindrical cavity expansion within a limitless

half space was initially investigated by Lame (1852). In this

contribution the soil surrounding the cavity exhibited linear

elastic behaviour and was assumed to be weightless, homo-

geneous, and isotropic.

It is difficult to examine all of the studies which have dealt with

the study of the expansion of a cylindrical cavity in an infinite

medium. However, it is possible to classify these studies accord-

ing to the type of contribution, namely analytical and numerical

methods. The analytical methods were developed by assuming

that the medium around the cavity followed various behaviour

laws such as linear elastic, elastic perfectly plastic, either by not

taking account of volume variation (Chadwick, 1959; Gibson and

Anderson, 1961; Hill, 1950; Menard, 1957) or by considering

volume variation (Carter et al., 1986; Hughes et al., 1975;

Ladanyi, 1963; Manassero, 1989; Mecsi, 1991; Salencon, 1966;

Vesic, 1972; Yu and Houlsby, 1991). In the analytical contribu-

tions, hypotheses were adopted for the soil behaviour either in

small strains (Bishop et al., 1945; Gibson and Anderson, 1961;

Hill, 1950; Hughes et al., 1975; Ladanyi, 1963; Menard, 1957;

Vesic, 1972; Windle and Wroth, 1977) or in large strains (Cao et

al., 2002; Carter et al., 1986; Chadwick, 1959; Yu and Houlsby,

1991, etc.). Note that Hughes et al. (1975) proposed neglecting

the elastic strain with respect to the plastic component, whereas

Carter et al. (1986) demonstrated that for a strain level beyond

10%, the elastic strain component should not be neglected. Based

on yield design theory approaches, Bouassida and Frikha (2007)

focused on the theoretical determination of the extreme net

pressure which results from lateral expansion exerted within an

infinite half-space in cylindrical as well as spherical cavities

(Frikha and Bouassida, 2013).

Frikha and Bouassida (2013) investigated the problem of ex-

panded cylindrical cavities in a homogeneous, isotropic and

weightless medium. Elastoplastic behaviour was assumed by

taking account of plastic volume variation. This latter was

described by a plastic potential of flow characterised with a

coefficient of compressibility, k. Therefore, an analytical limit

pressure was obtained as a function of mechanical characteristics

and the coefficient of compressibility of the expanded medium.

Stone column installation

A A A A A A

A–A A–A A–A

Probe insertion Lateral expansion of clayey soil Final stone column

Figure 1. Schematic of the vibro-replacement installation of

stone column (Frikha et al., 2013)

2

Ground Improvement Prediction of stone column ultimatebearing capacity using expansion cavitymodelFrikha and Bouassida

Offprint provided courtesy of www.icevirtuallibrary.comAuthor copy for personal use, not for distribution

3. Empirical approachesSeveral empirical correlations have evolved by using the limit

lateral pressure (derived from pressuremeter data tests) for the

determination of undrained shear strength of purely cohesive

soils. The main primary investigations have been performed by

Amar and Jezequel (1972), Amar et al. (1991), Cassan (1978),

Lukas and LeClerc de Bussy (1976), Marsland and Randolph

(1977), Martin and Drahos (1986) and Menard (1957). The

undrained shear strength is derived from a linear relationship with

the limit net pressure defined by

p�l ¼ pl � p01:

where p0 represents the initial horizontal stress at rest before the

execution of cavity and pl is the lateral limit pressure. This

relationship leads to

cU ¼p�l�2:

A number of recommendations for the value of � are available.

(� ¼ 5.5 (Menard, 1957); � ¼ 8 and 15 (Cassan, 1978); � ¼ 6.8

(Marsland and Randolph, 1977); � ¼ 5.1 (Lukas and LeClerc de

Bussy, 1976); � ¼ 10 (Martin and Drahos, 1986); � ¼ 3.3 and 12

(Clarke, 1995)). The most commonly used value of � is 5.5.

Using yield design theory, Bouassida and Frikha (2007) found a

good agreement with the correlation proposed by Menard (1957)

(� ¼ 5.4).

Amar and Jezequel (1972) suggested Equation 3 when

p�l . 300 kPa

cU ¼p�l10þ 25 kPa3:

Baguelin et al. (1978) suggested a non-linear relationship be-

tween cU and p�l

cU ¼ 0.67( p�l )0.754:

Table 1 summarises different correlations that depend on the

tested soil.

4. Analytical approachesThis section exploits some analytical methods to determine the

limit pressure of the cylindrical cavity with the assumption of

varied behaviour laws for the medium around the cavity. The

main parameters used are the undrained shear strength and the

Menard modulus E. Poisson’s ratio is taken to be � ¼ 0.5.

