外文翻译-孔隙水压力作用下土坡的极限分析

附：英文翻译LIMITANALYSISOFSOILSLOPESSUBJECTEDTOPORE-WATERPRESSURESByJ.KimR.salgado,assoicitemember,ASCE,andH.S.,member,ASCEABSTRACT:thelimit-equilibriummethodiscommonly,usedforslopestabilityanalysis.However,itiswellknownthatthesolutionobtainedfromthelimit-equilibriummethodisnotrigorous,becauseneitherstaticnorkinematicadmissibilityconditionsaresatisfied.Limitanalysistakesadvantageofthelower-andupper-boundtheoremofplasticitytoproviderelativelysimplebutrigorousboundsonthetruesolution.Inthispaper,threenoddedlineartriangularfiniteelementsareusedtoconstructbothstaticallyadmissiblestressfieldsforlower-boundanalysisandkinematicallyadmissiblevelocityfieldsforupper-boundanalysis.Byassuminglinearvariationofnodalandelementalvariablesthedeterminationofthebestlower-andupper-boundsolutionmaybesetupasalinearprogrammingproblemwithconstraintsbasedonthesatisfactionofstaticandkinematicadmissibility.Theeffectsofpro-waterpressureareconsideredandincorporatedintothefinite-elementformulationssothateffectivestressanalysisofsaturatedslopemaybedone.Resultsobtainedfromlimitanalysisofsimpleslopeswithdifferentground-waterpatternsarecomparedwiththoseobtainedfromthelimit-equilibriummethod.INTRODUCTIONStabilityanddeformationproblemingeotechnicalengineeringareboundary-valueproblem;differentialequationsmustbesolvedforgivenboundaryconditions.Solutionsarefoundbysolvingdifferentialequationsderivedfromconditionofequilibrium,compatibility,andtheconstitutiverelationofthesoil,subjectedtoboundarycondition.Traditionally,insoilmechanics,thetheoryofelasticityisusedtosetupthedifferentialequationsfordeformationproblems,whilethetheoryofplasticityisusedforstabilityproblems.Toobtainsolutionforloadingsrangingfromsmalltosufficientlylargetocausecollapseofaportionofthesoilmass,acompleteelastoplasticanalysisconsideringthemechanicalbehaviorofthesoiluntilfailuremaybethoughtofasapossiblemethod.However,suchanelastoplasticanalysisisrarelyusedinpracticeduetothecomplexityofthecomputations.Fromapracticalstandpoint,theprimaryfocusofastabilityproblemisonthefailureconditionofthesoilmass.Thus,practicalsolutionscanbefoundinasimplermannerbyfocusingonconditionsatimpendingcollapse.Stabilityproblemofnaturalslopes,orcutslopesarecommonlyencounteredincivilengineeringprojects.Solutionsmaybebasedontheslip-linemethod,thelimit-equilibriummethod,orlimitanalysis.Thelimit-equilibriummethodhasgainedwideacceptanceinpracticeduetoitssimplicity.Mostlimit-equilibriummethodarebasedonthemethodofslices,inwhichafailuresurfaceisassumedandthesoilmassabovethefailuresurfaceisdividedintoverticalslices.Globalstatic-equilibriumconditionsforassumedfailuresurfaceareexamined,andacriticalslipsurfaceissearched,forwhichthefactorofsafetyisminimized.Inthedevelopmentofthelimit-equilibriummethod,effortshavefocusedonhowtoreducetheindeterminacyoftheproblemmainlybymakingassumptionsoninter-sliceforces.However,nosolutionbasedonthelimit-equilibriummethod,noteventhesocalled“rigorous”solutionscanberegardedasrigorousinastrictmechanicalsense.Inlimit-equilibrium,theequilibriumequationsarenotsatisfiedforeverypointinthesoilmass.Additionally,theflowruleisnotsatisfiedintypicalassumedslipsurface,norarethecompatibilityconditionandpre-failureconstitutiverelationship.Limitanalysistakesadvantageoftheupper-andlower-boundtheoremsofplasticitytheorytoboundtherigoroussolutiontoastabilityproblemfrombelowandabove.Limitanalysissolutionsarerigorousinthesensethatthestressfieldassociatedwithalower-boundsolutionisinequilibriumwithimposedloadsateverypointinthesoilmass,whilethevelocityfieldassociatedwithanupper-boundsolutioniscompatiblewithimposeddisplacements.Insimpleterms,underlower-boundloadings,collapseisnotinprogress,butitmaybeimminentifthelowerboundcoincideswiththetruesolutionliescanbenarroweddownbyfindingthehighestpossiblelower-boundsolutionandthelowestpossibleupper-boundsolution.Forslopestabilityanalysis,thesolutionisintermsofeitheracriticalslopeheightoracollapseloadingappliedonsomeportionoftheslopeboundary,forgivensoilpropertiesand/orgivenslopegeometry.Inthepast,forslopestabilityapplications,mostresearchconcentratedontheupper-boundmethod.Thisisduetothefactthattheconstructionofproperstaticallyadmissiblestressfieldsforfindinglower-boundsolutionsisadifficulttask.Mostpreviousworkwasbasedontotalstresses.Foreffectivestressanalysis,itisnecessarytocalculatepore-waterpressures.Inthelimit-equilibriummethod,pore-waterpressuresareestimatedfromground-waterconditionssimulatedbydefiningaphreaticsurface,andpossiblyaflownet,orbyapore-waterpressureratio.Similarmethodscanbeusedtospecifypore-waterpressureforlimitanalysis.Theeffectsofpore-waterpressurehavebeenconsideredinsomestudiesfocusingoncalculationofupper-boundsolutionstotheslopestabilityproblem.MillerandHamiltonexaminedtwotypesoffailuremechanism:(1)rigidbodyrotation;and(2)acombinationofrigidrotationandcontinuousdeformation.Pore-waterpressurewasassumedtobehydrostaticbeneathaparabolicfreewatersurface.Althoughtheircalculationsledtocorrectanswers,thephysicalinterpretationoftheircalculationofenergydissipation,wherethepore-waterpressureswereconsideredasinternalforcesandhadtheeffectofreducinginternalenergydissipationforagivencollapsemechanism,hasbeendisputed.Pore-waterpressuresmayalsoberegardedasexternalforce.InastudybyMichalowski,rigidbodyrotationalongalog-spiralfailuresurfacewasassumed,andpore-waterpressurewascalculatedusingthepore-waterpressureratioru=u/ǐz,whereu=pore-waterpressure,ǐ=totalunitweightofsoil,andz=depthofthepointbelowthesoilsurface.Itwasshowedthatthepore-waterpressurehasnoinfluenceontheanalysiswhentheinternalfrictionangleisequaltozero,whichvalidatestheuseoftotalstressanalysiswithΦ=0.Inanotherstudy,Michalowskifollowedthesameapproach,exceptfortheuseoffailuresurfacewithdifferentshapestoincorporatetheeffectofpore-waterpressureonupper-boundanalysisofslopes,thewritersarenotawareofanylower-boundlimitanalysisdoneintermofeffectivestresses.Thisisprobablyduetotheincreasedinconstructingstaticallyadmissiblestressfieldsaccountingalsoforthepore-waterpressures.Theobjectivesofthispaperare(1)presentafinite-elementformulationintermsofeffectivestressesforlimitanalysisofsoilslopessubjectedtopore-waterpressures;and(2)tochecktheaccuracyofBishop’ssimplifiedmethodforslopestabilityanalysisbycomparingBishop’ssolutionwithlower-andupper-boundsolution.Thepresentstudyisanextensionofpreviousresearch,whereBishop’ssimplifiedlimit-equilibriumsolutionsarecomparedwithlower-andupper-bundsolutionsforsimpleslopeswithoutconsideringtheeffectofpore-waterpressure.Inthepresentpaper,theeffectofpore-waterpressureisconsideredinbothlower-andupper-boundlimitanalysisunderplane-strainconditions.Pore-waterpressuresareaccountedforbymakingmodificationstothenumericalalgorithmforlower-andupper-boundcalculationsusinglinearthree-nodedtrianglesdevelopedbySloanandSloanandKleeman.Tomodelthestressfieldcriterion,flowoflinearequationsintermsofnodalstressesandpore-waterpressures,orvelocities,theproblemoffindingoptimumlower-andupper-boundsolutionscanbesetupasalinearprogrammingproblem.Lower-andupper-boundcollapseloadingsarecalculatedforseveralsimpleslopeconfigurationsandgroundwaterpatterns,andthesolutionsarepresentedintheformofchart.LIMITANALYSISWITHPORE-WATERPRESSUREAssumptionsandTheirimplementationLimitanalysisusesanidealizedyieldcriterionandstress-strainrelation:soilisassumedtofollowperfectplasticitywithanassociatedflowrule.TheassumptionofperfectplasticityexpressesthepossiblestatesofstressintheformF()=0(1)WhereF()=yieldfunction;and=effectivestresstensor.AssociatedflowruledefinestheplasticstrainratebyassumingtheyieldfunctionFtocoincidewiththeplasticpotentialfunctionG,fromwhichtheplasticstrainratecanbeobtainedthough（2）where=nonnegativeplasticmultiplierratethatispositiveonlywhenplasticdeformationsoccur.Eq.(2)isoftenreferredtoasthenormalitycondition,whichstatesthatthedirectionofplasticstrainrateisperpendiculartotheyieldsurface.Perfectplasticitywithanassociatedwithverylargedisplacementsareofconcern.Inaddition,theoreticalstudiesshowthatthecollapseloadsforearthslopes,wheresoilsarenotheavilyconstrained,arequiteinsensitivetowhethertheflowruleisassociatedornon-associated.PrincipleofVirtualWorkBoththelower-andupper–boundtheoremsarebasedontheprincipleofvirtualwork.Thevirtualworkequationisapplicable,giventheassumptionofsmalldeformationsbeforecollapse,andcanbeexpressedaseither（3）Or（4）Where=boundaryloadings;=bodyforcesnotincludingseepageandbuoyancyforces;=bodyforcesincludingseepageandbuoyancyforces;=totalstresstensorinequilibriumwithand;=effectivestresstensorinequilibriumwithand;=Kroneckerdelta;p=pore-waterpressure;and=strainratetensorcompatiblewiththevelocityfield.Thereisnoneedfor,,andtoberelatedtoandinanyparticularwayfor(3)or(4)representtherateoftheexternalwork,whiletheright-handsidesrepresenttherateoftheinternalpowerdissipationdonebytheassumedstressfieldandexternalloadsontheassumedstrainandvelocityfields.Thedifferencebetween(3)and(4)isthewaytoincorporatetheeffectofpore-waterpressure:thepore-waterpressuresareconsideredasinternalforce,reducingtheinternalpowerdissipation,in(3),whiletheyareconsideredexternalforcein(4).Bytakingadvantageofthenormalitycondition,itcanbeeasilyshownthatelasticstressandstrainhavenoinfluenceonthecollapseload;thatis,onlyplasticdeformationoccursduringplasticflow,and=.Thismakeslimitanalysisasimplemethodtosolvestabilityproblems,withoutlossofrigor,assumingrigidperfectplasticity.Lower-boundTheoremIfthestressfieldwithinthesoilmassisstableandstaticallyadmissible,thencollapsedoesnotoccur;thatis,thetruecollapseloadisdefinitelygreaterthantheappliedload.Thiscanbewrittenintheformofthevirtualworkequation,using(3),as(5)Where=staticallyadmissiblestressfieldinequilibriumwiththetractionandbodyforcenotincludingtheseepageandbuoyancyforce;=actualstress;=actualstainrate;and=velocityfields.In(5),theinequalityisduetotheprincipleofmaximumplasticdissipation,accordingtowhichtheactualstrainratefieldisalwayslargerthantherateofworkdoneontheactualstrainratefieldbyastressfieldnotcausingcollapse.In(5),onlytheequilibriumconditionandthestressboundaryconditionsnottakenintoaccount.Thebestlowerboundtothetruecollapseloadcanbefoundbyanalyzingvarioustrialstaticallyadmissiblestressfields.中文对照翻译：孔隙水压力作用下土坡的极限分析摘要：极限平衡法一般用于土坡的稳定性分析。然而，众所周知的是，从极限分析法中获得的解是不严密的，因为它既不满足静态的允许条件，又不满足动态的允许条件。极限分析法充分利用了塑性体的上下边界原理，在求真实解中提供了一个相对简单但又严密的边界。在这篇文章中，三点确定的三角形三边有限元法被利用与构造在下边界分析中的静态允许应力场和上边界分析中的速度场。通过假设三角形顶点的线变量和元素变量，真实解应该是一个线形的约束问题。在静态和动态的条件都满足的基础上，真实解应该处在上下边界所得的解之间。在有限元公式中，要考虑包括了孔隙压力的影响，以便使饱和土坡的有效应力分析可以得出。作者对从不同地下水形式下简单土坡的极限分析所得的结果与极限平衡法中所得的结果作了比较。概述：稳定性和变形问题在全球技术工程领域是一个边界值问题。微分方程必须用给定的边界条件来解决。通过解决由平衡协调条件以及沙土的本构关系推出的微分方程，从而得到边界条件下的解。按照传统的说法，在土力学中，弹性理论是用来建立变形微分方程的，就象塑性理论是用来建立稳定性问题的微分方程一样。为了获得这个解，荷载由小到大变化，直到足够大引起部分土体的滑坡。作为土体破坏的力学行为，完整的弹塑性分析以为是一个可能的方法。然而，这样一个弹塑性分析方法很少应用于实际问题当中，因为他的计算机太过复杂。站在实践的立场上，稳定性的最基本关注点应该是土体破坏条件。因此，真实解应该是通过关注即将发生的破坏条件的一个简单的方法中得来。自然土坡、填方土坡和挖方土坡的稳定性问题是土木工程领域碰到的最常见的问题。求解通常建立在滑移线方法上，极限平衡方法或极限分析法的基础上。由于它的简单，在实践中，极限平衡法是最被广泛使用的。极限平衡法大部分建立在分块理论的基础上，在这种理论中，假设有一个破坏的滑动面，而且在此之上的土体被划分为若干垂直土条，整个静态平衡条件下假设的失稳表面是被确定的，一个临界的滑动破裂面必须要找到，因为它的安全因数最小。在极限平衡法的发展过程中，要努力去做的是怎么降低通过内力假设的不确定性。但是，没有一种解的得来是建立在这样的极限分析法的基础上，甚至在严格的力学意义上讲，它都不算一个严密的解。在极限平衡法中，平衡方程并不是对土体的每一点都适用的。另外，在典型的假定滑动面方法中，流动法则是不满足的，同样，协调性条件和破坏前的本构关系也是不满足的。极限分析法充分利用了边界理论，得出了相对应用于上下边界的两个严密的解。极限分析法在以下两种意义上是严密的，一是土体在外加荷载作用下的平衡，下边界解所对应的应力场；二是与外加位移相协调，上边界所对应的速度场。就简单而言，下边界荷载作用下，滑动不会发生，但是如果下边界受到外加荷载的作用，则滑动可能立即发生。同样，在上边界作用外加荷载，滑坡也会立即发生。通过寻找下边界的最大可能解和上边界的最小可能解，真实解存在于他们之间的范围内。对于土坡稳定性问题，给定土体的性质或几何尺寸的基础上，知道土坡发生滑动的临界高度和发生部分滑坡的临界荷载才能得出解来。在过去，对于土坡稳定性的应用，大多数研究工作都集中在上边界法上，这是因为通过求解下边界适合静态允许应力场方程的解是一项很困难的任务。大多数先前的工作都基于总应力之上。对于有效应力的分析，考虑孔隙水压力的作用是很有必要的。在极限平衡法中，孔隙水压力是通过限定一个地下水表面和一个可能的流动网或者通过一个孔隙水压力比率模拟地下水条件推测出来的。相似的方法可以用于明确说明孔隙水压力作用下的极限分析。在大量的实践中，孔隙水压力的影响被看作集中考虑在土坡稳定性问题的上边界解上。Miller和Hamilton两人实验出了两种力学破坏类型：（1）刚体旋转；（2）刚体旋转和持续变形相结合。孔隙水压力被假定成流体静力学下的一个抛物线型的自由水表面，尽管他们的研究得出了正确的答案，但是，从物理学上解释他们的研究，在能量消散上是有争议的。在他们的理论中，孔隙水压力被看作是内力，在给定的滑坡机理下，它对降低内部能量消散是有影响的。孔隙水压力也可以看作是一种外力。在Michalowsk的研究中，假设刚体是沿着螺旋线破坏的。孔隙水压力被考虑用孔隙水压力比来表示：这里，u是孔隙水压力；是沙土的比重；z是土体表面以下的深度。它表示孔隙水压力在内部摩擦力等于零时的分析没有影响，这就证实了用总应力分析时。在另一项研究中，除了用不同形状的破裂面结合分块分析法时，Michalowski秉承了相同的方法。当这种努力结合孔隙水压力作用在土坡上边界的影响中，作者没有意识到就有效应力而言有任何的下边界极限分析要做。这可能因为在考虑孔隙水压力的情况