BOLTON,M.D.(Geotechnique 36, No.I, 65-78 The strength and dilatancy of sands M.D.BOLTON Extensivedata of the strength and dilatancy oil7 sanW inaxisymmetric or plane strain at different densities and confiningpressuresarecollated. The whichisshearingatcon-stant volumeisprincipally a functionof mineralogy and canreadilybedeterminedexperimentallywithina marginof about1',beingroughly33'for 40"forfeldspar.Thetoitsrateof dilationandthence-lOTtS-relativedensityanamcaiietreCtm'Siiess, combinedina newrelativedilatancyindex. Thedata of "';.,,,- "';,"in triaxialorplanestrainareseparatelyfittedwithina typicalmarginofabout2",thoughthestrengthof certainsandsisunderpredictedintheI()()()"'IOOOO leN/m'rangeowingtothecontinueddilationoftheir crush-resistantgrains.Thepracticalconsequencesof thesenewcorrelationsareassessed,withregardtoboth laboratory and fieldtesting procedures. L'auteuranalysedenombreusesdonneesconcernantla resistanceetladilatance de17sablessousdeformation planeouaxisymetriquepourdifferentesdensiteset pressionsd'etreinte.L'anglederesistanceaucis-aillementdansI'etat critiqued'unsolsoumisau cisaillementIivolumeconstantestprincipalementune fonetiondelamineralogieetpeutsedeterminerfacile-menta I"pres, commeayantunevaleurd'environ33' pourIequartzet40"pourIefeldspath.L'anglesupple-mentairedecisaillementd'unsoldensedependa sa vitessededilationdonedesadensiterelativeetdela contrainte effective moyenne,combineesdansunnouvel indicede dilatancerelative.Lesdonneesde -endeformationplane autriaxiale sont separeesparune margede2approximativement,bienquelaresistance decertainssablessoitsousestimeedansunefourchette deI()()()"'IOOOOIeN/m'enraisondelapoursuitedela provoqueeparI'ecrasementdeleursgrainsresistants. L'articleevaluelesconsequencespratiquesdecosnou-vellescorrelationsencequiconcernelesmethodes d'essaienlaboratoire etinsitu. KEYWORDS:friction;sands;sheartests;soilproper-ties; statisticalanalysis. INTRODUCTION FollowingtheearlyworkofD.W.Taylor,the strengthanddilatancyofsoilsreceivedagreat dealof attentioninthe1960s.Severalschoolsof thOUghtdeveloped,thedifferencesbetweenwhich DiscussiononthisPapercloseson1July1986.For further detailssee insidebackcover. Cambridge UniversityEngineering Depanment. 65 weremoreaccentuatedthanthepointsin common.Therehasfollowedament anlOngstthoseresearcilipTthe. stiengthof sOlr .... ........... .. .... _-_ .... -_ .... _..... (a)thatsecant,ratherthantangent,'values should be thebasisfordiscussion (b)that dilatancytowardscriticalstatesiscentral to anunderstanding of soilbehaviour (c)that both effective stressand soil density affect therateof dilatancy of soilsandtherebytheir strength parameters. Thatthisunderstandinghasfailedtopermeate morewidelyintopracticecanpartlybeblamed onthestructureofthehistoricargumen t,which revolvedaroundthetheoreticalrelationshiJL betweenstrengthanddilatancy.Sincepractition-erswouldusually be inthe positioneither of mea-suringbothorguessingboth;thisaspectofthe dispute must have seemed sterile. Thefailure- tobridgethegapbetweenresearch andpracticehasmanyseriousconsequences, however. Engineers often do not appreciate (a)thatasecant'valuederivedfromasingle triaxialtestonasinglespecimencanoffera conservativeparameterfordesignifthe testing conditions are carefully chosen (b)thatthefullrangeofsoilstrengthscanbe expressedintermsof thevariationof (secant) 'with density and stress (c)thattheconventionaltangentparameters(c', ')canonlydescribethefullrangeofsoil strengthsifbothareallowedtovarywi th density and stress (d)thatignoranceoftheforegoingcanleadto significanterrorsinpredictingultimate bearing stresses, for example. .TheObjectivesQLthiLl''''!Re!