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Creep improvement factors for vibro-replacement design

Sexton, B. G., Sivakumar, V., & McCabe, B. A. (2017). Creep improvement factors for vibro-replacement design.Proceedings of the ICE - Ground Improvement, 170(1), 35-56.

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Creep improvement factors forvibro-replacement design&1 Brian G. Sexton BE, PhD, MIEI

Geotechnical Engineer, AGL Consulting, Dublin, Ireland; formerlyCollege of Engineering and Informatics, National University of Ireland,Galway, Ireland

&2 Vinayagamoothy Sivakumar MSc, DIC, PhD, PGCHETReader, School of Natural and Built Environment, Queens Universityof Belfast, Belfast, Northern Ireland

&3 Bryan A. McCabe BA, BAI, PhD, CEng, Eur Ing, FIEILecturer, College of Engineering and Informatics, NationalUniversity of Ireland, Galway, Ireland (corresponding author:[email protected])

1 2 3

Although the vibro-replacement stone column technique is being deployed increasingly in soft cohesive soil depositsin which creep settlements may be significant/dominant, the majority of existing stone column settlement designmethods are either non-specific or pertain to primary settlement only. Consequently, in the absence of furtherguidance, designers sometimes apply the same settlement improvement factor to creep settlements that they haveestimated for primary settlements. In this paper, Plaxis 2D finite-element analyses carried out in conjunction with theelasto-viscoplastic soft soil creep model have indicated that settlement improvement factors are lower when creep isconsidered and therefore the design of stone columns ignoring creep is unconservative. These analyses were used toestablish the impact of a range of relevant variables on ‘primary’, ‘total’ and ‘creep’ settlement improvement factors,leading to the development of a simplified empirical approach for predicting creep settlement improvement factorsfor use in conjunction with an existing primary settlement design method.

NotationA cross-sectional area of soil unit treated with

granular materialAc cross-sectional area of granular columnAc/A area-replacement ratioCc compression indexCs swelling indexCα coefficient of secondary compression/creep

coefficientc′ effective cohesionDc column diameterEref reference modulusEoed oedometric modulusEur unload–reload modulusE50 secant/triaxial moduluse void ratioep end-of-primary (EOP) void ratioe0 initial void ratioH layer thicknessK0 coefficient of lateral earth pressure at restK0nc coefficient of lateral earth pressure in the normally

consolidated conditionk constant dependent on column arrangement

(square, triangular or hexagonal)

k, kx, ky permeability, horizontal permeability, verticalpermeability

m power dictating the stress dependency of soilstiffness (hardening soil model)

n settlement improvement factor, n= δuntreated/δtreatedncreep ‘creep’ settlement improvement factornprimary ‘primary’ settlement improvement factorntotal ‘total’ settlement improvement factor

(i.e. primary + creep)n2 Priebe’s (1995) settlement improvement factorp, p′ mean principal total stress, mean principal

effective stresspref reference pressurepa applied load/load levelpp preconsolidation stress/pressure (three

dimensional)q deviatoric stressRc column radiuss column spacingt end timet0 time at which creep beginsγ bulk unit weightΔSc vertical displacement of columnΔSs vertical displacement of soil

35

Ground ImprovementVolume 170 Issue GI1

Creep improvement factors forvibro-replacement designSexton, Sivakumar and McCabe

Proceedings of the Institution of Civil EngineersGround Improvement 170 February 2017 Issue GI1Pages 35–56 http://dx.doi.org/10.1680/jgrim.16.00029Paper 1600029Received 13/09/2016 Accepted 06/01/2017Keywords: columns/foundations/granular materials

ICE Publishing: All rights reserved

Downloaded by [ Queens University Belfast - Periodicals] on [22/03/17]. Copyright © ICE Publishing, all rights reserved.

mailto:bryan.�mccabe@�nuigalway.�ie

Δσ load levelδ settlementκ* swelling indexλ* compression indexμ* creep coefficient/indexν Poisson ratioνur Poisson ratio for unloading–reloadingσc stress in the columnσp preconsolidation stress/pressure (one dimensional)σs stress in the soilσ′yy effective vertical stressσ0 initial effective stress/pressure (one dimensional)ϕ′ friction angleψ dilatancy angle

1. IntroductionThe majority of research conducted to date into the behaviourof vibro-replacement stone columns has focused on their effec-tiveness in improving bearing capacity or reducing primarysettlement (e.g. Ambily and Gandhi, 2007; Barksdale andBachus, 1983; Black et al., 2011; Mitchell and Huber, 1985;Watts et al., 2000). Stone columns accelerate consolidation,meaning that creep may contribute a significant proportionof post-construction settlement. Given that the magnitude ofcreep settlements may be very significant in some soft organicssoils (e.g. Simons and Som, 1970), it is surprising that theability of stone columns to arrest long-term creep settlementshas received scant attention to date.

Settlement design methods typically involve the direct predic-tion of a settlement improvement factor, n= δuntreated/δtreated,defined as the ratio of the settlement of untreated ground (i.e.no columns) divided by the settlement of ground treatedwith stone columns. This settlement improvement factor can,in turn, be used to predict the settlement of treated ground.Other than an analytical formulation by Madhav et al. (2009,2010) accounting for creep settlements, the majority of theanalytical methods for determining n values are either non-specific or pertain to primary settlement only (e.g. Castroand Sagaseta, 2009; Priebe, 1995; Pulko et al., 2011), and inthe absence of further guidance, designers sometimes applythese methods to estimate the improvement to both creep andprimary settlements.

Sexton and McCabe (2012, 2013, 2015) carried out numericalstudies using a soil model based on the isotache concept (thesoft soil creep (SSC) model) in conjunction with the Plaxis 2Dfinite-element (FE) program (Brinkgreve et al., 2011) to gaugethe influence of creep on stone column settlement performancein a soft single-layer soil profile. It was concluded thatcolumns arrested long-term creep settlement but not to thesame extent as primary settlement. Similar conclusions wereobtained by Sexton and McCabe (2016) and Sexton et al.(2016) for a multi-layer soil profile; the soil model used

in the latter study incorporated anisotropy, bonding anddestructuration. Test data to support these numerical trends(at least in a qualitative sense) are available from

& laboratory testing in a modified triaxial cell carried outby Moorhead (2013)

& back-analyses of two serviceability failures by Pugh (2016),from which it was concluded that while the primarysettlements of the treated ground were in keeping withthose predicted by Priebe’s (1995) analytical method(with a friction angle of 40° assumed for the columnmaterial), the reduction in creep settlement offered bystone columns was actually quite limited.

