Numerical simulations of stone columninstallation

Jorge Castro and Minna Karstunen

Abstract: This paper describes the results of numerical simulations investigating the installation effects of stone columnsin a natural soft clay. The geometry of the problem is simplified to axial symmetry, considering the installation of one col-umn only. Stone column installation is modelled as an undrained expansion of a cylindrical cavity. The excess pore pres-sures generated in this process are subsequently assumed to dissipate towards the permeable column. The process issimulated using a finite element code that allows for large displacements. The properties of the soft clay correspond toBothkennar clay, modelled using S-CLAY1 and S-CLAY1S, which are Cam clay–type models that account for anisotropyand destructuration. Stone column installation alters the surrounding soil. The expansion of the cavity generates excesspore pressures, increases the horizontal stresses of the soil, and most importantly modifies the soil structure. The numericalsimulations performed allow quantitative assessment of the post-installation value of the lateral earth pressure coefficientand the changes in soil structure caused by column installation. These effects and their influence on stone column designare discussed.

Key words: stone columns, installation, numerical modelling, anisotropy, destructuration.

Resume : Cet article decrit les resultats de simulations numeriques visant a investiguer les effets de l’installation de colon-nes ballastees dans de l’argile molle naturelle. La geometrie du probleme est simplifiee sous forme de symetrie axiale, enconsiderant l’installation d’une seule colonne. L’installation de la colonne ballastee est modelisee en tant qu’une expansionnon-drainee d’une cavite cylindrique, ainsi la pression interstitielle excessive generee est dissipee vers la colonne per-meable. Le processus est simule a l’aide d’un code par elements finis qui permet les grands deplacements. Les proprietesde l’argile molle correspondent a l’argile Bothkennar, qui est modelisee avec S-CLAY1 et S-CLAY1S, qui eux sont desmodeles de type Cam-clay considerant l’anisotropie et la destructuration. L’installation des colonnes ballastees altere le solenvironnant. L’expansion de la cavite genere des pressions interstitielles excessives, augmente les contraintes horizontalessur le sol, et modifie de facon importante la structure du sol. Les simulations numeriques effectuees ont permis d’evaluerla valeur du coefficient de pression laterale des terres post-installations ainsi que les variations dans la structure du sol cau-sees par l’installation des colonnes. Ces effets et leur influence sur la conception des colonnes ballastees sont discutes.

Mots-cles : colonnes ballastees, installation, modelisation numerique, anisotropie, destructuration.

[Traduit par la Redaction]

IntroductionStone columns are a ground improvement technique that

not only increases the overall strength and stiffness of afoundation system, but also modifies the properties of thesoil surrounding the columns. The design of stone columnsis usually based on their performance as rigid inclusions(Balaam and Booker 1981; Barksdale and Bachus 1983;Priebe 1995; Castro and Sagaseta 2009) and the alterationin the surrounding soil caused by column installation iscommonly not considered. However, the installation effects,

whether they are positive, negative or negligible, are one ofthe major concerns for an accurate design (Egan et al.2008).

Field measurements (Watts et al. 2000; Watts et al. 2001;Kirsch 2004; Gab et al. 2007; Castro 2008) have shownsome of the effects of column installation, such as increasein pore pressures and horizontal stresses, and remoulding ofthe surrounding soil caused by vibrator penetration. How-ever, based on these measurements it is difficult to achieveconclusions that can be used in stone column design, be-cause these measurements relate to a specific case andhence, cannot be generalized in a straightforward manner.There have also been attempts to investigate these effectsthrough physical modelling of the process by means of cen-trifuge testing (Lee et al. 2004; Weber et al. 2010), but thesoils used are reconstituted and hence, not representative ofnatural clays.

Numerical modelling is a useful tool that may help to de-rive some conclusions or recommendations about installationeffects for column design, if the assumptions made in themodel are validated by experimental measurements. Few at-tempts (Kirsch 2006; Guetif et al. 2007) have been made inthis field. In both cases, the soil model used was very sim-

Received 27 October 2009. Accepted 21 February 2010.Published on the NRC Research Press Web site at cgj.nrc.ca on30 September 2010.

J. Castro1,2 and M. Karstunen. Department of CivilEngineering, University of Strathclyde, John Anderson Building,107 Rottenrow, Glasgow G4 0NG, UK.

1Corresponding author (e-mail: castrogj@unican.es).2Present address: Group of Geotechnical Engineering,Department of Ground Engineering and Materials Science,University of Cantabria, Avda. de Los Castros, s/n, 39005Santander, Spain.

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plistic and not representative of real soil behaviour: elastic –perfectly plastic with a nonassociated Mohr–Coulomb fail-ure criterion. Thus, for example, it was not possible to ac-count for any hardening of the soil due to installation.