_..firstly;'ioClanlythe conceptsof frictionanddilat-ancyinrelationtotheselectionofstrength parametersfordesign;secondly,tointroducea newrelativedilatancyindexandtodemonstrate itsuseinthepredictionof the behaviourof sands atfailureinrelationtotheavailablepublished data;thirdlytoconsidertheimplicationsofthe newcorrelationstotheproceduresoflaboratory and fieldtesting. Mostof theconceptswhicharetobe discussed canbeappliedwithequalforcetoclaysasto 66BOLTON 6 '0= 069 ui- 138KN/m2 '0= 069 ui= 5870kN/ m2p c 0 -2 _--c -1P * 0 '.j C 20 2468 t ,:%(,:% la)Ib) Fig.1.Slress-slrainbehaviourof densesandinplanecompression(a) allow stress and (b).1 high stress (Barden et 1969) .. __a,__ __ __'.... =448 .,I.(de.) SIn'Pmax=---dY'3p:.if;max =147 (O",'/O",')m"- 1 (0" ,'/0" ,')mox+ I (dedde')mox+1 (dedde')mu- I (I) (2) Itshouldparticularlybenotedthat' herein referstothe 'secant' angle of shearing obtainedby droppingatangentfromthe originontoasingle Mohr circleof effective stress. 2 ., r :kN/m2 STRENGTHANDDILATANCYOFSANDS la) dy/2 r :kN/ m2 10000 Ib) dy/2 Fig.2.Mohrcirclesofstressandstrainincrementsofdensesandatmaximum stress ratio inFig.1 (a)at lowstres.! and Asimilarplane strainteston densesandunder extremestressesmimicsthenormalbehaviourof veryloosesandandappearsinFig.I(b),withthe dataforpointCattl =8%presentedinMohr circlesinFig.!iy.eIJ____ __cQ."- tractsratherthandilates_Theendstateofthe whol e saiTip"ie-ai-ci;analogous tothatwhich can beobservedinthethinrupture zonesof the dilat-ant sample,towards acriticalstate 4>;,;,=35 if;1t=0 Themechanicalsignificanceoftheangleof dilationinaplane strain deformationcanbestbe appreciatedbyassumingthattheMohrcircleof plane strainincrementsinFig.2(a) can be applied tothe caseof direct shear showninFig.3.If rigid blocksofnon-failingsoilareassumedtobound thethinuniformly straining rupturezone ZZ,this mustmeanthatforcompatibilityZZmustbea zero extension line so that ,.,-. ..,..de,=0(3) -- ------, I la)Iz T "I Ie) dz I dy Y, Idyy: (b) dy/2 Fig. 3.Augle of dilation'" inplane shear 67 68BOLTON o tan If "'" ,':;0Fig. 4.Logarithmic spiral slip surface withintherupture zone.Also dz dy=-"y soinFig.3(b) dy y dedy tan0/1=--' =-dyY'dz (4) The angleof dilati;)ilTsthen seeninFig.3(b)to beequaltotheinstantaneousangleof motionof theslidingblocksrelativetotherupture surface. Figure4demonstratesthattheconsequenceof theassumptionof aconstant angleof dilationon aslipsurfaceisthereplacementof slipcirclesby thelogarithmicspiralslipsurfacebetweenrigid zones, since fora smallangleAOB=dO sothat On integrating CAB=0/1 CB=dr AC=rdO dr tan0/1 =r dO dr - = tanrf; dO r ,= roexp(0tan0/1)(5) (6) wherer = '0at0 =O. At0/1 = 0thewell-known slipcircler'= roisindicated.Thelargeris0/1 the largeristhe'downstream'radiusoftherupture lineforagiven'upstream' excitationatr= ro .If arelativerotationofthetwosupposedlyrigid zonesthenoccursduringcollapse,theensuing downstreamdisplacementsaresimplypro-portionaltotheradiusandarethereforecorre-spondingly increasedasrf; increases.Inpractice, it isknownthatfailurescanoccurprogressively, withdifferentmobilizationsof strengthanddilat-ancyatvariouslocationsalongadevelopingslip surface. Ifrupturefigurescomprisingamosaicof slidingwedgeswithplanefacesarealternatively assumed,theeffectonthestability calculationsof increasing0/1 atconstant1>' iseasilyshowneither tobezeroifthemosaicisstillcapableof sliding inthemannerintendedortocauseaseizureof themosaicduetoenhancedwedgingwhichcan makeapreviouslycriticalslipmechanismkine-maticallyinadmissible.Dependingon theparticu-largeometryoftheproblemtherefore,theeffect ofincreasing0/1 atconstant1>' mayeitherbe neutral or beneficialtothe overallstability.A