In this paper, a parametric study is carried out using FE analysisto establish the soil parameters that have the largest influenceon ‘primary’, ‘creep’ and ‘total’ (i.e. primary+ creep) settlementimprovement factors and the stress transfer process from soil tocolumn due to creep. Based on the FE output, a simple empiri-cal approach to account for creep in the design of end-bearinggranular columns has been developed, which is amenable toroutine use by practising engineers. The approach can be used inconjunction with an existing primary settlement design methodthat captures all the key features of primary settlementbehaviour.

2. Stone column settlement design methods

2.1 The ‘unit cell’Although a small number of settlement design methods havebeen developed using plane strain (e.g. Van Impe and De Beer,1983) or homogenisation techniques (e.g. Lee and Pande,1998; Schweiger and Pande, 1986), the majority have beendeveloped based on the unit-cell concept (e.g. Balaam andBooker, 1981; Castro and Sagaseta, 2009; Pulko et al., 2011).The unit-cell approach is used to model an infinite grid ofregularly-spaced columns subjected to a uniform load (e.g.Figure 1), and is valid for large loaded areas except along theperiphery. The amount of stone replacement is quantifiedusing a dimensionless area-replacement ratio, Ac/A (Equation 1),where Ac denotes the cross-sectional area of the granularcolumn (=πDc

2/4, where Dc is the column diameter), andA denotes the cross-sectional area of its attributed ‘unit cell’(dependent on the centre-to-centre column spacing, s, and thecolumn arrangement, quantified by k, see Figure 1).

1:AcA

¼ 1k

Dcs

� �2

2.2 Elastic against elastic–plastic methodsAnalytical settlement design methods tend to be based oneither elastic or elastic–plastic theory. Elastic–plastic methods,which are typically based on the Mohr–Coulomb failure

36




criterion, are preferable to purely elastic methods because thelatter tend to over-predict settlement improvement factors,especially for soft soils. Elastic methods do not account foryielding of the column material, so they over-predict the stressconcentration factor (SCF= σc/σs, where σc is the stress in thecolumn and σs is the stress in the soil). For elastic methodsthat consider vertical deformation only, the SCF is equal tothe ratio of the oedometric moduli of the column and soilmaterials (Ec/Es); elastic methods that consider both radial andvertical deformation yield slightly lower SCF values.

Sexton et al. (2014) used FE analyses to review numerous pre-diction approaches for primary settlement with a view to estab-lishing their merits in capturing a range of features of stonecolumn behaviour. Axisymmetric unit-cell analyses were con-ducted on end-bearing columns using the elasto-plastic hard-ening soil (HS) model. Predicted n values and SCFs obtainedusing the methods of Castro and Sagaseta (2009) and Pulkoet al. (2011) offered the best agreement with the FE analyses,capturing the effect of a wide range of input variables, includ-ing applied load (pa), the friction (ϕ′c) and dilatancy (ψc)angles of the column material and post-installation lateralearth pressure coefficient (K ).

2.3 Review of Madhav et al. (2009, 2010)The method developed by Madhav et al. (2009, 2010) is basedon an extension of an earlier method for primary settlementdeveloped by Shahu et al. (2000), also based on the unit-cellassumption. However, in contrast to the methods reviewed inSexton et al. (2014), the method developed by Shahu et al.(2000) incorporates a granular mat in order to combat/reduce the high stress concentrations that occur near the top ofthe granular columns. Shahu et al. (2000) have used elastictheory to calculate the settlement of the granular column

(e.g. Equation 2), where ΔSc and Δh denote the settlementsand thicknesses of each element of the column, respectively.

2: ΔSc ¼ σcEc Δh

Conventional e− log σ theory is used to calculate the settle-ment of each sub-layer of the soil (ΔSs) (e.g. Equation 3),where σ0 is the initial overburden stress at the centre of eachelement and e0 is its initial void ratio. The soil is assumed tobe normally consolidated. In each element, the vertical dis-placement of the column is equal to the vertical displacementof the soil (ΔSc =ΔSs), leading to Equation 4. The methoddoes not consider radial displacement. An iterative procedurecan then be used to solve for σc and σs in each element bycombining Equations 4 and 5 (Equation 5 describes the equili-brium of vertical stresses in each element).

3: ΔSs ¼ Cc1þ e0 Δh log 1þσsσ0

� �

4: σc ¼ Cc1þ e0 Ec log 1þσsσ0

� �

5: pa ¼ σcðAc=AÞ þ σsð1� Ac=AÞ

Madhav et al. (2009, 2010) account for creep based on the prin-ciple that the stress on the soil will decrease as it creeps (Δσs)while the column (which does not creep) will assume the surplus

Triangular grid

k = (2√3)/πSquare grid

k = 4/π

s

s

Column (diameter = Dc, area = Ac)

Unit cell (area A)

(b)(a)

Hexagonal grid

k = (3√3)/π

(c)

s

Figure 1. Typical column grids encountered in practice: (a) triangular, (b) square and (c) hexagonal

37




load (Δσc), as illustrated in Equation 6. A similar load transfermechanism has also been suggested by Mitchell and Kelly(2013). As a result of the stress unloading, the soil becomes over-consolidated. This overconsolidation effect is additional toBjerrum’s (1967) overconsolidation effect due to ageing.

6: pa ¼ ðσc þ ΔσcÞðAc=AÞ þ ðσs � ΔσsÞð1� Ac=AÞ

The settlements of the ith layer of the granular column andsoil are given by Equations 7 and 8, respectively, where the netsettlement during creep is calculated as the creep settlementminus the rebound due to unloading (ep denotes the end-of-primary (EOP) void ratio, t0 denotes the time at which creepbegins and t denotes the end time). Using the compatibilitycondition (ΔSc =ΔSs), the additional stress carried by thecolumn during creep is calculated using Equation 9. An itera-tive procedure can then be used to calculate Δσc and Δσs ineach element by combining Equations 6 and 9.

