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Numerical simulations of stone column installation

by

Jorge Castro (1)(*) and Minna Karstunen (2)

Department of Civil Engineering

University of Strathclyde

John Anderson Building

107 Rottenrow

Glasgow G4 0NG, United Kingdom

Tel.: +44 141 548 3252

Fax: +44 141 553 2066

e-mail: (1)castrogj@unican.es (2)minna.karstunen@strath.ac.uk

(*) Corresponding Author

Date: October 2009

Number of words: 6,257

Number of tables: 3

Number of figures: 19

(*) Current address and affiliation of the corresponding author:

Group of Geotechnical Engineering

Department of Ground Engineering and Materials Science

University of Cantabria

Avda. de Los Castros, s/n

39005 Santander, Spain

Tel.: +34 942 201813

Fax: +34 942 201821

e-mail: castrogj@unican.es

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ERRATA:

Curve for S-CLAY1S in Figure 5 is not correct.

Corrected Figure 5:

0

1

2

3

4

05 1015 20 25

S-CLAY1

S-CLAY1S

rc=0.4 m

Distance to column axis, r / rc

Effective horizontal stress, σ'x / σ'x0

That also affects Figure 6 and Figure 7:

0

1

2

3

4

051015

S-CLAY1

S-CLAY1S

K0=0.544

K=1

rc=0.4 m

Distance to column axis, r / rc

Lateral earth pressure coefficient, K / K0

1.0

1.2

1.4

1.6

1.8

2.0

05 1015 2025

S-CLAY1

S-CLAY1S

Kirsch (2006) - Field 1

Kirsch (2006) - Field 2

rc=0.4 m

Distance to column axis, r / rc

Lateral earth pressure coefficient, K / K0

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Abstract

The paper describes the results of numerical simulations investigating the installation

effects of stone columns in a natural soft clay. The geometry of the problem is

simplified to axial symmetry, considering the installation of one column only. Stone

column installation is modelled as an undrained expansion of a cylindrical cavity. The

excess pore pressures generated in this process are subsequently assumed to dissipate

towards the permeable column. The process is simulated using a finite element code

that allows for large displacements. The properties of the soft clay correspond to

Bothkennar clay, which is modelled using S-CLAY1 and S-CLAY1S, which are Cam

clay type models that account for anisotropy and destructuration. Stone column

installation alters the surrounding soil. The expansion of the cavity generates excess

pore pressures, increases the horizontal stresses of the soil and most importantly

modifies the soil structure. The numerical simulations performed allow quantitative

assessment of the post installation value of the lateral earth pressure coefficient and the

changes in soil structure caused by column installation. These effects and their influence

on stone column design are discussed.

Keywords: stone columns, installation, numerical modelling, anisotropy,

destructuration.

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Introduction

Stone columns are a ground improvement technique, which not only increases the

overall strength and stiffness of the foundation system, but also modifies the properties

of the soil surrounding the columns. Design of stone columns is usually based on their

performance as rigid inclusions (Balaam and Booker 1981; Barksdale and Bachus 1983;

Priebe 1995; Castro and Sagaseta 2009) and the alteration caused in the surrounding soil

by column installation is commonly not considered. However, the installation effects,

whether they are positive, negative or negligible, are one of the major concerns for an

accurate design (Egan et al. 2008).

Field measurements (Watts et al. 2000; Watts et al. 2001; Kirsch 2004; Gäb et al. 2007;

Castro 2008) have shown some of the effects of column installation, like the increase of

pore pressures and horizontal stresses, and the remoulding of the surrounding soil

caused by the vibrator penetration. However, based on these measurements it is difficult

to achieve conclusions that can be used in stone column design, because they relate to a

specific case and hence cannot be generalised in a straightforward manner. There have

also been attempts to investigate these effects through physical modelling of the process

by means of centrifuge testing (Lee et al. 2004; Weber et al. 2006), but the soils used

are reconstituted and hence not representative of natural clays.

Numerical modelling is a useful tool that may well help to derive some conclusions or

recommendations of installation effects for column design, if the assumptions made in

the model are validated by experimental measurements. Few attempts (Kirsch 2006;

Guetif et al. 2007) have been made in this field. In both cases, the soil model used was

very simplistic, and not representative of real soil behaviour: elastic-perfectly plastic

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with a non-associate Mohr-Coulomb failure criterion. Hence, for example, it was not

possible to account for any hardening of the soil due to installation.