safe strategymustthereforebetoerr,ifatall,by underestimating both1>' andrf; inany subsequent analysis. Agreatdealof attentionhasbeenfocusedon therelationbetween1>' andrf;,andinparticular between1>.....and0/1 mn'Thestress-dilatan9: theory of Rowe (De1962,1969)hasprovedtopossessgreatexplana-torypower,aswellasacloseapproachtothe publisheddata.Sincetheintentionhereisprin-cipallytointroduceandcorrelatedata,itis unnecessarytoexpandindetailonthetheory.It willbevaluable,however,toemployalessrigor-ousbutsimilarapproachtodevelopanexpres-sionwhichdeviatesonlyslightlyfromRowe'sin itsestimateof'appropriatetoaplaneshear test. Supposethat istheangleofshearing observedina shear tesL.t1sQillaPse !mQ!I_gIUSJ.J2ein a Now Fig: 5,thatthe same soil istesteddense,sothatoverridingatpointsof contactmustoccurunlesstheparticlescrush. Supposethattheparticlesabovetheoverallzero-extensionlineZZformonerigidzonesliding upwardsat0/1 overtherigidzonebeneath,in accordancewiththeexternalobservationofa dilatancy angle 0/1. Assumethattheangleof shear-ingdevelopedontheinclinedmicrofacetsSS,on whichthereiszerodilation,remainsat;,,,. Sincealltheslidingnowtakesplaceonsurfaces paralleltoSSitispermissibletoviewthe observedangleofshearing'ontherupture surfaceascomprisingthetwocomponents;,,, andrf; asshown 11>'=1>;,,,+rf; (7) L.____ . ____._, STRENGTHANDDILATANCYOFSANDS69 Fig.5.The sawblades modelof dilatancy asanelementaryfriction-dilatancyrelationin whichnoattempthasbeenmadeeithertoopti-mizethefailuremechanismortocorrelatethe directionsofprincipalstressandstrain increments. Figure6comparesequation(7)withRowe's stress-dilatancy relation forplane strain (8) intheparticular case1>;,,,= 33corresponding to thetypicalvalueforquartzsands,andusing equations(1)and(2)for1>' and0/1 respectively.It willbeseenthatequation(7)overestimates 1>' - 1>;,;,comparedwithequation(8)byabout 20%_Inotherwords,Rowe'sstress-dilatancy relationshipforplaneshear,overtherangeof0/1showninFig_6,isoperationallyindistinguishable from (9) Theadvantageofhavingdevelopedexpressions suchas(7)and (9)isthat anyangleof shearing in excessof thefrictionangleof looseearthisseen tobedue solelytothe geometryof thevolumetric expansionwhichisnecessarybeforeshearing can take place_ EFFECTSOFDENSITYANDCONFINING STRESS Theaccepteddefinitionforthestateofcom-paction of granular materials isrelative density (10) whereem .. isdefinedasthevoidsratioachieved inquicklyinverting ameasuring cylinder contain-ingthedrysoilandem;nisthatachievedunder optimalvibrationofacompactivemassunder saturatedconditionsandwithoutcausingcrush-ing.It hasgenerallybeenfound(Cornforth,1973L thatrelativedensityoffersasupeiTof--correTatlon comparedwithvoidsratioforthestrengthof sands,presumably sinceitcompensates foreffects of particlegrading andshape whichinfluence em .. andemino If soilparticleswereperfectlystrongandrigid, thetendencytowardsdilationwouldbesolelya functionofthedensityandarrangementofthe particlestructure.However,experimentswith steelshot,whichwereeffectivelyunbreakablebut weredeformableplastically,haveshown(Bishop. 1972)thatincreasedconfiningpressureleadsto reducedangleofshearing.Ithasalsobeen demonstrated(Bishop,1972;Billam,1972;Vcsic &Clough,1968)bytestsongranular soilsatele-vatedpressuresthatparticlecrushingoccurs, therebyreducingtheobservedmaximumangles ofdilationandshearing,rf; m ..and:.",[ora giveninitialdensity.Soilparticlesmaycrush beforethey override. Nocompletelyconsistenttreatmentofboth densityandconfiningpressurehaspreviously beenundertakeninproducing empiricalrelations bywhich1>:' ..couldbepredicted.Thisisnow attempted.Toachieveagivenangleof dilatancy, itisargued,theparticlestructureshouldbothbe