7: ΔSc ¼ σc þ ΔσcEc Δh

8:ΔSs ¼ Cc1þ e0 Δh log 1þ

σsσ0

� �þ Cα1þ ep Δh log

tt0

� �

� Cs1þ ep Δh log

σsσs � Δσs

� �

9:

Δσc ¼"

Cc1þ e0 log 1þ

σsσ0

� �þ Cα1þ ep log

tt0

� �

� Cs1þ ep log

σsσs � Δσs

� �#� Ec � σc

As the Madhav et al. (2009, 2010) method is based on elastictheory, n values will be over-predicted unless adequate thick-ness of granular mat is provided. Additionally, the methodis based on the assumption that primary consolidation isfinished before creep begins (creep ‘hypothesis A’), which hasbeen denounced in recent years; creep ‘hypothesis B’ (alsoknown as the isotache concept, e.g. Degago et al., 2011),which models creep occurring concurrently with primary con-solidation, is preferred. The Madhav et al. (2009, 2010)method also assumes that the soil is initially normally con-solidated so that only Cc influences primary settlement, asshown in Equation 3 (Cs is only taken into account whenconsidering the unloading from soil to column due to creep,e.g. Equation 8). Finally, the method necessitates an iterativesolution technique, whereas a closed-form solution, the goal ofthe study reported in this paper, would be preferable from adesigner’s standpoint.

3. Numerical modelling preliminaries

3.1 Soil modelsThe FE analyses in this paper have been carried out inconjunction with the Plaxis 2D FE program. The elasto-viscoplastic SSC model (Vermeer and Neher, 1999; Vermeeret al., 1998), based on the isotache concept, is used to modelthe host clay, and the elasto-plastic HS model (Schanz et al.,1999) is used to model the granular column material.

3.2 Range of soft clay soil parametersSoil parameters from a selection of well-researched soft claysites have been compiled to help identify suitable parameterranges for the study. Strength and compressibility parametershave been summarised in Table 1; the compressibility par-ameters have been used to develop the plot of creep ratio,(λ*−κ*)/μ*, against μ* in Figure 2 (λ*, κ* and μ* are the isotro-pic equivalents of the one-dimensional (1D) compression, Cc,swelling, Cs and creep indices, Cα, respectively). The propertiesof Bothkennar clay in Scotland may be considered as anapproximate ‘mid-range’ of the selection in Figure 2, so itsproperties (Table 2) are adopted as the ‘base case’ for the para-metric study.

The Bothkennar clay parameters in Table 2 are based on thoseused by Killeen and McCabe (2014), which are in turn basedon ICE (1992). However, as creep was not considered byKilleen and McCabe (2014), the incremental load testing pro-gramme carried out by Nash et al. (1992) was used to deriveadditional parameters necessary for the SSC model. The soilparameters have been comprehensively validated in Sexton(2014) both by using the Plaxis ‘soil test’ facility to simulatethe triaxial soil test data published in ICE (1992) and by simu-lating two field tests (described by Jardine et al. (1995)) on anunreinforced rigid pad footing at the Bothkennar site.

3.3 The numerical modelA standardised stratigraphy as shown in Figure 3 was preferredfor the parametric study, enabling the effect of changing asingle parameter to be examined. This profile (crust + clay) waspreferred to a single-layer profile because clay sites reported inthe literature generally include a stiff crust (formed by weather-ing and groundwater level fluctuations). The crust parametersin Table 2 are based on those used by Killeen and McCabe(2014), who identified the important influence of the crust onthe mechanism of stone column behaviour; a stiff crust tendsto confine columns in the upper layers, forcing bulging tooccur in the deeper layers, which in turn enhances the load-carrying capacity of columns.

Axisymmetry has been used to model the problem for the pur-poses of this numerical study (Figure 3). The vertical bound-aries of the ‘unit cell’ are restrained laterally and the base isfixed in both the vertical and lateral directions. The left-handside boundary of the unit cell shown in Figure 3 is a symmetry

38




boundary. In this study, a column diameter of 600 mm hasbeen adopted (column radius, Rc = 300 mm), typical of thatconstructed in soft cohesive soil deposits using a 430 mmdiameter poker.

The parameters used for the granular column material, docu-mented in Table 3, have also been based on the work of Killeen

and McCabe (2014). The HS model accounts for the stressdependency of soil stiffness using a power law (Equation 10;Brinkgreve et al., 2011), where m denotes the power, and Eref

denotes the reference stiffness modulus corresponding to a refer-ence pressure, pref. According to Brinkgreve et al. (2011),m=1·0 should be used for soft soils to simulate logarithmiccompression behaviour, whereas m=0·5 is more suitable

0

5

10

15

20

25

30

35

40

45

0 0·005 0·01 0·015 0·02 0·025

(λ*

– κ*

)/µ*

µ*

Bothkennar (Scotland)

Väsby (Sweden)

Skå-Edeby (Sweden)

Bäckebol (Sweden)

Haney (Finland)

Hut (Finland)

Otanemi (Finland)

Drammen (Norway)

Ellingsrud (Norway)

Boston Blue clay (USA)

Avonmouth alluvium (UK)

Batiscan (Canada)

Saint-Herblain (France)

HKMD (China)

Tagus Basin (Portugal)

Bothkennar clay

Figure 2. (λ*−κ*)/μ* against μ* for worldwide clays

Table 1. Soil strength and compressibility parameters for soft clays

Soil type ϕ′s: deg St λ* κ* μ* (λ*−κ*)/μ* References

Bothkennar clay 34·0 5–13 0·162 0·023 0·0065 21·43 Hight et al. (1992); Killeenand McCabe (2014)

Swedish claysVäsby clay 30·0 — 0·326 0·038 0·0195 14·74 Degago (2011)Skå-Edeby clay 30·0 7–20 0·364 0·050 0·0185 16·96 Degago (2011); Lo and

Hinchberger (2006)Bäckebol clay — 25 0·410 0·010 0·0205 19·51 Yin and Graham (1989)

Finnish claysHaney clay 32·0 6–10 0·105 0·016 0·0020–0·0040 22·25–44·50 Vermeer and Neher (1999);

Yin and Karstunen (2011)Hut clay 30·0 10 0·089 0·015 0·0018 40·11 Suhonen (2010)Otanemi clay 27·7 7–14 0·150 0·015 0·0035 38·57 Neher et al. (2003)

Norwegian claysDrammen clay — 10–

120·158 0·004 0·0070 22·00 Yin and Graham (1994, 1996)

Ellingsrud clay — — 0·237 0·008 0·0103 22·26 Degago (2011)

Other claysBoston Blue clay (BBC) 30·0 7 0·052–0·196 0·013–0·049 0·0015–0·0056 26·25 Azzouz et al. (1990),