In this paper, numerical simulations of installation effects of stone columns are carried

out using two advanced constitutive models: S-CLAY1 (Wheeler et al. 2003) and S-

CLAY1S (Karstunen et al. 2005), which have been especially developed to represent

natural structured soft soils, a common type of soils to be treated with stone columns.

The numerical models account only for pure cavity expansion effects of installation, and

ignore e.g. the shearing and soil disturbance due to the penetration of the poker, the

vibration of the poker, etc. It is, however, thought that the main effect is the cavity

expansion and the advanced soil models allow, for the first time, quantitative

predictions of e.g. the influence of the cavity expansion on earth pressure at rest and the

soil structure.

Numerical model

The finite element code Plaxis v8 (Brinkgreve 2004) was used to develop a numerical

model of a reference problem to study installation effects of stone columns. The

installation of only one stone column was considered to simplify the problem to an

axisymmetric two dimensional geometry. In order to consider a realistic situation,

properties of Bothkennar clay were used for the soft soil. The Bothkennar soft clay test

site has been the subject of a number of comprehensive studies (Géotechnique

Symposium in print 1992). The soil at Bothkennar consists of a firm to stiff silty clay

crust about 1.0 m thick, which is underlain by about 19 m of soft clay. The ground

water level is 1.0 m below the ground surface. Typically to a structured soil the in situ

water content is close to the liquid limit.

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Stone columns have been applied in Bothkennar clay (Watts et al. 2001; Serridge and

Sarsby 2008) or other Carse clays (Egan et al. 2008). For the numerical model in this

paper, a column length of 10 m is used. The untreated clay underneath is not modelled,

because the installation effects in this part of the soil are not particularly significant and

furthermore, modelling the tip of the column may lead to some numerical instabilities.

The behaviour of Bothkennar clay was modelled using two advanced constitutive

models, namely S-CLAY1 (Wheeler et al. 2003) and S-CLAY1S (Karstunen et al.

2005). S-CLAY1 is a Cam clay type model with an inclined yield surface to model

inherent anisotropy, and a rotational component of hardening to model the development

or erasure of fabric anisotropy during plastic straining. The S-CLAY1S model accounts,

additionally, for interparticle bonding and degradation of bonds, using an intrinsic yield

surface and a hardening law describing destructuration as a function of plastic straining.

The models have been implemented as User-defined soil models in Plaxis.

The values for S-CLAY1 model parameters (soil constants) for Bothkennar clay were

calibrated by McGinty (2006) and are listed in Table 1. Hydraulic conductivity of

Bothkennar clay has been assumed to be anisotropic: the horizontal permeability is

assumed to be twice the vertical one. The initial state variables of Bothkennar clay are

taken from Vogler (2008) (Table 2). He obtained initial void ratios from laboratory tests

(Géotechnique Symposium in print 1992) and the initial inclination of the yield surface,

0

α , through considering the deposition history (see Wheeler et al. 2003 for details). The

additional parameters for S-CLAY1S, outlined in Table 3, were calibrated by McGinty

(2006). The initial bonding parameter, 1

0

−=

tS

χ

, agrees with the reported sensitivity

(Géotechnique Symposium in print 1992) of 85−=

tS

. For this constitutive model the

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slope of the post yield compression line, λ , corresponds to an intrinsic value,

iλ , which

can be obtained from oedometer tests on reconstituted samples. In contrast, for S-

CLAY1, the value for λ is determined from oedometer tests on intact soil samples.

The numerical model is 10 m high and 15 m wide (see Figure 1). Parametric studies

were carried out to check how wide the model should be to have a negligible influence

of the outer boundary. A width of 15 m was considered sufficient. Roller boundaries

were assumed on all sides to enable the soil to move freely due cavity expansion. The

finite element mesh is extra fine close to the column cavity, where the installation

effects are expected to be noticeable and mesh sensitivity studies were performed to

confirm the accuracy of the mesh.

Column installation is modelled as the expansion of a cylindrical cavity, which is

considered to occur in undrained conditions, because columns are usually installed in a

short period of time. The expansion of the cavity is modelled as a prescribed

displacement from an initial radius,

0 a , to a final one,

f a . Although there are other

possibilities to model the expansion of the cavity, such as applying an internal

volumetric strain, a prescribed displacement is superior to the other methods due to

numerical stability, as Kirsch (2006) has already pointed out.

