numerical study

foraWideRangeofSands

NobutakaYamamoto1;MarkF.Randolph2;andItaiEinav3

Abstract:Thispaperpresentsanumericalinvestigationoftheeffectoffoundationsizeontheresponseofshallowcircularfoundationsonsiliceousandcalcareoussands.ThestudyisbasedonthepredictivecapabilitiesoftheMIT-S1soilmodelforsimulatingboththecompressionandshearbehaviorsofnaturalsandsoverawiderangeofdensities,K0valuesandconfiningpressures.Thepaperhighlightsthevariationsinthedeformationmechanismsforthesiliceousandcalcareoussandscases.Theassessmentofthebearingcapacityfactor,N␥,isexamined,showingadramaticdecreaseinthevalueswithincreasingfoundationsizeforthecaseoffootingsoncalcareoussands,eventuallyconvergingtoaterminalN␥value.Atthisstagethesandresistanceisinsensitivetovariationsininitialdensityandfoundationsizebecausethesandtendstolooseitsinitialcharacteristicsduetograincrushing,leadingthematerialrapidlytowardultimateconditions.Inthesilicioussandcase,itisfoundthat,eventually,forextremelylargefootingdiameters,thedeformationmechanismprogressestowardapunchingshearmechanism,ratherthantheclassicalrapturepatternaccompaniedbysurfaceheaveasemployedincurrentbearingcapacityequations.Adimensionaltransitionbetweenthefailuremechanismscanclearlybedefined,referredtoasa“criticalsize”intheN␥–Drelationship.

DOI:10.1061/͑ASCE͒1090-0241͑2009͒135:1͑37͒

CEDatabasesubjectheadings:Sand;Calcareoussoils;Silica;Shallowfoundations;Sizeeffect;Finiteelementmethod;Numericalanalysis.

Introduction

Thebearingcapacityoffoundationsongranularmaterialshasbeenstudiedextensivelyasoneofthefundamentalproblemsofgeotechnicalengineering.ThemostcommonmethodtoestimatethebearingcapacityistheclassicalTerzaghiequationthatin-cludesthreefactors:theNcfactorforcohesion,theNqforembed-mentdepth,andtheN␥fortheself-weightcomponent.Thesedifferentfactorsaremodifiedfortheparticularloadingconditionandmaterialcaseinhand.TheN␥factorisofprimaryimportanceforshallowfoundationsonsandsbutitisextremelysensitivetovariationsinthematerialproperties.Earlyexperimentalstudiesofthisfactorinsandweremainlyconcernedwithrelativelysmallmodelfoundations,testedinnatural1gconditions.Itwasreal-izedthatN␥decreaseswithincreasingfootingwidthordiameter,andthisisnowwidelyrecognizedasthe“foundationsizeeffect”͑DeBeer1963͒.DeBeerreasonedthatthefoundationsizeef-fectresultsfromthefactthatthestrengthcriterionofsands

Engineer,AdvancedGeomechanics,4LeuraSt.,Nedlands,WA,6009,Australia;formerly,Ph.D.Student,CentreofOffshoreFoundationSystems,Univ.ofWesternAustralia,Crawley,WA6009,Australia.E-mail:nobutakay@ag.com.au2

Professor,CentreforOffshoreFoundationSystems,Univ.ofWesternAustralia,35StirlingHighway,Crawley,WA6009,Australia.E-mail:randolph@civil.uwa.edu.au3

SeniorLecturer,SchoolofCivilEngineeringJ05,Univ.ofSydney,Sydney,NSW2006,Australia.E-mail:I.Einav@civil.usyd.edu.au

Note.DiscussionopenuntilJune1,2009.Separatediscussionsmustbesubmittedforindividualpapers.ThemanuscriptforthispaperwassubmittedforreviewandpossiblepublicationonMarch15,2007;ap-provedonApril30,2008.ThispaperispartoftheJournalofGeotech-nicalandGeoenvironmentalEngineering,Vol.135,No.1,January1,2009.©ASCE,ISSN1090-0241/2009/1-37–45/$25.00.

1shouldbenonlinear,withthefrictionangledecreasingwithin-creasingstresslevel,ratherthanlinearasintheconventionallinearMohr–Coulombcriterion.Thenonlinearfailureenvelopearisesfromthestressdependencyofdilation,whichbyitselfarisesfromparticlerearrangementandcrushing͑LeeandSeed1967;Bolton1986͒.