Fatahi et al. (2012)Estuarine alluvium(Avonmouth, UK)

— — 0·110 0·013 0·0116 8·37 Nash and Ryde (2001)

Batiscan clay, Canada 25·0 2·5 0·140 0·013 0·0055 23·03 Yin and Karstunen (2011)Saint-Herblain clay, France 31·0 1·0 0·126 0·007 0·0074 16·12 Yin and Karstunen (2011)Hong Kong marinedeposits (HKMD)

31·0 1·6 0·079 0·018 0·0027 23·03 Yin and Karstunen (2011)

Tagus Basin soils, Portugal — — 0·121 0·029 0·0177 5·20 Lopes (2011)

St, sensitivity; ϕ′s, soil critical state friction angle

39




for sands and silts. The value of m=0·3 for the granularcolumn material quoted in Table 3 is based on Gäb et al.(2008), as are the values of E50

ref (triaxial/secant modulus at 50%of the maximum stress), Eoed

ref (oedometric modulus) andEurref (unload–reload modulus). For the HS model, E50

ref and Eoedref

control the positions of the shear hardening and volumetrichardening yield surfaces, respectively.

10: E ¼ Eref ppref

� �m

4. Numerical modelling

4.1 Analysis stagesFor the analyses carried out herein, the key stages are asfollows.

& The K0 procedure (Brinkgreve et al., 2011) is used togenerate the initial stresses; this method of initial stressgeneration can be used when all layers in the numericalmodel, including the water table, are horizontal. For the‘base case’ described in Section 4.3, K0 for the clay hasbeen calculated according to Equation 11 (Brinkgreveet al., 2011), where K0

nc denotes the coefficient of lateralearth pressure in the normally consolidated condition(K0

nc = 1−sin ϕ′s) and νuris the unload–reload Poisson ratio.A higher value of K0 = 1·0 has been used for the crust.

11: K0 ¼ Knc0 OCR�νur

1� νur ðOCR� 1Þ

& End-bearing columns are ‘wished-in-place’ (e.g.Killeen and McCabe, 2014), after which a plastic nil-stepis used to ensure any out-of-equilibrium stresses generatedas a result of the installation method are restored.

& A plate element is placed at the surface of the unit cell overthe soil and column; the plate acts as a loading platformand ensures the soil and column surfaces undergo equalsettlement.

& A 100 kPa load is applied in undrained conditions, afterwhich a consolidation period is allowed. Once the excesspore pressures have dissipated, no further settlement occursfor the case with the ‘almost zero’ creep coefficient.

4.2 Establishing the influence of creep on nThe SSC model is based on the aforementioned isotacheconcept (strain rate approach/hypothesis B). The concept wasfirst proposed by Šuklje (1957), and is based on a uniquerelationship between the creep rate, the current stress state andthe current strain or void ratio (ε′–σ′–ε), irrespective of the pre-vious loading history (Bodas Freitas, 2008). Therefore,Casagrande’s (1936) method cannot be used for separating theprimary and creep settlement components (and for derivingseparate n values) under a given load increment. To overcomethis, two sets of analyses are performed using the SSC modelto establish the influence of creep.

& For the first set of analyses, a standard creep coefficient isused for the soil. For these analyses, ‘total’ settlementimprovement factors (i.e. primary and creep together) canbe calculated according to the following equation

12: ntotal ¼δtotalðuntreatedÞδtotalðtreatedÞ

–1·00 mCrust

Clay

0·00 m

–10·00 m

Stone column

Fixed in radialdirection

Fixed in vertical andradial directions

Rc = 0·3 m

Unit cell radius

Symmetryboundary

Water tableat –1·00 m

Plate

Figure 3. Soil profile (crust + clay)/axisymmetric unit cell

Table 2. Material parameters for the crust and clay layers

Crust Clay

Depth: m 0·0–1·0 1·0–10·0γ: kN/m3 18·0 16·5OCR — 1·5POP 15·0 —K0 1·0 0·54ϕ′s: deg 34 34c’: kPa 3·0 1·0ψs: deg 0 0e0 1·0 2·0λ* 0·015 0·162κ* 0·002 0·023μ* 0·0006 0·0065νur 0·2 0·2kx: m/d 1�10−4 1�10−4ky: m/d 6·9�10−5 6·9�10−5

γ is the bulk unit weight; POP is the pre-overburden pressure= σp−σ0, where σpis the 1D preconsolidation stress and σ0 is the initial effective stress; OCR is theoverconsolidation ratio = σp/σ0; K0 is the at-rest coefficient of lateral earthpressure; c’ is the effective cohesion; ψs is the soil dilatancy angle; νur is thePoisson ratio for unloading–reloading; kx is the horizontal permeability; ky is thevertical permeability

40




‘Creep’ settlement improvement factors (ncreep) can also bederived from these analyses based on the slopes of thesettlement–log(time) plots after the complete dissipation ofexcess pore pressure (i.e. after EOP consolidation) accord-ing to Equation 13, where μ*untreated and μ*treated denote theslopes of the untreated and treated settlement–log(time)plots, respectively.

13: ncreep ¼ μ�untreated

μ�treated

& For the second set, a very low creep coefficient is used forthe soil (≈1% of the standard value), effectively removingthe creep contribution; ‘primary’ settlement improvementfactors can be calculated according to Equation 14.Numerical difficulties arise in Plaxis if μ*=0 is used.

14: nprimary ¼δprimaryðuntreatedÞδprimaryðtreatedÞ

For the second set, a very low creep coefficient is used forthe soil (≈1% of the standard value), effectively removing

Table 3. Material parameters for the granular column material

γ: kN/m3 ϕ′c: deg ψc: deg c‘: kPa Eoedref : MPa E50

ref: MPa Eurref: MPa pref: kPa m kx: m/d ky: m/d

19·0 45 15 1 70 70 210 100 0·3 1·7 1·7

(a)

(b)

0

0·2

0·4

0·6

0·8

1·0

1·2

1·4

0·1 1 10 100 1000 10 000 100 000 1 000 000

Sett

lem

ent:

m

Time: d

Time: d

Untreated

A/Ac = 10

A/Ac = 6

A/Ac = 3

0

0·5

1·0

1·5

2·0

2·5

0·1 1 10 100 1000 10 000 100 000 1 000 000

Sett

lem

ent:

m

Untreated

A/Ac = 10

A/Ac = 6

A/Ac = 3

Figure 4. Settlement against log(time): (a) very low creep coefficient and (b) standard creep coefficient

41




the creep contribution; ‘primary’ settlement improvementfactors can be calculated according to Equation 13.Numerical difficulties arise in Plaxis if μ*=0 is used.