In reality the cylindrical cavity is expanded from an initial cavity radius of zero while

the numerical calculations must necessarily begin with a finite cavity radius,

0 a , to have

finite circumferential strains. However, the authors have verified that this restriction

does not pose any inconsistency of the results. Carter et al. (1979) elegantly explain that

in plane strain the solution for expansion from a finite radius will ultimately furnish the

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solution to the expansion from zero initial radius. For an elastic-perfectly plastic

material, the effects of the cavity expansion are determined by the parameter

2

0

2

aaf−

,

once the limit internal pressure of the cavity has been reached. Carter et al. (1979)

decided to double the cavity size, because after that the internal pressure is within 6 per

cent of the ultimate limit pressure. A further expansion of the cavity was numerically

expensive and the increase gained in the solution accuracy is negligible. In the present

analysis, as the constitutive models used are much more complex, the solution for

doubling the cavity was compared with the solution that quadruples the cavity size.

Both simulations gave almost identical results, and therefore, the comments made for

the elastic-perfectly plastic model are also applicable to the advanced constitutive

models used (S-CLAY1 and S-CLAY1S). A typical column radius,

cr , of 0.4 m was

chosen. Consequently, initial cavity radius of 0.1 m and 0.23 m and final cavity radius

of 0.41 m and 0.46 m were, respectively, used to double and quadruple the size of the

cavity. The expansion of a cavity is assumed to model reasonably well the effects

caused by the installation of bottom feed vibro displacement columns in Bothkennar

clay, as the vibratory action of the probe is expected to have only a small influence on

soft structured soils.

The excess pore pressures generated in the expansion of the cavity are subsequently

assumed to dissipate towards the column and the surface. Since the analysis focuses on

the surrounding soil, there is no need to model the column material and therefore the

cavity is kept as a hole with infinite permeability during the consolidation phase. This

modelling technique has two drawbacks: the infinite permeability of the column and the

lack of interaction between soil and column during consolidation. However, firstly, the

column permeability is high enough in comparison with the soil one to be modelled as

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infinite. Secondly, the lateral displacement of the soil-column interface after the

installation of the column is pretty small and has only a small effect on the soil

properties. The lateral displacement of the soil-column interface could be represented by

means of a slight relaxation of the prescribed displacement if it were of interest.

To sum up, two calculation phases are performed after the generation of initial stresses:

the expansion of a cavity in undrained conditions followed by consolidation process.

The cavity expansion generates large strains, making necessary to account for large

displacements in the calculation. The “updated mesh” option in Plaxis software allow

for this kind of calculation. Despite the name, a large displacement calculation implies

considerably more than simply updating nodal coordinates (Brinkgreve 2004). This

updated Lagrangian formulation is described by McMeeking & Rice (1975). The co-

rotational rate of Kirchhoff stress (or known as Hill stress rate) is adopted. The details

on the implementation can be found in Van Langen (1991). In addition, the value of the

pore pressures was also updated in each step, even tough it is not particularly important

for this problem. In terms of controlling the solution of the non-linear problem with

Plaxis, the arc-length control was deactivated, the over-relaxation was set to 1.0 and the

step size parameter of the S-CLAY1 model was -0.5 to avoid numerical instabilities

with the User-defined soil model.

Pore pressures

Field measurements (Gäb et al. 2007; Castro 2008) clearly show that pore pressures

immediately increase during vibrator penetration. The pore pressures reach a peak

during column construction and are later on dissipated. The value of these peak pore

pressures and their dissipation are the first installation effect to be analysed.

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The excess pore pressures generated by column construction, u

∆ , are shown in Figure

2 for two different depths. Following common practice, the distance to column axis, r ,

is normalised by the column radius,

cr . Because the excess pore pressures increase with

the depth, two different depths, namely 3 and 7 m, were chosen for inspection. The

increase of excess pore pressures with depth has been also measured in field tests

(Castro 2008). The authors reckon that this phenomenon stems from the increase of

undrained shear strength with depth, which can be theoretically proven for an elastic-

perfectly plastic material in plane strain (Randolph et al. 1979). Although other authors

(Guetif et al. 2007) tend to normalise the pore pressures by their initial value, here the

excess pore pressures are normalised by the undrained shear strength,

uc , because it

allows for direct comparison between different depths, soil models and field

measurements (Figure 3). The normalised values of the excess pore pressure,

ucu/

∆

,

agree very well for all depths with the exception of the dry crust.