Equation 5 is derived from the work of Menard (1957), Bishop et

al. (1945) or Houlsby and Withers (1988), using ideal Mohr–

Coulomb elastic–plastic assumptions

p�l ¼ cU 1þ ln I r½ �5:

where

I r ¼E

2cU(1þ �)

� �6:

Cao et al. (2002) studies the non-linear elastic response prior to

yielding on the expansion of the cylindrical cavity in a soil that is

modelled as a non-linear modified Cam Clay material. Large

strain formulation is adopted for both the elastic and the plastic

regions. In the elastic region, the non-linear behaviour which

corresponds to a change in stiffness with strain can be expressed

by a power law function or by a hyperbolic stress–strain curve.

Using a power law function, the variation of shear stress � and

shear strain ª may be expressed as

� ¼ Glª�7:

cU Clay type Reference

p�l5.5

Soft clays Menard (1957)

p�l8

Firm to stiff clays Cassan (1978)

p�l15

Stiff to very stiff clays

p�l5.5

p�l . 300 kPa Amar and Jezequel

(1972)p�l10þ 25 kPa p�l . 300 kPa

p�l6.8

Stiff clays Marsland and

Randolph (1977)p�l5.1

Hard clays Lukas and LeClerc de

Bussy (1976)

p�l10

Stiff clays Martin and Drahos

(1986)

0.67(p�l )0.75

All clays Baguelin et al. (1978)p�l3.3

Soft clays Clarke (1995)

p�l12

Stiff clays

Table 1. Empirical relations between undrained shear strength

and the limit net pressure

3



where � is a non-linear elastic exponent (0 , � < 1) and Gs is

the secant shear modulus given by

Gs ¼ Glª��18:

Note that Gs equals the shear modulus of a linear material Gl if

� ¼ 1. Based on the power law function, the limit cavity pressure

is (Cao et al., 2002)

p�l ¼cU

�þ cU ln I r9:

Gupta (2000) analysed the expansion of a cylindrical cavity using

the above classical hypothesis. The sole difference was to assume

zero volume variation in the plastic zone; then the radial

displacement at the interface of the plastic and elastic zone is

R2c � R2

0 ¼ r2p � (rp � �rp)2

10:

where Rc, R0 and rp are the current and initial radii of the cavity,

respectively and the radius of the plastic zone

�rp ¼rp

2I r11:

After Gupta (2000), the limit net pressure is

p�l ¼ cU þ cU ln 4I2r =(4I r � 1)

� �12:

Frikha and Bouassida (2013) generalised the contribution of

Salencon (1966) related to an expanded cylindrical cavity within

a medium governed by a constitutive law with variable flow. This

problem was solved by dividing the medium around the cavity

into two zones. The first zone, close to the cavity border, was

assumed plastic with a variable flow law. The second zone was

assumed elastic. The plastic zone was divided into n flow zones,

each one being characterised by its own plastic radius ci and

coefficient of compressibility ki (i ¼ 1, n).

In the present study, the plastic zone only comprises two flow

zones (Figure 2). In the external zone II, the condition of no

volume variation at infinity is assumed (Salencon, 1966), then

there is no plastic volume variation, hence k2 ¼ 1. However, the

interior plastic zone I is defined by its potential flow (k1 ¼ k).

Frikha and Bouassida (2013) calculated the limit pressure by

introducing an elastoplastic model, governed by Mohr–

Coulomb’s strength criterion, with a variable plastic potential of

flow. During the column expansion, in the plastic zone, a ring of

soft soil around the column, with small thickness, is assumed to

behave in the drained condition due to the presence of drained

column material. In this zone, especially, the plastic volume

variation is not negligible, and an undrained behaviour is assumed

(then plastic volume variation is zero) in the exterior zone due to

the rapidity of column installation. The aimed prediction of

bearing capacity, stated above, is undertaken on the basis of an

analytical result detailed in Frikha and Bouassida (2013).

Furthermore, it is assumed that the ratio of the variation in time

of plastic radius (denoted Æ ¼ c1=c2) is constant. According to

these assumptions, in the case of purely cohesive soil, the limit

pressure is

p�l ¼ cU 1þ 2

k þ 1ln

E

4Æk�1cU(1� �2)

� �� 13:

In the case where deformation occurs without volume variation

(k1 ¼ k ¼ 1), from Equation 13, the limit pressure established by

Salencon (1966) is then obtained

p�l ¼ cU 1þ lnE

4cU(1� �2)

� � �

¼ cU 1þ lnI r

2(1þ �)

� � �14:

The coefficient of compressibility (k) depends on the angle of

dilation, denoted ł, and is written

k ¼ 1� sinł

1þ sinł15:

Elastic zone

Plastic zonesc1

C2Cavity

Zone I

Zone II

k1

k2 1�

Figure 2. Definition of zones around the cavity subjected to radial

expansion

4



5. Ultimate bearing capacity of an isolatedcolumn

5.1 Background

The analytical model of the interaction of a stone column and the

soil surrounding the stone column has a very complex stress

history and is difficult to determine.