4.3 ‘Base case’ parametersFor both sets of analyses, the parameters for the ‘base case’are based on those of Bothkennar clay documented in Table 2.The water table is located at a depth of 1·0 m. For the casewith a standard creep coefficient, the relative percentages ofprimary/creep settlement to the total settlement of untreatedground under pa = 100 kPa after 1, 10, 30, 100 and 1000 yearsare 79/21, 74/26, 71/29, 68/32 and 64/36, respectively, althougha wider range has been captured by the parametric study. Theanalyses have been carried out for 3

due to creep is more prevalent at closer spacings (i.e. A/Ac = 3).The mechanism of stress transfer from soil to column due tocreep is in qualitative agreement with that proposed byMadhav et al. (2009, 2010); the Madhav et al. (2009, 2010)and Shahu et al. (2000) predictions are superimposed onFigures 7(c), 7(d) and 8(b). For the elasto-plastic columnsmodelled in this FE study, the stress transfer process results inadditional column yielding and contributes to the lower ‘total’settlement improvement factors for the ‘with creep’ case.

5.4 Parametric studyThe parametric study assessed the effect of a range of differentsoil parameters (Cc or λ*, Cs or κ*, Cα or μ*, K0, ϕ′s) onnprimary, ntotal, ncreep and the average vertical effective stresses inthe soil and column. The parameters of the crust layer arefixed throughout, with only the relevant parameter in the

clay layer altered (the range of values examined span therange in Table 1). The influence of load level (pa) is alsoinvestigated.

5.4.1 Effect of K0The effect of K0 on nprimary, ntotal and ncreep is shown inFigure 9. The initial horizontal stresses generated in Plaxisincrease as K0 increases; intuitively the n values should increaseaccordingly because the larger horizontal stresses providemore lateral support to resist column bulging. Figure 9 indi-cates that nprimary increases as K0 increases but ncreepis relatively unchanged. Accordingly, percentage differencesbetween nprimary and ntotal increase as K0 increases. The averagevertical effective stresses (σ′yy) in the soil and column after 100years are compared in Figure 10 for the different K0 values. Asexpected, the vertical stresses (with and without creep) are

1

2

3

4

5

6

7

8

3 4 5 6 7 8 9 10

n

n

A/Ac

A/Ac

(a)

(b)

nprimary

ntotal

ncreep

63%

45%30%

1

10

100

3 4 5 6 7 8 9 10

nprimaryntotalncreepnprimary (Shahu et al., 2000)

ntotal (Madhav et al., 2009, 2010): t/t0 = 10

Figure 6. (a) nprimary, ntotal and ncreep, and (b) comparison with Madhav et al. (2009, 2010) and Shahu et al. (2000), t/t0 = end time/EOPtime

43




similar for all three K0 values, with a maximum difference of�5% over the range of K0 values considered for the ‘withoutcreep’ case.

5.4.2 Effect of μ*In this parametric study, the range of μ* values consideredhave varied from half to double that of the base case, resultingin creep ratios, (λ*−κ*)/μ*, of 42·8 and 10·7, respectively, span-ning the range in Figure 2. The creep ratio, (λ*−κ*)/μ*=5,reported by Lopes (2011) for the soft soils of the Tagus Basinin Portugal seems to be an outlier because the high μ*/λ* ratio

of 0·146 is out of kilter with the correlation proposed by Mesriand Godlewski (1977) encompassing a variety of natural soils.

The findings in Figure 11 indicate that ntotal reduces as μ*increases because the weighted effect of creep has more influ-ence (the nprimary values are unaffected). It is interestingto note that, at close spacings, the ncreep values increase mar-ginally for lower μ* values because full yielding of the granularmaterial does not take place. The average vertical effectivestresses in the soil and column after 100 years for the differentμ* values are compared with the ‘without creep’ case inFigure 12. For higher μ* values, additional stress is unloaded

Without creep

With creep

(a) (b)

(c) (d)

0

20

40

60

80

100

120

3 4 5 6 7 8 9 10

A/Ac

Without creep (Shahu et al., 2000)

With creep (Madhav et al., 2009, 2010): t/t0 = 10

0

200

400

600

800

1000

1200

1400

3 4 5 6 7 8 9 10

A/Ac



0

20

40

60

80

100

120

3 4 5 6 7 8 9 10

σ' yy

: kN

/m2

σ' yy

: kN

/m2

σ' yy

: kN

/m2

σ' yy

: kN

/m2

A/Ac

0

100

200

300

400

500

600

700

3 4 5 6 7 8 9 10

A/Ac

Without creep

With creep

Figure 7. Average σ′yy after 100 years: (a) soil, (b) stone column, (c) soil: comparison with Madhav et al. (2009, 2010) and Shahu et al.(2000) and (d) stone column: comparison with Madhav et al. (2009, 2010) and Shahu et al. (2000), t/t0 = end time/EOP time

(a) (b)

0

10

20

30

40

50

3 4 5 6 7 8 9 10

SCF



0

1

2

3

4

5

6

7

8

3 4 5 6 7 8 9 10

SCF

A/Ac A/Ac

Without creep

With creep

Figure 8. (a) SCFs after 100 years and (b) SCFs after 100 years: comparison with Madhav et al. (2009, 2010) and Shahu et al. (2000),t/t0 = end time/EOP time

44




1

2

3

4

5

6

7

8

9

10

3 4 5 6 7 8 9 10

n

A/Ac

nprimary (K0 = 1·0)nprimary (K0 = 0·7)nprimary (K0 = 0·54)ntotal (K0 = 1·0)ntotal (K0 = 0·7)ntotal (K0 = 0·54)ncreep (K0 = 1·0)ncreep (K0 = 0·7)ncreep (K0 = 0·54)

Figure 9. Effect of K0 on nprimary, ntotal and ncreep

(a)

(b)

40

50

60

70

80

90

100

110

3 4 5 6 7 8 9 10

A/Ac

Without creep (K0 = 0·54)

With creep (K0 = 0·54)





300

350

400

450

500

550

600

3 4 5 6 7 8 9 10

A/Ac







σ' yy

: kN

/m2

σ' yy

: kN

/m2

Figure 10. Effect of K0 on average σ′yy after 100 years: (a) soil and (b) stone column