The area affected by column installation is constant with depth, and clearly visible in

Figures 2 and 3. In this case, for Bothkennar clay, its value is around 13.5 times the

column radius. This radius of influence depends on the rigidity index,

rI , the quotient

between shear apparent modulus and undrained shear strength,

ucG/, and given that

both increase with depth in a similar way, linearly with

0' p , the radius of influence is

constant with depth. This is the radius of influence in terms of pore pressures and it may

well be different for other parameters, as it will be seen later on. The radius of influence

where excess pore pressures develop coincides with the extension of the plastic zone,

R, with a soil area that has reached the critical state.

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The differences between the two constitutive models used, S-CLAY1 and S-CLAY1S,

are very small; only close to the column, where the destructuration caused by the

expansion of the cavity is most evident, the S-CLAY1S model predicts slightly higher

excess pore pressures than S-CLAY1. Additionally, both models predict a nearly linear

decrease of the pore pressures with the distance to the column axis beyond about five

column radiuses, while the decrease is very steep close to the column. Consequently,

the shapes of the curves do not present a logarithmic decrease of the pore pressure with

the distance to the column axis, as predicted by the cavity expansion theory for an

elastic-perfectly plastic material (Randolph et al. 1979). Furthermore, the radius of

influence, R, using an elastic-perfectly plastic model would be 12.2cr (

rc

IrR

=

/

)

and the maximum value at the cavity wall would be 5

uc (

ru

Icu

ln

max

=

) because the

value of

rI for Bothkennar clay in this numerical model is about 150.

To highlight the influence of the soil anisotropy in the generation of excess pore

pressures during stone column construction, the Modified Cam-Clay (MCC) model was

also used, setting the initial anisotropy and the parameters of the rotational hardening

law equal to zero. Close to the cavity wall, the excess pore pressures are higher than

predicted by S-CLAY1, but they decrease quicker with the radius, resulting in a radius

of influence slightly higher than 11 column radiuses. The values calculated using the

MCC model were also used compared with the semi-analytical solution of Collins and

Yu (1996), showing a good agreement.

The numerical model suggests excess pore pressures in the same range as the values

measured in the field (Egan et al. 2008; Serridge and Sarsby 2008). However, the scatter

of the limited field measurements and the lack of detailed information make a thorough

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comparison impossible. The field measurements in overconsolidated clays (OCR>2)

(Castro 2008) give clearly lower values than in normally or slightly overconsolidated

clays (OCR<2) and therefore do not offer suitable comparisons. Field measurements

during pile driving (Poulos and Davis 1980) recorded higher excess pore pressures for

sensitive marine clay than for clays of low-medium sensitivity. However, the

differences are larger than computed in this case.

Pore pressure dissipation is outlined in Figure 4 corresponding to a depth of 7 m with S-

CLAY1 model. Dissipations at other depths and for S-CLAY1S follow similar trends as

the example drawn. The peak excess pore pressures generated near the column during

the undrained expansion of the cavity are quickly dissipated towards the column, i.e.

towards the internal permeable boundary in the numerical model. In fact, as the column

installation is not perfectly in undrained conditions and takes some time, field

measurements are expected to be more similar to the short time isochrones than to the

curve that corresponds to the undrained situation. For Bothkennar clay, which has a

very low permeability, the peak excess pore pressure reduces from roughly 120 kPa at 7

m depth to half, 60 kPa, in only 1 day. The results are in agreement with the

observations by Serridge and Sarsby (2008), although direct comparison is not possible

due to differences in column lengths. According to the numerical results excess pore

pressures need over 100 days to be fully dissipated owing to the low permeability of

Bothkennar clay.

Lateral earth pressures

Column installation evidently generates an increase in the horizontal stresses of the

surrounding soil. In fact, the positive effects of column installation in soft soils are due

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to the increase of effective horizontal stresses after the consolidation process that

follows the expansion of the cavity. For example, Priebe (1995) already assumed in his

analysis a value of the soil lateral earth pressure coefficient of 1, which is higher that the

initial value at rest for most soils. The lateral earth pressures clearly influence the

improvement factor achieved with a stone column treatment since it gives the amount of

lateral support for the column and influences its yielding. The K value is therefore an

important state parameter in stone column design.