In this paper, numerical simulations of installation effectsof stone columns are carried out using two advanced constit-utive models: S-CLAY1 (Wheeler et al. 2003) and S-CLAY1S (Karstunen et al. 2005), which have been devel-oped especially to represent natural structured soft soils, acommon type of soil to be treated with stone columns. Thenumerical models account only for pure cavity expansion ef-fects of installation, and ignore the shearing and soil disturb-ance due to the penetration of the poker, the vibration of thepoker, etc. It is, however, thought that the main effect iscavity expansion, and the advanced soil models do allow —for the first time — quantitative predictions of, e.g., the in-fluence of the cavity expansion on earth pressure at rest andthe soil structure.

Numerical modelThe finite element code Plaxis v8 (Brinkgreve 2004) was

used to develop a numerical model of a reference problemto study installation effects of stone columns. The installa-tion of only one stone column was considered, to simplifythe problem to an axisymmetric two-dimensional geometry.To consider a realistic situation, properties of Bothkennarclay were used for the soft soil. The Bothkennar soft claytest site has been the subject of a number of comprehensivestudies (Institution of Civil Engineers 1992). The soil atBothkennar consists of a firm to stiff silty clay crust about1.0 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, in a structured soil, the in situ water content isclose to the liquid limit.

Stone columns have been constructed in Bothkennar clay(Watts et al. 2001; Serridge and Sarsby 2008) or other Carseclays (Egan et al. 2008). For the numerical model in this pa-per, a column length of 10 m is used. The untreated clayunderneath is not modelled, because the installation effectsin this part of the soil are not particularly significant andfurthermore, modelling the tip of the column may lead tosome numerical instabilities.

The behaviour of Bothkennar clay was modelled usingtwo 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 inclinedyield surface to model inherent anisotropy, and a rotationalcomponent of hardening to model the development or era-sure of fabric anisotropy during plastic straining. The S-CLAY1S model accounts for, additionally, interparticlebonding and degradation of bonds, using an intrinsic yieldsurface and a hardening law describing destructuration as afunction of plastic straining. The models have been imple-mented as user-defined soil models in Plaxis.

The values for S-CLAY1 model parameters (soil con-stants) for Bothkennar clay were calibrated by McGinty(2006) and are listed in Table 1. Hydraulic conductivity ofBothkennar clay has been assumed to be anisotropic: thehorizontal permeability is assumed to be twice the verticalone. The initial state variables of Bothkennar clay are taken

from Vogler (2008) (Table 2). He obtained initial void ratiosfrom laboratory tests (Leroeuil et al. 1992; Nash et al.1992b) and the initial inclination of the yield surface, a0,through consideration of the deposition history (see Wheeleret al. 2003 for details). The additional parameters for S-CLAY1S, outlined in Table 3, were calibrated by McGinty(2006). The initial bonding parameter, c0 = St – 1, agreeswith the reported sensitivity (Hight et al. 1992; Nash et al.1992a) of St = 5–8. For this constitutive model, the slope ofthe post-yield compression line, l, corresponds to an intrin-sic value, li, which can be obtained from oedometer tests onreconstituted samples. In contrast, for S-CLAY1, the valueof l is determined from oedometer tests on intact soil sam-ples.

The numerical model is 10 m high and 15 m wide (seeFig. 1). Parametric studies were carried out to check howwide the model should be to have a negligible influence ofthe outer boundary. A width of 15 m was considered suffi-cient. Roller boundaries were assumed on all sides to enablethe soil to move freely due to cavity expansion. The finiteelement mesh was extra fine close to the column cavity,where the installation effects were expected to be noticeable,and mesh sensitivity studies were performed to confirm theaccuracy of the mesh.

Column installation is modelled as the expansion of a cy-lindrical cavity, which is considered to occur in undrainedconditions, because columns are usually installed in a shortperiod of time. The expansion of the cavity is modelled asa prescribed displacement from an initial radius, a0, to a fi-nal one, af. Although there are other possibilities for model-ling the expansion of the cavity, such as applying an internalvolumetric strain, a prescribed displacement is superior tothe other methods due to numerical stability, as Kirsch(2006) has already pointed out.