denseandnotsohighlystressedthatasperities shouldfractureinpreferencetooverriding_This suggestsarelationinwhichadensityparameter ismultipliedbyastressparameter.Sincethe logarithmof stresshasbeenshown(Billam,1972; Vesic&Clough,1968)toeffectalinearreduction 1>:' .. , an expression of the form ./,=AIIn(p;,;.,)(II) 'Ymax0p' Equation 17)50 deg Fig, 6.Stress-dilataocy relations 70BOLTON Table I.Sand data IdentificationNamed6 . :d,.:eminemaxReference mmmm ABrastedriver02901204707932-6Cornforth(1964,1973) BLimassolmarine0110003057\1834-4Cornforth(1973) CMerseyriver",02",0 1 0Montereyno.20",03",015 EMontereyno.0",0 5",03 FHamriver025 GLeightonBuzzard14/ 250-85 HWeilandriver014 IChattahoocheeriver0-47 JMol021 KBerlin025 LGuineamarine041 MPortlandriver036 NGlacialoutwashsand09 PKarlsruhemediumsand038 .- . -- Ii. Sacramento-r iver-" 022 SOttawasand076 isimplied,inwhichp'isthemeaneffectivestress atfailure, istheelevatedstress justsufficient toeliminatedilationbycrushingandAisacon-stant.However,thequantityofdatafor4>;'"is rathergreaterthanthatfor1/1m.. 'andthatfor triaxialtest sgreatly exceededthatforplane strain tests,andthereforeanapproachmustbeused whi chcan embracethese additional data. I! shouldberecognizedthattheangleof dilat-ancybecomesameaninglessparameterinan axisymmetrictriaxialcompressiontest,sincethe geometricalrelationshipofFig.2 appliesonlyto planestrain.IndeedaMohrcircleofstrain incrementsforthedensestsandintriaxialcom-pressionwouldindicatethatmeanstrainsinany verticalplanewerecontractileratherthandilat-ant,owingtotheimplieddistributionoflateral strainamongstalltheavailabledirections.I!is thereforenecessarytorevertto(- d,/de')muas theusefulmeasure of triaxialdilatancy rate. TableIassemblesthecharacteristicsofthe sandsusedinthecorrelationstudy,comprisinga largeproportionof therelevantdataavailablein theliteratureandemanatingfromwell-establishedlaboratories.Thedatareferto samplestestedincompressionwithaninitial height-to-widthratioof 2,unlessindicatedother-wise.Someresearchersfailedtodisclosethe preciseconditionsof thetestplatens,anditmust beassumedthatsomeof thevariationsinresults maybeaccountedforbyvariationsinendfric-tion.Thewiderangeofgradingclassificationsis madeevidentbythe d60andd 10sizeswhichhave beenquoted; estimateshavebeenmade incertain caseswhere theseparticular valueswerenot given 016 065 010 021 014 011 016 023 015 020 015 065 049082320Rowe (1969) Rowe&Barden(1964) 057078369Marachi,Chan, Seed&Duncan(1969) 057086370Lade&Duncan(1973) 05909233-0Bishop&Green(1965) 049079350Stroud(1971) 062094350Bardenetal.(1969) 061110325Vesic&Clough(1968) 056089325Ladanyi (1960) 046075)30DeBeer(1965) 052090330Cornforth(1973) 0631\0361Cornforth(1973) 041084370Hirschfield&Poulos(1964) 054082340Hettler (1981) 061 "-._-lee&Seed(1967)103333 049",0830-0Lee&Seed(1967) bytheresearchers.Figuresforem"andeml owere essentialtothestudy.Tavenas&LaRochelle (1972)haveshownthatvariationsinlimiting voidsratiosduetovariationsintechnique betweentheAmericanSocietyforTestingand Materials'methodsandthoseofKolbuszewski shouldnotusuallyexceed002.Thiswouldimply anexperimentalerrorbandforrelativedensity determinationsof the orderof!005. Thecriticalstateangleof shearingwascentral toarationalportrayalofthedata.Theideal methodof determinationforisfromextrapo-lationofaseriesofvaluesof4>;' ..recordedin compressiontestsonsoilsamplesof variousden-sities,inwhichtherateof dilationatfailurewas observed,sothatavalueconsistentwithzero dilationatfailurecouldbedetermined.Ineach casequotedinTableI,itwasfeltthatsufficient informationwasavailabletomakeajudgement on4>;",whichshouldnothavebeeninerrorby morethanI ' . Theobservedrangeforthesands