Distribution-of-earth-pressure-on-a-retaining-wall

DistributionofearthpressureonaretainingwallY.-Z.WANGOnthebasisofCoulomb'sconceptthattheearthpressureagainstthebackofaretainingwallisduetothethrustexertedbyaslidingwedgeofsoilbetweenthebackofthewallandaplanewhichpassesthroughthebottomedgeofthewallandhasaninclinationofè,adifferentialequationof®rstorderissetupbyconsideringtheequilibriumoftheforcesonanelementofthewedge.Atheoreticalresultfortheunitearthpressureonaretainingwallisobtained.AcomparisonismadebetweenCoulomb'sformulaandtheformulapresentedhere,andtheearthpressurecalculatedbytheformulapresentedhereisalsocomparedwithexperi-mentalobservations.KEYWORDS:earthpressure;theoreticalanalysis.SurlabaseduconceptdeCoulombindiquantquelapress-ionterrestresurl'arrieÁred'unmurdesouteÁnementestdueaÁlapousseÂeexerceÂeparunecaledesolglissantesitueÂeentreledosdumuretunplanquitraverselebasdumuravecuneinclinaisondeè,nouseÂtablissonsuneeÂquationdiffeÂrentielledupremierordreenconsideÂrantl'eÂquilibredesforcessuruneÂleÂmentdelacale.NousavonsobtenuunreÂsultattheÂoriquepourlapressionterrestreunitairesurlemurdesouteÁnement.NousavonsfaitunecomparaisonentrelaformuledeCoulombetlaformulepreÂsenteÂeicietnouscomparonsaussilapressionterrestrecalculeÂegraÃceaÁlaformulepreÂsenteÂeaveclesobservationsexpeÂrimentales.INTRODUCTIONThepressureagainstthebackofaretainingwallcausedbyback®llandsurchargeonthegroundsurfaceisaclassicalproblemofsoilmechanics.Coulomb's(1776)theoryandRankine's(1857)theoryarecommonlyappliedtothecalcula-tionoftheearthpressureonaretainingwall(Craig,1983).Rankine'stheoryisapplicabletothecalculationoftheearthpressureonaperfectlysmooth,verticalwall,butmostretainingwallsarefarfromfrictionless.Coulomb'sformulawasestab-lishedonthebasisoftheequilibriumoftheforcesonawholeslidingwedgebetweentheplaneoffailureandthebackfaceoftheretainingwall.Strictlyspeaking,Coulomb'stheoryisapplic-ableonlytothecalculationoftheresultanttotalearthpressureandcannotbeusedtodeterminethedistributionoftheearthpressure.Theapproximatedistributionoftheearthpressurecanbedeterminednumericallybycomputingtheresultantearthpressureatvariousdepthsalongthewall(Wu,1976).TheunitearthpressurecalculatedonthebasisofCoulomb'sandRanki-ne'stheoriesvarieslinearlywithdepth.However,alotoflaboratoryand®eldexperimentsshowthattheunitearthpressureiscurvilinearlydistributedonthebackofawall,forexample,theexperimentsofKankanina,Shirvy,KahanaandCaharly(Kerlyn,1977).Adifferentinterpretationisgivenoftheexperimentalresults,butasolutionfortheunitearthpressurehasnotbeenachieved.InthispaperonthebasisofCoulomb'sconceptthattheearthpressureagainstthebackofaretainingwallisduetothethrustexertedbyaslidingwedgeofsoilbetweenthebackofthewallandaplanewhichpassesthroughthebottomedgeofthewallandhasaninclination(relativetothehorizontal)ofè,abasicequationissetupbyconsideringtheequilibriumoftheforcesonanelementofthewedge.Itisa®rst-orderdifferentialequationfortheunitearthpressure.Thesolutionoftheequa-tionisobtained,givingatheoreticalresultfortheunitearthpressureonaretainingwall.BASICEQUATIONIthasbeenshown,fromavastamountofexperimentaldata,thataslidingsurfaceexistsinthesoilbehindaretainingwallunderthelimitequilibriumcondition,andthesurfacecanbeapproximatedbyaplanewhichpassesthroughthebottomedgeofthewallandhasaninclination(relativetothehorizontal)ofè.Thetriangularmassofsoilbetweenthisplaneoffailureandthebackfaceofthewallisreferredtoastheslidingwedge.Onthebasisofexperimentalresults,itisassumedthattheearthpressureagainstthebackofthewallisduetothethrustexertedbytheslidingwedgewhenthewallmovesforward.Now,takingtheslidingwedgeasanisolatedunit,asshowninFig.1(a),anelementofthicknessdyistakenfromthewedgeatadepthybelowthegroundsurface.Theforcesonthiselementincludetheverticalpressurepyonthetopoftheelement,theverticalreactionpy‡dpyonthebottomoftheelement,thehorizontalreactionpxoftheretainingwall,theshearô1betweentheback®llandthebackoftheretainingwall,thenormalreactionrofthesoilatrest,theshearô2betweentheslidingback®llandtheremainingback®llatrest,andtheweightdWoftheelement,asshowninFig.1(b).Theshearingforcesonthetop83Wang,Y.-Z.(2000).GeÂotechnique50,No.1,83±88Manuscriptreceived3September1998;revisedmanuscriptaccepted18May1999.Discussiononthispapercloses31August2000;forfurtherdetailsseep.ii.TianjinUniversity.Fig.1.Analyticmodel(a)qABHydyC(b)pypy1dpydWrpxô1ô2andbottomoftheelementareneglected,consideringthatthewedgeslidesasawhole.Itcanbeshownfromtheequilibriumconditionofthehorizontalforcesontheelement,thatpxdy‡ô2dysinècosèÿr3dysinècos(908ÿè)ˆ0(1)Thiscanbewrittenaspx‡ô2cotèÿrˆ0(2)Thefollowingequationcanbeobtainedfromtheequilibriumconditionoftheverticalforcesontheelement:py(Hÿy)cotè‡dWÿ(py‡dpy)(Hÿyÿdy)3cotèÿô1dyÿô2dysinèsinèÿr3dysinècosèˆ0(3)Omittingthedifferentialtermsofsecondorder,thiscanbesimpli®edtodpydyˆã‡1Hÿy[pyÿrÿ(ô1‡ô2)tanè](4)whereãistheunitweightoftheback®ll.LetpxˆKpyô1ˆpxtanä(5)ô2ˆrtanjwhereKisthelateralpressurecoef®cient,äisthefrictionalanglebetweenthebackofthewallandtheback®llandjistheinternalfrictionangleoftheback®ll.Substitutingequation(5)intoequation(2),itcanbe