In reality, the cylindrical cavity is expanded from an ini-tial cavity radius of zero, while the numerical calculationsmust necessarily begin with a finite cavity radius, a0, tohave finite circumferential strains. However, the authorshave verified that this restriction does not produce any in-consistency in the results. Carter et al. (1979) elegantly ex-plain that in plane strain the solution for expansion from afinite radius will ultimately furnish the solution to the ex-pansion from zero initial radius. For an elastic – perfectlyplastic material, the effects of the cavity expansion are de-termined by the parameter a2

f � a20, 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 in-ternal pressure is within 6% of the ultimate limit pressure. Afurther expansion of the cavity was numerically expensiveand the increase gained in the solution accuracy is negli-gible. In the present analysis, as the constitutive modelsused are much more complex, the solution for doubling thecavity was compared with the solution that quadruples thecavity size. Both simulations gave almost identical resultsand therefore, the comments made for the elastic – perfectlyplastic model are also applicable to the advanced constitu-tive models used (S-CLAY1 and S-CLAY1S). A typical col-umn radius, rc, of 0.4 m was chosen. Consequently, initialcavity radii of 0.1 and 0.23 m and final cavity radii of 0.41and 0.46 m were used to double and quadruple, respectively,the size of the cavity. The expansion of a cavity is assumed

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to model reasonably well the effects of the installation ofbottom-feed vibro displacement columns in Bothkennarclay, as the vibratory action of the probe is expected tohave only a small influence on soft structured soils.

The excess pore pressures generated by the expansion ofthe cavity are subsequently assumed to dissipate towardsthe column and the surface. As the analysis focuses on thesurrounding soil, there is no need to model the column ma-terial and therefore, the cavity is kept as a hole with infinitepermeability during the consolidation phase. This modellingtechnique has two drawbacks: the infinite permeability ofthe column and the lack of interaction between soil and col-umn during consolidation. However, firstly, the column per-meability is high enough in comparison with the soilpermeability to be modelled as infinite. Secondly, the lateraldisplacement of the soil–column interface after the installa-tion of the column is quite small and has only a minimal ef-fect on the soil properties. The lateral displacement of thesoil–column interface could be represented by means of aslight relaxation of the prescribed displacement, if it wereof interest.

To sum up, two calculation phases are performed after thegeneration of initial stresses: the expansion of a cavity inundrained conditions followed by the consolidation process.The cavity expansion generates large strains, making it nec-essary to account for large displacements in the calculation.The ‘‘updated mesh’’ option in Plaxis software allows forthis kind of calculation. Despite the name, a large displace-ment calculation implies considerably more than simply up-dating nodal coordinates (Brinkgreve 2004). This updatedLagrangian formulation is described by McMeeking andRice (1975). The co-rotational rate of Kirchhoff stress (alsoknown as Hill stress rate) is adopted. The details of the im-plementation can be found in Van Langen (1991). In addi-tion, the values of the pore pressures were also updated ineach step even though it is not particularly important forthis problem. In terms of controlling the solution of the non-linear problem with Plaxis, the arc-length control was deac-tivated, the overrelaxation was set to 1.0, and the step sizeparameter of the S-CLAY1 model was –0.5 to avoid numer-ical instabilities with the user-defined soil model.

Pore pressuresField measurements (Gab et al. 2007; Castro 2008) clearly

show that pore pressures increase immediately during vibra-tor penetration. The pore pressures reach a peak during col-umn construction and are later dissipated. The values ofthese peak pore pressures and their dissipations are the firstinstallation effect to be analysed.

The excess pore pressures generated by column construc-tion, Du, are shown in Fig. 2 for two different depths. Fol-lowing common practice, the distance to the column axis, r,is normalized by the column radius, rc. Because the excesspore pressures increase with the depth, two different depths,namely 3 and 7 m, were chosen for inspection. The increaseof excess pore pressures with depth has also been measuredin field tests (Castro 2008). The authors think that this phe-nomenon stems from the increase of undrained shearstrength with depth, which can be theoretically proven foran elastic – perfectly plastic material in plane strain (Ran-dolph et al. 1979). Although other authors (Guetif et al.2007) tend to normalize the pore pressures by their initialvalue, here the excess pore pressures are normalized by theundrained shear strength, cu, because it allows for directcomparison between different depths, soil models, and fieldmeasurements (Fig. 3). The normalized values of the excesspore pressure, Du/cu, agree very well for all depths with theexception of the dry crust.

The area affected by column installation is constant withdepth, and clearly visible in Figs. 2 and 3. In this case, forBothkennar clay, its value is around 13.5 times the columnradius. This radius of influence depends on the rigidity in-dex, Ir, which is the quotient between shear apparent modu-lus and undrained shear strength, G/cu, and given that bothincrease with depth in a similar way, i.e., linearly with theinitial effective mean pressure, p00, the radius of influence isconstant with depth. This is the radius of influence in termsof pore pressures and it may well be different for other pa-

Table 1. S-CLAY1 parameters for Bothkennar clay.