Thereareseveralnumericalapproachesforassessingthein-fluenceofN␥usingnonassociatedconstitutivemodels͑i.e.,mod-elsthatincorporateadilationanglethatisnotequaltothefrictionangle͒.FrydmanandBurd͑1997͒andEricksonandDrescher͑2002͒studiedtheeffectofthedilationangleonN␥forstripandcircularfootings,respectively,usinganonassociatedMohr–Coulombmodel.Theyfoundthattheeffectofdilationangleisnegligibleatlowfrictionangles,butquiteimportantforfrictionanglesgreaterthan35°,especiallyforthecaseofroughcircularfootings.However,thesepreviousnumericalstudiesarelimitedbythefactthattheMohr–Coulombmodelcannotcapturesuffi-cientlywellthestressanddensitystatedependencyofsandbe-haviororthecompressibilityofsands.

Theintentionoftheworkreportedherewastoperformamorecomprehensivenumericalstudythataccountsformanymorebe-havioralaspectsofsand.ForthatpurposetheMIT-S1constitutivemodel͑PestanaandWhittle1999͒wasadoptedasthatmodelhassufficientcomplexitytosimulatebothcompressionandshearbe-haviorsofnaturalsandsoverawiderangeofdensities,K0values,andconfiningpressuresusingasinglesetofmodelparametersofagivensandtype.

Insummary,thispaperpresentsanumericalinvestigationofthefoundationsizeeffectinthecaseofshallowcircularfootingsonsiliceousandcalcareoussandsusingtheMIT-S1model.Fol-lowingabriefdescriptionontheMIT-S1model,thestrengthcharacteristicsofsiliceousandcalcareoussandsarediscussedinthecontextofdrainedtriaxialsheartestresults.Theeffectsare

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Table1.IndexPropertiesofSoils

Siliceous

Property

Toyourasand

Goodwynsand

DogsBaysanda

Calcareous

GoodwynsiltCalciumcarbonate͑94%͒

Skeletalgrain

2.770.03452.401.21

MineralogyQuartz,feldspar,magnetiteCalciumcarbonate͑94%͒Calciumcarbonate͑98%͒GrainshapeSubangularSkeletalgrainSkeletalgrain

2.652.722.75Specificgravity,Gs

0.16–0.200.1–0.20.2Meanparticlesize,D50͑mm͒

1.3–1.710–152.06Coefficientofuniformity,Cu

0.982.32–1.972.21–1.83Maximumvoidratio,emax

0.61–0.581.41–0.941.48–0.98Minimumvoidratio,emin

aPropertiesofDogsBaysandwerereassessedanddifferentfromthevalueprovidedbyPestana͑1994͒.

thentranslatedtoexplainthefoundationproblemforbothtypesofsand,followedbyadiscussionofthefoundationsizeeffectintermsoftheN␥factor.

ModelingSandsUsingtheMIT-S1Model

FulldetailsoftheMIT-S1modelcanbefoundinPestanaandWhittle͑1999͒.AccordingtoPestanaetal.͑2002͒,themodeliscapableofsimulatingmanybehavioralcharacteristicsofsandbehavior,includingnonlinearityofthecompressioncurvesandcriticalstatelinesone–lnpЈplots,thedilatancybehaviorofsands,andthevariationofpeakfrictionangleasafunctionofstresslevelanddensity.Themodelcancapturearangeofdifferentcharacteristicsofbothcompressibleandincompressiblegranularmaterialsthroughappropriateadjustmentofthemodelparameters.

TheMIT-S1modelrequires13inputparameterstomodelthebehavioroffreshlydepositedsand͑whichisthetypeofsandthispaperisconcernedwith͒.AccordingtoPestanaandWhittle͑1999,2002͒theseparameterscanbedeterminedfromstandardlaboratorytests.

Thispaperfocusesontwodifferenttypesofsands,Toyourasiliceoussand͑fromJapan͒,andGoodwyncalcareoussand͑fromtheNorthWestShelfofAustralia͒.ThemodelparametersforthesesandsweredeterminedinPestanaetal.͑2002͒andYamamotoetal.͑2008͒.ThemodelparametersforDogsBay

Table2.MIT-S1ModelParametersforVariousSoils

calcareoussandandGoodwyncalcareoussiltarealsoprovidedtoenableacompletediscussiononthefoundationsizeeffect.ThephysicalpropertiesofthesandsandthesiltaresummarizedinTable1,andthemodelinputparametersaregiveninTable2.TheparticlesizedistributionsforToyourasiliceoussand͑Ishihara1993͒,DogsBaycalcareoussand͑Coop1990͒,Goodwyncalcar-eoussand͑Ismail2000͒,andGoodwyncalcareoussilt͑Finnie1993͒areshowninFig.1.Asmaybeseen,theDogsBaycalcar-eoussandhaslargerparticlesthantheToyourasiliceoussand.Further,itisnotedthattheGoodwynsandisrelativelywellgradedwith30%finescontent.CompressionBehavior