45




1

2

3

4

5

6

7

8

3 4 5 6 7 8 9 10

n

A/Ac

nprimaryntotal (µ* = 0·0032)ntotal (µ* = 0·0065)ntotal ( µ* = 0·0130)ncreep (µ* = 0·0032)ncreep (µ* = 0·0065)ncreep (µ* = 0·0130)

Figure 11. Effect of μ* on ntotal and ncreep

(a)

(b)

40

50

60

70

80

90

100

110

3 4 5 6 7 8 9 10

A/Ac

A/Ac

Without creep

With creep (µ* = 0·0032)



300

350

400

450

500

550

600

3 4 5 6 7 8 9 10

Without creep




σ' yy

: kN

/m2

σ' yy

: kN

/m2

Figure 12. Effect of μ* on average σ′yy after 100 years: (a) soil and (b) stone column

46




from the soil (Figure 12(a)) and transferred to the column(Figure 12(b)). Consequently, the SCFs increase as μ* increases(Figure 13). These trends are in qualitative agreement withthose of Madhav et al. (2009, 2010).

5.4.3 Effect of κ*The effects of both halving and doubling the default value ofκ* (to 0·012 and 0·046) on nprimary, ntotal and ncreep are shownin Figure 14. Again, these κ* values are towards the lower andhigher end, respectively, of those quoted in Table 1. Lower κ*values result in higher nprimary and ntotal values, although thelatter do not increase to the same extent as the former becausethe ncreep values are relatively unaffected by κ*. The higher nvalues at lower κ* values can be explained as follows: for lowerκ* values, the settlements of untreated and treated groundreduce. However, the settlement of untreated ground will onlyreduce marginally (lightly overconsolidated soil), whereas the

settlement of treated ground will experience more of a reductionas the columns reduce the stress carried by the soil (resulting inan overconsolidation effect) so that κ* has more of an influence.Accordingly, the denominator (n= δuntreated/δtreated) reduces morethan the numerator and so n increases.

The average vertical effective stresses in the soil and column forthe three different κ* values without and with creep are plottedin Figure 15. The amount of stress transferred from the soil tothe column due to creep is relatively unaffected by κ*; this indi-cates that the load transfer process from soil to column due tocreep is not influenced by the unload–reload index.

5.4.4 Effect of λ*The effect of λ* on nprimary, ntotal and ncreep is shown inFigure 16. Values of λ*>0·162 have not been studied as thesoil profile would be too soft to provide sufficient lateral

0

1

2

3

4

5

6

7

8

9

3 4 5 6 7 8 9 10

SCF

A/Ac

Without creep

With creep ( µ* = 0·0032)



Figure 13. Effect of μ* on the SCFs after 100 years

1

2

3

4

5

6

7

8

9

3 4 5 6 7 8 9 10

n

A/Ac

nprimary (κ* = 0·012)nprimary (κ* = 0·023)nprimary (κ* = 0·046)ntotal (κ* = 0·012)ntotal (κ* = 0·023)ntotal (κ* = 0·046)ncreep (κ* = 0·012)ncreep (κ* = 0·023)ncreep (κ* = 0·046)

Figure 14. Effect of κ* on nprimary, ntotal and ncreep

47




(a)

(b)

40

50

60

70

80

90

100

110

3 4 5 6 7 8 9 10

A/Ac

Without creep (κ* = 0·012)

With creep (κ* = 0·012)





300

350

400

450

500

550

600

3 4 5 6 7 8 9 10

A/Ac







σ' yy

: kN

/m2

σ' yy

: kN

/m2

Figure 15. Effect of κ* on average σ′yy after 100 years: (a) soil and (b) stone column

1

2

3

4

5

6

7

8

3 4 5 6 7 8 9 10

n

A/Ac

nprimary (λ* = 0·162)nprimary (λ* = 0·122)nprimary (λ* = 0·081)ntotal (λ* = 0·162)ntotal (λ* = 0·122)ntotal (λ* = 0·081)ncreep (λ* = 0·162)ncreep (λ* = 0·122)ncreep (λ* = 0·081)

Figure 16. Effect of λ* on nprimary, ntotal and ncreep

48




support for granular columns without some form of geotextileencasement. Lower λ* values correspond to higher oedometricsoil moduli; accordingly Ec/Es reduces, resulting in lower nvalues. The average vertical effective stresses in the soil andcolumn after 100 years for the different λ* values, compared inFigure 17, indicate that the stresses are relatively independentof λ* for the range of values considered in this study (softercreep-prone soils).

5.4.5 Effect of ϕ′sThe friction angle of the Bothkennar Carse clay (ϕ′s = 34°) ishigher than that of other soft clays reported in the literature(Table 1), attributable to the significant amount of angular siltparticles and the relatively high organic content (Allman andAtkinson, 1992). The effect of ϕ′s on n (in the range 26°–34°)is shown in Figure 18; both nprimary and ntotal increase margin-ally as ϕ′s increases (ncreep is unaffected). The increases are

attributable to an increased K0, which is automatically updatedwhen ϕ′s is changed (e.g. Equation 13 with K0

nc = 1−sin ϕ′s). Theaverage vertical effective stresses in the soil and column after100 years are unaffected (e.g. Figure 19 (as was the case forthe different K0 values, see Figure 10)).

5.4.6 Effect of paThe effect of load level on nprimary, ntotal and ncreep is presentedin Figure 20. The range of load levels considered is basedon the ranges quoted by Castro and Sagaseta (2009), Ellouzeand Bouassida (2009) and Mitchell and Huber (1985). Atlower load levels, n values are larger; stone columns are moreeffective because there is less yielding. As a result, ncreep valuesat pa = 50 kPa are larger than those at either pa = 100 kPa orpa = 150 kPa, which are almost the same. The average verticaleffective stresses in both the soil and column increase as paincreases (e.g. Figure 21, after 100 years). The stress

(a)

(b)

40

50

60

70

80

90

100

110

3 4 5 6 7 8 9 10

A/Ac

Without creep (λ* = 0·162)

With creep (λ*= 0·162)


With creep (λ* = 0·122)



300

350

400

450

500

550

600

3 4 5 6 7 8 9 10

A/Ac







σ' yy

: kN

/m2

σ' yy

: kN

/m2

Figure 17. Effect of λ* on average σ′yy after 100 years: (a) soil and (b) stone column