Depth (m) g (kN/m3) kh (m/s) kv (m/s) k n’ l M m b

0–1 18.0 2.8� 10–9 1.4� 10–9 0.02 0.2 0.48 1.4 30 0.941–10 16.5 1.4� 10–9 0.7� 10–9 0.02 0.2 0.48 1.4 30 0.94

Table 2. S-CLAY1 initial state variables.

Depth (m) e0 a0 OCR POP (kPa) K0

0–1 1.1 0.539 — 30 1.351–10 2 0.539 1.5 — 0.544

Table 3. S-CLAY1S additional parameters.

Depth (m) li c0 a b0–1 0.18 5 11 0.21–10 0.18 5 11 0.2

Fig. 1. Model geometry and finite element mesh.

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rameters, as will be seen later on. The radius of influencewhere excess pore pressures develop coincides with the ex-tension of the plastic zone, R, with a soil area that hasreached the critical state.

The differences between the two constitutive models used,S-CLAY1 and S-CLAY1S, are very small; only close to thecolumn, where the destructuration caused by the expansionof the cavity is most evident, the S-CLAY1S model predictsslightly higher excess pore pressures than S-CLAY1. Addi-tionally, both models predict a nearly linear decrease of thepore pressures with the distance to the column axis beyondabout five column radii, while the decrease is very steepclose to the column. Consequently, the shapes of the curvesdo not present a logarithmic decrease of the pore pressurewith the distance to the column axis, as predicted by thecavity expansion theory for an elastic – perfectly plastic ma-terial (Randolph et al. 1979). Furthermore, the radius of in-fluence, R, using an elastic – perfectly plastic model wouldbe 12.2rc (R=rc ¼

ffiffiffiIr

p) and the maximum value at the cavity

wall would be 5cu(Dumax/cu = lnIr) because the value of Irfor Bothkennar clay in this numerical model is about 150.

To highlight the influence of the soil anisotropy in thegeneration of excess pore pressures during stone columnconstruction, the modified Cam clay (MCC) model was alsoused, setting the initial anisotropy and the parameters of therotational hardening law equal to zero. Close to the cavitywall, the excess pore pressures are higher than predicted byS-CLAY1, but they decrease more quickly with the radius,resulting in a radius of influence slightly higher than 11 col-umn radii. The values calculated using the MCC model werealso compared with the semi-analytical solution of Collinsand Yu (1996), showing a good agreement.

The numerical model suggests excess pore pressures inthe same range as the values measured in the field (Egan etal. 2008; Serridge and Sarsby 2008). However, the scatter ofthe limited field measurements and the lack of detailed in-formation make a thorough comparison impossible. Thefield measurements in overconsolidated clays (OCR > 2)(Castro 2008) give clearly lower values than in normally orslightly overconsolidated clays (OCR < 2) and therefore, donot offer suitable comparisons. Field measurements duringpile driving (Poulos and Davis 1980) recorded higher excesspore pressures for sensitive marine clay than for clays oflow–medium sensitivity. However, the differences are largerthan computed in this case.

Pore pressure dissipation is outlined in Fig. 4, correspond-ing to a depth of 7 m with the S-CLAY1 model. Dissipa-tions at other depths and for S-CLAY1S follow similartrends as the example drawn. The peak excess pore pres-sures generated near the column during the undrained ex-pansion of the cavity quickly dissipate towards the column,i.e., towards the internal permeable boundary in the numeri-cal model. In fact, as the column installation is not in per-fectly undrained conditions and takes some time, fieldmeasurements are expected to be more similar to the shorttime isochrones than to the curve that corresponds to the un-drained situation. For Bothkennar clay, which has a verylow permeability, the peak excess pore pressure reduces

Fig. 2. Excess pore pressures generated by stone column installa-tion at (a) 3 m depth and (b) 7 m depth.

Fig. 3. Normalized excess pore pressures generated by stone col-umn installation.

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from roughly 120 kPa at 7 m depth to half, 60 kPa, in only1 day. The results are in agreement with the observations bySerridge and Sarsby (2008), although direct comparison isnot possible due to differences in column lengths. Accordingto the numerical results, excess pore pressures need over100 days to be fully dissipated owing to the low permeabil-ity of Bothkennar clay.

Lateral earth pressuresColumn installation evidently generates an increase in the

horizontal stresses of the surrounding soil. In fact, the posi-tive effects of column installation in soft soils are due to theincrease of effective horizontal stresses after the consolida-tion process that follows the expansion of the cavity. For ex-ample, Priebe (1995) already assumed, in his analysis, avalue 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 improve-ment factor achieved with a stone column treatment as itgives the amount of lateral support for the column and influ-ences its yielding. The K value is, therefore, an importantstate parameter in stone column design.

The predicted effective horizontal stresses after consolida-tion are shown in Fig. 5. They are normalized by their initialvalues to remove the influence of depth. S-CLAY1 and S-CLAY1S show very different responses. The destructurationthat takes place near the column, which can only be mod-elled using S-CLAY1S, limits significantly the increase ofhorizontal stresses. The plot of the coefficient of lateralearth pressure (Fig. 6) additionally includes the influence ofthe vertical stresses, which also change, mainly close to thecolumn. Between four and eight column radii from the col-umn axis, the curves show a plateau with a nearly constantvalue of the lateral earth pressure coefficient. This will bethe value that should be used for the stone column design,as long as the pore pressures generated during columnconstruction have been dissipated. With S-CLAY1, thepost-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 installa-tion has a negative effect not only on the undrained shearstrength, but also on the increase of the lateral confinementof the column.

As far as the authors are aware, the only published fieldmeasurements of the post-installation lateral earth pressurecoefficient were done by Kirsch (2004, 2006) at two differ-ent field sites. The soil of the first field site was a silty claywith a relatively high initial lateral earth pressure coefficientat rest, K0 = 0.91, while the second trial was done in a siltysand with K0 = 0.57. The columns were constructed usingthe bottom-feed vibro displacement method and their diame-ter was 0.8 m. Despite the differences between the two fieldsites, the same range of values and pattern of variation withthe distance to the column axis of the normalized lateralearth pressure coefficient were found. The values calculatedwith the numerical model presented in this paper for theBothkennar clay field site have very similar trends (Fig. 7).

Fig. 4. Isochrones of pore pressure dissipation for S-CLAY1 at 7 mdepth.

Fig. 5. Effective horizontal stresses after consolidation.

Fig. 6. Increase of lateral earth pressure coefficient due to columninstallation.

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The dip close to the column of the values measured byKirsch (2006) may have been caused by remoulding and dy-namic effects that are not considered in the presented nu-merical model, while the dip of the computed values isnotably smaller and caused only by the destructuration ofthe soft soil and the increase of the vertical stress.

DestructurationThe main goal of using an advanced constitutive model

such as S-CLAY1S was to study the installation effects ofstone columns on the structure of the surrounding soil.Some field measurements (Watts et al. 2000; Castro 2008;Serridge and Sarsby 2008) provide an alert on the reductionof the undrained shear strength caused by the installation ofvibro displacement columns in sensitive soft soils. There-fore, it would be very desirable to be able to account forthis effect in the column design.

Figure 8 shows the predicted decrease of the bonding pa-rameter, c, of S-CLAY1S, as a result of column installation,which is directly linked to the sensitivity of the soil. The re-duction in the values suggests strain-softening, from a peakvalue of the undrained strength to the respective remouldedvalue when c is equal to zero. Additional numerical studiesdemonstrated that the initial value of the bonding parameterhas no influence on the process and therefore, the bondingparameter is normalized by its initial value in Fig. 8. Themajor changes are limited to the area near the column, andfor example, beyond four column radii, the reduction iswithin 10%. The results suggest that the main part of the de-structuration is caused by the undrained expansion of thecavity and the consolidation process has minimal influence.In a sensitive soil, the destructuration caused immediatelyafter column installation will reduce the apparent undrainedshear strength of the soil, but during the consolidation itsvalue will increase again as a consequence of the increaseof the effective mean stress and the limited destructurationcaused during consolidation.

Although it is difficult to have extensive and reliable fielddata on the destructuration or reduction in undrained shear

strength, Roy et al. (1981) measured a good set of valuesimmediately after pile driving in soft sensitive marine clay,namely Saint-Alban clay, and report the variation of the nor-malized in situ vane strength with the radial distance. Thedecrease of the undrained shear strength measured in thefield is compared with the decrease of the bonding parame-ter in Fig. 9. Despite the scatter of the field measurements,the agreement is very good. Contrary to pile driving, wherethe main interest is in the soil at the pile wall, in the case ofstone columns, the average value between columns is mostimportant. For practical purposes in stone column design, areduction of 15%–20% of the initial value can be used fornormal stone column spacings. Similar reductions of the insitu vane strength (15%) were measured in the middles ofpile groups (Fellenius and Samson 1976; Bozozuk et al.1978). Roy et al. (1981) concluded that the radius of influ-ence of the destructuration is smaller than the radius of in-fluence of the excess pore pressures. The numerical resultsillustrate that this is true for practical purposes because thedestructuration developed in the outer part of the criticalstate area is negligible.