Fig.2showstheMIT-S1predictionsofthecompressioncurvesofbothsiliceousandcalcareoussands.Theinitialdensitiesandcur-vatureofthecompressioncurvesvarysignificantly,butthemodelcapturesthesevariationswell.Calcareoussandshavehigherini-tialvoidratiosandgreaterreductionofvolumethansiliceoussands.Thecriticalstatelinesofthesandsarealsosignificantlydifferent,butagainthemodelpredictsthemnicely.ShearBehavior

Fig.3showstheMIT-S1predictionsfordrainedisotropicallyconsolidatedsheartestsonsiliceousandcalcareoussandswithdifferentinitialdensitiesbutthesameconfiningstress͑100kPa͒.

Siliceous

TesttypeCompressiontest

Symbol␳cprefЈ␪K0NC␮0Ј␻␾cs␾mrЈnpm␺Cb␻s

Physicalmeaning

Compressibilityatlargestresses͑LCCregime͒

ReferencestressatunityvoidratiofortheH-LCC͑kPa͒

Firstloadingcurvetransitionparameter

K0intheLCCregime

Poisson’sratio

ParameterfornonlinearPoisson’sratio

Criticalstatefrictionangle͑°͒

Peakfrictionangleasafunctionofvoidratio͑°͒

ConstantofpeakfrictionangleGeometryofboundingsurfaceRateofevolutionofanisotropySmallstrainstiffnessparameterSmallstrainnonlinearityparameter

Toyourasanda0.3705,500

0.2000.4900.2331.0031.028.52.450.5550.07502.50

GWsandb0.3502,500

0.9000.4900.1502.0039.660.02.000.3550.04503.00

Calcareousb

DBsandb0.3504,000

0.4000.5100.2001.0046.080.02.000.5550.07502.50

GWsiltb0.2502,000

0.9000.4500.2002.0040.072.02.000.3050.04503.0

K0consolidationtest

Sheartest

Sheartestwithlocalmeasurementsystems

ab

Pestana͑1994͒.

GW=Goodwyn;DB=DogsBay.

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100500

SiliceoussandsToyourasand(Ishihara,1993)CalcareoussandsDogsBaysand(Coop,1990)Goodwynsand(Ismail,2000)Goodwynsilt(Finnie,1993)Line

400Deviatoricstress,q(kPa)80Percentagefiner60300

e00.950.900.800.700.60

40200

20100

01E-41E-30.010.11SandFineMediumCoarse10Grainsize(mm)ClaySiltGravel00

(a)

CIDtests

Toyourasiliceoussandp'0=100kPa

5

10

15

20

25

30

ASTM(D422;D653)Shearstrain,Hs(%)

500

CIDtestsGoodwyncalcareoussandp'0=100kPaTriaxialTeste0=1.1(Finnie,1993)Fig.1.Particlesizedistributionsforsands

Deviatoricstress,q(kPa)Insiliceoussand,densersamplesexhibitaclearpeakstressatrelativelysmallstrainlevels,whereasnopeakstressisfoundforloosersamples.Ontheotherhand,allcalcareoussamplesshowcontractivebehavior,althoughtheexperimentalresponsefromFinnie͑1993͒forrelativedenseGoodwynsandshowsaslightpeakatsmallstrainlevels.

SensitivityStudyoftheMIT-S1Parameters

Asmentionedearlier,theMIT-S1modelrequires13modelpa-rameterstodefinethebehaviorofsand.Althoughtheparametersshouldbespecifiedprecisely,theparticularshallowfootingprob-leminthispapertendstobedominatedbyonlyafewparameters.Yamamoto͑2006͒carefullyinvestigatedtheeffectofthedifferentmodelparametersontheresponseofshallowcircularfootingsonsiliceousandcalcareoussands.Asummaryofthesesensitivity

400

300

200

Line100

00(b)

e01.901.751.601.451.301.1030510152025Shearstrain,Hs(%)

Fig.3.Triaxialdrainedsheartestsresultsforsiliceousandcalcareoussands:͑a͒siliceoussand;͑b͒calcareoussand

2MIT-S1predictionVoidRatio,e10.90.80.70.60.50.40.310100100010000100000MeanEffectiveStress,p'(kPa)SiliceousconsolCalcareousconsolSiliceousCSLCalcareousCSLanalysesisgiveninTable3.Thecompressionparameters,prefЈand␪,andtheshearparameter,m,arethemostsignificant,whereastheremainingparametershavelesseffect.Itisfoundthattheshearparameters͑apartfromm͒havelittleeffectontheresultsforcalcareoussand,implyingthatthebearingresponseoncalcar-eoussandisdominatedmorebythecompressioncomponentthanbyshear.Hence,fortheshallowfootingproblemtherelativelylargenumberof13parameterswasreducedtoamoremanageablestudyinvolvingthreesignificantparameters.