49




1

2

3

4

5

6

7

8

9

3 4 5 6 7 8 9 10

n

A/Ac

nprimary (φ's = 26)nprimary (φ's = 30)nprimary (φ's = 34)ntotal (φ's = 26)ntotal (φ's = 30)ntotal (φ's = 34)ncreep (φ's = 26)ncreep (φ's = 30)ncreep (φ's = 34)

Figure 18. Effect of ϕ′s on nprimary, ntotal and ncreep

(a)

(b)

40

50

60

70

80

90

100

110

3 4 5 6 7 8 9 10

A/Ac

A/Ac

Without creep (φ's = 34)

With creep (φ's = 34)





300

350

400

450

500

550

600

3 4 5 6 7 8 9 10







σ' yy

: kN

/m2

σ' yy

: kN

/m2

Figure 19. Effect of ϕ′s on average σ′yy after 100 years: (a) soil, (b) stone column

50




1

2

3

4

5

6

7

8

9

10

11

3 4 5 6 7 8 9 10

n

A/Ac

nprimary ( pa = 50 kPa)nprimary ( pa = 100 kPa)nprimary (pa = 150 kPa)ntotal (pa = 50 kPa)ntotal (pa = 100 kPa)ntotal (pa = 150 kPa)ncreep (pa = 50 kPa)ncreep (pa = 100 kPa)ncreep (pa = 150 kPa)

Figure 20. Effect of pa on nprimary, ntotal and ncreep

(a)

(b)

30

50

70

90

110

130

150

3 4 5 6 7 8 9 10

A/Ac

A/Ac

Without creep ( pa = 50 kPa)

With creep ( pa = 50 kPa)





100

200

300

400

500

600

700

800

3 4 5 6 7 8 9 10


With creep (pa = 50 kPa)





σ' yy

: kN

/m2

σ' yy

: kN

/m2

Figure 21. Effect of pa on average σ′yy after 100 years: (a) soil and (b) stone column

51




transfer process from soil to column due to creep is more sig-nificant at pa = 50 kPa than at pa = 100 kPa or pa = 150 kPa.At pa = 50 kPa without creep, the column has not fully yieldedand so there is more scope for stress transfer from soil tocolumn due to creep.

6. Incorporating creep into thevibro-replacement design process

The FE output in this paper has been used to developan empirical design approach to incorporate creep into thevibro-replacement design process. This approach, developed

for end-bearing columns, is closed form, and can be used inconjunction with any pre-existing primary settlement designmethod, although as mentioned in Section 2.2, the methodsderived by Castro and Sagaseta (2009) and Pulko et al. (2011)are recommended.

The FE output presented in Section 5 is presented as a ratioof (ncreep−1)/(nprimary−1) against A/Ac in Figure 22. The ratio(ncreep−1)/(nprimary−1) is used instead of (ncreep)/(nprimary) toensure that the value of ncreep predicted using Equation 15 willalways be >1. In the interest of simplicity, the formula

(a) (b)

(c) (d)

(e) (f)

0

0·1

0·2

0·3

0·4

0·5

0·6

0·7

3 4 5 6 7 8 9 10

n cre

ep –

1/n

prim

ary

– 1

n cre

ep –

1/n

prim

ary

– 1

n cre

ep –

1/n

prim

ary

– 1

n cre

ep –

1/n

prim

ary

– 1

n cre

ep –

1/n

prim

ary

– 1

n cre

ep –

1/n

prim

ary

– 1

A/Ac A/Ac

A/Ac A/Ac

A/Ac A/Ac

λ* = 0·081 λ* = 0·122 λ* = 0·162

Equation 15

0

0·1

0·2

0·3

0·4

0·5

0·6

0·7

3 4 5 6 7 8 9 10

κ* = 0·012 κ* = 0·023 κ* = 0·046

Equation 15

0

0·1

0·2

0·3

0·4

0·5

0·6

0·7

3 4 5 6 7 8 9 10

µ* = 0·0032 µ* = 0·0065 µ* = 0·0130

Equation 15

Equation 15

Equation 15

Equation 15

0

0·1

0·2

0·3

0·4

0·5

0·6

0·7

3 4 5 6 7 8 9 10

K0 = 0·54 K0 = 0·7 K0 = 1·0

0

0·1

0·2

0·3

0·4

0·5

0·6

0·7

3 4 5 6 7 8 9 10

φ's = 26 φ's = 30 φ's = 34

0

0·1

0·2

0·3

0·4

0·5

0·6

0·7

3 4 5 6 7 8 9 10

pa = 50 kPa pa = 100 kPa pa = 150 kPa

Figure 22. (ncreep−1)/(nprimary−1) against A/Ac: (a) influence of λ*, (b) influence of κ*, (c) influence of μ*, (d) influence of K0, (e) influenceof ϕ′s and (f) influence of pa

52




developed for ncreep is a function of A/Ac only. The ratio islowest at A/Ac = 3 (i.e. larger differences between nprimary andncreep) and increases as A/Ac increases. In general, Equation 15(superimposed on Figure 22) provides a good and slightly con-servative match to the FE data in the majority of cases; thedeviation at pa = 50 kPa occurs due to the absence of yielding.This design equation will only be applicable to creep-pronesoils (λ* ≳0·8, see Table 1), for which the influence of modularratio (i.e. λ*) is small.

15:ðncreep � 1Þðnprimary � 1Þ ¼ 0�225þ 0�01 ðA=AcÞ

Having established an expression for ncreep , ntotal can be calcu-lated as a weighted average of nprimary as predicted by a pre-existing settlement design method and ncreep (from Equation 15),as shown in the following equation

16: ntotal ¼ nprimaryw1 þ ncreepw2

where w1 and w2 are weighting factors dependent on the per-centages of primary and creep settlement anticipated in anequivalent untreated profile, which depend on the creep ratio,(λ*−κ*)/μ*, of the untreated soil profile. The percentages canbe worked out using the 1D compression formulae outlined inEquation 17 (for σ0 +Δσ< σp), Equation 18 (σ0 +Δσ> σp) andEquation 19 (creep component), where Δσ denotes the load,H is the layer thickness.