The radial stresses in the cavity could be of a high order

depending on the level of lateral deformation and the vibro-

compaction energy. After stone column installation, an annulus

of soil in the immediate vicinity of the stone column–soil

interface gains strength as consolidation takes place after installa-

tion. The extent of gain varies according to the distance from the

soil–stone column interface and is also dependent on the

consumption of the stone and the corresponding lateral displace-

ment.

There is a change in the stress conditions starting from the

initial Ko state in which the direction of the major principal

stress is vertical to a final axisymmetric state of stress in

which the maximum principal stress is in a horizontal radial

direction.

Depending on the type of column, based on the earlier work of

Datye (1982), three modes of failure can be foreseen, as detailed

in Soyez (1985): lateral expansion, generalised shearing in the

case of an end-bearing column, and punching failure in the case

of a floating column. The ultimate bearing capacity of an isolated

column is investigated here by combining a state of stress with a

failure mechanism of lateral expansion in the cylindrical cavity,

which reproduces the stone column installation in a purely

cohesive soft soil.

The proposed design approach is based on an initial categorisa-

tion of the soil zones into elastic and plastic zones (Datye, 1982).

The share of the load associated with the stone columns is

estimated by using equilibrium methods and a preliminary

evaluation is made of the hazard of the stone column yield in

different layers considering the in situ undrained strength and

overburden pressure in the different layers. The ultimate capacity

of the stone columns has generally been significantly higher than

the values estimated according to the parameters of the soil

(Datye and Madhav, 1988).

The isolated column is subjected to passive triaxial compression;

hence the ultimate bearing capacity as vertical stress is (Soyez,

1985)

qu ¼ Kppl16:

Kp ¼ tan2(�=4þ �=2) is the coefficient of passive pressure.

pl is the limit pressure in surrounding soil.

Otherwise by use of empirical and analytical equations (Equa-

tions 2, 5, 9, 12, 13 and 14), Equation 16 can be written

qu ¼ Kp( p0 þ LcU)17:

in which L is a multiplier.

The limit net pressure p�l can then be written as follows

p�l ¼ L:cU18:

5.2 Application with proposed model

Using different empirical approaches, the value of multiplier L

equals the coefficient � that can be varied from 3.3 to 15 (Table 1).

Using the described analytical approaches, the following values

of L may occur.

(a) The model of Menard (1957), Gibson and Anderson (1961)

and Bishop et al. (1945) give a limit net pressure equal to

p�l ¼ cU 1þ ln I r½ �19:

Combining Equation 17 with Equation 19, it becomes

L ¼ 1þ ln I r20:

In the case of compressible material � ¼ 0.5, the model of

Salencon (1966) is similar to that of Gibson and Anderson

(1961); the coefficient Ir will then be equal to

I r ¼E

2(1þ �)cU

¼ E

4(1� �2)cU

¼ E

3cU21:

(b) In the model of Gupta (2000), the multiplier L equals

L ¼ 1þ ln 4I2r =(4I r � 1)

� �22:

(c) In the model of Cao et al. (2002), the multiplier L equals

L ¼ 1

�þ ln I r23:

(d ) The model of Frikha and Bouassida (2013) can be considered

with the two laws of plastic flow described above. The

multiplier L is equal for a Poisson ratio ı ¼ 0.5 to

5



L ¼ 1þ 2

k þ 1Ln

Ir

Æk�1

� 24:

In the following, the calibration of the model with two plastic

flow zones was undertaken, to determine the bearing capacity of

an isolated stone column.

6. Calibration of the suggested modelIt is very difficult to verify the assumed hypothesis through the

observed behaviour of the zone of interest because of the

disturbance caused during the installation of the stone columns

(Datye and Madhav, 1988). One must therefore rely on semi-

empirical methods and use results of load tests to evaluate the

adopted parameters.

Bergado and Lam (1987) tested 13 in situ stone column models

(G1 to G13); each model was characterised by specified grain-

size column material and process of installation. Table 2 presents

the recorded ultimate bearing capacity of all column models and

the angle of friction of column material.