EffectsofStressLevel,Density,and

CompressibilityontheStrengthCharacteristicsofSands

Theeffectsofstresslevel,density,andcompressibilityareofgreatimportanceforassessingthebehaviorofsands.Theeffectscanbecapturedthrougharelationshipbetweenthepeakfrictionangle,␾Ј0,andvoidp,theinitialmeaneffectivestressatfailure,pЈratio,e.

Fig.4illustratestherelationshipbetween␾Ј0andeforthep,pЈ

Fig.2.Consolidationcurvesandcriticalstatelinesforsiliceousand

calcareoussands

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Table3.SignificanceoftheMIT-S1ModelParametersintheShallowFoundationAnalysisSymbolSiliceoussandCalcareoussandϫ᭺᭺—ϫϫϫ⌬ϫ᭺ϫϫϫϫ␳c᭺prefЈ᭺␪—K0NCϫ␮0Јϫ␻ϫ␾cs᭺␾mrЈ᭺np᭺mϫ␺Cb⌬⌬␻sNote:᭺=significant,⌬=slight,ϫ=negligible.14CIDtests1210I'p-I'cs(deg.)PredictionSiliceousSandCalcareousSandTriaxialTest(Goodwyn)e0=1.1e0=1.40.6=e0820101.3=e01.451.60.91000.80.7100010000ToyourasiliceousandGoodwyncalcareoussands.Thepeakfric-tionanglesatlowerstresslevelsforToyourasiliceoussandareinitiallyonlyweaklydependentontheincreaseinpressure,butthisdependencythenstrengthenstoarapidreductionwithin-creasingconfiningpressure.Athigherstresslevels,thepeakfric-tionangleseventuallyconvergetothecriticalstatevalues͑i.e.,␾ЈЈ͒at“criticalstresses,”assuggestedbyVesicandCloughp=␾cs

͑1968͒.Itisnoticedthatthecriticalstressdecreasesasthedensitydecreases.Thepeakfrictionanglesforcalcareoussandsalsode-pendonthecombinedinfluenceofeandpЈ0.However,theyre-ducerapidlywithincreasingpЈ0,evenatlowstresslevel.Thecriticalstressesforcalcareoussandsaresignificantlylowerthanforsiliceoussands.ThreetriaxialcompressiontestresultsusingGoodwynsandareplottedinFig.4,onefore0=1.1͓fromFinnie͑1993͔͒andtwoothersfore0=1.4͓fromSharma͑2004͔͒.TheMIT-S1predictionsunderestimatethepeakfrictionanglesforthesedata,whichisconsistentwiththeslightpeakindeviatorstressobservedintriaxialtests͑Fig.3͒.

Thevariationofpeakfrictionangleraisesquestionsontheapplicabilityofconventionalbearingcapacitytheories,whicharebasedonconstantfrictionanglewithdepth͑normalizedbyfoun-dationsize͒.Forexample,ananalysisofa10mdiameterfoun-dationwithpracticalsettlementlimitsof5–10%offoundationdiameter͑orwidth͒maybebasedoninitialstressesof40kPa͑multiplyinghalfofthediameter,5m,byasoileffectiveunitweightof8kN/m3͒.However,whenthesamesettlementlevelisappliedtoa100mdiameterfoundation,thecorrespondingstresslevelissimplytentimes͑400kPa͒thatforthe10mdiameterfooting.Atthatstresslevel,thepeakfrictionanglesarenolongerconstantwithdepth.Thepeakfrictionanglesforcalcareoussandsareobviouslynotconstantat40kPa,thusforthissandthecon-ventionalbearingcapacityformulasdonotfitevenforamoderatefoundationsize.

Initialmeaneffectivestress,p'0(kPa)

Fig.4.Peakfrictionangleandinitialstaterelationshipsforsiliceousandcalcareoussands

hasbeentakententimeshigher,avoidingtheneedtomodifythefinite-elementmeshes.Thustheincreaseinthefoundationsizeissimulatedsimplybyincreasingtheinitialstressgradient.Pressure–DisplacementCurves

Fig.5͑a͒showsN␥and␦/Drelationshipsfor100mdiametersmoothandroughfootingsonsiliceoussand,withthe10mdi-ameterresultsalsoplottedforcomparison.Thebearingresponseofthelargescaleroughfootingshowsnopeakvaluebutratherincreasescontinuouslywithincreasingpenetrationdepth.Thisisbecausethecompressioncomponentofthematerialdominatesthebearingresponseasthefoundationsizeincreases.Forthe100mdiametersmoothfootingcase,however,anultimatebear-ingcapacityisstillobservedalthoughitneedsmuchlargerverti-caldisplacementthanforthesmallfooting.Thisappearstobebecausethedeformationmechanismforsiliceoussandprogres-sivelyshiftstowardpunchingshearwithincreasingsizeoffoun-dation.Itisworthnotingthattheeffectofroughnessforlargerfoundationsismuchsmaller.