In the comparison shown in Fig. 9, the influence of theeffective mean pressure on the undrained shear strength isnot taken into account in the bonding parameter. The varia-tions of the mean pressures are shown in Fig. 10. An estima-tion of the undrained shear strength immediately aftercolumn installation is derived from the values of thesemean pressures, given the linear variation of cu withp’OCRa and assuming an approximate value of a = 0.7(Fig. 11).

The values predicted by S-CLAY1 are also plotted inFig. 11 to highlight the influence of destructuration andchanges in effective mean pressure. The decrease of the un-drained shear strength close to the cavity wall is caused bythe loss of apparent bonding, while in contrast beyond fivecolumn radii, the slight decrease of the undrained shearstrength is due to the decrease of the effective mean pres-sure. Obviously, the relative relevance of both factors de-pends on the soil sensitivity, which was assumed to be 6 inthis analysis.

Fig. 7. Comparison of lateral earth pressure coefficient computednumerically and measured in the field after column installation.

Fig. 8. Destructuration caused by column installation.

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As was mentioned previously, after consolidation the ef-fective mean pressure increases (Fig. 12) and consequentlythe undrained shear strength follows suit (Fig. 13). The ex-pansion of the cavity wall increases considerably the effec-tive horizontal stresses after consolidation (Fig. 5) and alsothen the effective mean pressure, which exceeds the initialoverconsolidation pressure within five column radii.

The undrained shear strength after consolidation is esti-mated in the same way as for the undrained case. Figure 13confirms that for a nonsensitive soil, the undrained shearstrength increases after the dissipation of the excess porepressures generated during the cavity expansion. However,the undrained shear strength of a sensitive soil can still belower than the initial value. The increase of the undrainedshear strength after consolidation predicted by the MCCmodel is also drawn in Fig. 13 and is comparable with theresults obtained by Randolph et al. (1979) for Boston Blueclay. The results of both calculations are very similar andthe maximum value at the cavity wall is the same (1.6),

although the specific shape of both curves is slightly differ-ent.

Anisotropy

The initial cross-anisotropy of Bothkennar clay is also al-tered by stone column installation, which changes the incli-nation of the yield surface (Fig. 14). No distinction is madebetween S-CLAY1 and S-CLAY1S, because both modelspredict the same results when dealing with anisotropy. Thearea of influence is limited to 10 column radii and the con-solidation process only modifies the anisotropy of the sur-rounding soil nearer than five column radii. Although Fig. 9tries to show the erasure of cross-anisotropy, the scalar pa-rameter of the inclination of the yield surface, a, increasesnear the column due to the development of anisotropy in an-other direction. Therefore, it is more convenient to plot thecomponents of the fabric tensor, a (Fig. 15). Because of theaxial symmetry, polar coordinates are used. The decrease of

Fig. 9. Destructuration computed numerically compared with fieldmeasurements after pile driving.

Fig. 10. Effective mean pressures after column installation at 7 mdepth.

Fig. 11. Decrease of the undrained shear strength after column orpile installation.

Fig. 12. Effective mean pressures after consolidation at 7 m depth.

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az from 1.36 to 0.86 at the cavity wall after consolidationshows clearly the erasure of cross-anisotropy. On the otherhand, the lateral strains caused by the cavity expansion leadto an increase of ar while aq keeps roughly constant, whichshows the development of anisotropy towards planes that areperpendicular to the radial direction. This rotation of theyield surface from the vertical axis to the radial one isshown in Fig. 16, where the inclination vector is viewedfrom the isotropic axis (ar = aq = az). As one gets closer tothe cavity wall, the yield surface rotates toward the radialaxis following a nearly straight line. The rotation getsgreater after consolidation. The rotation along a nearlystraight line is what causes the slight decrease of the scalarparameter of the yield surface inclination, a, and its later in-crease.

One of the special features of the S-CLAY1 model is thatat critical states it predicts a unique inclination of the yieldcurve. This explains the constant value of anisotropy nearthe column after the undrained expansion of the cavity. Dur-

ing the undrained expansion of the cavity, if there are novertical displacements, the effective intermediate stress isequal to the average value of the effective major and minorstresses, s 02 ¼ ðs 01 þ s 03Þ=2. So, at critical state the stress ra-tio vector is ½s 0r s 0q s 0z� ¼ ½1þM=2 1�M=2 1� and theinclination vector is ½ar aq az� = ½1þM=6 1�M=6 1�,which agrees quite well with the calculated values, exceptfor the small discrepancies caused by the vertical displace-ments of the numerical model. The scalar anisotropy param-eter, a, is

ffiffiffi3

pM=6 for this stress path. It is noted that for

triaxial compression, a = M/3 at critical state. The calculatedvalue (0.5) is higher than the theoretical one (0.4) becausethe ar value (1.29) is slightly higher than the theoreticallypredicted one (1.23) for plane strain conditions along thevertical direction.