The empirical and the analytical models can be calibrated using

the above results of Bergado and Lam (1987).

The calibration of the equations aims to determine the following

quantities

j the best value of multiplier L using empirical results

j the best fit of coefficient � in Equation 23 from the Cao et al.

(2002) model

j the values of Æ and k in Equation 24 from the Frikha and

Bouassida (2013) model.

The multiplier L using models of Gibson and Anderson (1961)

and Gupta (2000) can be derived directly from the soil parameters

E and cU:

For each column model, the recorded ultimate bearing capacity is

identified as that predicted by the analytical solution or the

empirical one whereas the value of p0 is calculated (at depths

equal to two column diameters) from full-scale data recorded on

isolated column models.

Figure 3 compares the empirical results and measured normalised

bearing capacity (qu/cU) as a function of the frictional angle of

stone material. It shows the empirical correlations over the

estimate (qu/cU) for reinforced clayey soil by a stone column.

The closed boundary of the measured bearing capacity of the

stone column is obtained in the range of values L ¼ 2.4 and

L ¼ 3.6. The best fit of the multiplier L (¼ �) as mentioned in

Figure 3 is equal to 3.1 with a coefficient of determination

R2 ¼ 0.49. The ultimate bearing capacity is then

qu

cU

¼ (3.1þ p0 ) Kp25:

The correlation in Equation 3 proposed by Amar and Jezequel

(1972) gives an underestimated bearing capacity because this

correlation is only valid for p�l . 300 kPa; that is not the case of

the considered experimental data (p�l , 300 kPa).

In turn, the non-linear empirical relation Equation 4, proposed by

Baguelin et al. (1978), overestimates the bearing capacity of the

isolated column. (Figure 3)

From Figure 3, the incompatibility between empirical results and

measurements from loading tests can be explained, the more

likely, by the uncertainty of the used correlation for the soft

Bangkok clay where Bergado and Lam (1987) performed their

trial tests.

Figure 6 shows the predictions of ultimate bearing capacity by

several analytical models. It can be noted that the models of

Gibson and Anderson (or Salencon) and Gupta lead to compar-

able results which overestimate the normalised bearing capacity

of an isolated column.

As a result of this overestimation of qu, the best fit of the

multiplier L using Equation 23 derived from Cao et al. (2002)

gives � ¼ 1. Hence, the solution proposed by Gupta (2000) gives

comparable results to those obtained by Gibson and Anderson

(1961). Furthermore, the nonlinear elasticity assumption by Cao

et al. (2002) remains quite comparable with the classical linear

elastic model.

The calibration of coefficients Æ and k (defined by Frikha and

Bouassida, 2013) under necessary condition Æ . 0, was carried

out by using KaleidaGraph software. Successive iterations, by

increments equal to 0.1, for Æ in the interval (0.1, 1) were

performed to fit as close as possible to the experimental ultimate

Column models G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13

�: degrees 39.1 38.4 37.2 37.0 36.0 37.6 35.1 36.2 35.6 37.4 37.9 42.5 44.7

Bearing capacity: kN 35.0 32.5 32.5 32.5 30.0 30.0 22.5 22.5 20.0 32.5 30.0 35.0 37.5

Table 2. Ultimate bearing capacity of isolated column models

(Bergado and Lam, 1987)

6



bearing capacity. Then, the interval of variation for parameter k

was identified, namely 0 , k < 0.75. Figure 4 illustrates the

evolution of parameter Æ plotted against parameter k as a result

of the best calibration of suggested model which led, after linear

regression with coefficient of determination R2 ¼ 0.98, to

Æ ¼ �0.1812k þ 0.140826:

Otherwise, in terms of dilation angle, substituting Equation 15

into Equation 26 gives

Æ ¼ �0.18121� sinł

1þ sinłþ 0.1408

27:

Figure 5 illustrates the variations of Æ and k parameters,

according respectively to Equations 15 and 27, plotted against ł.

Substituting Equation 26 in Equation 13 and by setting � ¼ 0.5,

in the case of saturated incompressible medium, it gives

qu ¼ Kp p0 þ cU 1þ 2

1þ k

��

3 lnE

3(�0.1812k þ 0.1408)k�1 3 cU

� ��28:

Equation 28 which holds for 0 , k < 0.75 leads to 0 < Æ < 0.14.

Furthermore, from condition Æ . 0 it follows that ł > 88

(k < 0.75). The angle of dilation, which is usually in the range

[�58; 158], is measured from the undrained triaxial shear test. Thus,

the calibration margins are 0.4 < k < 0.75 and 88 < ł < 158. For

the extreme values ł ¼ 88 and 158 (i.e. k ¼ 0.75 and k ¼ 0.4) the

corresponding values are Æ ¼ 0.005 and Æ ¼ 0.068, respectively.