Thebearingresponsesoncalcareoussandwithdifferentfoun-dationsizesshowsimilartrendsbutthe100mdiameterfounda-tionshowsamorelinearresponse͓Fig.5͑b͔͒,andthemobilizedN␥forthe100mcaseissmaller.DeformationMechanisms

Asdescribedearlier,atransformationinthemechanismsfromsmalltolargefoundationsmaybeseen,inparticularforroughfootingsonsiliceoussand.Fig.6͑a͒showsthatatapenetrationof10%ofthediametertheamountofsurfaceheavereducessignifi-cantlywithincreasingdiameter.However,forthesmoothfootinganalysis͓Fig.6͑b͔͒,aclassicalrupturefailurepatternwithsurfaceheaveisstillevidentforthe100mdiametercalculationsalthoughmoreobviousdownwarddeformationsareexhibitedatshallowerpenetration.

Theincrementaldisplacementvectorsfor10and100mdiam-eterfootingsoncalcareoussandshowalmostidenticaldefor-

ResponsesofShallowFoundationsonSandsThefollowingdescribesnumericalresultsfortheresponseof10and100mdiameterfootingsonsiliceousandcalcareoussands.Initialvoidratiosatthegroundsurface,e0,andeffectiveunitweights,␥Ј,are0.8͑dense͒and8kN/m3forthesiliceoussand,and1.3͑mediumdense͒and7kN/m3forthecalcareoussand.Tocarryoutthe100mdiameteranalyses,theeffectiveunitweight

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MobilizedBearingResistance,NJ=2qb/J'D100

80

ShallowfoundationanalysisToyourasiliceoussandK0=1.0,e0=0.8J'=8kN/m3SmoothRough60

D=10m40

20

D=100m005101520(a)

Displacement/Diameter,G/D(%)

100

ShallowfoundationanalysisGoodwyncalcareoussanddense(e0=1.3),K0=1.0J'=7kN/m3MobilizedBearingResistance,NJ=2qb/J'D80

60

SmoothRoughFig.6.Incrementaldisplacementvectorsafterapenetrationof10%ofthediameter:͑a͒siliceoussand,rough,D=10m͑left͒,100m͑right͒;͑b͒siliceoussand,smooth,D=10m͑left͒,100m͑right͒;and͑c͒calcareoussand,smooth,D=10m͑left͒,100m͑right͒

40

D=10m20

D=100malsoallowsapossibledeductionofthedimensionaltransitionbetweendilativeandcontractiveresponsesofthesoil.

200051015SiliceousSand

Fig.7summarizesthebearingresponsefromanalyseswithdif-ferentfootingsizesoffullysmoothshallowcircularfootingsonsiliceoussand,byplottingtheN␥–␦/Drelationshipsfore0=0.8͑loose͒,andN␥-␦/Dfore0=0.65͑dense͒,where␦denotesthefootingdownwarddisplacement.TheeffectofthefoundationsizehasbeenrecognizedexperimentallywiththemobilizedN␥de-creasingwithincreasingdiameter͑e.g.,DeBeer1963͒,butwithexperimentalevidenceonlyoverarelativelysmalldiameterrange.ThenumericalpredictionsusingtheMIT-S1modelsuggestthatthefoundationsizeeffectexistsforlargerfoundationsaswell.

Moreover,asexpected,atransitionfromdilativetocontractivedeformationscanbeseenasthefoundationsizeincreases.Thesmallerfootingstendtoshowdilativebehaviorwithclearpeakstress,whereasthelargerfoundationspresentcontractivere-sponseandexhibitlowermobilizedN␥values.Thisisalsore-flectedfromtheresultsofdrainedtriaxialtestswithdifferentinitialvoidratiosasshowninFig.3.Fig.8showsN␥–Drela-tionshipsforlooseanddensesiliceoussands.TwoN␥valuesareshown,onecorrespondingtothepeakbearingresistance͑ifoneexists͒andtheothercorrespondingto␦/D=10%͑shownonlyifN␥keepsincreasingfor␦/Dgreaterthan10%͒.Thetwo

(b)

Displacement/Diameter,G/D(%)

Fig.5.Shallowfoundationresponsesforsiliceousandcalcareoussands:͑a͒siliceoussand;͑b͒calcareoussand

mationpatternsatallpenetrationlevels͓Fig.6͑c͔͒.Thesoilbe-neaththefootingscompressalmostinaone-dimensionalverticalmanner.