Fig. 13. Estimated undrained shear strength after consolidation.

Fig. 14. Variation of the inclination of the yield surface due to col-umn installation.

Fig. 15. Changes in anisotropy caused by column installation.

Fig. 16. Rotation of the yield surface caused by column installation.

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These numerical simulations predict a systematic distor-tion of the soil’s fabric after column installation as has beenmeasured in the field by Bond and Jardine (1991) after theinstallation of closed-ended steel piles in heavily overconso-lidated London clay. Fellenius and Samson (1976) alsomeasured the inclination of bedding planes that were ini-tially horizontal after the installation of displacement pilesin sensitive clay. As a rough comparison, Fellenius andSamson (1976) measured an inclination of 278 of the ini-tially horizontal bedding planes for soil samples at a radialdistance of four pile radii, and for that distance the numeri-cal simulation estimates an inclination of 398. Although thecomparison is for different soils, it seems that the valuesmeasured in the field are somehow lower than the valuescomputed numerically for an ‘‘ideal’’ process of cavity ex-pansion.

Stress paths

The stress paths followed during the undrained expansionof the cavity help to provide an understanding of some ofthe installation effects commented upon previously. Thestress paths of a point near the cavity wall (at r = 1.05r0and 7 m depth) are plotted in a p–q diagram and in the p

plane (hydrostatic axis viewpoint, s 0r ¼ s 0q ¼ s 0z) in Fig. 17.During the initial elastic part, the stress paths follow a

straight line, vertical in the p–q diagram and horizontal inthe p plane because the increment of the effective verticalstress is equal to the average value of the increments of theeffective radial and hoop stresses. In this elastic part, thereare no changes of pore pressure and soil structure. Whenthe yield surface is inclined (S-CLAY1 and S-CLAY1S), itis reached earlier than for the MCC model; this is clearly

Fig. 17. Stress paths during undrained cavity expansion near the cavity wall (at r = 1.05r0 and 7 m depth): (a) modified Cam clay ; (b) S-CLAY1; (c) S-CLAY1S. Note that total stress values do not include ambient pore pressure. CSL, critical state line.

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visible in the p plane and leads to a rotation of the yield sur-face towards the radial axis and changes in anisotropy. Theexcess pore pressures generated during this rotation of theyield surface are small ( p–q diagram). Once the yield sur-face is reached, the stress paths bend toward the criticalstate point that is in the yield surface at the maximum devia-toric stress and in the horizontal line aligned with the originin the p plane. Almost all the excess pore pressure is gener-ated after reaching this point.

When soil destructuration is included (S-CLAY1S), theyield surface shrinks towards the intrinsic one and therefore,the stress paths make a small loop ( p–q diagram) and goalong a horizontal line towards the origin (p plane). IfFig. 16 is compared with Fig. 17b, it can be seen that thestress path followed makes the yield surface rotate towardsthe radial axis.

Ground displacementsAlthough the ground displacements caused by column con-

struction are not particularly relevant to the stone column de-sign, they are briefly discussed in this section for the sake ofcompleteness. The radial displacements after the undrainedexpansion of the cavity are shown in Fig. 18. They do notdepend on the soil model and the theoretical solution is wellknown in plane strain conditions along the vertical direction.It is easily obtained applying conservation of volume

½1� r2 � r20 ¼ a2

f � a20 ¼ r2

c

where r is the location of a point initially situated at r0.The values shown in Fig. 18 are taken from a depth of 7 m,

but it is observed that vertical displacements have very littleinfluence below a depth of 3 m. The accuracy of the numer-ical model is revealed when compared with the theoreticalsolution. The results match very well, except for the smalldifferences derived from the situation of the outer boundarythat imposes a null radial displacement at r = 15 m. The ra-dial displacements that take place during consolidation arevery small and therefore, they are not analyzed.

The vertical displacements at the surface are clearly more

interesting and may be of interest when dealing with foot-ings (Egan et al. 2008). The undrained heave is independentof the soil model and it is shown in Fig. 19. Near the col-umn, there are few differences depending on the initial cav-ity radius, a0, that is chosen. The values shown in Fig. 19are for a0 = 0.1 m. The shallow strain path method (SSPM)(Sagaseta and Whittle 2001) predicts the ground movementscaused by the installation of driven (or jacked) piles in clayand therefore, it is also plotted for comparison. The resultsmatch reasonably well. The numerical model predictsslightly lower values of the heave close to the pile becausethe cavity is expanded from a finite radius. On the otherhand, the numerical model computes a slightly higher heaveat the outer part because the model is not infinite in exten-sion.