17: δ ¼ H Cs1þ e0 log

σ0 þ Δσσ0

� �

18: δ ¼ H Cs1þ e0 log

σpσ0

� �þH Cc

1þ e0 logσ0 þ Δσ

σp

� �

19: δ ¼ H Cα1þ e0 log

tt0

� �

7. Comparison of a new empirical approachwith Moorhead (2013)

To the authors’ knowledge, there are no field studies explicitlyisolating creep settlements with which to test the applicability ofEquation 15. However, Equation 15 is considered in Figure 23in the context of the laboratory measurements reported byMoorhead (2013). Moorhead’s (2013) testing programmeinvolved the use of stone columns to treat two different types ofsoil, considered to have insignificant (kaolin) and significant(Belfast soft silt, known as ‘sleech’) creep potential, respectively.Both rigid raft and isolated foundation scenarios were con-sidered. For the isolated foundation loading scenario modelled,the initial conditions for the ‘sleech’ layer were different for theuntreated and treated cases (for the reinforced case, theundrained shear strength of the clay bed was 20% larger),leading to over-predicted settlement improvement factors.Accordingly, these data are not included in Figure 23.

Although the laboratory results clearly exhibit significantscatter, nine of the 12 datapoints in Figure 23 for the raftloading scenario (overconsolidated condition, ‘OC’) are ingood agreement with Equation 15. This provides some confi-dence in the applicability of Equation 15 to soft soils, althoughfurther validation is required, ideally from full-scale researchtrials or stone columns in service.

8. ConclusionsAxisymmetric FE analysis techniques have been used to studythe impact of creep on the behaviour of stone columns in softcohesive soil deposits.

0

0·5

1·0

1·5

2·0

2·5

0 1 2 3 4 5 6

n cre

ep –

1

nprimary – 1

Kaolin (OC)

Sleech (OC) Equation 15 with A/Ac = 6·67

Figure 23. Comparison of Equation 15 with data from Moorhead (2013)

53




The FE analyses have indicated that n values in creep-pronesoils will be over-predicted unless creep is accounted for indesign. In addition, the findings have highlighted that theMadhav et al. (2009, 2010) formulation, which is based onelastic theory, will lead to over-predicted n values in soft soils.The Madhav et al. (2009, 2010) formulation also predicts a soiloverconsolidation effect due to the stress transfer process fromsoil to column while the soil creeps (additional to Bjerrum’s(1967) ageing effect). However, the FE analyses in this paper(which account for plasticity) suggest that this soil overconsoli-dation effect will not benefit the combined soil–column systembecause the surplus load transferred to the stone column, whichhas already reached its active state/yield point, inducesadditional yielding.

The influence of a practical range of soil parameters on nprimary,ntotal and ncreep has been studied to assess the applicability of thefindings to a spectrum of soft clays. The influence of the differentparameters on the average vertical effective stresses in the soiland column for A/Ac = 3, 6 and 10 has also been studied. Theoutcomes may be summarised as follows

& Regardless of the parameters adopted, the presence ofcreep gives rise to lower n values than if only primaryconsolidation was considered.

& The magnitude of vertical stress transferred from soilto column due to creep is more pronounced in soils withhigher μ* values because additional yielding is induced inthe columns.

& The amount of stress transferred from soil to columndue to creep is largely independent of λ*, κ* and ϕ′s.

& The stress transfer due to creep is more significantat the lower load level considered in this studybecause full yielding of the granular material is notinduced for the ‘without creep’ case and so there ismore scope for stress transfer from soil to column dueto creep.

The FE output has been used to develop a simple empiricalapproach to incorporate potential improvement to creep settle-ments into the vibro-replacement design process. The approachis applicable to end-bearing columns, with spacings in therange 3

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1. Introduction2. Stone column settlement design methods2.1 The 18unit cell 19Equation 12.2 Elastic against elastic 13plastic methods2.3 Review of Madhav etal. (2009, 2010)Equation 2Equation 3Equation 4Equation 5Figure 1Equation 6Equation 7Equation 8Equation 9

3. Numerical modelling preliminaries3.1 Soil models3.2 Range of soft clay soil parameters3.3 The numerical modelFigure 2Table 1Equation 10

4. Numerical modelling4.1 Analysis stagesEquation 114.2 Establishing the influence of creep on nEquation 12Figure 3Table 2Equation 13Equation 14Table 3Figure 44.3 18Base case 19 parameters

5. Parametric study results5.1 18Base case 19 time 13settlement behaviour5.2 18Base case 19 settlement improvement factors5.3 18Base case 19 soil and column stressesFigure 55.4 Parametric study5.4.1 Effect of K0

Figure 65.4.2 Effect of ;C*

Figure 7Figure 8Figure 9Figure 10Figure 11Figure 125.4.3 Effect of ;A*5.4.4 Effect of ;B*

Figure 13Figure 14Figure 15Figure 165.4.5 Effect of =5 32s5.4.6 Effect of pa

Figure 17Figure 18Figure 19Figure 20Figure 21

6. Incorporating creep into the vibro-replacement design processFigure 22Equation 15Equation 16Equation 17Equation 18Equation 19

7. Comparison of a new empirical approach with Moorhead (2013)8. ConclusionsFigure 23

ACKNOWLEDGEMENTSREFERENCESAllman and Atkinson 1992Ambily and Gandhi 2007Azzouz et al. 1990Balaam and Booker 1981Barksdale and Bachus 1983Bjerrum 1967Black et al. 2011Bodas Freitas 2008Brinkgreve et al. 2011Casagrande 1936Castro and Sagaseta 2009Degago 2011Degago et al. 2011Ellouze and Bouassida 2009Fatahi et al. 2012Gäb et al. 2008Hight et al. 1992ICE (Institution of Civil Engineers) 1992Jardine et al. 1995Killeen and McCabe 2014Kok Shien 2013Lee and Pande 1998Lo and Hinchberger 2006Lopes 2011Madhav et al. 2009Madhav et al. 2010McCabe et al. 2009Mesri and Godlewski 1977Mitchell and Huber 1985Mitchell and Kelly 2013Moorhead 2013Nash 2001Nash and Ryde 2001Nash et al. 1992Neher et al. 2003Priebe 1995Pugh 2016Pulko et al. 2011Schanz et al. 1999Schweiger and Pande 1986Sexton 2014Sexton and McCabe 2012Sexton and McCabe 2013Sexton and McCabe 2015Sexton and McCabe 2016Sexton et al. 2014Sexton et al. 2016Shahu et al. 2000Simons and Som 1970Suhonen 2010�0uklje 1957Van Impe and De Beer 1983Vermeer and Neher 1999Vermeer et al. 1998Watts et al. 2000Yin and Graham 1989Yin and Graham 1994Yin and Graham 1996Yin and Karstunen 2011

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