EffectofFoundationSizeonBearingCapacityFactor,N␥

Thefollowingexplorestheeffectoffoundationsizeonthemo-bilizedbearingresistancefactor,N␥.Thiseffectoffoundationsizehasbeenexplainedpreviouslyasduetothestressdependencyofgranularmaterials͑DeBeer1963;HettlerandGudehus1988;Kusakabeetal.1991͒,ormorepreciselyonthestressdependencyofthepeakfrictionangles.ThenumericalinvestigationusingtheMIT-S1modelprovidesfurtherexplanationsofthiseffectand

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MobilizedBearingResistance,NJ=2qb/J'DCircularfootingToyourasiliceoussand57J'=8kN/m3MobilizedBearingResistance,NJ=2qb/J'D100

2001801601401201008060402001e0=0.8PeakG/D=10%e0=0.6580

K0=0.49e0=0.65SmoothCircularfoundationsToyourasiliceoussandCentrifugetests(Okamuraetal.,1997)3e0=0.65,J'=9.7kN/m60

15102030406070100Diameter(m)40

CriticalsizeDcr20

Dcr10100005101520(a)

100

Displacement/Diameter,G/D(%)CircularfoundationsToyourasiliceoussand80

Diameter,D(m)

Fig.8.N␥andDrelationshipsforsiliceoussand

MobilizedBearingResistance,NJ=2qb/J'DJ'=8kN/m3K0=0.49e0=0.8Smooth60

Diameter(m)540

ceoussandperformedbyOkamuraetal.͑1997͒.Unfortunately,thefinite-elementresultscouldnotbeobtainedforsmalldiam-etersowingtonumericalinstabilityforthehighdilationratesassociatedwithshearingatlowstresslevels.However,bothre-sultsshowthereductionofN␥withincreasingdiameter.CalcareousSand

103020

2040507080100005101520(b)

Displacement/Diameter,G/D(%)

Fig.7.Effectoffoundationsizeforshallowcircularfootingsonsiliceoussand:͑a͒dense;͑b͒loose

N␥valuesmergeatabout20mdiameterforloose͑e0=0.8͒samplesand60mdiameterfordense͑e0=0.65͒samples͑indi-catedbyarrows͒andthiswillbedefinedasthetransitiondiam-eterpointfromdilativetocontractiveresponse.Thisdiametermaybereferredtoasa“criticalsize,”Dcr,whichbasicallyfol-lowsthesameconceptbehindthedefinitionofthe‘criticalstress’byVesicandClough͑1968͒,asdescribedbefore.

Kimuraetal.͑1985͒suggestedthattheN␥valuereduceswithreductionindensity.Fig.8showsthegreatvariationwithdensityoverawiderangeoffoundationsize.Thefactordiminishesrap-idlywithincreasingfoundationdiameterforsmalldiameters,buttheeffectreducesatlargerdiameters͑notingthelogarithmicscaleoftheplot͒.

Fig.8alsocomparesthenumericalresultswithcentrifugemodeltestsforcircularfootings͑D=1.5–3m͒onToyourasili-

Fig.9showsbearingresponsesoffullysmoothshallowcircularfootingsoncalcareoussandwithK0=1fortworepresentativedensities͑e0=1.3fordenseore0=1.9forloose͒,appliedoverawiderangeofdiameters͑1–100m͒.Itisnoticedthattheeffectoffoundationsizeanddensityareverystrongforsmallerdiameters.Fig.10plotsN␥–Drelationshipsforcalcareoussand.AdditionalanalysestothoseinFig.9wereundertakenwithidenticalsoilparametersapartfromtakingK0=0.49,andthoseresultsareshowninFig.10alongsidethoseforK0=1.TherateofdecreaseofN␥withincreasingfoundationsizebecomesgraduallylowerforlargerfoundationsizesandforloosesamplestheN␥valuesbecomenearlyconstantfordiametersofmorethan30m.Physi-calmodelresultsfromFinnieandRandolph͑1994͒arealsoshown,andthoughtheseshowsomedecreaseinN␥withincreas-ingfoundationsize,therateofdecreaseisnotasdramaticasforthenumericalresults.Itmayalsobeseenthatthenumericalre-sultsgivehigherN␥values,foragivenvoidratio,thanthosereportedbyFinnieandRandolph,inspiteofgivinglowerpeakfrictionanglesfortriaxialtests͑seeFigs.3and4earlier͒.Again,thisemphasizestheimportanceofthesoilcompressibilityinthebearingresponse.