The heave after consolidation is still important (Fig. 19).As expected, the S-CLAY1S model gives a greater settle-ment during consolidation than the S-CLAY1 model does.It is worth remembering that these values are for the instal-lation of only one column, and group effects are not consid-ered.

ConclusionsFinite element simulations were conducted to quantita-

tively study some of the installation effects of stone columnsin structured soft soils. The installation of only one columnin Bothkennar soft clay was used as a reference problem.Two advanced constitutive soil models were used for Both-kennar clay, namely S-CLAY1 and S-CLAY1S, which madepossible an accurate estimation of the installation effects.

Stone column installation, which was modelled as theexpansion of a cavity in undrained conditions, generatedexcess pore pressures in the surrounding soil that were laterdissipated towards the column. The excess pore pressuresincreased with depth in a similar way to the undrained shearstrength. On the contrary, the radius of influence, i.e., thearea where the pore pressures increase, was constant withdepth, in this case it was equal to 13.5 times the columnradius.

Fig. 18. Radial displacement after undrained expansion of the cavity. Fig. 19. Surface heave caused by column installation.

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Additionally, the installation increased the horizontalstresses and after the excess pore pressure dissipation, thelateral earth pressure coefficient was roughly 1.4 times theinitial value at rest, which means an increase of the columnlateral constraint and therefore, of the improvement factor.

The destructuration caused by column installation erasesall the interparticle bonding at the column interface. How-ever, the destructuration is limited to the soil very close tothe column, nearer than four to five column radii for Both-kennar clay. The average reduction of the undrained shearstrength for normal stone column spacings may be assumedas roughly 15%–20%. Finally, the initial horizontal aniso-tropy changes systematically towards planes perpendicularto the radial axis.

AcknowledgementsThe work presented was carried out as part of a Marie

Curie Research Training Network project ‘‘Advanced Mod-elling of Ground Improvement on Soft Soils (AMGISS)’’(MRTN-CT-2004-512120) supported by the European Com-mission through the programme ‘‘Human Resources andMobility’’ and Marie Curie Industry – Academia Partner-ships and Pathways project on ‘‘Modelling Installation Ef-fects in Geotechnical Engineering (GEO-INSTALL)’’(PIAP-GA-2009-230638). The first author was a ResearchFellow appointed by the AMGISS network and also receiveda grant from the Spanish Ministry of Education (Ref. FPUAP 2005-0195).

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List of symbols

a absolute effectiveness of destructuration in S-CLAY1S

a0 initial cavity radiusaf final cavity radiusb relative effectiveness of destructuration in S-

CLAY1Scu undrained shear strength

cu0 initial undrained shear strengthe0 Initial void ratio (state parameter)G shear modulusIr rigidity index (= G/cu)K lateral earth pressure coefficient

K0 initial lateral earth pressure coefficient at restkh permeability in horizontal directionkv permeability in vertical directionM slope of the critical state line

OCR overconsolidation ratioOCR0 initial overconsolidation ratio

POP pre-overburden pressurep total mean stressp’ effective mean stressp0m preconsolidation pressure (state parameter)p0mi intrinsic preconsolidation pressure (state parameter;¼ p0m=ð1þ cÞ)

p00 initial effective mean stressq deviatoric stressR radius of influencer distance from column axis

r0 initial distance from column axisrc column radiusSt sensitivity

Du excess pore pressureDumax maximum excess pore pressure

a fabric tensora inclination of the yield surface in the q–p’ plane

(state parameter)a0 initial inclination of the yield surface in the q–p’

planeai component of the fabric tensor (i direction)ar component of the fabric tensor (radial direction)ar0 initial component of the fabric tensor (radial direc-

tion)az component of the fabric tensor (vertical direction)az0 initial component of the fabric tensor (vertical di-

rection)aq component of the fabric tensor (hoop direction)aq0 initial component of the fabric tensor (hoop direc-

tion)b relative effectiveness of rotational hardening in S-

CLAY1g unit weightdr radial displacementdz vertical displacementk slope of swelling line from e–lnp’ diagraml slope of post-yield compression line from e–lnp’

diagramli slope of intrinsic post-yield compression line from

e–lnp’ diagramm absolute effectiveness of rotational hardening in S-

CLAY1n’ Poisson’s ratiop hydrostatic axis viewpoints 01 effective major principal stresss 02 effective intermediate principal stresss 03 effective minor principal stresss 0r effective radial stresss 0x effective horizontal stresss 0x0 initial effective horizontal stresss 0z effective vertical stresss 0q effective hoop stressc amount of bonding in S-CLAY1S (state parameter)c0 initial amount of bonding in S-CLAY1S

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