wasfrom320'to370',thehighervaluesalways pertainingtosandswhichweresaidtocontaina significantproportionoffeldspar,thelower valuespertainingtoquartzsands.Thiswascon-sistentwithpreviousfindings(Koerner,1970; Lee,1966)that forfelspathic sandswasof the order of 40". Aninitialperusalofthetestdatashowedthat equation(II)wasinsufficientasameasureof dilatancypotential,sincep;",appearedtoreduce withreducingrelativedensity,andatanincreas-ingrateatverysmallrelativedensiti es.Arelative dilatancy index of theform IR=lo(Q-In p')- R(12) STRENOTIfANDDILATANCYOFSANDS71 wastherefore studied, sincethis functionreturns a zeroval ue at some pressure P;d, suchthat R InP;d,=Q-] o (13) (Here,thevalueforQwilldependontheunits takenforp' :kilo newtonspermetresquaredwill beusedhere).Thisexpressionhastheadvantage ofensuringthatzerodilatancyisachievedata criticaleffectivestresswhichitself reducesstrong-lywhentherelativedensity 10takessmallvalues. Beyondtheseconsiderations,however,anypref-erenceforaparticulardefinitionofarelative dilatancy index must be entirely empirical. ItwasfoundthatvaluesQ= 10andR= 1 createdadefinition forarelative dilatancy index IR=lo(lO-In p')- 1(14) whichapparentlyofferedauniquesetof correla-tionsforthedilatancy-relatedbehaviourof each ofthesandsinlaboratoryelementtests.Thefol-lowingcorrelationswerefoundtobeavailablein therange 0;""wascommonlyobservedaroundthebest estimate for . ThebestinterpretationofI R valueswhichare calculatedtobenegative,owing eithertoextreme loosenessor tohighconfining stressisthatacon-siderablecontractioninvolume(andconsequent increaseinI D) willtake placebefore4>;,;0 ismobi-lizedatI R =0afterlargeshearstrains.Expres-sion(14)givesanindicationofthevalueofp' (p;,;"say)atwhichIR=0anddilationissup-pressedforanyinitialrelativedensity:In= 10- 1/10 ,However,aperusalofFig.10will '"w " Empiricalrelalion (16) 12\withto=1 0 oSacramentoriversand 10 oOttawasand 8 0 o ;6 I o0E o 4 o 12 10 '" 8 o

I6 a indicatethatsignificanterrorsmayariseinthe predictionof p;,;,duetothelogarithmic scale. Thereissomeevidenceintheliteraturethat certainuniform,roundedsandsarelittleaffected byconfiningpressureslessthanabout1000 kN/m2,afterwhichtheybegintocrushrelatively swiftly.ThisisshowninFig.11(a)wherethetri-axialdataofLee&Seed(1967)forsubangular Sacramentoriversandfollowtheempiricalrela-tion(16)incontrastwithOttowa sandwhichwas foundnottocrushsignificantlyuptopressuresof 4000kN/m2andwhosedatalieupto5'above theempiricalpredictioninthemedium-to-high stressrange.Fig.II(b)forKarlsruhesand o'D ::>;086 a'0=1-00 o Empirical relation(16) 'for'0=0-86 o OL-_________________O__ 1001000010 000 0 p':kN/m2

p':kN/m2 (a) (b) Fig.II.(a)Triaxialtestdatafortwodensesandsateleva ledslresses (Lee&Seed,1967)and(b)Iriaxialtesl dalafor dense Karlsruhe sand (P) at various stresses (Helder,1981) 74BOLTON 20 of/>ma'l.' - q,> c: rilaQ' 8l/1max '" " tJ GLeightonBuzzard sandat'0=087 HRiverWeilandsandal' 0=0-69 .E16 ....., o G 6GEmpiricalrelation (15) -g 12 n18H 100 p', kN/m' o Fig.12.PlanestraindataforsandsinTable1failingatvarious mean effective stresses (Hettler,1981)demonstratesasimilartendencyin auniformroundedsandnotedforitsparticle strength (Drescher &Vardoulakis,1982). Figure12introducesthe rather sparse available informationofplanestraintestsconductedat variousstresses.Resultsinthemiddlebandof stresseswerebroadlyasexpected,butthelow stresstestonsandGinthesimpleshearappar-atusandone of thehighstresstestsonsandHin aplanecompressionapparatusdeviatedquite markedly inopposite senses.Bothtestsmustha ve beendifficulttocarryoutsatisfactorily,forcon-trasting reasons. Forthelowstresssheartest,theeffectsof any stressandstrainnonuniformitiescouldhave