InFig.9,noneoftheanalysesexhibitsaclearultimatestate.Thecalculationfora1mdiameterfoundationondensecalcare-oussandwasterminatedatabout15.5%normalizeddisplace-ment,atwhichstagetheincrementaldisplacementvectorswereasshowninFig.11.Theseindicateasignificantcomponentofsurfaceheaveadjacenttothefooting,asinaclassicalrupturefailurepattern.Itmaybeconcludedthatthecriticalfoundationsizeforthedensecalcareoussandmaybeestimatedasabout1m.

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MobilizedBearingResistance,NJ=2qb/J'D150

100

ShallowfoundationanalysisGoodwyncalcareoussand80

J'=7.0kN/m3MobilizedBearingResistance,NJ=2qb/J'DShallowfoundationanalysisGoodwyncalcareoussande0=1.3(Dense)K0=1.0100

Diameter(m)1J'=7kN/m31.523SmoothG/D=10%e0=1.3Centrifugetest(Finnie,1993)e0=1.3K0=1.0K0=0.4960

40

e0=1.950

5710301002070155020

0051015200110100(a)

150

Displacement/Diameter,G/D(%)ShallowfoundationanalysisGoodwyncalcareoussande0=1.9(Loose)K0=1.0100

Diameter,D(m)

Fig.10.N␥andDrelationshipsforcalcareoussand

MobilizedBearingResistance,NJ=2qb/J'DJ'=7kN/m3Diameter(m)50

11.52357151030207050100152000510surface.Thevaluesreducestronglywithincreasingdiameter,butstilllieabovethecomputedvaluesfortheoverlappingdiameterrangeof2–3m.Thus,althoughthegeneraltrendsaresimilarfortheexperimentalandnumericalresults,itisdifficulttodemon-stratecompleteconsistency.

ThecalcareoussiltanalysesarebasedonextremelylowprefЈand␪valuesandleadtoverylowN␥valuesevenforsmallfoundationsizes͑seeFig.13͒.TheN␥valuesforloosesamples͑e0=2.7͒,inparticular,areessentiallyindependentofthefounda-tionsize.Physicalmodelresults͑FinnieandRandolph1994͒liebetweenthenumericalpredictionsoflooseanddensestates.TheexperimentaldataalsorevealedthattheN␥valuesforcalcareoussiltareinsensitivetothefoundationsize.

TheN␥–Dcurvesforalltheabove-presentedmaterialsarecomparedinFig.13.Forsmalldiameters,DogsBaysandhasthehighestbearingcapacity,whereastheGoodwynsiltgivesthelow-est,althoughitshouldbenotedthatresultsforToyourasandare

(b)

Displacement/Diameter,G/D(%)

Fig.9.Effectoffoundationsizeforshallowcircularfootingsoncalcareoussand:͑a͒dense;͑b͒loose

N␥–DRelationshipforVariousSands

Theinvestigationoftheeffectoffoundationsizehasalsobeenconductedwithrespecttotwoothertypesofcalcareoussoils,namelyDogsBaycalcareoussandandGoodwyncalcareoussilt.TheMIT-S1modelparametersforthesandandsiltaretabulatedinTable1.Effectiveunitweightsweresetto7kN/m3fortheDogsBaysandand6kN/m3fortheGoodwynsilt.

TheN␥valuesfromtheanalysesforDogsBaysandareshowninFig.12.Thecomputedfactorishighforsmalldiameters,evenincomparisonwiththoseforsiliceoussandshowninFig.8.Thisappearsrelatedtothehighervaluesof␾mrЈandnp͑i.e.,higherfrictionangles͒andhigherprefЈ͑i.e.,higherstiffness͒.Thecom-putedresultsarestillmuchlowerthantheexperimentalresultsfromKlotzandCoop͑2001͒,althoughthesearetakenfromtheend-bearingresistanceofjackedpiles,extrapolatedbacktothe

Fig.11.Incrementaldisplacementvectorsfor1mdiameterfootingondensecalcareoussand͑␦/D=15.5%͒

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MobilizedBearingResistance,NJ=2qb/J'D800

Dense(e0=1.4)ShallowfoundationanalysisDogsBaycalcareoussandCentrifugetests(Klotz&Coop,2001)3Jdry=16.0kN/mMobilizedBearingResistance,NJ=2qb/J'D1000100

Shallowfoundationanalysis80

DogsBaysandGoodwynsand60

Toyourasand40

Goodwynsilt20

SmoothG/D=10%K0=1.0Dense600

Medium(e0=1.5)400

Loose(e0=1.7)1.6e0=1.9110Diameter,D(m)