0

5

10

15

20

25

30

35

40

25 30 35 40 45

qc

uu

/

Frictional angle of ballast : degreesφ

cU �p*

10l

� 25 kPa

c pU l0·750·67( *)�

� 3·1�

� 2·4�

� 3·6�

Amar and Jézéquel (1972)

Baguelin . (1978)et al

Measured

cU �p*l�

� 3·6�

� 3·1�

� 2·4�

Figure 3. Normalised ultimate bearing capacity plotted against

friction angle of column material (empirical and experimental

results)

0

0·02

0·04

0·06

0·08

0·10

0·12

0·14

0 0·1 0·2 0·3 0·4 0·5 0·6 0·7 0·8

α

k

α 0·1812 0·14080·9805

� ��

kR

Figure 4. Calibration result of Æ and k parameters from

KaleidaGraph software

7



After Equations 13 and 28 note that if the dilation angle increases,

the ratio of plastic radii increases as well. Applying Equation 28

for the two extreme cases k ¼ 0.75 and k ¼ 0.4 gives

qu(k¼0.75)¼Kp p0þcU 1þ1.14lnE

11.28cU

� � �� 29a:

qu(k¼ 0.4)¼Kp p0þ cU 1þ1.43lnE

15.05cU

� � �� 29b:

Figure 6 shows the variation of (qu/cU) ratio, plotted against the

friction angle of column material, predicted by various models

with recorded measurements and those derived from Equations

29a and 29b.

Compared to in situ measurements, it appears that the two

proposed models lead to reliable estimates of ultimate bearing

capacity of an isolated column model.

The parametric study shows that the suggested model is suitable

for Æ ¼ 0.005 and Æ ¼ 0.068. It is then noticed that the volume

variation, characterised by a high coefficient of compressibility,

occurs at a maximum about 6.8% of the radius of the plastic zone

surrounding the column. Furthermore, the radial expansion occurs

in the undrained condition, and then involves negligible volume

variation. In conclusion, the plastic behaviour accompanied by

high dilation, even if occurring in a small area of soft clay, played

a positive role in efficient prediction of bearing capacity of an

isolated stone column after comparison with recorded in situ

measurements.

7. ConclusionIn the present study, based on existing lateral expanded cavity

studies, some formulations have been presented for estimating the

ultimate bearing capacity of an isolated stone column installed in

purely cohesive soft clay. This problem was investigated by

combining a state of stress and a failure mechanism of lateral

expansion in a cylindrical cavity to simulate the stone column

installation in a purely cohesive soft soil.

Analytical methods and correlations were used to predict the limit

lateral pressure of the cylindrical cavity by assuming different

behaviour laws of the medium surrounding the cavity. Using the

predicted limit lateral pressure, the ultimate bearing capacity of

an isolated stone column was derived.

The empirical and the analytical models were calibrated using the

recorded ultimate bearing capacity of in situ columns having

variable friction angles.

Empirical results led to an overestimated prediction of the

ultimate bearing capacity with respect to measurements recorded

from full-scale loading tests on isolated stone column models.

Such incompatibility was explained by the uncertainty of used

correlation for the tested soft clay.

The various analytical models also overestimate the bearing

capacity of the isolated stone column. The use of non-linear

variation of shear modulus G (Cao et al., 2002) does not seem to

have any influence on the predicted limit pressure.

Two main parameters (coefficient of compressibility k and the

ratio of the plastic radius variation Æ) of the considered elasto-

plastic model (Frikha and Bouassida, 2013) have been calibrated.

This calibration made it possible to assess the proposed model.

�0·10

�0·08

�0·06

�0·04

�0·02

0

0·02

0·04

0·06

0·08

0·10

�15 �10 �5 0 5 10 15 20 25

α

ψ: degrees(a)

0

0·2

0·4

0·6

0·8

1·0

1·2

1·4

�15 �10 �5 0 5 10 15 20 25

k

ψ: degrees(b)

Figure 5. Variation of Æ and k parameters plotted against the

angle of dilation

8



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Gupta (2000)

Measured

5

10

15

20

25

30

35

25 30 35 40 45

qc

uu

/

Frictional angle of ballast : degreesφ

Figure 6. Normalised ultimate bearing capacity plotted against

friction angle of column material (analytical and experimental

results)

9



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Prediction of Stone Column Ultimate Bearing Capacity Using Expansion Cavity Model