The predicted effective horizontal stresses after consolidation are shown in Figure 5.

They are normalised by their initial values to remove the influence of the depth. S-

CLAY1 and S-CLAY1S show very different responses. The destructuration that takes

places near the column, which can only be modelled using S-CLAY1S, limits

significantly the increase of horizontal stresses. The plot of the coefficient of lateral

earth pressure (Figure 6) additionally includes the influence of the vertical stresses,

which also change, mainly close to the column. Between 4 and 8 column radiuses from

the column axis, the curves show a plateau with a nearly constant value of the lateral

earth pressure coefficient. This will be the value that should be used for the stone

column design, as long as the pore pressures generated during column construction have

been dissipated. With S-CLAY1 the post installation lateral earth pressure coefficient is

nearly 1 while this value is clearly lower using S-CLAY1S, which illustrates that the

destructuration caused by column installation has a negative effect not only in the

undrained shear strength, but also in the increase of the lateral confinement of the

column.

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As far as the authors are aware, the only published field measurements of the post

installation lateral earth pressure coefficient were done by Kirsch (2004, 2006) in two

different field sites. The soil of the first field site was a silty clay with a relatively high

initial lateral earth pressure coefficient at rest,

91 . 0

0=

K

, while the second trial was

done in a silty sand with

57 . 0

0=

K

. The columns were constructed by the bottom feed

vibro displacement method and their diameter was 0.8 m. Despite the differences

between the two field sites, the same range of values and the same pattern of variation

with the distance to the column axis of the normalised lateral earth pressure coefficient

were found. The values calculated with the numerical model presented in this paper for

the Bothkennar clay field site have very similar trends (Figure 7).

Destructuration

The main goal of using an advanced constitutive model such as S-CLAY1S was to

study the installation effects of stone columns in the structure of the surrounding soil.

Some field measurements (Watts et al. 2000; Serridge and Sarsby 2008; Castro 2008)

alert on the reduction of the undrained shear strength caused by the installation of vibro

displacement columns in sensitive soft soils. Therefore, it would be very desirable to be

able to account for this effect in the column design.

Figure 8 shows the predicted decrease of the bonding parameter χ of S-CLAY1S, as a

result of column installation, which is directly linked to the sensitivity of the soil. The

reduction in the values suggests strain softening, from a peak value of the undrained

strength to the respective remoulded value when χ is equal to zero. Additional

numerical studies made demonstrated that the initial value of the bonding parameter has

no influence on the process, and therefore the bonding parameter is normalized by its

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initial value in Figure 8. The major changes are limited to the area near the column, and

for example, beyond 4 column radiuses, the reduction is within 10 per cent. The results

suggest that the main part of the destructuration is caused by the undrained expansion of

the cavity and the consolidation process has little influence. In a sensitive soil, the

destructuration caused just immediately after column installation will reduce the

apparent undrained shear strength of the soil, but during the consolidation its value will

increase again, as a consequence of the increase of the mean effective stress and the

limited destructuration caused during consolidation.

Although it is difficult to have extensive and reliable field data on the destructuration or

reduction in undrained shear strength, Roy et al. (1981) measured a good set of values

immediately after pile driving in soft sensitive marine clay, namely Saint-Alban clay,

and report the variation of the normalized in situ vane strength with the radial distance.

The decrease of the undrained shear strength measured in the field is compared with the

decrease of the bonding parameter in Figure 9. Despite the scatter of the field

measurements, the agreement is very good. Contrary to pile driving, where the main

interest is on the soil at the pile wall, in the case of stone columns, the average value

between columns is most important. For practical purposes in stone column design, a

reduction of 15-20% of the initial value can be used for normal stone column spacings.

Similar reductions of the in situ vane strength (15%) were measured in the middle of

piles groups (Fellenius and Samson 1976; Bozozuk et al. 1978). Roy et al. (1981)

concluded that the radius of influence of the destructuration is smaller than the radius of

influence of the excess pore pressures. The numerical results illustrate that this is true

for practical purposes because the destructuration developed in the outer part of the

critical state area is negligible.