200

NumericalanalysisK0=1.0J'=7.0kN/m300110100100(a)

Diameter,D(m)

100

Shallowfoundationanalysis80

DogsBaysandSmoothG/D=10%K0=1.0Loosenotavailableatsmallerdiameters.Thedifferenttrendsofthecalcareousmaterialsareevidentandresultfromchangesinthecompressionparameters,prefЈand␪,whichprimarilycontrolthebearingresponseforcalcareousmaterials͑seeTable3͒.ItisalsofoundthattheN␥valuesfordifferentcalcareousmaterialsanddensitiesreducewithincreasingdiameterandmergetoasome-whatuniformN␥͑intherange5–10͒,independentofthedensity,foundationsize,andmaterialtype.Ontheotherhand,theN␥valuesforlargefoundationsonsiliceoussandaresignificantlylargerthanthoseforcalcareoussoils͑Fig.14͒.

MobilizedBearingResistance,NJ=2qb/J'DFig.12.N␥andDrelationshipsforDogsBaycalcareoussand

60

40

GoodwynsandToyourasand20

Goodwynsilt0110100100MobilizedBearingResistance,NJ=2qb/J'DShallowfoundationanalysisGoodwyncalcareoussiltJ'=6.0kN/m3(b)

Diameter,D(m)

80

Fig.14.N␥andDrelationshipsfordifferenttypesofsoils:͑a͒dense;͑b͒loose

K0=1.0G/D=10%60

Centrifugetests(Finnie,1993)e0=2.2-1.7Limitations

Theprincipallimitationoftheanalysesconductedisthatthefinite-elementresultsforsmallerdiameterfoundationsonsili-ceoussandcouldnotbeobtainedduetocalculationinstability.Onepossiblereasonisthehighlydilativeresponseofsilicasandatloweffectivestresslevels,inconjunctionwithextremelylargedeformationsattheedgeoffootingsduringloading.Duetotherupturetypeoffailurepattern,neighboringelementimmediatelyinsideandoutsidethefootingshowdownwardandupwarddefor-mations,respectively,whichledtoterminationofthesolutionduetotheextremelyhighdisplacementgradient.Thesmallerthefoundationsize͑soloweffectivestresses͒,themoresignificantthisissuebecame.Bycontrast,thefailuremechanismforcalcar-eoussandsgavedownwarddeformationsjustbeyondtheedgeofthefootings.

40

e0=1.720

e0=2.70110Diameter,D(m)

Fig.13.N␥andDrelationshipsforGoodwyncalcareoussilt

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Conclusions

Thispaperhaspresentedanumericalexaminationoftheeffectoffoundationsizeonshallowfoundationresponses.Thedefor-mationpatternsbeneaththefootingwereexaminedbasedonfinite-elementanalysesusingtheMIT-S1model,highlightingthekinematicdifferencesbetweentheresponseofsiliceousandcal-careoussands,withaclassicalrupturefailurepatternfortheformer,butapunchingshearfailurepatternforthelatter.Theinvestigationintermsofthebearingcapacityfactor,N␥,indicatesthatthecalculatedN␥valuesforlooseGoodwyncalcareoussandandsiltarefoundtoberelativelyconstant,independentofthefoundationdiameter,whereasfordenseGoodwynsandandsiltandDogsBaysand,N␥wasfoundtodecreasemarkedlywithincreasingfoundationsize.TheN␥valuesforcalcareousmaterialseventuallyconvergetoaterminalvaluewithinarelativelynarrowrange.

Itwasalsofoundthatthedeformationmechanismforsiliceoussandtransformsslightlytowardapunchingshearmechanismforverylargefoundationsizes.Atransitiondiametersizebetweendeformationmechanismscorrespondingtocontractiveordilativebehaviors,i.e.,acriticalsizeDcr,wasidentifiedforbothsiliceousandcalcareoussands.Thestudysuggeststhattheconventionalbearingcapacityanalysesmaybeapplicableforfoundationsizeslessthanthecriticalsize,whereasalternativeapproaches,focus-ingmainlyonthesoilcompressibility,areneededforshallowfoundationsgreaterthanthecriticalsize.AlternativeapproacheshavebeenproposedinthethesisofYamamoto͑2006͒andwillbesummarizedinaseparatepaper͑Yamamotoetal.2008͒.

Acknowledgments

Theworkpresentedinthispaperformspartoftheresearchac-tivitiesoftheCentreforOffshoreFoundationSystems͑COFS͒,establishedundertheAustralianResearchCouncil’sResearchCentresProgram.Thewriterswouldliketogratefullyacknowl-edgeProfessorAndrewWhittlefromMITforsupplyinganimple-mentedversionoftheMIT-S1modelinABAQUS.

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