beenstrong.Thesandwasexceptionallybrittle andthereforesubjecttoprogressiveruptureas4>' coulddropfromat least52" to35". Furthermore, theaccuratemeasurementof stressesassmallas 15kN/ m'presentsseveredifficulties.If theresult canbereliedon,however,itmightindicatean absolutelimittotherateofdilationofadense sandirrespectiveofhowsmallthestressesare. Untilgoodlowstressdataareavailablefora particular sand,itwouldbeprudent tosetalimit of4tothemagnitudeoftherelativedilatancy indexI R'irrespectiveoflargervaluescalculated byequation(14).Thiswouldhavetheeffectof limiting 4>'.n"-to20inplane strainand12 intriaxialstrain,valueswhichhavebeenattained butnotexceededinthetestprogrammesrecalled here. FortheplanecompressiontestonsoilHat 6000kN/m2,theunexpectedlyhighrateof dilat-ancyatfailurewasmatchedbyanevenswifter rateof softeningtoacriticalstate:therequired boundary displacementwasabout onehalf apar-ticlediameter.Thissuggeststhatthesandwasin asomewhatunstablestatewithinitstest chamber,andmaycorrespondtotheremarks. madeearlieraboutthesuddendegradationof someuniformsandswhichhaddisplayedrela-tivelylittleprogressivecrushing.Whetherornot theextramid-rangestrengthwasduetothis,or to someother cause,itmightbeprudenttodisre-garditfordesignpurposes. Itshouldberecalledthatthebulkof dataso farpresentedderivefromcompressiontestswith H/D =2andsomeundeterminedplatenfriction. Previousstudiesareambiguouswithregardto theeffectofreducingfrictiononsampleswith HID =2.Bishop&Green(1965),inanexhaus-tivestudyof Ham sand, concludethattheeffectis negligible.Rowe&Barden(1964)foundthat, acrossafullrangeofdensitiesforMerseysand, 4>'measured withlubricatedplatensonsamples withH/D =1wereabout25"lessthanvalues measuredwithconventionalplatensonsamples withH/D =2.Testsconductedonbothdense andlooseKarlsruhesandwithlubricatedplatens onsampleswithH/ D =1/ 3indicated(Drescher &Vardoulakis,1982)that4>;""wasabout3less thanintheconventionaltest.Sincethemajority ofengineersareusedtoworkingwithtestdata derivedintheconventionalwaythesediscrep-anciesare,perhaps,ofinterestprincipallytothe researchworker. Noattempthasbeenmadeheretoextendthe discussiontotestsintriaxialextension, or totests witharbitraryintermediatestressratios.These arehabituallyfoundtoofferstrengthsthatare intermediatetotriaxialcompressionandplane strainvalues,thedifferencesbetweenwhichhave beenshownheretorelatesolelytodilatancy STRENGTHANDDILATANCYOFSANDS75 effectsinthesoil.I! maybesufficienttonotethat thesoilgrainsinatriaxialtesthaveconsiderably greaterfreedomtodeviatelaterallythanthosein aplanestraintest,andthatinthesecircum-stancestheenhancementof strengthduetodilat-ancy mighthavebeenexpectedto be smaller . IMPLICATIONSFORLABORATORY SOILTESTING TheconceptofarelativedilatancyindexI Rwhichiscoupledwithratesofdilatancy,and relatedtotheextraangleof shearingabovecp'. hasprovedpowerfulinorganizingthedataof'l.+ contrastingsandsfromaroundtheworldtested independentlyby12groupsof workers.Thepar-ticularnumericalvaluesquotedinexpressions (14HI7)havebeencapableofretrospectively fittingtheconventionaldataofshearingresist-anceinthe100-1000 kN/ m2rangeof meaneffec-tivestresswithatypicaldepartureof1or2" onceagoodvaJuefor hasbeenobtained. Thedatathemselvesmakeclearthatafurther1or 2errorincp:""mightbeincurredif were estimatedfromtheshearingstrengthofasingle veryloosesample.Cornforth(1973)hasreported thattheangleof reposeof alooselytippedheap of drymaterialsubjectedtoexcavationof thetoe providedasimplebenchtestfor which shouldsimiliulyofferanaccuracyofabout1". Thelikelyerrorinsimplytaking =33fora sandwhichhasbeenidentifiedascomprising chieflyquartz grainsisapparently alsoonly of the order of 1"or 2". Althoughthelimitingbehaviour of sands inthe range10'_104 kN/m2hasalsobeenfitted,the uncertaintiesaregreater.Samplesdeformand contractsubstantiallyundertest,andtherele-vanceofinitialrelativedensitymustbeques-tioned,asmusttherelevanceofthestrength parameter itself sincetheshear strainsrequiredto mobilize4>;,;\arelarge(Vesic&Clough,1968). The analogy withtheengineering problemsposed bysoft claysinthesmall stressrangeisappealing. Thissuggeststhatfurtherstudyofthebehaviour of sandsunder extremepressuresintermsof their instantaneous(e,4>', p')statesisnecessaryif more accurateandcompletepredictionsarerequired. Thepresentcorrelationsunderestimatethe strengthsof someuniformwell-roundedsandsin thisstressrange:thesematerialsdonotsuffer progressivecrushingbuttendtodegraderather suddenly at some very large stress. The sandspresentedinthisstudy aremainlyof quartzorfeldspar.Thepresenceofsubstantial proportionsofmica,calciteorothermaterials wouldbebound to affectbothand the crush-ingwhichreducesIRathighstresses.Billam's (1972)triaxialdataforthedilatancyandstrength Table 2 QGraintype 10Quartzandfeldspar 8Limestone 7Anthracite 55Chalk ofinitiallydensesamplesofgranulatedchalk, crushedanthraciteandlimestonesandindicate thattheY'eachgainroughly7of shearingresist-ancepertenfold stressreduction,according preci selywiththegradient of lhe10=IlineinFig.10. Whatwasshown,however,wasthatreducingthe crushingstrengthofthegrainsreducedthecriti calmeaneffectivestressatwhichdilatancywas suppressed.ThisimpliesthatparameterQin equation(12)shouldbereducedforsoilsof weakergrains.Billam'sdata,assumingthatthey relateto'soils'withI D =I,isroughlyconsistent withthevaluesinTable 2. IMPLICATIONSFORLABORATORY REPORTS A minimallaboratoryreport couldcomprise (a)sourceof materialandassessmentof mineral, ogy (b)grading curve (c)emuande ml by astandardmethod (d)angle of repose of dry heap (e)insitudensity. Dependingonthesignificanceof theproblemin hand,thefollowingadditionaltriaxialcompres sionteststakeninorderwouldconfirmtherel e vanceorneedforrecalibrationoftheempirical relations proposed above (f) loose soilata cellpressure circa500 kN/m2 (g)dense soilata cellpressure circa150 kN/ m2 (h)dense soilat the highestavailable cellpressure (i)other relevant tests. I! shouldthenbepossible,withadegreeof con fidencethatisproportionaltothenumber of tests conducted,topredictthepeakstrengthofany elementofthesoilwhosedensityandstressis known. Analternative consistent approachwouldbeto testasinglesoilsamplefromeachzoneata densitythatislowerthananywhichcouldrea sonablybeanticipatedinthefieldandatacon finingpressurethatislargerthananywhich couldbeexperienced.Theresultingsecantvalue of4>:....couldthenbeusedconservativelyin designforthewholezone,providedthatpro gressivefailurecouldnotoccur.If thereweresig-76BOLTON N 4 E Z ::;3 500 400 200 100 ' D=1 34 d:MN/m2 (a) 100200300400500600700 d:kN/m2 (b) Fig.13.Mohrenvelopesderivedfromempiricalrelations fortriaxialtests onquartz sands nificantdoubtsaboutthelooseststateinthefield orthepossibilityofstrainsoftening,theuseof ;';1 wouldbenecessary.Inthiscase,thetesting programmecouldsimplyconsistof items(d)and ( f). Ifrequired,the'intrinsiccurve'onaT-(]' diagramcanbegeneratedusingequation(14) witheitherequation(15)or(16)dependingon whetheraplanestrainortriaxialenvelopeis required.It maybeassumed.withsufficientaccu-racy,thatthepointofcontactofaMohrcircle withthe intrinsic curveis(T'= p',T= p'tan'.nn.Dependingontherequiredscale,curvesofthe formofFig.l3(a)orFig.13(b)canthenbepro-ducedforany givenf o. It istheneasyto confirm thattheconventionaltangentparameters(c' ,')cannotbefittedwithoutseriousmisrepresenta-tion.Atleastthreepairsofsuchvalues,corre-spondingtolowstress,mediumstressandhigh stressranges,wouldbenecessarytocapturethe data of sand ateachrelative density. The fittingof commontangentstoMohrcirclesmusttherefore beregardedasaquiteinferiormethodof presen-tationofsoilstrengthresults.Furthermore,the confusiongeneratedbydescribinganysuch tangentangleasthe'angleofinternalfriction' mustbe deploredasvigorously asthe description oftheinterceptc'asthe'truecohesion'.The detailedstudiesbroughttogetherheretellan unambiguousstoryoffrictionanddilation relatedtosecant' valuesand'"valuesdeter-mined separately for each soiltest. IMPLICATIONSFORFIELDTESTING Althoughtheforegoinghasbeenconcerned withthecollationof laboratorytestdata,ithasa strongimplicationfortheinterpretationoffield tests.Thenewcorrelationsmakeclearthatthere shouldnotbeaone-to-onecorrespondence betweenthestrengthofgranularsoilsandtheir relativedensity,evenatagivenstresslevel.A particularangleofshearingresistance(orpen-etrometerreading)couldindicate,forexample, eitheradensequartz sandwithlow ;,i( andhigh "'rnnvaluesoralessdensefelspathicsandwith higher;,;, andlow'" rnnvalues.Thepotential confusionbetweenthesealternativesmightbe mostsignificantinapredictionofearthquake-inducedexcesspore waterpressures.Therelative density hasbeen shownto correlate onlywiththe dilatantstrengthcomponent,whichcanonlybe resolvedfromthepeakstrength(orpenetrometer reading)if;'i' isseparatelydeterminedusinga disturbedsample, for example. Itmaythereforebeconsideredpreferableto measuretheangleof dilation'" rnndirectly,using aself-boringpressuremeterafterthefashionof Hughes,Wroth&Windle(1977).Thedirectcor-relationinequation (15)between"'rnnandfoat a givenstresslevelshouldthenleadto amorereli-ableestimateofrelativedensitythanwould otherwisebeavailable.Theattempttorelate cyclicporepressuregenerationdirectlytothe measuredangleofdilationofthesoil(Vaid, Byrne&Hughes,1981)shouldproveequally fruitfulinstudies of liquefaction susceptibility. The empiricalrelations (14)-(17)provide acon-sistenttreatmentofstressleveleffectsbywhich thedataof allpenetrationtestscanbenormal-ized.Previoustreatmentsoftheproblemof curvedstrengthenvelopesCSaligh,1976)may therebyberecastasproblemsof'.n..varying withstress(DeBeer,1970),andappropriate methodsof calculationdeveloped.Theprospects foraconsistentapproachtoreductionsinN. withdepthor N,withfoundationwidtharethen evident.Thefirstobjectiveshouldbetousethe newrelationstopredictidealizedconepenetrat-ionresistanceprofileswhichcouldbecompared withthoseobtainedinpractice(Mitchell& Lunne,1978),sothatlaboratoryandfielddata can be harmonized. STRENGTHANDDILATANCYOFSANDS77 CONCLUSIONS It hasbeendemonstratedthatsecantanglesof shearingarerequiredinarationalapproachto thestrengthanddilatancyof sands.'.n.. - ; ~ , i (and;,,, isafunctionof mineralogy.Typicalvaluesfor;,,, are33fora quartz sandand40 forafelspathicsand,though simplelaboratorytestsshouldbeusedasa routinemeasure.Itwouldbeprudentnotto invokeI. valuesinexcessof 4withouttestcor-roborationof the extreme dilatant strengthwhich wouldbeimplied.All'.n.. values inexcess of ;",aretransitorywithlargerstrainsandsubjectin designcalculationstoanassurancethatprogress-ivefailurewillnotoccur.Negativevaluesof J Rindicatethatlargecontractilestrainswilloccur before ;,,, can be mobilized: Thepossibilitynowexists.ofrationalizingthe variousad hocstresslevelcorrectionswhichhave emergedinpenetrationtestingandpiledesign. Clarificationsoftherelativeproportionsoffric-tionanddilatantstrengthcomponentsarelikely tobeparticularlyimportantininvestigationsof the liquefaction potential. 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